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Table 1: Oscillation amplitudes of a π+superscript𝜋\pi^{+}italic_π start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT with different projected energies.

2.3×10−102.3superscript10102.3\times 10^{-10}2.3 × 10 start_POSTSUPERSCRIPT - 10 end_POSTSUPERSCRIPT

1.3×10−231.3superscript10231.3\times 10^{-23}1.3 × 10 start_POSTSUPERSCRIPT - 23 end_POSTSUPERSCRIPT

4.0×10−234.0superscript10234.0\times 10^{-23}4.0 × 10 start_POSTSUPERSCRIPT - 23 end_POSTSUPERSCRIPT

Figure 1: Proper time oscillation of a π0superscript𝜋0\pi^{0}italic_π start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT particle with ω0=2.0×1023⁢s−1subscript𝜔02.0superscript1023superscript𝑠1\omega_{0}=2.0\times 10^{23}s^{-1}italic_ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = 2.0 × 10 start_POSTSUPERSCRIPT 23 end_POSTSUPERSCRIPT italic_s start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT and T0=5.0×10−24⁢ssubscript𝑇05.0superscript1024𝑠T_{0}=5.0\times 10^{-24}sitalic_T start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = 5.0 × 10 start_POSTSUPERSCRIPT - 24 end_POSTSUPERSCRIPT italic_s. The internal time never travels back to the past.

B

Encouraged by that, the S-wave phase shifts were calculated for D⁢π𝐷𝜋D\piitalic_D italic_π scattering with JP=0+superscript𝐽𝑃superscript0J^{P}=0^{+}italic_J start_POSTSUPERSCRIPT italic_P end_POSTSUPERSCRIPT = 0 start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT and D∗⁢πsuperscript𝐷𝜋D^{*}\piitalic_D start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT italic_π scattering with JP=1+superscript𝐽𝑃superscript1J^{P}=1^{+}italic_J start_POSTSUPERSCRIPT italic_P end_POSTSUPERSCRIPT = 1 start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT, focusing on isospin 1/2121/21 / 2 channels where resonances appear.

Following the Lüscher method, we first extracted the discrete energies of the D(∗)⁢πsuperscript𝐷𝜋D^{(*)}\piitalic_D start_POSTSUPERSCRIPT ( ∗ ) end_POSTSUPERSCRIPT italic_π system with zero total momentum in a finite box. The energy levels were obtained using quark-antiquark and two-meson operators in the correlation functions. The Lüscher formula then renders the phase shift for levels in the elastic region.

Assuming a localized interaction, the energy levels of a two-hadron system in a finite box are related to the scattering phase shift in the elastic region Luscher:1985dn ; Luscher:1986pf ; Luscher:1990ux ; Luscher:1991cf . One first needs to determine the energy levels E𝐸Eitalic_E of the two-hadron system on the lattice. We choose the total momentum of the system to be zero in this simulation, so the lattice frame coincides with in the center of momentum (CM) frame, and both hadrons have momentum p∗superscript𝑝p^{*}italic_p start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT. In this case we avoid the complications that arise for the extraction of the S-wave from simulations with non-zero total momentum caused by mD(∗)≠mπsuperscriptsubscript𝑚𝐷subscript𝑚𝜋m_{D}^{(*)}\neq m_{\pi}italic_m start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( ∗ ) end_POSTSUPERSCRIPT ≠ italic_m start_POSTSUBSCRIPT italic_π end_POSTSUBSCRIPT Fu:2011xz ; Leskovec:2012gb . In the exterior region, where the interaction is negligible,

Calculations are carried out using a lattice simulation with dynamical u,d𝑢𝑑u,ditalic_u , italic_d quarks. Correlation functions are constructed with quark-antiquark interpolators and, for the first time, also with D⁢π𝐷𝜋D\piitalic_D italic_π or D∗⁢πsuperscript𝐷𝜋D^{*}\piitalic_D start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT italic_π interpolators to study the relevant scattering channels. Our aim is to describe the two observed broad states D0∗⁢(2400)superscriptsubscript𝐷02400D_{0}^{*}(2400)italic_D start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( 2400 ) and D1⁢(2430)subscript𝐷12430D_{1}(2430)italic_D start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( 2430 ) as resonances, so we simulate D⁢π𝐷𝜋D\piitalic_D italic_π and D∗⁢πsuperscript𝐷𝜋D^{*}\piitalic_D start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT italic_π scattering and extract the corresponding phase shifts for the first time. The S-wave phase shift for D⁢π𝐷𝜋D\piitalic_D italic_π scattering is extracted using Lüscher’s formula and a Breit-Wigner fit of the phase shift renders the D0∗⁢(2400)superscriptsubscript𝐷02400D_{0}^{*}(2400)italic_D start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( 2400 ) resonance mass and width.

The main results for the D𝐷Ditalic_D meson spectrum are compiled in Fig. 11, where resonance masses for scalar and axial mesons are shown together with our results for other JPsuperscript𝐽𝑃J^{P}italic_J start_POSTSUPERSCRIPT italic_P end_POSTSUPERSCRIPT. The latter were calculated using just quark-antiquark operators and by equating the masses to the energy levels. Overall good agreement with experimental values of the well established states was obtained. Furthermore additional energy levels were observed in the vicinity of some of the resonances discovered recently by the BaBar collaboration delAmoSanchez:2010vq .

A

𝔄∈Obj⁢(𝔡⁢𝔭)𝔄Obj𝔡𝔭\mathfrak{A}\in\mathrm{Obj}(\mathfrak{dp})fraktur_A ∈ roman_Obj ( fraktur_d fraktur_p ) stands for 𝔄𝔄\mathfrak{A}fraktur_A is an object of the category 𝔡⁢𝔭𝔡𝔭\mathfrak{dp}fraktur_d fraktur_p in Cor.0.4.5, while

B∈𝔭⁢𝔡⁢𝔭B𝔭𝔡𝔭\mathrm{B}\in\mathfrak{pdp}roman_B ∈ fraktur_p fraktur_d fraktur_p stands for BB\mathrm{B}roman_B is an element of the set defined in Def.0.4.8.

Let 𝔅∈𝔡⁢𝔭𝔅𝔡𝔭\mathfrak{B}\in\mathfrak{dp}fraktur_B ∈ fraktur_d fraktur_p such that G𝔅subscriptG𝔅\mathrm{G}_{\mathfrak{B}}roman_G start_POSTSUBSCRIPT fraktur_B end_POSTSUBSCRIPT is a groupoid whose inversion map is continuous. Define

𝔅∈𝔡⁢𝔭𝔅𝔡𝔭\mathfrak{B}\in\mathfrak{dp}fraktur_B ∈ fraktur_d fraktur_p stands for 𝔅𝔅\mathfrak{B}fraktur_B is an element of the set defined in Def.0.4.1.

for all 𝔄,𝔅∈𝔡⁢𝔭𝔄𝔅𝔡𝔭\mathfrak{A},\mathfrak{B}\in\mathfrak{dp}fraktur_A , fraktur_B ∈ fraktur_d fraktur_p and (f,T)∈Mor𝔡⁢𝔭⁢(𝔄,𝔅)𝑓𝑇subscriptMor𝔡𝔭𝔄𝔅(f,T)\in\mathrm{Mor}_{\mathfrak{dp}}(\mathfrak{A},\mathfrak{B})( italic_f , italic_T ) ∈ roman_Mor start_POSTSUBSCRIPT fraktur_d fraktur_p end_POSTSUBSCRIPT ( fraktur_A , fraktur_B ).

C

Direct real photons serves as the most versatile tools to study relativistic heavy ion collisions because they are produced by various mechanisms during the whole space-time history of the interacting system. The observation of an unexpectedly large direct photon flow v2subscript𝑣2v_{2}italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT (essentially the same magnitude as hadrons) is a big challenge and so far eludes full and coherent explanationPHENIX:2011oxq ; PHENIX:2015igl .

Large v2subscript𝑣2v_{2}italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT of direct photons may imply an additional source at very late stage, for example,

At present, not all processes of photon production in QGP and HG phase are amenable to a calculation of viscous (shear and bulk) corrections.

Suppose a strong emission at early time from the collision system at the high temperature, and a non-zero flow velocity

It is believed that thermal photons are emitted during the whole evolution of the collision systems, especially stronger at higher temperature at early stage, ie, in QGP phase, but large v2subscript𝑣2v_{2}italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT usually indicates late production at hadronic phase.

D

In view of recent experimental progress with ultra-cold atoms forming a Bose-Einstein condensate, double-well systems are one of the most commonly exploited schemes studied Andrews ; Smerzi ; Milburn ; Menotti ; Meier ; Shin ; Albiez ; Levy ; Salgueiro ; Simon ; Liu . Typically, in this context one assumes that weakly interacting bosons occupying different wells can be described with two independent single-particle orbitals and that the dynamics is governed by two mechanisms: contact two-body interactions acting locally and single-particle tunneling between wells. Then, in the mean-field limit, a corresponding Gross-Pitaevskii equation is introduced and numerically solved for different initial conditions Raghavan ; Ostrovskaya ; Ananikin . Generalized two-mode models, taking into account additional terms originating from long-range interactions or occupation-dependent tunnelings, are also considered in the literature and relevant corrections to the dynamics are studied Lahaye ; Adhikari ; Bruno . Although the validity of these simplified two-mode models was confirmed experimentally for weak interactions between particles, they were extended beyond the range of their applicability and adopted for strongly interacting systems, i.e. in situations when the local interaction energy is much larger than the single-particle tunneling energy. For example, it was shown that for initially imbalanced occupations the dynamics is heavily affected by strong interactions DuttaS . Unfortunately, the validity of the model used was not discussed and its predictions were not compared with the exact dynamics governed by a general model.

In this article, we have studied the dynamical properties of two ultra-cold bosons confined in a one-dimensional double-well potential initially occupying the lowest state of a chosen site. We compare the exact dynamics governed by a full two-body Hamiltonian with two simplified two-mode models. In particular, we compared the evolution of particle density and spatial correlations between particles. We show that for a shallow barrier and strong enough interactions the simplified models break down and the correct multi-orbital description cannot be substituted with a two-mode model even if all appropriate interaction terms are taken into account. The fundamental difference between the exact and two-mode descriptions emerges when inter-particle correlations are considered. For example, the evolution of the probability that both bosons are found in opposite wells of the potential crucially depends on couplings to higher orbitals of an external potential. This fact sheds some light on recent theoretical results and opens some perspectives for further experimental explorations.

The time evolution of the probability 𝒫⁢(t)𝒫𝑡{\cal P}(t)caligraphic_P ( italic_t ) may shed some light on the differences between evolutions governed by different Hamiltonians in the strong interaction limit. This comes from the fact that for the simplest case (13), single-particle tunnelings is strongly suppressed by the conservation of energy. The energy difference between the initial state (10) and a state in which bosons occupy different wells is equal to U𝑈Uitalic_U, and is much larger than the energy gain from the tunneling, J𝐽Jitalic_J. Therefore, the dynamics is governed effectively by the second-order process in the tunneling, i.e. bosons tunnel between wells mainly in pairs Folling . This is visible in the evolution of the probability (16) (black curve in Fig. 4) – it is close to 00 at all moments. Situation may change for extended models (4) and (3) where other processes are taken into account. For example, in the case of the extended two-mode model (4) the density-induced tunnelings give additional contributions to single-particle tunneling and they effectively support the breaking of a bosonic pair. On the other hand, a pair-tunneling term supports an ordinary second-order tunneling of a composed pair. After all, as it is seen from numerical results, for strong enough interactions the on-site energy U𝑈Uitalic_U always dominates over other terms and the probability (16) is close to 00 also for the extended two-mode model (4) (blue line in Fig. 4).

In view of recent experimental progress with ultra-cold atoms forming a Bose-Einstein condensate, double-well systems are one of the most commonly exploited schemes studied Andrews ; Smerzi ; Milburn ; Menotti ; Meier ; Shin ; Albiez ; Levy ; Salgueiro ; Simon ; Liu . Typically, in this context one assumes that weakly interacting bosons occupying different wells can be described with two independent single-particle orbitals and that the dynamics is governed by two mechanisms: contact two-body interactions acting locally and single-particle tunneling between wells. Then, in the mean-field limit, a corresponding Gross-Pitaevskii equation is introduced and numerically solved for different initial conditions Raghavan ; Ostrovskaya ; Ananikin . Generalized two-mode models, taking into account additional terms originating from long-range interactions or occupation-dependent tunnelings, are also considered in the literature and relevant corrections to the dynamics are studied Lahaye ; Adhikari ; Bruno . Although the validity of these simplified two-mode models was confirmed experimentally for weak interactions between particles, they were extended beyond the range of their applicability and adopted for strongly interacting systems, i.e. in situations when the local interaction energy is much larger than the single-particle tunneling energy. For example, it was shown that for initially imbalanced occupations the dynamics is heavily affected by strong interactions DuttaS . Unfortunately, the validity of the model used was not discussed and its predictions were not compared with the exact dynamics governed by a general model.

It is quite obvious that for sufficiently strong interactions between particles any two-mode model has to break down. This comes from the observation that interactions always introduce some multi-particle correlations that exist locally at each site of the potential. Therefore, a simple assumption that all particles occupying a single site can be described with a single effective orbital cannot be valid. The situation is very similar to the problem of ultra-cold bosons confined in a single harmonic trap. As it was shown in the case of two strongly interacting bosons in a harmonic trap, a single-mode description is not valid Sowinski2010 . In the case studied here, whenever interactions are significantly larger than typical tunneling energies between wells, local correlations induced by interactions are produced much faster than correlations between wells. Just from this simple observation it is quite obvious that any two-mode approximation is inconsistent.

D

λc±=12⁢(γ±γ2+4⁢(σ+ζ))superscriptsubscript𝜆𝑐plus-or-minus12plus-or-minus𝛾superscript𝛾24𝜎𝜁\lambda_{c}^{\pm}=\frac{1}{2}\left(\gamma\pm\sqrt{\gamma^{2}+4(\sigma+\zeta)}\right)italic_λ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ± end_POSTSUPERSCRIPT = divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( italic_γ ± square-root start_ARG italic_γ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + 4 ( italic_σ + italic_ζ ) end_ARG )

ζ>γ24𝜁superscript𝛾24\zeta>\frac{\gamma^{2}}{4}italic_ζ > divide start_ARG italic_γ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 4 end_ARG

Expanding (γ2−4⁢ζ>0superscript𝛾24𝜁0\gamma^{2}-4\zeta>0italic_γ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - 4 italic_ζ > 0)

γ2<4⁢ζsuperscript𝛾24𝜁\gamma^{2}<4\zetaitalic_γ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT < 4 italic_ζ, the expanding eigenvalues λeR±i⁢λeIplus-or-minussuperscriptsubscript𝜆𝑒𝑅𝑖superscriptsubscript𝜆𝑒𝐼\lambda_{e}^{R}\pm{i}\lambda_{e}^{I}italic_λ start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_R end_POSTSUPERSCRIPT ± italic_i italic_λ start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_I end_POSTSUPERSCRIPT are complex,

Expanding (γ2−4⁢ζ<0superscript𝛾24𝜁0\gamma^{2}-4\zeta<0italic_γ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - 4 italic_ζ < 0)

B

LagoudakisNphys2008 ; RoumposNphys2011 ; NardinNphys2011 ; SanvittoNphot2011 ; DominiciSA2015 ; BoulierPRL2016 ; caputo2016topological ; caputo2019josephson

and f1subscript𝑓1f_{1}italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT is the spin-conserved and spin-exchange polariton-polariton

and f2subscript𝑓2f_{2}italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT are the same-spin and cross-spin nonradiative loss rates, respectively.

σ=±𝜎plus-or-minus\sigma=\pmitalic_σ = ± representing the spin state of polaritons with effective mass

In the absence of external magnetic field the “spin-up” and “spin-down” states σ=±𝜎plus-or-minus\sigma=\pmitalic_σ = ± of noninteracting polaritons, or their linearly

D

In the case of the harmonic oscillator, the partner Hamiltonians are related by a shift in the spectrum, which allows the retrieval of all eigenvalues. The present paper shows algebraic conditions which provide the full benefit of the ladder structure for a class of systems that we call coupled SUSY which acts as a new, elementary generalization of the harmonic oscillator. Generalizing the harmonic oscillator algebra in various ways is not a new concept. Many such generalizations involve q𝑞qitalic_q-deformations [3, 5, 19, 24], commutators involving powers of operators or deformed commutators [18, 27, 30] that are of interest in quantum gravity, work with structure functions [6], or involve general energy eigenvalue structures [35].

