Mathematical Physics
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Showing new listings for Monday, 25 November 2024
- [1] arXiv:2411.14800 [pdf, html, other]
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Title: Fixed Points of Completely Positive Trace-Preserving Maps in Infinite DimensionComments: 17 pages LaTeX, 2 figure filesSubjects: Mathematical Physics (math-ph); Quantum Physics (quant-ph)
Completely positive trace-preserving maps $S$, also known as quantum channels, arise in quantum physics as a description of how the density operator $\rho$ of a system changes in a given time interval, allowing not only for unitary evolution but arbitrary operations including measurements or other interaction with an environment. It is known that if the Hilbert space $\mathscr{H}$ that $\rho$ acts on is finite-dimensional, then every $S$ must have a fixed point, i.e., a density operator $\rho_0$ with $S(\rho_0)=\rho_0$. In infinite dimension, $S$ need not have a fixed point in general. However, we prove here the existence of a fixed point under a certain additional assumption which is, roughly speaking, that $S$ leaves invariant a certain set of density operators with bounded ``cost'' of preparation. The proof is an application of the Schauder-Tychonoff fixed point theorem. Our motivation for this question comes from a proposal of Deutsch for how to define quantum theory in a space-time with closed timelike curves; our result supports the viability of Deutsch's proposal.
- [2] arXiv:2411.14841 [pdf, html, other]
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Title: Entropic Fluctuation Theorems for the Spin-Fermion ModelComments: 39 pages, 1 figure, related to arXiv:2409.15485Subjects: Mathematical Physics (math-ph)
We study entropic fluctuations in the Spin-Fermion model describing an $N$-level quantum system coupled to several independent thermal free Fermi gas reservoirs. We establish the quantum Evans-Searles and Gallavotti-Cohen fluctuation theorems and identify their link with entropic ancilla state tomography and quantum phase space contraction of non-equilibrium steady state. The method of proof involves the spectral resonance theory of quantum transfer operators developed by the authors in previous works.
- [3] arXiv:2411.14987 [pdf, html, other]
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Title: Twice Fourier transformable measures and diffraction theoryComments: 46 pagesSubjects: Mathematical Physics (math-ph)
Mathematical diffraction theory has been developed since about 1995. Hof's initial approach relied on tempered distributions in euclidean space. Nowadays often the Fourier theory by Argabright and Gil de Lamadrid is used, which applies to appropriate measures on locally compact abelian groups. We review diffraction theory using Wiener amalgams as test function spaces. For translation bounded measures, this unifies and simplifies the former two approaches. We treat weighted versions of Meyer's model sets as examples.
New submissions (showing 3 of 3 entries)
- [4] arXiv:2411.14556 (cross-list from math.PR) [pdf, html, other]
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Title: Emergence in graphs with near-extreme constraintsSubjects: Probability (math.PR); Mathematical Physics (math-ph)
We prove the existence of an infinite number of distinct phases in the Strauss model of graphs with edge and triangle constraints.
- [5] arXiv:2411.14573 (cross-list from quant-ph) [pdf, html, other]
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Title: Capacity-Achieving Entanglement Purification Protocol for Pauli Dephasing ChannelComments: 19 pages, 11 figures, 2 tablesSubjects: Quantum Physics (quant-ph); Mathematical Physics (math-ph)
Quantum communication facilitates the secure transmission of information and the distribution of entanglement, but the rates at which these tasks can be achieved are fundamentally constrained by the capacities of quantum channels. Although quantum repeaters typically enhance these rates by mitigating losses and noise, a simple entanglement swapping protocol via a central node is not effective against the Pauli dephasing channel due to the additional degradation introduced by Bell-state measurements. This highlights the importance of purifying distributed Bell states before performing entanglement swapping. In this work, we introduce an entanglement purification protocol assisted by two-way classical communication that not only purifies the states but also achieves the channel capacities. Our protocol uses an iterative process involving CNOT gates and Hadamard basis measurements, progressively increasing the fidelity of Bell states with each iteration. This process ensures that the resulting Bell pairs are perfect in the limit of many recursive iterations, making them ideal for use in quantum repeaters and for correcting dephasing errors in quantum computers. The explicit circuit we propose is versatile and applicable to any number of Bell pairs, offering a practical solution for mitigating decoherence in quantum networks and distributed quantum computing.