The remainder of this paper is organized as follows. In Section 2, we develop the coupled SUSY structure which expands the relationship among the QMHO, SUSY, and the corresponding coupled SUSYs. In Section 3, we establish eigenvalues and eigenfunctions of the corresponding coupled SUSY. In Section 4, we establish the connections between coupled SUSY and other oscillator systems, namely Schwinger’s approach to the two-particle QMHO system, and the 𝔰⁢𝔲⁢(1,1)𝔰𝔲11\mathfrak{su}(1,1)fraktur_s fraktur_u ( 1 , 1 ) approach to the QMHO. In Section 5, we develop the coherent states for coupled SUSY. Due to the algebraic structure of coupled SUSY, a more complex coherent state structure exists than what has been found previously for either SUSY or the QMHO. In Section 6, we derive some uncertainty principles associated to coupled SUSY via ladder operators, generalizing the traditional Heisenberg uncertainty principle. In doing so, we discover new quantum operators corresponding to the classical Lagrangian and dilation. In Section 7, we show that harmonic oscillators can be realized as special classes of coupled SUSYs, suggesting that coupled SUSYs may have utitlity in quantum field theory and elementary particles.

Coupled SUSY has elements of both of the above approaches to the QMHO. The ladder operators in all three cases are second order; in coupled SUSY, the ladder operators are a combination of the operators coming from two separate Hamiltonians; and coupled SUSY retains the 𝔰⁢𝔲⁢(1,1)𝔰𝔲11\mathfrak{su}(1,1)fraktur_s fraktur_u ( 1 , 1 ) structure implicitly built into the QMHO.

In a future work, the authors intend to explore broken coupled SUSY, the nature of the coupled SUSY algebra, and the spectral theory of the coupled SUSY momentum and position operators. The ladder structure for coupled SUSY appears to have a natural relationship with the group of Lorentz transformations SO⁡(2,1)SO21\operatorname{SO}(2,1)roman_SO ( 2 , 1 ),

By considering the QMHO in the context of SUSY, the coupled SUSY structure unifying the QMHO and SUSY was developed. Coupled SUSY has many of the desirable properties of both: true ladder operators exist, there are two sectors, and charge operators exist between the sectors. The existence of true ladder operators led to a richer theory for coherent states than what has existed in the past, namely applying a charge operator led to an intertwining in the coherent states structures in both sectors. Moreover, focusing on the case of unbroken coupled SUSY gave some results regarding the spectrum of coupled SUSY Hamiltonians as well as uncertainty principles which generalize the Heisenberg uncertainty principle. A strength of the theory is its background-independence, e.g., the above may apply to any L2⁢(Ω)superscript𝐿2ΩL^{2}(\Omega)italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( roman_Ω ) space. ΩΩ\Omegaroman_Ω may simply be a locally compact group or even have a manifold structure. Unbroken coupled SUSYs for which −γ=δ𝛾𝛿-\gamma=\delta- italic_γ = italic_δ can be realized as harmonic oscillators and as such coupled SUSY may be a better model for elementary particles than traditional supersymmetric quantum mechanics. Moreover, coupled SUSY is inherently a multi-particle and multi-dimensional theory and does not suffer some of the pitfalls that traditional SUSY quantum mechanics has for multiple particles or multiple dimensions [21, 25, 22]. It has also been shown that spurious states exist for sector two tensor Hamiltonians for tensorial systems [9] In spite of these drawbacks, tensorial multidimensional SUSY quantum mechanics has been used to determine excited-state energies and wavefunctions using adiabatic switching [10] in addition to the construction of sector one states via imaginary time propagation [8].

A

S=S⁢(N,T,V)𝑆𝑆𝑁𝑇𝑉S=S(N,T,V)italic_S = italic_S ( italic_N , italic_T , italic_V ) of this antihydrogen gas system is practically zero (in accord

system together with the theoretical assumption of the validity of C⁢P⁢T𝐶𝑃𝑇CPTitalic_C italic_P italic_T

In our opinion the ultimate validity or invalidity of the Proposal is an experimental question; it can be surely decided by

Regarding its current theoretical status, proving or disproving the Proposal using the apparatus of theoretical physics and mathematics is

Regarding 𝒮mattersubscript𝒮matter{\mathscr{S}}_{\rm matter}script_S start_POSTSUBSCRIPT roman_matter end_POSTSUBSCRIPT its individual ball constituents

C

{2}...a_{2n}\right\rangle| italic_ψ ⟩ = ∑ start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_a start_POSTSUBSCRIPT 2 italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_T start_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT … italic_a start_POSTSUBSCRIPT 2 italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT | italic_a start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT … italic_a start_POSTSUBSCRIPT 2 italic_n end_POSTSUBSCRIPT ⟩

We call the tensor T𝑇Titalic_T perfect if the state is maximally entangled across any bipartition cut of the set of 2⁢n2𝑛2n2 italic_n spins into two sets of n𝑛nitalic_n spins. This expansion can be rewritten (by the maps of bra into ket states) in various forms and for the 6 spins code we have the operators

It is well known that the holographic principle implies that a theory of gravity in a bulk space is dual to a quantum field theory on a boundary. In the AdS/CFT context this is translated into a duality between a weakly coupled gravity in the bulk and a strongly coupled conformal field theory on the boundary. In order for this duality to be meaningful we need to relate the bulk operators to boundary operators. This mapping has however some surprising aspects. Of course, while introducing the bulk space we identify a new radial dimension from the boundary towards the bulk. It has been shown in [27], [28] that such a radial dimension can be seen as a renormalisation group scale. We can see the radial coordinate of a spacetime with asymptotically AdS geometry as a flow parameter of the boundary field theory. Recent research has completed the idea that geometry, seen as an emergent property, is related to quantum entanglement in the sense that the geometry in the bulk can be expressed in terms of the entanglement structure of the boundary quantum field theory. Describing quantum field theory by non-geometric means is highly complicated, therefore its connection to geometry offered a new understanding of various quantum field theoretical phenomena [29], [30]. The connection to geometry has been made clear already by the introduction of the Ryu-Takayanagi formula [31] and its covariant counterpart [32]. This formula is known to acquire corrections by various local and non-local terms. Such terms, derived also in [33] can be seen by means of so called holographic quantum correction codes. Such codes not only demonstrate the idea that entanglement is a source from where geometry emerges, but also allows us to better understand various prescriptions of the AdS/CFT dictionary. Such a construction is based on a tensor network which is expressed in terms of polygons that are uniformly tiling the bulk space. The terminology here will become that of quantum information theory, and hence we will have physical quantum information units encoding the information of logical quantum states. The physical variables associated to the quantum code will be on the boundary while the logical operators reside in the bulk. Holographic codes allow us to explicitly compute the mapping between boundary and bulk and hence to derive the dictionary of AdS/CFT. In essence, local operators in the bulk theory are being mapped into non-local operators of the bulk. This allows us to connect bulk geometry to the entanglement structure of the boundary quantum field theory. The bulk Hilbert space or the code space is a proper subspace of the boundary Hilbert space preserved by the bulk operators. The idea of reconstructing bulk operators on the boundary is based on the AdS-Rindler reconstruction described in the introduction. The ambiguity of the reconstruction prescription has been resolved by making use of some form of redundancy. Either we considered the highly different boundary operators as being different physical representations of the same type of action on the code subspace, or we considered the distinction in the boundary theory as given by the redundant description given by gauge invariance. However, the usual gauge invariance on the boundary seemed problematic. Extending it to the gauge invariance of double field theory however may have certain benefits. The fact that operators residing in the causal wedge of a certain boundary region A𝐴Aitalic_A can be reconstructed on the boundary is well known. However, if the the boundary region A𝐴Aitalic_A is a union of two or more disconnected components, then the domain of the bulk from where operators can be reconstructed on the boundary region A𝐴Aitalic_A increases. This introduces the so called entanglement wedge E⁢(A)𝐸𝐴E(A)italic_E ( italic_A ) which may extend further into the bulk and from which bulk operators may be reconstructed on the disconnected region A𝐴Aitalic_A. Moreover, entangled pairs in the bulk with one of the members inside the entanglement wedge of the region A𝐴Aitalic_A and the other outside, will contribute to the entropy of the region A𝐴Aitalic_A and hence to the entanglement shared by A𝐴Aitalic_A and its complement A¯¯𝐴\bar{A}over¯ start_ARG italic_A end_ARG. This means that there should be operators in A𝐴Aitalic_A capable of detecting the member of the pair inside the entanglement wedge E⁢(A)𝐸𝐴E(A)italic_E ( italic_A ). Given the general entropy formula

The implementation of bulk fields in the holographic tensor network has been discussed in [26]. There, a generalisation of the holographic quantum error correction code for bulk gauge fields is presented. As most of the novel aspects of double field theory are revealed in the generalised gauge transformations they introduce, understanding briefly how gauge fields and gauge invariance are implemented in a holographic quantum error correction code seems essential. New degrees of freedom on the links of the holographic tensor network are being introduced to that end, and additional connections to further copies of the holographic code are implemented by suitable isometries. In the case of double field theory such degrees of freedom on the links must be extended even further, considering the topological properties of T-duality. In the non-doubled case boundary regions allow the reconstruction of bulk algebras with central elements in the interior edges of the entanglement edge. In the case of double field theory bulk algebras are further extended leading to new error correction codes, previously unavailable. A tensor network has an upper bound for the amount of entanglement the state described by it can have, and this is based on the minimal cut dividing the network. This upper bound is saturated for connected regions in certain classes of holographic states. The case of planar graphs with non-positive curvature has been described in [17]. In order for a circuit interpretation of a network of perfect tensors to be valid, according to [17] it must satisfy three criteria. The first is the covering criterium, namely that to each edge (contracted or uncontracted index) is assigned a directionality. This condition is required in order to interpret the direction in which each tensor in the network processes information and hence to meaningfully define the input and the output indices. The second condition is the so called flow condition, which implies that each tensor has an equal number of incoming and outgoing indices. This is required for the interpretation that every tensor is a unitary gate. The last condition, namely the acyclicity condition is however rather special. It is already noted in [17] that this condition is non-local and is demanded so that the order of the application of the operations in the network to be consistent. Inconsistencies in this interpretation would be the presence of a closed time-like curve in the circuit picture. The assumptions made in [17] require for the graph to be, first, a planar embedding, namely the tensor network to be laid out in a planar form, the boundary of the network being a simple boundary of the embedding. Second, the tensors are required to be perfect, having an even number of legs and being unitary along any balanced distribution of legs. Finally, the network was expected to represent an AdS bulk and hence corresponding to a network equivalent of the AdS negative curvature. This implies that the distance function between two nodes of the network has no local maxima away from the boundary. Thinking in terms of the acyclic condition, which is a non-local property, it has been shown in [17] that the presence of a cycle implies the existence of an interior local maximum for the labelling. The proof goes, according to [17] as follows. Let there be a cycle C𝐶Citalic_C in the construction of the tensor network. The node label values immediately in the interior of the loop will be larger or smaller (depending on the orientation of C𝐶Citalic_C) than those immediate to the exteriors. In the case of C𝐶Citalic_C counterclockwise we may chose a node in the interior of C𝐶Citalic_C with the lowest possible label. In the case of this note, the label is smaller than those of all its neighbours including those in the exterior of C𝐶Citalic_C, which means it contradicts the assumption that it is defined based on the graph distance function and its properties. In the clockwise case, we can chose a node in the interior of C𝐶Citalic_C with the largest possible label. In this case it represents an interior maximum for the distance function and hence the surface homeomorphic to the disc cannot be negatively curved. Fascinatingly enough, precisely this acyclicity condition cannot uphold in the case of T-duality in the bulk. But it is well known from [35] and [36] that the T-dual of the AdS spacetime is the de-Sitter spacetime and hence abandoning the acyclic condition introduces into our network the cosmologically relevant de-Sitter space. In order for this to become clear let me first follow the results of ref. [26] regarding the inclusion of gauge fields in the tensor network representation, so that in the next chapter I can bring plausibility arguments for the statement above. In the case of lattice gauge theories, there appears the requirement of additional degrees of freedom on the links of the discrete graph model in order to describe the associated gauge fields. The various holonomies arise as paths through the lattice and the Gauss constraint provide a valid gauge interpretation. In the context of the holographic code in order to treat bulk gauge theories we need to introduce additional degrees of freedom on the links of the tensor network corresponding to the pentagon code. The tensor associated to the pentagon tiling has a total of six indices, five of them being associated to the network and away from the boundary they are connected to nearby tensors. Every such tensor also has an uncontracted index associated with local bulk degree of freedom. When T𝑇Titalic_T is a perfect tensor, it describes an isometry from any three legs to the others, then an operator 𝒪𝒪\mathcal{O}caligraphic_O acting on any bulk input may be transported along three of the output legs to the three neighbouring tensors. This procedure together with the negative curvature assumption allows us to transport local bulk operators up to the boundary because each tensor has at least three legs pointing towards the boundary. The additional degrees of freedom modelling bulk gauge fields can be introduced by adding a three index tensor Gi⁢j⁢ksubscript𝐺𝑖𝑗𝑘G_{ijk}italic_G start_POSTSUBSCRIPT italic_i italic_j italic_k end_POSTSUBSCRIPT

the first term is the leading Ryu-Takayanagi term, which is local on the entangling surface and state independent, the second term is the bulk entropy in the entanglement wedge E⁢(A)𝐸𝐴E(A)italic_E ( italic_A ) defined by the Ryu-Takayanagi minimal surface and therefore generally non-local and non-linearly dependent on the bulk state, and the third term is an additional quantum correction to the Ryu-Takayanagi area which is both local on the minimal surface and linear in the bulk state. The last term appears to also originate from a quantum error correction code which is based on the operators 𝒪𝒪\mathcal{O}caligraphic_O associated precisely to the boundary between the entanglement wedge of the boundary area A𝐴Aitalic_A and that of its complement A¯¯𝐴\bar{A}over¯ start_ARG italic_A end_ARG [34]. Once the form of this correction was known, it has been noticed that it can be derived from holographic quantum error correction codes as well [26]. The operators 𝒪𝒪\mathcal{O}caligraphic_O must be reconstructible from both A𝐴Aitalic_A and A¯¯𝐴\bar{A}over¯ start_ARG italic_A end_ARG and hence the operator itself must lie in the centre of either reconstructed algebra. Terms like the first and the third in the entropy formula are related to aspects of the code derived from the values of the operators in this centre [26]. The minimal area computation in the Ryu Takayanagi formalism is translated into the calculation of the so called ”greedy” surface [26] in the context of holographic codes. MERA tensor networks also realise a hyperbolic geometry and entropy bounds as those found in holographic discussions. The description of such tensor networks rely on the so called perfect tensors which arise in the expansion of a pure state describing 2⁢n2𝑛2n2 italic_n v𝑣vitalic_v-dimensional spins in a suitable basis

A

4⁢cos2⁡(π3)=14superscript2𝜋314\cos^{2}\big{(}\frac{\pi}{3}\big{)}=14 roman_cos start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( divide start_ARG italic_π end_ARG start_ARG 3 end_ARG ) = 1 hence the trivial case n=3𝑛3n=3italic_n = 3 in the

function f:M→ℝ:𝑓→𝑀ℝf:M\rightarrow{\mathbb{R}}italic_f : italic_M → blackboard_R which is (i) strictly

4444-manifolds into the real interval [0,1)⊂ℝ01ℝ[0,1)\subset{\mathbb{R}}[ 0 , 1 ) ⊂ blackboard_R. But 𝖬𝖺𝗇4superscript𝖬𝖺𝗇4{\mathsf{M}}{\mathsf{a}}{\mathsf{n}}^{4}sansserif_Man start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT

(0,1]⊂ℝ01ℝ(0,1]\subset{\mathbb{R}}( 0 , 1 ] ⊂ blackboard_R. Consequently they are always strictly positive

Examples in the simply connected case are M=S4,ℝ4,R4𝑀superscript𝑆4superscriptℝ4superscript𝑅4M=S^{4},{\mathbb{R}}^{4},R^{4}italic_M = italic_S start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT , blackboard_R start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT , italic_R start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT (this latter is

C

This point was examined also in Appendix A and Appendix B. In Appendix A we confirmed that the result given in the main text (where we did not fix a specific hypersurface), is indeed reproduced when a specific choice of hypersurface is done. The specific hypersurface and discretisation considered in Appendix A is that of christodoulou_planck_2016 , which this work follows up on. In Appendix B we show that any boost angle between two timelike vectors will scale monotonically with X≡T/m𝑋𝑇𝑚X\equiv T/mitalic_X ≡ italic_T / italic_m, as well as with T𝑇Titalic_T and m𝑚mitalic_m separately. Since the transition concerns essentially the flipping of the sign of the extrinsic curvature encoded in the boost angles at the sphere ΔΔ\Deltaroman_Δ, this gives a further argument that one can indeed hope to arrive at scaling estimates, such as the one we arrive at here, independently from the choice of hypersurface by considering geometrical properties of the exterior spacetime.

A first attempt to implement this program concretely was given in christodoulou_planck_2016 using the Lorentzian EPRL amplitudes in the context of covariant Loop Quantum Gravity (LQG). In this work, we complete the calculation laid out in christodoulou_planck_2016 . We estimate that the crossing time scales linearly with the mass. We show that in our setting and approximation the lifetime depends on the spread of the quantum state. The choice of a balanced semiclassical state gives an exponential scaling of the lifetime in the mass squared. The calculation laid out in christodoulou_planck_2016 was built on a 2-complex without bulk faces. Our analysis is limited to this case, of 2-complexes without bulk faces. This is a significant limitation and in particular it is not clear how our estimates will be affected under refinements. The calculation presented here should be understood as a step towards a fuller calculation which should involve large 2-complexes with interior faces and ideally also a refinement procedure to explore the continuum limit.

In Section II we discuss how estimates for these Tcsubscript𝑇𝑐T_{c}italic_T start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT and τ𝜏\tauitalic_τ can be read from the transition amplitude (23), a functional of the boundary geometry. The estimates for the characteristic time scales should be independent from the choice of interior boundary ℬℬ\mathcal{B}caligraphic_B: the predictions of quantum theory are independent from where we set the boundary between the quantum and the classical systems, provided that the choice is such that the classical system does not include parts where quantum phenomena cannot be disregarded.

The independence of our estimates on the choice of ℬℬ\mathcal{B}caligraphic_B is an encouraging sign. But, we warn the reader this is a first result and not a generic conclusion, that should be read in the context of the specific task set out in this manuscript: to estimate the scaling of the timescales we have defined with the mass.

The calculation we presented completes the task set out in christodoulou_planck_2016 and is based on the techniques detailed in gravTunn . We have defined and discussed the timescales characterizing the geometry transition of a trapped to an anti–trapped region and provided estimates using covariant Loop Quantum Gravity. The crossing time Tcsubscript𝑇𝑐T_{c}italic_T start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT characterizes the duration of the process. We find that quantum theory suggests that it scales linearly with the mass. The lifetime τ𝜏\tauitalic_τ is a much larger time scale, which we interpret as the time at which it becomes likely that the transition takes place. The geometry transition is governed by the boundary data on the ‘corner’ of the lens region and may be understood as coming down to flipping the sign of the extrinsic curvature. One significant improvement from the calculation set out in christodoulou_planck_2016 is that we arrive at estimates of the crossing time and lifetime without fixing a specific boundary hypersurface.

C

In our large-Nfsubscript𝑁fN_{\rm f}italic_N start_POSTSUBSCRIPT roman_f end_POSTSUBSCRIPT QED calculation, these

to one of the e→e⁢e¯⁢e→𝑒𝑒¯𝑒𝑒e\to e\bar{e}eitalic_e → italic_e over¯ start_ARG italic_e end_ARG italic_e diagrams of

momentum fractions in the produced e⁢e¯𝑒¯𝑒e\bar{e}italic_e over¯ start_ARG italic_e end_ARG pair are yesubscript𝑦𝑒y_{e}italic_y start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT and

be related to the e→e⁢e¯⁢e→𝑒𝑒¯𝑒𝑒e{\to}e\bar{e}eitalic_e → italic_e over¯ start_ARG italic_e end_ARG italic_e diagrams

Each of these diagrams contain a virtual e⁢e¯𝑒¯𝑒e\bar{e}italic_e over¯ start_ARG italic_e end_ARG loop.

D

\Sigma,\Gamma,\phi}\left((\vec{q},q),(\vec{\lambda},0)\right).script_V start_POSTSUBSCRIPT roman_Σ , roman_Γ , italic_ϕ end_POSTSUBSCRIPT ( over→ start_ARG italic_q end_ARG , over→ start_ARG italic_λ end_ARG ) ≃ script_V start_POSTSUBSCRIPT roman_Σ , roman_Γ , italic_ϕ end_POSTSUBSCRIPT ( ( over→ start_ARG italic_q end_ARG , italic_q ) , ( over→ start_ARG italic_λ end_ARG , 0 ) ) .

One of the main ingredients in the proof of this theorem is the connectedness of the ind-group MorΓ⁢(Σ∗,G)subscriptMorΓsuperscriptΣ𝐺{\rm Mor}_{\Gamma}(\Sigma^{*},G)roman_Mor start_POSTSUBSCRIPT roman_Γ end_POSTSUBSCRIPT ( roman_Σ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT , italic_G ) consisting of ΓΓ\Gammaroman_Γ-equivariant morphisms from Σ∗superscriptΣ\Sigma^{*}roman_Σ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT to G𝐺Gitalic_G (cf. Theorem 9.5), where Σ∗superscriptΣ\Sigma^{*}roman_Σ start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT is a ΓΓ\Gammaroman_Γ-stable affine open subset of ΣΣ\Sigmaroman_Σ. Another important ingredient is the Uniformization Theorem for the stack of 𝒢𝒢\mathscr{G}script_G-torsors on the parahoric Bruhat-Tits group scheme 𝒢𝒢\mathscr{G}script_G due to Heinloth [He]; in fact, its parabolic analogue (cf. Theorem 11.3). Finally, yet another ingredient is the splitting of the central extension of the twisted loop group G⁢(𝔻q×)Γq𝐺superscriptsuperscriptsubscript𝔻𝑞subscriptΓ𝑞G(\mathbb{D}_{q}^{\times})^{\Gamma_{q}}italic_G ( blackboard_D start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT start_POSTSUPERSCRIPT × end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT roman_Γ start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT end_POSTSUPERSCRIPT over

In fact, a stronger version of Propagation Theorem is proved (cf. Theorem 4.3 and Corollary 4.5 (b)). Even though, we generally follow the argument given in [Be, Proposition 2.3], in our equivariant setting we need to generalize some important ingredients.

L^⁢(𝔤,Γq)≥0^𝐿superscript𝔤subscriptΓ𝑞absent0\hat{L}(\mathfrak{g},\Gamma_{q})^{\geq 0}over^ start_ARG italic_L end_ARG ( fraktur_g , roman_Γ start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT ≥ 0 end_POSTSUPERSCRIPT. To prove this, follow the same argument as in [LS, Proof of Proposition 4.7] and the construction of the projective representation of G⁢((zq))Γq𝐺superscriptsubscript𝑧𝑞subscriptΓ𝑞G((z_{q}))^{\Gamma_{q}}italic_G ( ( italic_z start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ) ) start_POSTSUPERSCRIPT roman_Γ start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT end_POSTSUPERSCRIPT given by (subsequent) Theorem 10.3.

We continue to have the same assumptions on G,Γ,Σ𝐺ΓΣG,\Gamma,\Sigmaitalic_G , roman_Γ , roman_Σ as in the beginning of Section 9. We recall some results due to Heinloth [He] (conjectured by Pappas-Rapoport [PR1, PR2]) only in the generality we need and in the form suitable for our purposes. In particular, we recall the uniformization theorem due to Heinloth for the parahoric Bruhat-Tits group schemes 𝒢𝒢\mathscr{G}script_G in our setting. We introduce the moduli stack 𝒫⁢a⁢r⁢b⁢u⁢n𝒢𝒫𝑎𝑟𝑏𝑢subscript𝑛𝒢\mathscr{P}arbun_{\mathscr{G}}script_P italic_a italic_r italic_b italic_u italic_n start_POSTSUBSCRIPT script_G end_POSTSUBSCRIPT of quasi-parabolic 𝒢𝒢\mathscr{G}script_G-torsors over Σ¯¯Σ\bar{\Sigma}over¯ start_ARG roman_Σ end_ARG and construct the line bundles over

B

^{2}\Phi_{Q}\Bigg{\}}.+ 40 roman_Φ start_POSTSUBSCRIPT italic_Q end_POSTSUBSCRIPT roman_csc start_POSTSUPERSCRIPT 5 end_POSTSUPERSCRIPT roman_Φ start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT roman_cot roman_Φ start_POSTSUBSCRIPT italic_Q end_POSTSUBSCRIPT [ 12 roman_Φ start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT roman_cos roman_Φ start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT - 9 roman_sin roman_Φ start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT - roman_sin ( 3 roman_Φ start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT ) ] - 224 roman_Φ start_POSTSUBSCRIPT italic_Q end_POSTSUBSCRIPT roman_cot roman_Φ start_POSTSUBSCRIPT italic_Q end_POSTSUBSCRIPT roman_csc start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT roman_Φ start_POSTSUBSCRIPT italic_Q end_POSTSUBSCRIPT } .

The integral over the three-dimensional angle θ𝜃\thetaitalic_θ can then be performed analytically, which yields a more manageable expression,

As mentioned in the main text, the starting point in our N3LO computation is the two-loop HTL pressure as written down in eq. (34) of ref. [27], where we convert the sum-integrals into ordinary 3+1 dimensional integrals because we work at T=0𝑇0T=0italic_T = 0. The full expression is rather unwieldy when written in full, and is not reproduced here, but we note that the simplifications outlined in the main text make extracting the double logarithm significantly easier. An additional useful result is that, in the notation of ref. [27], the propagator ΔX=𝒪⁢(αs)subscriptΔ𝑋𝒪subscript𝛼𝑠\Delta_{X}=\mathcal{O}(\alpha_{s})roman_Δ start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT = caligraphic_O ( italic_α start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ), which allows us to discard a number of higher-order terms.

where x≡P/Q𝑥𝑃𝑄x\equiv P/Qitalic_x ≡ italic_P / italic_Q and θ𝜃\thetaitalic_θ is the angle between 𝒑𝒑\boldsymbol{p}bold_italic_p and 𝒒𝒒\boldsymbol{q}bold_italic_q.

We present here some details of the calculation discussed in the main text. In particular, to carry out the four-momentum integrations such as ∫d4⁡Psuperscriptd4𝑃\int\operatorname{d}\!^{4}P∫ roman_d start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT italic_P, we find it very useful to change variables from (P0,|𝒑|)superscript𝑃0𝒑(P^{0},|\boldsymbol{p}|)( italic_P start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT , | bold_italic_p | ) to (P,ΦP)𝑃subscriptΦ𝑃(P,\Phi_{P})( italic_P , roman_Φ start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT ), where P𝑃Pitalic_P is the magnitude of the Euclidean four-vector and ΦPsubscriptΦ𝑃\Phi_{P}roman_Φ start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT is the four-dimensional polar angle, tan⁡ΦP=|𝒑|/P0subscriptΦ𝑃𝒑superscript𝑃0{\tan\Phi_{P}=|\boldsymbol{p}|/P^{0}}roman_tan roman_Φ start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT = | bold_italic_p | / italic_P start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT. The particular expression that we use in these coordinates is only valid for 0≤ΦP≤π/20subscriptΦ𝑃𝜋2{0\leq\Phi_{P}\leq\pi/2}0 ≤ roman_Φ start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT ≤ italic_π / 2, but due to the symmetry of the self energy under P0↦−P0maps-tosuperscript𝑃0superscript𝑃0P^{0}\mapsto-P^{0}italic_P start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT ↦ - italic_P start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT, cf. eqs. (5) and (6) of the main text, one may use the measure

A

To verify the extended exclusion statistics, we compute the statistical weight in an extensionHu et al. (2017, 2018) of the Levin-Wen modelLevin and Wen (2005)—a discrete topological quantum field theory (TQFT) for topological phases, and derive the corresponding statistics parameters. Namely, we adopt the extended Levin-Wen model of topological orders with gapped boundaries recently constructed by two of usHu et al. (2017, 2018) , and compute the state counting of multiple anyon excitations in this model. The state counting so obtained leads to the extended exclusion statistics of the anyons under consideration.

We develop a systematic basis construction for systems of non-Abelian anyons on a disk with a gapped boundary. From the basis construction, we can directly read off the corresponding statistics parameters and hence derive the statistical weight.

The computing results motivate us to develop a trustworthy universal method of basis construction of any multi-anyon Hilbert space on any Riemann surface with boundaries. Using this method of basis, we can easily read off the statistical weight being studied. The method of basis also reveals the true meaning of pseudo-species and how boundary conditions affect the pseudo-species. Our main results are as follows.

It will turn out that method 2) counting by state basis is the most convenient and systematic method to extract the extended statistical weight and deepen our understanding of pseudo-species.

The paper is structured as follows. Section 2 introduces our extended anyonic exclusion statistics, accompanied by briefing of the concept of anyon exclusions statistics. Section 3 computes the state counting and derive the statistical weight of doubled Fibonacci anyons on a disk with a gapped boundary, using the extended Levin-Wen model. Motivated by the results of Section 3, Section 4 systematically constructs the bases of the Hilbert spaces of Fibonacci system on a disk, from which the statistics parameters can be immediately read off. Section 5 addresses the physical meaning of pseudo-species and the boundary effect on state counting and pseudo-species. Section 6 generalizes the story to surfaces with multiple gapped boundaries by two topological operations. Section 7 brings up an important equivalence relation between statistical weight and fusion basis. Appendices collect certain reviews, details, and more examples, including an Abelian example—the ℤ2subscriptℤ2\mathbb{Z}_{2}blackboard_Z start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT toric code order, which further corroborate our results.

B

LE⁢H⁢(h):=(Scalh−2⁢Λ)⁢νhassignsubscript𝐿𝐸𝐻ℎsubscriptScalℎ2Λsubscript𝜈ℎL_{EH}(h):=({\rm Scal}_{h}-2\Lambda)\nu_{h}italic_L start_POSTSUBSCRIPT italic_E italic_H end_POSTSUBSCRIPT ( italic_h ) := ( roman_Scal start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT - 2 roman_Λ ) italic_ν start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT with cosmological constant

where N0subscript𝑁0N_{0}italic_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT denotes the unit normal to ∂M𝑀\partial M∂ italic_M with respect to the

ℝ=⨆x∈ℝ{x}.ℝsubscriptsquare-union𝑥ℝ𝑥{\mathbb{R}}=\bigsqcup_{x\in{\mathbb{R}}}\{x\}\>\>\>.blackboard_R = ⨆ start_POSTSUBSCRIPT italic_x ∈ blackboard_R end_POSTSUBSCRIPT { italic_x } .

is valid hence θC=(4⁢Λ−2⁢Λ)⁢νh⁢(X,⋅)=2⁢Λ⁢νh⁢(X,⋅)subscript𝜃𝐶4Λ2Λsubscript𝜈ℎ𝑋⋅2Λsubscript𝜈ℎ𝑋⋅\theta_{C}=(4\Lambda-2\Lambda)\nu_{h}(X,\cdot)=2\Lambda\nu_{h}(X,\cdot)italic_θ start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT = ( 4 roman_Λ - 2 roman_Λ ) italic_ν start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT ( italic_X , ⋅ ) = 2 roman_Λ italic_ν start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT ( italic_X , ⋅ ) and (ii)

Λ∈ℝΛℝ\Lambda\in{\mathbb{R}}roman_Λ ∈ blackboard_R and consider its variation with respect to a 1111-parameter

D

The papers [34, 11, 12, 51] also prove stabilization theorems in specific examples which are similar in spirit to and provide inspiration for the one in this paper.

Moreover, an admissible almost complex structure for the smoothed product is no longer split, so we cannot quite argue via the projection to the ℂℂ\mathbb{C}blackboard_C factor.

the relevant curves in X^^𝑋\widehat{X}over^ start_ARG italic_X end_ARG are in correspondence with curves in ℝ×Yℝ𝑌\mathbb{R}\times Yblackboard_R × italic_Y.

For one, the product X×ℂ𝑋ℂX\times\mathbb{C}italic_X × blackboard_C is not strictly speaking a symplectic filling, due to its noncompactness. If we replace ℂℂ\mathbb{C}blackboard_C with the two-ball B2⁢(a)superscript𝐵2𝑎B^{2}(a)italic_B start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_a ) of area a𝑎aitalic_a, then X×B2⁢(a)𝑋superscript𝐵2𝑎X\times B^{2}(a)italic_X × italic_B start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_a ) is compact but has corners. We can smooth the corners to obtain an honest symplectic filling, but the smoothing process is not entirely canonical. In particular, different choices of smoothings lead to different Reeb dynamics.

Naively, the main point for stabilization is that, for a split almost complex structure, curves in the product X×ℂ𝑋ℂX\times\mathbb{C}italic_X × blackboard_C are given by a product of curves in each factor, and curves in ℂℂ\mathbb{C}blackboard_C effectively must be constant.

D

As seen in Fig. 1(c), the triplet stars are mapped to the prime meridian at t=2⁢k⁢π/4𝑡2𝑘𝜋4t=2k\pi/4italic_t = 2 italic_k italic_π / 4 , the 90∘⁢W⁢(E)superscript90𝑊𝐸90^{\circ}W(E)90 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT italic_W ( italic_E ) meridian at t=(2⁢k+1)⁢π/4𝑡2𝑘1𝜋4t=(2k+1)\pi/4italic_t = ( 2 italic_k + 1 ) italic_π / 4 (k𝑘kitalic_k is integer).

Moreover, focusing on one period of time, all stars of the mutiplet will jointly form a pattern that is plane symmetry around the prime meridian plane (Fig. 2(b)).

Fixed the phase parameter φ𝜑\varphiitalic_φ at a special value, since stars of the pseudo spin are independent of time evolution, they are fixed at particular positions (Fig. 1(d)). Meanwhile, each set of the triplet stars is plane symmetry around the equatorial plane and the prime meridian plane at a whole period T=π𝑇𝜋T=\piitalic_T = italic_π, and each set still keeps the rotational symmetry at every single time and phase parameter value.

Fixed the time at a special value, we can distinctly find that one set of the triplet stars and the others are 180∘superscript180180^{\circ}180 start_POSTSUPERSCRIPT ∘ end_POSTSUPERSCRIPT rotational symmetric around the x𝑥xitalic_x-axis (the intersecting line of the prime meridian plane and the equatorial plane) (Fig. 1(b)).

Figure 1 shows the stars of the two spin-1/2121/21 / 2 particles represented in Bloch sphere with real phase parameter φ𝜑\varphiitalic_φ, time evolution and δ=0𝛿0\delta=0italic_δ = 0. Without time evolution, because the roots (z1subscript𝑧1z_{1}italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT) in our example are always real, with the variety of the real phase parameter φ𝜑\varphiitalic_φ, each set of the triplet stars is mapped to the north or south prime meridian. Intuitively, the stars of the pseudo spin only cover the north prime meridian.

B

(a), 9⁢μ9μ9\upmu9 roman_μs (b), 18⁢μ18μ18\upmu18 roman_μs (c), 45⁢μ45μ45\upmu45 roman_μs (d), 65⁢μ65μ65\upmu65 roman_μs

at 0⁢μ0μ0\upmu0 roman_μs (a), 9⁢μ9μ9\upmu9 roman_μs (b), 45⁢μ45μ45\upmu45 roman_μs (c), 65⁢μ65μ65\upmu65 roman_μs (d).

9⁢μ9μ9\upmu9 roman_μs (b), 18⁢μ18μ18\upmu18 roman_μs (c), 45⁢μ45μ45\upmu45 roman_μs (d), 65⁢μ65μ65\upmu65 roman_μs (e),

at 0⁢μ0μ0\upmu0 roman_μs (a), 9⁢μ9μ9\upmu9 roman_μs (b), 45⁢μ45μ45\upmu45 roman_μs (c), 65⁢μ65μ65\upmu65 roman_μs (d)

(a), 9⁢μ9μ9\upmu9 roman_μs (b), 18⁢μ18μ18\upmu18 roman_μs (c), 45⁢μ45μ45\upmu45 roman_μs (d), 65⁢μ65μ65\upmu65 roman_μs

B

If (d1−z1)⁢(z1−y1)=0subscript𝑑1subscript𝑧1subscript𝑧1subscript𝑦10(d_{1}-z_{1})(z_{1}-y_{1})=0( italic_d start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ( italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) = 0 and (d2−z2)⁢(z2−y2)=0subscript𝑑2subscript𝑧2subscript𝑧2subscript𝑦20(d_{2}-z_{2})(z_{2}-y_{2})=0( italic_d start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT - italic_z start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ( italic_z start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT - italic_y start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) = 0, as there is no pairing, we may assume that d1=z1subscript𝑑1subscript𝑧1d_{1}=z_{1}italic_d start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and z2=y2subscript𝑧2subscript𝑦2z_{2}=y_{2}italic_z start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_y start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT (or z1=y1subscript𝑧1subscript𝑦1z_{1}=y_{1}italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and d2=z2subscript𝑑2subscript𝑧2d_{2}=z_{2}italic_d start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_z start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT, which is treated similarly), so z1=d1subscript𝑧1subscript𝑑1z_{1}=d_{1}italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_d start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and x2=d2subscript𝑥2subscript𝑑2x_{2}=d_{2}italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_d start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT are fixed, z2subscript𝑧2z_{2}italic_z start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT has O⁢(N3)𝑂subscript𝑁3O(N_{3})italic_O ( italic_N start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) choices and x1=y1subscript𝑥1subscript𝑦1x_{1}=y_{1}italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT has O⁢(N2)𝑂subscript𝑁2O(N_{2})italic_O ( italic_N start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) choices, which implies #⁢S≲N2⁢N3less-than-or-similar-to#𝑆subscript𝑁2subscript𝑁3\#S\lesssim N_{2}N_{3}# italic_S ≲ italic_N start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_N start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT.

If moreover ιa=−subscript𝜄𝑎\iota_{a}=-italic_ι start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT = -, then we have the stronger bound

(a) Suppose ι3=+subscript𝜄3\iota_{3}=+italic_ι start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = +, then we have that

If N(1)∼Nasimilar-tosuperscript𝑁1subscript𝑁𝑎N^{(1)}\sim N_{a}italic_N start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT ∼ italic_N start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT and ιa=−subscript𝜄𝑎\iota_{a}=-italic_ι start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT = -, then we have

(b) Suppose ι3=−subscript𝜄3\iota_{3}=-italic_ι start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = -, then similarly we have that

D

Notation: For the rest of this section, we will use the following notation: Arg⁢(w)Arg𝑤\text{Arg}(w)Arg ( italic_w ), Argm⁢(w)subscriptArg𝑚𝑤\text{Arg}_{m}(w)Arg start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_w ) and Logm⁢(w)subscriptLog𝑚𝑤\text{Log}_{m}(w)Log start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_w ) are defined as in Definition 3.4.4; ArgP⁢(w)subscriptArg𝑃𝑤\text{Arg}_{P}(w)Arg start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT ( italic_w ) and LogP⁢(w)subscriptLog𝑃𝑤\text{Log}_{P}(w)Log start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT ( italic_w ) are the principal branches (i.e. with ArgP⁢(w)∈(−π,π]subscriptArg𝑃𝑤𝜋𝜋\text{Arg}_{P}(w)\in(-\pi,\pi]Arg start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT ( italic_w ) ∈ ( - italic_π , italic_π ]). When we switch coordinates to z=1/w𝑧1𝑤z=1/witalic_z = 1 / italic_w, we denote by Logp⁢(z)subscriptLog𝑝𝑧\text{Log}_{p}(z)Log start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( italic_z ) and Argp⁢(z)subscriptArg𝑝𝑧\text{Arg}_{p}(z)Arg start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( italic_z ) the branches with [−π,π)𝜋𝜋[-\pi,\pi)[ - italic_π , italic_π ). We then have the relations Logp⁢(z)=−LogP⁢(w)subscriptLog𝑝𝑧subscriptLog𝑃𝑤\text{Log}_{p}(z)=-\text{Log}_{P}(w)Log start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( italic_z ) = - Log start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT ( italic_w ) and Argp⁢(z)=−ArgP⁢(w)subscriptArg𝑝𝑧subscriptArg𝑃𝑤\text{Arg}_{p}(z)=-\text{Arg}_{P}(w)Arg start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( italic_z ) = - Arg start_POSTSUBSCRIPT italic_P end_POSTSUBSCRIPT ( italic_w ).

Now we finally define the electric and magnetic twistor coordinates. We will verify that the magnetic coordinate has the appropriate jumps on l±⁢(−2⁢i⁢m)subscript𝑙plus-or-minus2i𝑚l_{\pm}(-2\mathrm{i}m)italic_l start_POSTSUBSCRIPT ± end_POSTSUBSCRIPT ( - 2 roman_i italic_m ), matching with the corresponding jumps of the Ooguri-Vafa coordinates (see Proposition 2.2). For the definition of the magnetic coordinate, we will need to assume for now that the Stokes matrix element given by b⁢(ξ)𝑏𝜉b(\xi)italic_b ( italic_ξ ) does not vanish for ξ∈ℍ−m𝜉subscriptℍ𝑚\xi\in\mathbb{H}_{-m}italic_ξ ∈ blackboard_H start_POSTSUBSCRIPT - italic_m end_POSTSUBSCRIPT (this will be shown in Section 3.7).

For completeness, and because we will need it in the asymptotic computation for the magnetic twistor coordinate, we write the other normalization constants βi⁢(tj,ξ)subscript𝛽𝑖subscript𝑡𝑗𝜉\beta_{i}(t_{j},\xi)italic_β start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_t start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT , italic_ξ ). These are:

Here we develop some of the preliminary notation and computations that we will need for the main asymptotic computation of the twistor magnetic coordinate.

We then conclude that the magnetic coordinate has the same jumps as the magnetic coordinate of the Ooguri-Vafa space.

C

Waves in context types i and iii are narrowband (clear sinusoidal waveforms in time-domain data), whereas the waves in context type ii are a superposition of waves, without clear sinusoidal waveforms. We do not elaborate further on the whistler-mode waves in context types i or ii, because the near-fc⁢esubscript𝑓𝑐𝑒f_{ce}italic_f start_POSTSUBSCRIPT italic_c italic_e end_POSTSUBSCRIPT waves observed by Solar Probe are not limited to association with shocks or stream interaction regions, and they are narrowband rather than broadband.

The near-fc⁢esubscript𝑓𝑐𝑒f_{ce}italic_f start_POSTSUBSCRIPT italic_c italic_e end_POSTSUBSCRIPT wave observations from Parker Solar Probe presented here are distinct from the vast majority of those reported in these prior studies in several regards: (i) the wave frequencies are considerably higher, centered on 0.7 fc⁢esubscript𝑓𝑐𝑒f_{ce}italic_f start_POSTSUBSCRIPT italic_c italic_e end_POSTSUBSCRIPT or fc⁢esubscript𝑓𝑐𝑒f_{ce}italic_f start_POSTSUBSCRIPT italic_c italic_e end_POSTSUBSCRIPT in Parker Solar Probe data, compared to 0.1<f/fc⁢e<0.30.1𝑓subscript𝑓𝑐𝑒0.30.1<f/f_{ce}<0.30.1 < italic_f / italic_f start_POSTSUBSCRIPT italic_c italic_e end_POSTSUBSCRIPT < 0.3 reported in prior studies (Lengyel-Frey et al., 1996; Lin et al., 1998; Moullard et al., 2001; Lacombe et al., 2014; Kajdič et al., 2016; Stansby et al., 2016; Tong et al., 2019), (ii) the waves are electrostatic up to the sensitivity of the FIELDS data, whereas most prior studies identified whistler-mode waves using exclusively magnetic field data (Beinroth & Neubauer, 1981; Lacombe et al., 2014; Kajdič et al., 2016; Tong et al., 2019), (iii) the waves observed by Parker Solar Probe are both narrow band and frequently observed, observed up to 30% of the time when magnetic field conditions favorable to wave growth exist (prior studies of non-turbulence whistler-mode waves show much lower detection rates (e.g. (Lacombe et al., 2014; Tong et al., 2019))), and (iv) the near-fc⁢esubscript𝑓𝑐𝑒f_{ce}italic_f start_POSTSUBSCRIPT italic_c italic_e end_POSTSUBSCRIPT waves observed by Parker Solar Probe often include electron Bernstein modes, which have previously been reported in the solar wind only near shocks (Wilson et al., 2010) or in conjunction with the AMPTE Li ion release (Baumgaertel & Sauer, 1989).

Whistler-mode waves in the free solar wind are often thought to be generated by electron temperature anisotropy and/or heat flux instabilities (those involving a beaming component) (Gary et al., 2005; Shaaban et al., 2018). Many numerical studies have focused on this issue (e.g. (Vocks et al., 2005; Saito & Gary, 2007; Seough et al., 2015) and references therein), and data analyses of the radial evolution of solar wind electron distribution functions are frequently interpreted in terms of whistler-mode waves driving electron scattering (Walsh et al., 2013; Graham et al., 2017; Berčič et al., 2019). Further, observed correlations between whistler-mode wave detection and properties of electron distribution functions have been reported (Kajdič et al., 2016; Stansby et al., 2016; Tong et al., 2019).

The amplitude and number of detections of these near-fc⁢esubscript𝑓𝑐𝑒f_{ce}italic_f start_POSTSUBSCRIPT italic_c italic_e end_POSTSUBSCRIPT waves increases significantly as Parker Solar Probe approaches the Sun, suggesting that they play a role in the evolution of electron populations in the near-Sun plasma environment. Correlations are observed between the detection of these waves and properties of solar wind electrons, which are composed of a core, halo, and strahl (the population of electrons escaping the solar corona) (Montgomery et al., 1968; Feldman et al., 1975; Pilipp et al., 1987; Maksimovic et al., 2005). These correlations suggest that these waves play a role in regulating the solar heat flux carried by electrons. Finally, the detection of these waves is found to be strongly correlated with the presence of low-turbulence radial solar wind magnetic fields.

Previous observations of plasma waves in the solar wind near fc⁢esubscript𝑓𝑐𝑒f_{ce}italic_f start_POSTSUBSCRIPT italic_c italic_e end_POSTSUBSCRIPT focused on whistler-mode waves. These waves were reported in three primary contexts: (i) associated with solar wind plasma boundaries such as shocks (e.g. (Wilson et al., 2009; Ramírez Vélez et al., 2012) and references therein) and stream interaction regions (Beinroth & Neubauer, 1981; Lin et al., 1998; Lengyel-Frey et al., 1996; Breneman et al., 2010), (ii) associated with the turbulent cascade of magnetic field fluctuations (e.g. (Lengyel-Frey et al., 1996; Bruno & Carbone, 2013; Narita et al., 2016) and references therein), and (iii) present in the free solar wind (Zhang et al., 1998; Lacombe et al., 2014; Stansby et al., 2016; Tong et al., 2019).

B

_{12}\mathbf{e}_{23}+b_{13}\mathbf{e}_{13}+b_{23}\mathbf{e}_{23}+b_{123}Isansserif_B = italic_b start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_b start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT bold_e start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + italic_b start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT bold_e start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + italic_b start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT bold_e start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_b start_POSTSUBSCRIPT 12 end_POSTSUBSCRIPT bold_e start_POSTSUBSCRIPT 23 end_POSTSUBSCRIPT + italic_b start_POSTSUBSCRIPT 13 end_POSTSUBSCRIPT bold_e start_POSTSUBSCRIPT 13 end_POSTSUBSCRIPT + italic_b start_POSTSUBSCRIPT 23 end_POSTSUBSCRIPT bold_e start_POSTSUBSCRIPT 23 end_POSTSUBSCRIPT + italic_b start_POSTSUBSCRIPT 123 end_POSTSUBSCRIPT italic_I and 𝖠𝖠\mathsf{A}sansserif_A is

Expanding 𝖠2superscript𝖠2\mathsf{A}^{2}sansserif_A start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT in components and equating (real) coefficients at

equation 𝖠2=𝖡superscript𝖠2𝖡\mathsf{A}^{2}=\mathsf{B}sansserif_A start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = sansserif_B and for numerical checks in general.

coefficients at same basis elements in 𝖠2=𝖡superscript𝖠2𝖡\mathsf{A}^{2}=\mathsf{B}sansserif_A start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = sansserif_B now we obtain the

the square 𝖠2superscript𝖠2\mathsf{A}^{2}sansserif_A start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT has been expanded in the orthonormal basis where

A

It is common to try to avoid such changes in artificial agents, machines, and industrial processes. When something changes, the entire system is taken offline and modified to fit the new situation. This process is costly and disruptive; adaptation similar to that in nature might make such systems more reliable and long-term, and thus cheaper to operate.

It is common to try to avoid such changes in artificial agents, machines, and industrial processes. When something changes, the entire system is taken offline and modified to fit the new situation. This process is costly and disruptive; adaptation similar to that in nature might make such systems more reliable and long-term, and thus cheaper to operate.

While natural systems cope with changing environments and embodiments well, they form a serious challenge for artificial systems. For instance, to stay reliable over time, gas sensing systems must be continuously recalibrated to stay accurate in a changing physical environment. Drawing motivation from nature, this paper introduced an approach based on continual adaptation. A recurrent neural network uses a sequence of previously seen gas recordings to form a representation of the current state of the sensors. It then modulates the skill of odor recognition with this context, allowing the system to adapt to sensor drift. Context models can thus play a useful role in lifelong adaptation to changing environments in artificial systems.

Experiments in this paper used the gas sensor drift array dataset [7]. The data consists of 10 sequential collection periods, called batches. Every batch contains between 161161161161 to 3,60036003{,}6003 , 600 samples, and each sample is represented by a 128-dimensional feature vector; 8 features each from 16 metal oxide-based gas sensors. These features summarizing the time series sensor responses are the raw and normalized steady-state features and the exponential moving average of the increasing and decaying transients taken at three different alpha values. The experiments used six gases, ammonia, acetaldehyde, acetone, ethylene, ethanol, and toluene, presented in arbitrary order and at variable concentrations. Chemical interferents were also presented to the sensors between batches, and the time between presentations varied, both of which contributed to further sensor variability. The dataset thus exemplifies sensor variance due to contamination and variable odor concentration in a controlled setting.

Sensor drift in industrial processes is one such use case. For example, sensing gases in the environment is mostly tasked to metal oxide-based sensors, chosen for their low cost and ease of use [1, 2]. An array of sensors with variable selectivities, coupled with a pattern recognition algorithm, readily recognizes a broad range of odors. The arrangement is called an artificial nose since it resembles the multiplicity of sensory neuron types in the nasal epithelium. However, while metal oxide-based sensors are economical and flexible, they are unstable over time. Changes to the response properties of sensors make it difficult to detect and identify odors in the long term, and sensors have to be recalibrated to compensate [3]. Recalibration requires collecting and labeling new samples, which is costly because a skilled operator is needed, and challenging because the experimental conditions need to be controlled precisely [3]. Recalibrating a model with unlabeled examples, called semisupervised learning, is a possible alternative but difficult to establish in practice.

D

Qubits can be manipulated in a desired fashion by excellent architect design in several physical devices, such as, quantum dots, cavity quantum electrodynamics, superconducting devices, Majorana fermions [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]. Manipulation of qubits in these devices seems promising in that one can make quantum logic gates and memory devices for various quantum information processing applications. Such devices require sufficiently short gate operation time combined with long coherent time [13, 14, 15, 16]. When a qubit is operated on by a classical bit, then its decay time is given by a relaxation time which is also supposed to be longer than the minimum time required to execute one quantum gate operation.

In most cases, compared to coherent time, the dephasing time of qubits in presence of noise is reduced by several orders of magnitude due to coupling of qubits to the environment. The reduction of dephasing time depends on the specific dynamical coupling sequence from where the principle of quantum mechanics is inevitably lost. Therefore, one might need to decouple the qubits from the environment and may consider more robust topological method to preserve a quantum state against noise, enabling robust quantum memory[17].

Hence, to make quantum computers, one needs to find an efficient and experimentally feasible algorithm that overcome the issues of undesired interactions of qubits to the surrounding environment. Interactions of qubits to the surrounding environment destroy the quantum coherence that lead to generate errors and loss of fidelity. In quantum computing language, this phenomenon is called decoherence. For example, experimental observations reported that in GaAs quantum dots, decoherence time, T2⋆≈10⁢n⁢ssuperscriptsubscript𝑇2⋆10𝑛𝑠T_{2}^{\star}\approx 10nsitalic_T start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT ≈ 10 italic_n italic_s and coherent time, T1≈0.1⁢m⁢ssubscript𝑇10.1𝑚𝑠T_{1}\approx 0.1msitalic_T start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≈ 0.1 italic_m italic_s, whereas for Si, T2⋆≈100⁢n⁢ssuperscriptsubscript𝑇2⋆100𝑛𝑠T_{2}^{\star}\approx 100nsitalic_T start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ⋆ end_POSTSUPERSCRIPT ≈ 100 italic_n italic_s and T1≈0.1⁢m⁢ssubscript𝑇10.1𝑚𝑠T_{1}\approx 0.1msitalic_T start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≈ 0.1 italic_m italic_s [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30]. There are several possible ways to overcome the issues of decoherence, as for example, fidelity recovery by applying error-correcting codes, decoherence free subspace coding, noiseless subsystem coding, dynamical decoupling from hot bath, numerical design of pulse sequences, that is more robust to experimental

Considering ay⁢(t)=az⁢(t)=0subscript𝑎𝑦𝑡subscript𝑎𝑧𝑡0a_{y}(t)=a_{z}(t)=0italic_a start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT ( italic_t ) = italic_a start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ( italic_t ) = 0 in 2 and utilizing Eq. 13, the influence of π𝜋\piitalic_π, CORSPE, SCORPSE, symmetric and assymetric pulses on the error correction and fidelity measurement of qubit under random telegraph noises is shown in Figs. (4) and (5). Here δ=0.125⁢am⁢a⁢x𝛿0.125subscript𝑎𝑚𝑎𝑥\delta=0.125a_{max}italic_δ = 0.125 italic_a start_POSTSUBSCRIPT italic_m italic_a italic_x end_POSTSUBSCRIPT, δ=0.25⁢am⁢a⁢x𝛿0.25subscript𝑎𝑚𝑎𝑥\delta=0.25a_{max}italic_δ = 0.25 italic_a start_POSTSUBSCRIPT italic_m italic_a italic_x end_POSTSUBSCRIPT and δ=0.5⁢am⁢a⁢x𝛿0.5subscript𝑎𝑚𝑎𝑥\delta=0.5a_{max}italic_δ = 0.5 italic_a start_POSTSUBSCRIPT italic_m italic_a italic_x end_POSTSUBSCRIPT are chosen(a),(b) and (c) of Figs. (4) and (5), respectively. In the regime of vanishing noise correlation time (i.e.,τc→0i.e.,\tau_{c}\rightarrow 0italic_i . italic_e . , italic_τ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT → 0), error is minimum shown in Fig. 4(a,b,c) or Fidelity is maximum shown in Fig. 5)(a,b,c) for π𝜋\piitalic_π pulse followed by SCORPE, CORPE, symmetric and asymmetric pulses. The small error or large fidelity for p⁢i𝑝𝑖piitalic_p italic_i pulse in the qubit operation is due to the fact that the single qubit under π−limit-from𝜋\pi-italic_π -pulse does not have enough time to drift into the direction of densely populated random noise (see Fig. 2). Note that the noise function is very dense in the vicinity of zero correlation time (see Fig.2) while noise function has no jumps in the vicinity of infinite correlation time (e.g., density of noise jumps decreases as τcsubscript𝜏𝑐\tau_{c}italic_τ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT increases from 2 to 3). The energy amplitude of the noise functions rapidly changes its sign between ±Δplus-or-minusΔ\pm\Delta± roman_Δ. Hence, in the vicinity of zero correlation time, the error is smaller or the fidelity is larger to be about 99%percent\%% for π−limit-from𝜋\pi-italic_π -pulse than all the other pulses (e.g., scorpe, corpe, symmetric and asymmetric).

a single bit-flip computational basis states in a noisy environment. The present work seek to identify different regimes of operating parameters in the designed control pulses that eliminate the series of phase and dynamical errors and induce the recovery of high fidelities. The calculations are restricted to only eliminate the phase errors, which are more robust due to stochastic time-varying amplitudes, appear in the model Hamiltonian. More precisely, the designed pulses are π𝜋\piitalic_π, CORSPE, SCORPSE, symmetric and asymmetric acting on a qubit in a random telegraph noise environment. Then checking the quantum gate errors at various noise correlation times as well as various energy amplitudes of noise strength as an indication of most efficient way to perform algorithm for quantum bit operations. The results of calculations shows that when the qubits are driven by pulses in the x-direction and the noises act in the z-direction then the symmetric pulse sequence along with small energy amplitude of noise strength, (i.e., Δ≈0.125⁢ℏ/am⁢a⁢xΔ0.125Planck-constant-over-2-pisubscript𝑎𝑚𝑎𝑥\Delta\approx 0.125\hbar/a_{max}roman_Δ ≈ 0.125 roman_ℏ / italic_a start_POSTSUBSCRIPT italic_m italic_a italic_x end_POSTSUBSCRIPT) induce less systematic errors than all the other pulses (π𝜋\piitalic_π, CORPSE, SCORPSE and asymmetric pulses). Such error analysis in the qubits is useful for the laboratory experiments operating at low temperatures where one can correct the systematic errors in a more efficient way by designing additional quantum gates in a physical device . On the other hand for the case of strong noise environments (i.e., energy amplitude of noise strength, Δ≈0.25⁢ℏ/am⁢a⁢xΔ0.25Planck-constant-over-2-pisubscript𝑎𝑚𝑎𝑥\Delta\approx 0.25\hbar/a_{max}roman_Δ ≈ 0.25 roman_ℏ / italic_a start_POSTSUBSCRIPT italic_m italic_a italic_x end_POSTSUBSCRIPT), may suitable for the experiments operating at room temperature, CORPSE pulse is the most efficient way to reduce the error.

A

σMp2⁢(1+1/z)⁢(1+2⁢z)1+z.𝜎superscriptsubscript𝑀𝑝211𝑧12𝑧1𝑧\displaystyle\frac{\sigma}{M_{p}^{2}}\frac{(1+1/z)(1+2z)}{1+z}\,.divide start_ARG italic_σ end_ARG start_ARG italic_M start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG divide start_ARG ( 1 + 1 / italic_z ) ( 1 + 2 italic_z ) end_ARG start_ARG 1 + italic_z end_ARG .

In Figures 3 and 4, we plot the free energy F𝐹Fitalic_F, Eq. (29), the expected value of energy ⟨E⟩delimited-⟨⟩𝐸\langle E\rangle⟨ italic_E ⟩, Eq. (30), the entropy S𝑆Sitalic_S, Eq. (31), and the specific heat CVsubscript𝐶𝑉C_{V}italic_C start_POSTSUBSCRIPT italic_V end_POSTSUBSCRIPT, Eq. (32) of the black branes in asymptotically Lifshitz spacetimes in κ𝜅\kappaitalic_κ-Horndeski gravity as functions of the temperature T𝑇Titalic_T for various values of the parameters z𝑧zitalic_z and γ𝛾\gammaitalic_γ, respectively.

Finally, the the positivity of the expected value of the energy ⟨E⟩delimited-⟨⟩𝐸\langle E\rangle⟨ italic_E ⟩ and the specific heat CVsubscript𝐶𝑉C_{V}italic_C start_POSTSUBSCRIPT italic_V end_POSTSUBSCRIPT indicate at least local stability and the negativity of the free energy F𝐹Fitalic_F shows that the system has global stability both compatible with a large black hole (brane) mass Hawking:1982dh ; Rostami:2019xrx .

Then, we see that the ratio of the free energy F𝐹Fitalic_F and the expected value of energy ⟨E⟩delimited-⟨⟩𝐸\langle E\rangle⟨ italic_E ⟩, is simply given by

The free energy of the black hole, its entropy, and internal energy satisfy the thermodynamic relation F=⟨E⟩−T⁢S𝐹delimited-⟨⟩𝐸𝑇𝑆F=\langle E\rangle-TSitalic_F = ⟨ italic_E ⟩ - italic_T italic_S. Considering the value of critical exponent as z=1𝑧1z=1italic_z = 1, this gives us ⟨E⟩∼T3similar-todelimited-⟨⟩𝐸superscript𝑇3\langle E\rangle\sim T^{3}⟨ italic_E ⟩ ∼ italic_T start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT that is similar to the case of energy of the thermal radiation in Hawking:1982dh . Actually, for z=1𝑧1z=1italic_z = 1 and z=2𝑧2z=2italic_z = 2, we recover the results of Hawking:1982dh and Dimopoulos:2001hw , respectively.

C

The probes number N𝑁Nitalic_N and the twist angle θ𝜃\thetaitalic_θ only impact on the amplitude of the rescaled concurrence [Fig. 3(c)].

Times in correspondence to the Lee-Yang zeros are the centers of all the vanishing domains of the rescaled concurrence at low temperature.

Times in correspondence to the Lee-Yang zeros are the centers of all the vanishing domains of the rescaled concurrence at low temperature.

There is a one-to-one mapping between the zeros (centers of vanishing domains) of coherence (concurrence) and the Lee-Yang zeros as the temperature approaches zero.

Moreover, the centers of all the concurrence vanishing domains are corresponding to the Lee-Yang zeros.

A

\underline{\operatorname{\mathcal{IC}}}_{\mathcal{M}_{\mathbf{f},\mathbf{d}}(Q)}roman_ϕ start_POSTSUPERSCRIPT roman_mon end_POSTSUPERSCRIPT start_POSTSUBSCRIPT start_OPFUNCTION caligraphic_T roman_r end_OPFUNCTION ( italic_W start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ) start_POSTSUBSCRIPT bold_d end_POSTSUBSCRIPT end_POSTSUBSCRIPT under¯ start_ARG caligraphic_I caligraphic_C end_ARG start_POSTSUBSCRIPT caligraphic_M start_POSTSUBSCRIPT bold_f , bold_d end_POSTSUBSCRIPT ( italic_Q ) end_POSTSUBSCRIPT is trivial. It follows from the existence of the isomorphism (75) that these dimensions are the same, and so the monodromy is trivial, and isomorphism (74) follows.

The construction and results of the present paper can be applied in nonabelian Hodge theory, and the construction of “higher genus” BPS lie algebras, since they concern any category for which the moduli of objects is locally modeled by moduli stacks of modules for preprojective algebras. Here we briefly explain one such application, referring the reader to [Dav21, DHM22] for further details.

Outside of cases with zero Lie bracket, no examples of BPS Lie algebras have been calculated before this paper; we start to remedy this situation here. We identify the zeroth cohomologically graded piece of 𝔤ΠQ𝒮superscriptsubscript𝔤subscriptΠ𝑄𝒮\mathfrak{g}_{\Pi_{Q}}^{\operatorname{\mathcal{S}}}fraktur_g start_POSTSUBSCRIPT roman_Π start_POSTSUBSCRIPT italic_Q end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT caligraphic_S end_POSTSUPERSCRIPT (for various choices of 𝒮𝒮\operatorname{\mathcal{S}}caligraphic_S) with various variants of the Kac–Moody Lie algebras associated to the underlying graph of Q𝑄Qitalic_Q (see §6). In particular, we provide the first examples of nonabelian BPS Lie algebras.

We find that, in contrast to Ngô’s result, the subvarieties Zisubscript𝑍𝑖Z_{i}italic_Z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT appearing in our analogue of (2) are almost never dense in the base. In this paper we will use a sheaf-theoretic lift of cohomological Hall algebra structures to describe the summands that do not have full support, and write a precise conjectural description of the direct image 𝙹𝙷∗⁢𝔻⁢ℚ¯𝔐⁢(ΠQ)subscript𝙹𝙷𝔻subscript¯ℚ𝔐subscriptΠ𝑄\mathtt{JH}_{*}\mathbb{D}\underline{\mathbb{Q}}_{\mathfrak{M}(\Pi_{Q})}typewriter_JH start_POSTSUBSCRIPT ∗ end_POSTSUBSCRIPT blackboard_D under¯ start_ARG blackboard_Q end_ARG start_POSTSUBSCRIPT fraktur_M ( roman_Π start_POSTSUBSCRIPT italic_Q end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT in terms of convolution tensor products of shifted intersection complexes (Conjecture 7.7). The derived global sections of the summands in the decomposition (1) will turn out to be tightly connected to the theory of BPS Lie algebras, certain Lie algebras that arise in the study of cohomological Hall algebras and algebras of BPS states [HM98, KS11, Dav17, DM20]: see Theorem B for the main result in this direction.

We recall a general construction, producing modules for cohomological Hall algebras out of moduli spaces of framed quiver representations, see [Soi16] for related examples and discussion.

D

The spherical angles (θ,ϕ)𝜃italic-ϕ(\theta,\phi)( italic_θ , italic_ϕ ) are the angles of one of the two decay vectors in the GJ frame.

This final state has been extensively studied in the past as the golden decay mode of the Higgs boson and in searches for physics beyond the Standard Model Keung et al. (2008); Gao et al. (2010); Bolognesi et al. (2012); Modak et al. (2016); Berge et al. (2015).

We note that a negligibly small polarization is measured in the prompt production of charmonium (”head on” collisions) Aaij et al. (2013a); Chatrchyan et al. (2013); Aaltonen et al. (2012); Aaij et al. (2013b); Sirunyan et al. (2018).

An angular analysis can help to distinguish different scenarios Dong et al. (2021); Wang et al. (2021); Karliner and Rosner (2020); Gong et al. (2022) and knowledge of the quantum numbers of the states can elucidate the mechanism for the binding of four charm quarks Liu et al. (2019); Weng et al. (2021); Chen et al. (2020); An et al. (2022); Becchi et al. (2020).

the heavy quarkonium system Godfrey and Olsen (2008); Aaij et al. (2015, 2019); Guo et al. (2018), and beyond Aaij et al. (2020a, b, c, 2023a, 2023b, 2022a, 2022b).

B

(tot)}}]].divide start_ARG italic_d end_ARG start_ARG italic_d italic_t end_ARG over^ start_ARG italic_ρ end_ARG start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT = - divide start_ARG 1 end_ARG start_ARG roman_ℏ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT italic_d italic_s tr start_POSTSUBSCRIPT g end_POSTSUBSCRIPT [ italic_H start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT (int) end_POSTSUPERSCRIPT , [ italic_H start_POSTSUBSCRIPT italic_t - italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT (int) end_POSTSUPERSCRIPT , over^ start_ARG italic_ρ end_ARG start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT (tot) end_POSTSUPERSCRIPT ] ] .

Refs. (louisell1970quantum, ; breuer2002theory, ; gardiner2004quantum, ; schlosshauer2007decoherence, )

only if the gravitational field exhibits bonafide quantum features (bose2017spin, ) 111The results of bose2017spin were first reported in a conference talk in Bangalore ICTS ., see also marletto2017gravitationally .

The underlying mechanism for the quantum entanglement of masses (QGEM) has been analyzed within perturbative quantum gravity (Marshman:2019sne, ; bose2022mechanism, ; Vinckers:2023grv, ; Carney_2019, ; Carney:2021vvt, )

waves and several different theoretical approaches have been considered (calzetta1994noise, ; anastopoulos1996quantum, ; anastopoulos2013master, ; riedel2013evidence, ; blencowe2013effective, ; suzuki2015environmental, ; de2015decoherence, ; oniga2016quantum, ; oniga2017quantum, ; quinones2017quantum, ; vedral2020decoherence, ; xu2020toy, ).

A

}\rfloor}italic_M start_POSTSUBSCRIPT italic_n , 0 end_POSTSUBSCRIPT ⊃ italic_M start_POSTSUBSCRIPT italic_n , 1 end_POSTSUBSCRIPT ⊃ italic_M start_POSTSUBSCRIPT italic_n , 2 end_POSTSUBSCRIPT ⊃ … ⊃ italic_M start_POSTSUBSCRIPT italic_n , ⌊ ( FRACOP start_ARG italic_n end_ARG start_ARG 2 end_ARG ) ⌋ end_POSTSUBSCRIPT

Notice that while {H1⁢(Xn)/[Vn,1]}n≥2={Vn,2}n≥2subscript/subscript𝐻1subscript𝑋𝑛delimited-[]subscript𝑉𝑛1𝑛2subscriptsubscript𝑉𝑛2𝑛2\{{\raisebox{1.79997pt}{$H_{1}(X_{n})$}\left/\raisebox{-1.79997pt}{$[V_{n,1}]$%

{H1⁢(Xn)/[Vn,1]}n≥2={Vn,2}n≥2=:V*,2\{{\raisebox{1.79997pt}{$H_{1}(X_{n})$}\left/\raisebox{-1.79997pt}{$[V_{n,1}]$%

Therefore, it is natural to define the representations Vn,p=Mn,p/Mn,p+1subscript𝑉𝑛𝑝/subscript𝑀𝑛𝑝subscript𝑀𝑛𝑝1V_{n,p}={\raisebox{1.79997pt}{$M_{n,p}$}\left/\raisebox{-1.79997pt}{$M_{n,p+1}%

{Vn/Kn}n∈ℕsubscript/subscript𝑉𝑛subscript𝐾𝑛𝑛ℕ\{{\raisebox{1.79997pt}{$V_{n}$}\left/\raisebox{-1.79997pt}{$K_{n}$}\right.}\}%

C

Weissman (2021); Giron and Lebed (2020); Richard (2020); Chao and Zhu (2020); Wang et al. (2021a); Yang et al. (2020b); Maciuła et al. (2021); Karliner and Rosner (2020); Wang (2021); Dong et al. (2021a); Feng et al. (2022); Zhao et al. (2020); Gordillo et al. (2020); Faustov et al. (2020); Weng et al. (2021); Zhang (2021); Zhu (2021); Guo and Oller (2021); Feng et al. (2021); Cao et al. (2021); Gong et al. (2022); Wan and Qiao (2021); Huang et al. (2021); Zhao et al. (2021); Goncalves and Moreira (2021); Albuquerque et al. (2021); Faustov et al. (2021); Ke et al. (2021); Liang et al. (2021); Yang et al. (2021); Mutuk (2021); Li et al. (2021); Majarshin et al. (2022); Pal et al. (2021); Dong et al. (2021b); Wang et al. (2021b); Zhuang et al. (2022); Asadi and Boroun (2022); Kuang et al. (2022); Wu et al. (2022); Liang and Yao (2022); Szczurek et al. (2022); Wang and Liu (2022); Zhou et al. (2022); Chen et al. (2022); Biloshytskyi et al. (2022); An et al. (2023); Zhang et al. (2022); Faustov et al. (2022); abu shady et al. (2022); Lu et al. (2023); Kuang et al. (2023); Agaev et al. (2024); Feng et al. (2023); Sang et al. (2023); Kuchta (2023); Wang et al. (2023).

Iwasaki (1975); Chao (1981); Ader et al. (1982); Zouzou et al. (1986); Heller and Tjon (1985); Lloyd and Vary (2004); Barnea et al. (2006); Vijande et al. (2009); Berezhnoy et al. (2012); Wu et al. (2018); Chen et al. (2017); Karliner et al. (2017); Bai et al. (2019); Wang (2017); Richard et al. (2017); Anwar et al. (2018); Debastiani and Navarra (2019); Richard et al. (2018); Esposito and Polosa (2018); Wang and Di (2019); Liu et al. (2019b); Wang et al. (2019); Chen (2020); Lundhammar and Ohlsson (2020).

Weissman (2021); Giron and Lebed (2020); Richard (2020); Chao and Zhu (2020); Wang et al. (2021a); Yang et al. (2020b); Maciuła et al. (2021); Karliner and Rosner (2020); Wang (2021); Dong et al. (2021a); Feng et al. (2022); Zhao et al. (2020); Gordillo et al. (2020); Faustov et al. (2020); Weng et al. (2021); Zhang (2021); Zhu (2021); Guo and Oller (2021); Feng et al. (2021); Cao et al. (2021); Gong et al. (2022); Wan and Qiao (2021); Huang et al. (2021); Zhao et al. (2021); Goncalves and Moreira (2021); Albuquerque et al. (2021); Faustov et al. (2021); Ke et al. (2021); Liang et al. (2021); Yang et al. (2021); Mutuk (2021); Li et al. (2021); Majarshin et al. (2022); Pal et al. (2021); Dong et al. (2021b); Wang et al. (2021b); Zhuang et al. (2022); Asadi and Boroun (2022); Kuang et al. (2022); Wu et al. (2022); Liang and Yao (2022); Szczurek et al. (2022); Wang and Liu (2022); Zhou et al. (2022); Chen et al. (2022); Biloshytskyi et al. (2022); An et al. (2023); Zhang et al. (2022); Faustov et al. (2022); abu shady et al. (2022); Lu et al. (2023); Kuang et al. (2023); Agaev et al. (2024); Feng et al. (2023); Sang et al. (2023); Kuchta (2023); Wang et al. (2023).

To this end, in addition to employing the values of the wave function at the origin taken from References liu et al. (2020); Lü et al. (2020),

Bedolla et al. (2020); Deng et al. (2021); liu et al. (2020); Yang et al. (2020a); Wang (2020); Jin et al. (2020); Becchi et al. (2020); Lü et al. (2020); Wang et al. (2020); Albuquerque et al. (2020); Sonnenschein and

A

   B−32⁢(Lμ+Lτ)𝐵32subscript𝐿𝜇subscript𝐿𝜏B-\frac{3}{2}(L_{\mu}+L_{\tau})italic_B - divide start_ARG 3 end_ARG start_ARG 2 end_ARG ( italic_L start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT + italic_L start_POSTSUBSCRIPT italic_τ end_POSTSUBSCRIPT )

7.0×10−277.0superscript10277.0\times 10^{-27}7.0 × 10 start_POSTSUPERSCRIPT - 27 end_POSTSUPERSCRIPT

6.6×10−276.6superscript10276.6\times 10^{-27}6.6 × 10 start_POSTSUPERSCRIPT - 27 end_POSTSUPERSCRIPT

7.3×10−277.3superscript10277.3\times 10^{-27}7.3 × 10 start_POSTSUPERSCRIPT - 27 end_POSTSUPERSCRIPT

7.2×10−277.2superscript10277.2\times 10^{-27}7.2 × 10 start_POSTSUPERSCRIPT - 27 end_POSTSUPERSCRIPT

D

He is supported by NSF grant DMS-1512908151290815129081512908 and a Miller Professorship from the Miller Institute for Basic Research in Science.

specify Brownian LPP, the integrable model of our choice, and its dynamical perturbation; and state our main result, regarding this dynamics, pinning down the transition from stability to chaos of the geodesic.

Brownian LPP is our model of study because it is the unique integrable LPP model for which the scaled energy profile has been shown to closely resemble Brownian motion and to which Chatterjee’s theory of superconcentration and chaos applies.

The concerned LPP model is not integrable, and, in this principal case, our approach couples this LPP model to an integrable one, namely Poissonian LPP,

The use of Fourier theory for deriving low supercritical overlap, namely Theorem 3.1(2), as well as for proving an assertion, Proposition 4.2, of subcritical weight stability, which we have seen to be a cornerstone of our proposed proof approach to proving the high subcritical overlap. As already alluded to in Section 2.3, this necessitates the extension of the tools from Chatterjee’s theory of chaos and superconcentration to the setting of Brownian LPP.

B

λ2λ1=v2v1subscript𝜆2subscript𝜆1subscript𝑣2subscript𝑣1\frac{\lambda_{2}}{\lambda_{1}}=\frac{v_{2}}{v_{1}}divide start_ARG italic_λ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG start_ARG italic_λ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG = divide start_ARG italic_v start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG start_ARG italic_v start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG

1 demonstrates the example c∝a−1/2proportional-to𝑐superscript𝑎12c\propto a^{-1/2}italic_c ∝ italic_a start_POSTSUPERSCRIPT - 1 / 2 end_POSTSUPERSCRIPT,

and the variation rule c∝a−1/2proportional-to𝑐superscript𝑎12c\propto a^{-1/2}italic_c ∝ italic_a start_POSTSUPERSCRIPT - 1 / 2 end_POSTSUPERSCRIPT for the VSL universe.

would vary as c∝a−1/2proportional-to𝑐superscript𝑎12c\propto a^{-1/2}italic_c ∝ italic_a start_POSTSUPERSCRIPT - 1 / 2 end_POSTSUPERSCRIPT.

in the velocity of light in the c∝a−1/2proportional-to𝑐superscript𝑎12c\propto a^{-1/2}italic_c ∝ italic_a start_POSTSUPERSCRIPT - 1 / 2 end_POSTSUPERSCRIPT fashion.

A

}\cos\phi\right)}{1+e^{-2\left|\alpha\right|^{2}}\cos\phi},divide start_ARG | italic_α | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( 1 - italic_e start_POSTSUPERSCRIPT - 2 | italic_α | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT roman_cos italic_ϕ ) end_ARG start_ARG 1 + italic_e start_POSTSUPERSCRIPT - 2 | italic_α | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT roman_cos italic_ϕ end_ARG ,

\left|\alpha\right|^{2}}\cos\phi},divide start_ARG - italic_i italic_α italic_e start_POSTSUPERSCRIPT - 2 | italic_α | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT roman_sin italic_ϕ end_ARG start_ARG 1 + italic_e start_POSTSUPERSCRIPT - 2 | italic_α | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT roman_cos italic_ϕ end_ARG ,

\alpha\!^{2}\right)\!\approx\!4\!+\!4N\alpha^{2}\mu^{2}\tau^{2}\!,4 + 4 italic_α start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_N italic_θ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - 2 + 2 roman_coth italic_α start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ≈ 4 + 4 italic_N italic_α start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_μ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_τ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ,

\alpha^{2}caligraphic_F start_POSTSUBSCRIPT italic_m italic_a italic_x end_POSTSUBSCRIPT = 4 + 8 italic_α start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( 2 + roman_coth italic_α start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ≈ 4 + 24 italic_α start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT

If we take α=α*𝛼superscript𝛼\alpha=\alpha^{*}italic_α = italic_α start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT, then we have the QFI

D

With these ingredients, we can evaluate the functional derivative δ⁢D+⁢(a)/δ⁢E⁢(x)𝛿subscript𝐷𝑎𝛿𝐸𝑥\delta D_{+}(a)/\delta E(x)italic_δ italic_D start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ( italic_a ) / italic_δ italic_E ( italic_x ) solving (25) with standard integration routines, in our case taken from the GNU Science Library. The derivatives α⁢(k,a)𝛼𝑘𝑎\alpha(k,a)italic_α ( italic_k , italic_a ) and κ⁢(k,a)𝜅𝑘𝑎\kappa(k,a)italic_κ ( italic_k , italic_a ) of the power spectrum with respect to the scale factor a𝑎aitalic_a and the wave number k𝑘kitalic_k, defined in (15) and (17), also need to be calculated numerically except for the derivative of the linear power spectrum with respect to a𝑎aitalic_a.

It is interesting to consider an Einstein-de Sitter universe as a reference cosmology. It contains only matter with critical density. Since all the input functions are then known analytically, we can compare the analytic solution for the functional derivative in such a model universe with the numerical results. The characteristic functions in this universe are

This analytic result is shown in the top panel of Fig. 4. It is opposed there to the numerical result for an Einstein-de Sitter universe in the centre panel for comparison, and to the functional derivative of D+subscript𝐷D_{+}italic_D start_POSTSUBSCRIPT + end_POSTSUBSCRIPT obtained by evaluating (28) numerically as described in the following subsection, shown in the bottom panel. The analytical and the numerical results for the Einstein-de Sitter universe are identical, thus confirming our numerical implementation. Despite the difference in the cosmological background evolution between the left and central panels compared to the right panel, the results are quite similar.

As for the Einstein-de Sitter result, the functional derivative of the normalised growth factor is positive on the entire domain. This illustrates that, with the growth factor fixed to unity today, an increase in the expansion function causes an increase in structure growth: if structures are to reach their present amplitude in a more rapidly expanding universe, they need to grow faster against the more rapidly expanding background. In the Einstein-de Sitter universe, structure growth is somewhat delayed compared to our actual universe, which expands more rapidly than Einstein-de Sitter at late times. Once more, this reflects the fact that structures need to grow earlier in a universe expanding more quickly if their present amplitude is fixed.

Figure 4: Functional derivative of the normalised growth factor with respect to the expansion function for two different cases. Upper panel: analytic result for an Einstein-de Sitter universe for reference; central panel: numerical result for an Einstein-de Sitter universe; lower panel: numerical result obtained using an empirically constrained expansion function.

A

The last query complexity gap is between measurements. Understanding why process tomography fails at the if clause (it fails at the (1/d)1𝑑(1/d)( 1 / italic_d )th power for a similar reason) lead us to define modified tasks which are achievable: The entangled and random if clause. If simpler implementations exist, the entangled if clause could serve as a subroutine in larger algorithms. The random if clause reveals the complexity gap between measurements: its query complexity is infinite in the one-success-outcome model, but finite (at most exponential) in a model with exp⁡(n)𝑛\exp(n)roman_exp ( italic_n ) success outcomes (Table 2). Are there algorithms requiring fewer queries and fewer outcomes? An optimal algorithm will exactly quantify the advantage of many outcomes – finding one remains an open question.

Z.G. and Y.T. thank Dorit Aharonov for the supervision and support. All authors thank Amitay Kamber for sketching the more elementary proof of 1, Mateus Araújo for enlightening discussions of the tomography issue, Alon Dotan, Itai Leigh, Giulio Gasbarri and Michalis Skotiniotis for useful feedback, and an anonymous referee from the QIP conference for suggesting an alternative approach via S⁢U⁢(d)𝑆𝑈𝑑SU(d)italic_S italic_U ( italic_d ) covering spaces (Appendix C is our take on this approach). This work was supported by the Simons Foundation (Grant No. 385590), by the Israel Science Foundation (Grants No. 2137/19 and No. 1721/17), by the European Commission QuantERA grant

The first and the most surprising result shows that the quantum-circuit complexity of the if clause is infinite. We show that no matter how many times a postselection circuit queries U𝑈Uitalic_U and U†superscript𝑈†U^{\dagger}italic_U start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT, it cannot implement the if clause with a nonzero success probability for every U∈U⁢(d)𝑈𝑈𝑑U\in U(d)italic_U ∈ italic_U ( italic_d ) – not even approximately! Surprisingly, process tomography suggested above fails for the if clause, and only works for a relaxed variant of the task. We prove this limitation of process tomography directly. Second, we give a different proof that implementing the fractional power U1dsuperscript𝑈1𝑑U^{\frac{1}{d}}italic_U start_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG italic_d end_ARG end_POSTSUPERSCRIPT for all U∈U⁢(d)𝑈𝑈𝑑U\in U(d)italic_U ∈ italic_U ( italic_d ) is impossible, from any number of queries to U𝑈Uitalic_U and U†superscript𝑈†U^{\dagger}italic_U start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT. Our contradiction is independent of the particular d𝑑ditalic_dth root function. Third, we show that quantum circuits and process matrices fail to neutralize some specific numbers of queries. Last, we prove related limitations regarding the transpose and inverse tasks.

The rest of the paper is organized as follows. Section II defines oracle computation using functions on d𝑑ditalic_d-dimensional unitaries, U∈U⁢(d)𝑈𝑈𝑑U\in U(d)italic_U ∈ italic_U ( italic_d ). Section III proves the if-clause impossibility and the process tomography limitation by exploiting the continuity of algorithms and the topology of the space U⁢(d)𝑈𝑑U(d)italic_U ( italic_d ) (1). Section IV uses this topological approach to prove results regarding the neutralization, 1/d1𝑑1/d1 / italic_dth power, transpose, and inverse. In Section V we emphasize that our method applies to the worst-case models with the exception of linear optics. We discuss the cause and the significance of this exception. Then we discuss relaxed causality and measurements.

The if-clause (m=1𝑚1m=1italic_m = 1) impossibility is immediate. In addition to the following full proof of Theorem 1, the appendix contains two more proofs for only the exact, ϵ=0italic-ϵ0\epsilon=0italic_ϵ = 0, impossibility. The “operational” proof in Appendix B reaches a contradiction by using the supposed cϕmsubscriptsuperscript𝑐𝑚italic-ϕc^{m}_{\phi}italic_c start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_ϕ end_POSTSUBSCRIPT algorithm as a building block in a larger circuit. The proof in Appendix C for only the m=1𝑚1m=1italic_m = 1 case 333We thank an anonymous referee from the QIP conference for observing that such a proof is possible. gives additional intuition: The special unitary group S⁢U⁢(d)𝑆𝑈𝑑SU(d)italic_S italic_U ( italic_d ) is a d𝑑ditalic_dth cover of P⁢U⁢(d)𝑃𝑈𝑑PU(d)italic_P italic_U ( italic_d ), the projective unitary group. This prevents the existence of a continuous map from a unitary superoperator to a matching operator, P⁢U⁢(d)→U⁢(d)→𝑃𝑈𝑑𝑈𝑑PU(d)\to U(d)italic_P italic_U ( italic_d ) → italic_U ( italic_d ), preventing an exact if-clause algorithm. The full proof below holds for approximations and relies on 1 proven next.

A

Trends in the work function based on which chemical species are present at the surface are shown in Figure 3. The fraction of surfaces with a low work function (<2.5absent2.5<2.5< 2.5 eV, i.e., roughly 1.5 times the standard deviation below average) is especially high for surfaces with alkali or alkaline metals present in the topmost atomic layer. Conversely, the fraction of surfaces with a high work function (>5.5absent5.5>5.5> 5.5 eV, i.e., roughly 1.5 times the standard deviation above average) is especially high for surfaces with halogens, carbon, nitrogen, sulfur, selenium or oxygen present in the topmost atomic layer (cf. Figure 3a). Surfaces with hydrogen present exhibit more of both low and high work functions than average, likely due to complex chemistries. The total number of surfaces (rather than fractions) are shown in Figure S4. Overall, 48.8% (29.5 %) of surfaces that exhibit a work function below 2.0 (2.5) eV contain either alkali or alkaline metals in the topmost atomic layer. Conversely, 54.1 % (38.0 %) of surfaces that exhibit a work function above 6.0 (5.5) eV contain either carbon, oxygen, or halogens in the topmost atomic layer.

The average work function is plotted as a heat-map based on the chemical species present in the topmost two atomic layers. The trends observed in Figure 3a are also seen in the average work function trend in Figure 3b. However, one can also observe trends based on combinations of chemical species in the topmost and second atomic layers. For example, the work function average is high for surfaces where halogens are present in the first or second layer. In contrast, the work function average is low for surfaces with alkali or alkaline metals present in the first layer and sulfur or selenium present in second layer – however, the work function average is high for the reversed layers (i.e., sulfur/selenium in topmost layer). Further trends are plotted in Figures S5 (barchart of average work function as a function of the chemical species present at the topmost layer) and S6 (heat-maps showing percentage and total number of low and high work function surfaces as a function of chemical species present in the top two layers).

Figure 3: Work function trends observed in the database. a Fraction of material surfaces that have a work function below 2.5 eV (purple) and above 5.5 eV (orange) is shown depending on which type of chemical species is present at the topmost surface. The dashed lines indicate the average fraction across the entire database regardless of chemical species at the surface. b Heat-map of the average work function plotted as a function of chemical species present at the topmost layer (vertical axis) and second atomic layer (horizontal axis). The color bar displays work functions below and above average as blue and red, respectively. Categories with a population of less than 10 surfaces have been left blank.

The observation that the distribution in work functions is near-Gaussian could indicate that the chemical space we chose was diverse enough to evenly sample work functions across possible values. The extended tail at the high work function end appears to be an artifact coming from ionically unrelaxed surfaces where a small, electronegative atom (e.g., oxygen or hydrogen) is cleaved at a large, unphysical distance (as discussed in the next section and corroborated by Figure S10). This might also be the case for the low work function tail but appears to be less pronounced. This artifact can be mitigated by ionically relaxing the surface slabs (see next section) and we expect this to result in an overall slightly narrower distribution. Interestingly, the work function distributions of binary and ternary compounds (and to a certain extent also the elemental crystals) have similar averages, standard deviations, and ranges. This may be explained by the observation that the work function is primarily determined by the chemical species present in the topmost layer at the surface (as discussed in the next paragraph), and will largely not depend on the total number of chemical species present in the entire unit cell. Moreover, the average work function of the database is lower than the average work function for the JARVIS database and C2DB (4.91 and 5.43 eV, respectively, cf. Table S1) while the standard deviations are somewhat similar (1.22 and 1.08 eV, respectively). The average cleavage energy of all asymmetric slabs (103.4 meV/Å2/\AA{}^{2}/ italic_Å start_FLOATSUPERSCRIPT 2 end_FLOATSUPERSCRIPT) is higher than the average for all symmetric slabs (88.0 meV/Å2/\AA{}^{2}/ italic_Å start_FLOATSUPERSCRIPT 2 end_FLOATSUPERSCRIPT). This is expected because this database is calculated for unrelaxed slabs and cleaving asymmetric slabs may lead to dangling atoms in nonphysical positions too far/close to the other surface atoms.

Trends in the work function based on which chemical species are present at the surface are shown in Figure 3. The fraction of surfaces with a low work function (<2.5absent2.5<2.5< 2.5 eV, i.e., roughly 1.5 times the standard deviation below average) is especially high for surfaces with alkali or alkaline metals present in the topmost atomic layer. Conversely, the fraction of surfaces with a high work function (>5.5absent5.5>5.5> 5.5 eV, i.e., roughly 1.5 times the standard deviation above average) is especially high for surfaces with halogens, carbon, nitrogen, sulfur, selenium or oxygen present in the topmost atomic layer (cf. Figure 3a). Surfaces with hydrogen present exhibit more of both low and high work functions than average, likely due to complex chemistries. The total number of surfaces (rather than fractions) are shown in Figure S4. Overall, 48.8% (29.5 %) of surfaces that exhibit a work function below 2.0 (2.5) eV contain either alkali or alkaline metals in the topmost atomic layer. Conversely, 54.1 % (38.0 %) of surfaces that exhibit a work function above 6.0 (5.5) eV contain either carbon, oxygen, or halogens in the topmost atomic layer.

A

Since V0subscript𝑉0V_{0}italic_V start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT is future-directed over 𝒮0subscript𝒮0\mathcal{S}_{0}caligraphic_S start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and

is open in 𝒬𝒬\mathcal{Q}caligraphic_Q and contains 𝒮0subscript𝒮0\mathcal{S}_{0}caligraphic_S start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT,

Then 𝒮τsubscript𝒮𝜏\mathcal{S}_{\tau}caligraphic_S start_POSTSUBSCRIPT italic_τ end_POSTSUBSCRIPT is an embedded submanifold of 𝒬𝒬\mathcal{Q}caligraphic_Q

natural inclusion of ι⁢(𝒮)𝜄𝒮\iota(\mathcal{S})italic_ι ( caligraphic_S ) in 𝒬𝒬\mathcal{Q}caligraphic_Q

The image ι⁢(𝒮)𝜄𝒮\iota(\mathcal{S})italic_ι ( caligraphic_S ) is an open submanifold of 𝒬𝒬\mathcal{Q}caligraphic_Q and

D

(for any λ∈ℝ∖{0}𝜆ℝ0\lambda\in\mathbb{R}\setminus\{0\}italic_λ ∈ blackboard_R ∖ { 0 } and K>0𝐾0K>0italic_K > 0).

RNsubscript𝑅𝑁R_{N}italic_R start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT is as in (1.8) with λ∈ℝ∖{0}𝜆ℝ0\lambda\in\mathbb{R}\setminus\{0\}italic_λ ∈ blackboard_R ∖ { 0 } and k=3𝑘3k=3italic_k = 3.

focusing Gibbs measure with an L2superscript𝐿2L^{2}italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT-cutoff:

L2superscript𝐿2L^{2}italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT-cutoff) for the Benjamin-Ono equation (1.19) with k=3𝑘3k=3italic_k = 3.

and thus are incompatible with the Wick-ordered L2superscript𝐿2L^{2}italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT-cutoff.

C

The μ𝜇\muitalic_μSR measurements were carried out at the General Purpose Spectrometer Amato et al. (2017) and the high transverse field spectrometer HAL9500 Scheuermann (2020), both located at PSI. We used a smaller piece of the same sample and applied the magnetic field along the c𝑐citalic_c axis. See Appendix C for details on the μ𝜇\muitalic_μSR measurements and data analysis.

We acknowledge DESY (Hamburg, Germany), a member of the Helmholtz Association HGF, for the provision of experimental facilities. Parts of this research were carried out at beamline BW5 at DORIS.

The sample was oxygenated at the University of Connecticut through a wet-chemical technique for several months Mohottala et al. (2006). The rod was cut into pieces to fit it in the cryo-magnets used in the neutron experiments. A small piece of oxygenated sample was measured with a VSM in an applied field of 20 Oe. It reveals a single transition for our La1.94Sr0.06CuO4+y sample, shown in Fig. 6 in blue dots.

We performed hard X-ray diffraction experiments at the BW5 beamline at HASYLAB, Deutsches Elektronen-Synchrotron, Hamburg, Germany. Here we looked for a signature of charge-density waves. The sample was cut into a thin slice (d=1.4𝑑1.4d=1.4italic_d = 1.4 mm) in order to reduce the absorption, meaning that 10 % of the incoming beam was transmitted through the sample.

Figure 5: Examples of the raw data from the X-ray diffraction experiments. No field-induced diffraction peak indicating charge density waves can be found in the data measured at 3 K and 10 T (gray points).

C

Although the computational cost of making adiabatic inspiral waveforms is high, the most expensive step of this calculation — computing a set of complex numbers which encode rates of change of (E,Lz,Q𝐸subscript𝐿𝑧𝑄E,L_{z},Qitalic_E , italic_L start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT , italic_Q), as well as the gravitational waveform’s amplitude — need only be done once. These numbers can be computed in advance for a range of astrophysically relevant EMRI orbits, and then stored and used to assemble the waveform as needed. The goal of this paper is to lay out what quantities must be computed, and to describe how to use such precomputed data to build adiabatic waveforms.

Our particular goal is to show how to organize and store the most important and useful data needed to assemble the waveforms in a computationally effective way. A key element of our approach is to view the complicated EMRI waveform as a sum of simple “voices.” Each voice corresponds to a mode (l,m,k,n)𝑙𝑚𝑘𝑛(l,m,k,n)( italic_l , italic_m , italic_k , italic_n ) representing a particular multipole of the radiation and harmonic of the fundamental orbital frequencies. The voice-by-voice decomposition was suggested long ago to one of us by L. S. Finn, and was first presented in Ref. [15] for the special case of spherical Kerr orbits (i.e., orbits of constant Boyer-Lindquist radius, including those inclined from the equatorial plane). This paper corrects an important error in Ref. [15] (which left out a phase that is set by initial conditions), and extends this analysis to fully generic configurations.

For all the cases we examine, we demonstrate how one can account for a system’s initial conditions by adjusting a set of phase variables which depend on the initial values of the anomaly angles that parameterize the system’s radial and polar motions. We calibrate our calculations in one case (presented in Sec. VII) by comparing to an EMRI waveform computed using a time-domain Teukolsky equation solver [18, 19]. Interestingly, in this case we find a small phase offset that, for most initial conditions, secularly accumulates, causing the waveforms computed with this paper’s techniques to dephase from those computed with the time-domain code by up to several radians over the course of an inspiral. We argue tat this is an artifact of the adiabatic approximation’s neglect of terms which vary on the slow timescale, and is not unexpected. We show in this comparison case that we can empirically compensate for much of the dephasing using an ad hoc correction that corrects some of the neglected “slow time” evolution. Though this correction is not rigorously justified, its form suggests that the dephasing may arise from a slow-time evolution of the phases variables which are set by the orbit’s initial conditions.

Although the computational cost of making adiabatic inspiral waveforms is high, the most expensive step of this calculation — computing a set of complex numbers which encode rates of change of (E,Lz,Q𝐸subscript𝐿𝑧𝑄E,L_{z},Qitalic_E , italic_L start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT , italic_Q), as well as the gravitational waveform’s amplitude — need only be done once. These numbers can be computed in advance for a range of astrophysically relevant EMRI orbits, and then stored and used to assemble the waveform as needed. The goal of this paper is to lay out what quantities must be computed, and to describe how to use such precomputed data to build adiabatic waveforms.

In this paper, we have shown how to to use precomputed frequency-domain Teukolsky equation solutions to make EMRI waveforms. In this way, one can compute and store in advance the most computationally expensive aspects of EMRI analysis, and then use the methods we describe to rapidly assemble waveforms using the stored data. We particularly emphasize the usefulness of the multivoice waveform structure, which facilitates identifying particularly important waveform multipoles and harmonics, both in the time and frequency domains.

A

For HGHG, the required harmonic number for reaching 13.55 nm is nineteenth. Owing to the large frequency up-conversion amplitude, the required energy modulation should be greater than 10 times of the energy spread. To achieve sufficient modulation of the electron beam with the 8-m-long modulator, the peak power of the seed laser is chosen to be 1 MW. Under this condition, the laser-induced energy modulation is 11.6 times of the energy spread. For a nominal HGHG whose modulator length is usually one Rayleigh length of the seed laser, simulation results show that the seed laser power needs to be larger than 640 MW to obtain the same energy modulation. Hereby the required peak power of the seed laser is reduced by almost three orders of magnitude with the proposed technique. The reduction in peak power can be traded for the increase of the repetition rate. For the direct-amplification method, the average power of the seed laser is 0.35 W under 1 MHz repetition rate. Such a power could be achieved by a commercial fiber laser system.

The long upstream modulator (two undulator segments) is used for seeding amplification and electron modulation, and the energy-modulated electron beam together with the amplified seed laser are then guided to the downstream elements for further beam manipulation and high harmonic generation through HGHG or EEHG processes. In the modulator, an externally coherent seed laser with a peak power larger than shotnoise is directly injected to interact with the electron beam. The long-distance modulation process can ensure sufficient energy exchange between the electron beam and the seed laser, as portrayed in Fig. 1(a)(I-III). In the initial stage of modulation, the power of the seed laser grows hardly during the laser-electron interaction process while a weak energy modulation is imprinted onto the electron beam. In the latter part, the lethargy regime of seeded FEL is overcome and there is an intense energy exchange between the laser and the electron beam. At this time, the electric field increases rapidly and produces a significant sinusoidal modulation on the electron beam. Although the power of the laser enters the exponential gain regime, it is far from saturation and the rotation of the phase space is almost not perceptible, which ensures that the energy modulation of the electron beam maintains sinusoidal. This laser-electron interaction mechanism realizes the direct amplification of seed laser and effectively sinusoidal energy modulation of the electron beam.

With the dispersion strength of two chicanes as 8.1 mm and 0.18 mm, respectively, the electron beam is highly micro-bunched at 6 nm and the maximum bunching factor of 6% is achieved. Phase space distribution of the electron beam at the entrance of radiator is presented in Fig. 4(b). The distribution is very close to that of a nominal EEHG, which indicates the feasibility of the setup in Fig. 4. The radiator is an in-vacuum undulator with a 0.87 T peak magnetic field at 11 mm gap. K parameter of the radiator is 3.3. The peak power of the radiation grows exponentially in the radiator and reaches saturation at 15 m, with a saturated power of 3.2 GW and and pulse duration of about 54 fs (FWHM). The pulse energy is 0.135 mJ. At saturation, the spectrum bandwidth is about 1/3320 (Δ⁢λ/λΔ𝜆𝜆\Delta\lambda/\lambdaroman_Δ italic_λ / italic_λ), 1.18 times to the Fourier transform limit. These results are shown in Fig. 4(c). With a repetition rate of 1 MHz, the average power of the FEL radiation at 6 nm reaches 135 W. The radiation performances like power and spectrum bandwidth of EEHG at 6 nm are better than those of HGHG at 13.5 nm, since the bunching factor is larger and the laser-induced energy spread is smaller. It should be emphasized that self-feedback mechanism also emerges in this case, which is a hint for the generation of stable output.

The phase space distribution is very close to that of a nominal HGHG, indicating that the direct-amplification process yields sinusoidal energy modulation. Maximum bunching factor of 3% is achieved at 13.55 nm with the density-modulated beam. The radiator is an in-vacuum undulator with a 1.4 T peak magnetic field at 6.7 mm gap. K parameter is 5.4. In the radiator, the peak power of the EUV radiation grows exponentially and reaches saturation at 20 m with a value of 1.6 GW and pulse duration of about 60 fs (FWHM). The pulse energy is 77 μ𝜇\muitalic_μJ. At this point, the spectrum bandwidth is about 1/1780 (Δ⁢λ/λΔ𝜆𝜆\Delta\lambda/\lambdaroman_Δ italic_λ / italic_λ), 1.25 times to the Fourier transform limit, which manifests the great longitudinal coherence of the radiation. These results are shown in Fig. 2(c). The average power of the FEL radiation at EUV reaches 77 W with 1 MHz repetition rate.

By setting the dispersion strength of the chicane as 34 μ𝜇\muitalic_μm, the energy-modulated electron beam becomes spatially density-modulated. Its phase space distribution is shown in Fig. 2(b).

D

The combination of (3.57) and (3.58) implies that ν≡0𝜈0\nu\equiv 0italic_ν ≡ 0, that is, that the cohomology class of ω1⁢(x1)subscript𝜔1subscript𝑥1\omega_{1}(x_{1})italic_ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) in Hq⁢(Zi)superscript𝐻𝑞subscript𝑍𝑖H^{q}(Z_{i})italic_H start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT ( italic_Z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) must vanish.

Combining all these lemmas yields the following local description of weighted L2superscript𝐿2L^{2}italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT-cohomology of a QFCQFC\operatorname{QFC}roman_QFC-metric.

To compute the weighted L2superscript𝐿2L^{2}italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT-cohomology in these local models, let us first assume that Bisubscript𝐵𝑖B_{i}italic_B start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT is a point. One then needs to compute the weighted L2superscript𝐿2L^{2}italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT-cohomology of the metric

It turns out that both of these intermediate results can be suitably adapted to study the Hodge cohomology of QFBQFB\operatorname{QFB}roman_QFB-metrics. First, to compute the weighted L2superscript𝐿2L^{2}italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT-cohomology of a QFBQFB\operatorname{QFB}roman_QFB-metric, we can in fact introduce the analog of fibered cusp metrics, namely the notion quasi-fibered cusp metrics (QFCQFC\operatorname{QFC}roman_QFC-metrics), a class of metrics conformally related to QFBQFB\operatorname{QFB}roman_QFB-metrics, see Definition 3.1 below for more details. Quasi-fibered cusp metrics should not be confused with the class of iterated fibered cusp metrics considered in [15, 26], which constitutes yet another way of generalizing the notion of fibered cusp metrics to stratified spaces of higher depth. In fact, one important difference is that as opposed to iterated fibered cusp metrics, QFCQFC\operatorname{QFC}roman_QFC-metrics do not admit nice local models in terms of warped products. In particular, the Künneth formula of Zucker for the L2superscript𝐿2L^{2}italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT-cohomology of a warped product cannot be used to compute the L2superscript𝐿2L^{2}italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT-cohomology of local models of QFCQFC\operatorname{QFC}roman_QFC-metrics. As described in (2.8) below, a good local model for QFBQFB\operatorname{QFB}roman_QFB-metrics is not given by a warped product, but by a subset of a Cartesian product. To compute the weighted L2superscript𝐿2L^{2}italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT-cohomology of such a local model, our main tool consists instead in using basic mapping properties of the exterior differential on a half-line. As in [23], we need to show that we can define weighted L2superscript𝐿2L^{2}italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT-cohomology using a sheaf of conormal forms, which can be achieved using a soft parametrix inverting the Hodge Laplacian of a QFBQFB\operatorname{QFB}roman_QFB-metric, for instance using the general pseudodifferential calculus of [5] for Lie structures at infinity. With this approach, we can in many cases identified weighted L2superscript𝐿2L^{2}italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT-cohomology groups with some intersection cohomology groups. In the following case, we can even compute the Hodge cohomology of a QFCQFC\operatorname{QFC}roman_QFC-metric (see also Corollary 3.14 below for more details).

This decay of L2superscript𝐿2L^{2}italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT-harmonic forms and our computation of the weighted L2superscript𝐿2L^{2}italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT-cohomology of a QFCQFC\operatorname{QFC}roman_QFC-metric

A

We also see a minimum value for the inversion temperature only for c𝑐citalic_c=0.01. On the other hand, for c=0.80𝑐0.80c=0.80italic_c = 0.80 and 1.0 there are minimum values for the pressure.

In the right panel of Fig. 1, we also see that these non-zero values for the pressure increase with increasing electric charge Q𝑄Qitalic_Q. So, besides the increase of the derivative of the inversion temperature, the increase of the electric charge Q𝑄Qitalic_Q also implies a small displacement of these curves in the direction of increasing pressure, in accordance with Eq. (32). In this panel, one also sees that for small values of the temperature and pressure the curves of constant charge intersect each other.

This shows that the pressure for zero temperature and fixed horizon radius increases with Q𝑄Qitalic_Q but decreases with a𝑎aitalic_a. Since ω𝜔\omegaitalic_ω considered here is negative, we also see that the pressure at zero temperature increases with c𝑐citalic_c and with the modulus of ω𝜔\omegaitalic_ω.

Notice that we do not see the interception of the curves for fixed ω𝜔\omegaitalic_ω and comparing the left and right panels one sees that if we increase the absolute value of ω𝜔\omegaitalic_ω the minimum pressure increases as well, also in agreement with Eq. (32).

In the right panel, when we increase c𝑐citalic_c the pressure increases and the temperature decreases, as in the left panel. In this case we do not see the interception of the curves.

C

Quantum non-locality demonstrated in EPR steering is strictly weaker than that in the Bell test. We may wonder if the above discussion of Maxwell demon applies to the case of the Bell test. The answer, unfortunately, is negative. To make it explicit, let’s consider a Bell game in which a third party, Charlie, who chooses collaboration with the demons, tries to simultaneously deceive Alice and Bob, who are separated by distant space, into believing that their received qubits are entangled. In order to achieve this goal, two demons have to be sent secretly to Alice and Bob, respectively. The operations of demons are almost the same in the case of EPR steering, i.e., they have access to measurement basis information and rotate the qubits before they are measured. The main issue, however, is how to rotate the qubits based on the basis information. In EPR steering, the demon can randomly rotate the qubit into one eigenstate of measurement basis due to the classical communication between Alice and Bob, and the demon could inform Alice of its operation. This is not possible in the Bell test, in which classical communication is prohibited. One might imagine that two demons could share a table list from the beginning, guiding them on how to operate based on the obtained basis information. That would require a table list long enough to cover all the runs, which is physically impossible. One may also imagine that the random generators of two demons are entangled, such that their operations are correlated. This can indeed be possible using the quantum circuit model. However, it is a logical circular argument because it uses the non-locality correlation of demons to replace the non-locality correlation between Alice and Bob in the Bell test. We thus conclude that there is no Maxwell demon loophole in the Bell test.

The Maxwell demon-assisted EPR steering can be demonstrated by programmable quantum processors, such as superconducting quantum computers Sup1 ; Sup2 ; Sup3 ; Sup4 . Here, we discuss in detail the quantum circuit realization as depicted in Fig. 2. Using this quantum circuit model, we establish the quantitative relationship between the work done by the demon and the quantum non-locality demonstrated in the EPR steering.

In conclusion, we have addressed the issue of simulating quantum non-locality through work. In the task of EPR steering, the Maxwell demon can be introduced in collaboration with Alice to deceive Bob using only local operations and classical communication. The existence of Maxwell demon-assisted EPR steering implies a new-type loophole, i.e., Maxwell demon loophole, which can only be closed by carefully monitoring heat fluctuations in the local environment by the participant. To give a quantitative relationship between quantum non-locality correlation in EPR steering and the work done by the demon, we construct a quantum circuit model of Maxwell demon-assisted EPR steering, which can be demonstrated in current quantum processors.

This work is organized as follows: We first introduce the Maxwell demon-assisted EPR steering and discuss the physical limitations of the Maxwell demon that could potentially close this new type of loophole. We then construct the quantum circuit model of Maxwell demon-assisted EPR steering and derive the formula describing the relation between the work done by the demon and quantum non-locality based on this model. Further discussion and conclusions are summarized in the last section.

Here, we take a different perspective and ask whether the quantum non-locality correlation can be simulated by the Maxwell demon. In the context of Einstein-Podolsky-Rosen (EPR) steering, we demonstrate the existence of a protocol in which Alice and the demon collaborate to cheat Bob. This form of Maxwell demon-assisted EPR steering implies that non-locality correlations, which refute the local hidden state model, can be simulated through work. It also means the existence of a new type of loophole in the EPR steering which we call Maxwell demon loophole. By applying Landauer’s erasure principle, we demonstrate that the only method to close this loophole involves Bob continuously monitoring the temperature fluctuation of the local environment. We construct the quantum circuit model for this Maxwell demon-assisted EPR steering, which is feasible to be demonstrated with current quantum processors. Based on this quantum circuit model, we derive the explicit formula that describes the relationship between quantum non-locality demonstrated in EPR steering and the work done by the demon.

B

Thus, a spacetime describing this system cannot be expected to contain Cauchy hypersurfaces with compact radiation content.

We also refer the reader to these papers for more general background and motivation for the study of late-time asymptotics.

See also §2.2 and §6 of [Keh21a] for a detailed discussion of this restricted to the spherically symmetric mode.

On the other hand, the data considered in [Keh21a] and the present paper have a clear physical motivation333In addition to the remarks at the beginning of the paper, see also sections 1 and 2.1 of [Keh21a] for a more detailed discussion of the physical motivation. and, thus, seem like a more reasonable starting point for the question of physically relevant late-time asymptotics.

Furthermore, we showed in [Keh21b] that the failure of peeling manifested by the early-time asymptotics (1.3) translates into logarithmic late-time asymptotics near i+superscript𝑖i^{+}italic_i start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT, providing evidence for the physical measurability of the failure of null infinity to be smooth. We will return to the discussion of late-time asymptotics in section 1.3.

C

A single SLIP measure is enough to provide necessary and sufficient conditions for any two generic pure n𝑛nitalic_n-qubit states to be SLOCC-equivalent.

In this paper, we show that, contrary to intuition, a single SLIP measure is enough to verify the SLOCC-equivalence between any two generic pure n𝑛nitalic_n-qubit states. By generic state, we mean the set of all pure states except of the measure zero subset with respect to the Haar measure. We achive this by using a mathematical trick, in particular, we use a SLIP measure defined for n𝑛nitalic_n qubit systems to verify equivalence of systems of n+1𝑛1n+1italic_n + 1 quibits. In essence, we look how the roots of the SLIP measure for those states behave under SLOCC. We show that if the states are SLOCC-equivalent, then the roots of the SLIP measure for each state must be related by a Möbius transformation, which is straightforward to verify. In particular, we use this procedure to show that the 3-tangle measure is enough to discriminate between generic 4-qubit states. Finally, we show how one may use the roots of a SLIP entanglement measure to obtain the normal form of a 4-qubit state which bypasses the possibly infinite iterative standard procedure.

Choose any SLIPnhsuperscriptsubscriptSLIP𝑛ℎ\text{SLIP}_{n}^{h}SLIP start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_h end_POSTSUPERSCRIPT entanglement measure of degree h≥3ℎ3h\geq 3italic_h ≥ 3 and two (n+1)𝑛1(n+1)( italic_n + 1 )-qubit states.

In the previous section we showed that in principle any SLIP measure of degree h≥3ℎ3h\geq 3italic_h ≥ 3 can be used to verify SLOCC-equivalence. In this section, we consider the special case when such a measure has degree h=4ℎ4h=4italic_h = 4.

Intuitively, a generic pure quantum state shall have exactly hℎhitalic_h distinct roots for each subsystem, where hℎhitalic_h is the degree of the SLIPnhsuperscriptsubscriptSLIP𝑛ℎ\text{SLIP}_{n}^{h}SLIP start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_h end_POSTSUPERSCRIPT measure. Thus, any such measure of degree h≤3ℎ3h\leq 3italic_h ≤ 3 should be sufficient to verify SLOCC-equivalence between generic quantum states. We confirmed this intuition by looking at several examples of degree-4444 measures. In particular, Eqs. 3 and 2 provides a family of SLIP measures for arbitrary number of qubits.

C

\omega_{E}}| ⟨ bold_italic_u start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT , bold_italic_u start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ⟩ | ≤ italic_N start_POSTSUPERSCRIPT - italic_ω start_POSTSUBSCRIPT italic_E end_POSTSUBSCRIPT end_POSTSUPERSCRIPT

Then, using (4.46), instead of (4.27), as an input we readily conclude the analogous versions of Lemmas 4.2–4.3. Finally, by Lemmas 4.2–4.3 we conclude the proof of (4.45) proceeding exactly as in (4.28)–(4.33).

Then proceeding as in (4.1)–(4.5), after replacing (4.3) with (4.47), we find that to conclude the proof of this proposition it is enough to prove

From now on we focus only on the precesses 𝝀xr⁢(t1)superscript𝝀subscript𝑥𝑟subscript𝑡1{\bm{\lambda}}^{x_{r}}(t_{1})bold_italic_λ start_POSTSUPERSCRIPT italic_x start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ( italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ), 𝝀~(r)⁢(t1)superscript~𝝀𝑟subscript𝑡1\widetilde{\bm{\lambda}}^{(r)}(t_{1})over~ start_ARG bold_italic_λ end_ARG start_POSTSUPERSCRIPT ( italic_r ) end_POSTSUPERSCRIPT ( italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) and so on the proof of Lemma 4.2. The proof to conclude Lemma 4.3 is completely analogous and so omitted. Combining (4.40) with another application of Proposition 4.6, this time for i0=jrsubscript𝑖0subscript𝑗𝑟i_{0}=j_{r}italic_i start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_j start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT and i=1𝑖1i=1italic_i = 1, we readily conclude that

Finally, by (4.6) together with (4.5) and Proposition 4.1 applied to H#:=Ht1#assignsuperscript𝐻#subscriptsuperscript𝐻#subscript𝑡1H^{\#}:=H^{\#}_{t_{1}}italic_H start_POSTSUPERSCRIPT # end_POSTSUPERSCRIPT := italic_H start_POSTSUPERSCRIPT # end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT we conclude the proof of Proposition 3.1.

A