- [6] arXiv:2411.14587 (cross-list from math.AP) [pdf, html, other]
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Title: Internal waves in a 2D subcritical channelComments: 26 pages, 2 figures. Comments are welcome!Subjects: Analysis of PDEs (math.AP); Mathematical Physics (math-ph); Classical Analysis and ODEs (math.CA); Spectral Theory (math.SP)
We study scattering and evolution aspects of linear internal waves in a two dimensional channel with subcritical bottom topography. We define the scattering matrix for the stationary problem and use it to show a limiting absorption principle for the internal wave operator. As a result of the limiting absorption principle, we show the leading profile of the internal wave in the long time evolution is a standing wave whose spatial component is outgoing.
- [7] arXiv:2411.14758 (cross-list from astro-ph.CO) [pdf, html, other]
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Title: Anisotropy in the cosmic acceleration inferred from supernovaeComments: 13 pages, 3 figures, 1 table. Invited proceedings for the Royal Society discussion meeting "Challenging the Standard Cosmological Model" [this https URL]. Submitted to Philosophical Transactions ASubjects: Cosmology and Nongalactic Astrophysics (astro-ph.CO); General Relativity and Quantum Cosmology (gr-qc); High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph)
Under the assumption that they are standard(isable) candles, the lightcurves of Type Ia supernovae have been analyzed in the framework of the standard Friedmann-Lemaître-Robertson-Walker cosmology to conclude that the expansion rate of the Universe is accelerating due to dark energy. While the original claims in the late 1990s were made using overlapping samples of less than 100 supernovae in total, catalogues of nearly 2000 supernovae are now available. In light of recent developments such as the cosmic dipole anomaly and the larger than expected bulk flow in the local Universe (which does not converge to the Cosmic Rest Frame), we analyze the newer datasets using a Maximum Likelihood Estimator and find that the acceleration of the expansion rate of the Universe is unequivocally anisotropic. The associated debate in the literature highlights the artifices of using supernovae as standardisable candles, while also providing deeper insights into a consistent relativistic view of peculiar motions as departures from the Hubble expansion of the Universe. The effects of our being `tilted observers' embedded in a deep bulk flow may have been mistaken for cosmic acceleration.
- [8] arXiv:2411.14769 (cross-list from nlin.CD) [pdf, html, other]
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Title: Kolmogorov Modes and Linear Response of Jump-Diffusion Models: Applications to Stochastic Excitation of the ENSO Recharge OscillatorComments: 20 pages, 9 figuresSubjects: Chaotic Dynamics (nlin.CD); Mathematical Physics (math-ph); Atmospheric and Oceanic Physics (physics.ao-ph)
We introduce a generalization of linear response theory for mixed jump-diffusion models, combining both Gaussian and Lévy noise forcings that interact with the nonlinear dynamics. This class of models covers a broad range of stochastic chaos and complexity for which the jump-diffusion processes are a powerful tool to parameterize the missing physics or effects of the unresolved scales onto the resolved ones.
By generalizing concepts such as Kolmogorov operators and Green's functions to this context, we derive fluctuation-dissipation relationships for such models. The system response can then be interpreted in terms of contributions from the eigenmodes of the Kolmogorov operator (Kolmogorov modes) decomposing the time-lagged correlation functions of the unperturbed dynamics. The underlying formulas offer a fresh look on the intimate relationships between the system's natural variability and its forced variability.
We apply our theory to a paradigmatic El Niño-Southern Oscillation (ENSO) subject to state-dependent jumps and additive white noise parameterizing intermittent and nonlinear feedback mechanisms, key factors in the actual ENSO phenomenon. Such stochastic parameterizations are shown to produce stochastic chaos with an enriched time-variability. The Kolmogorov modes encoding the latter are then computed, and our Green's functions formulas are shown to achieve a remarkable accuracy to predict the system's response to perturbations.
This work enriches Hasselmann's program by providing a more comprehensive approach to climate modeling and prediction, allowing for accounting the effects of both continuous and discontinuous stochastic forcing. Our results have implications for understanding climate sensitivity, detection and attributing climate change, and assessing the risk of climate tipping points. - [9] arXiv:2411.14824 (cross-list from math.AP) [pdf, html, other]
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Title: Spectral regularity with respect to dilations for a class of pseudodifferential operatorsComments: 10 pagesSubjects: Analysis of PDEs (math.AP); Mathematical Physics (math-ph)
We continue the study of the perturbation problem discussed in \cite{CP3} and get rid of the 'slow variation' assumption by considering symbols of the form $a\big(x+\delta\,F(x),\xi\big)$ with $a$ a real Hörmander symbol of class $S^0_{0,0}(\mathbb{R}^d\times\mathbb{R}^d)$ and $F$ a smooth function with all its derivatives globally bounded, with $|\delta|\leq1$. We prove that while the Hausdorff distance between the spectra of the Weyl quantization of the above symbols in a neighbourhood of $\delta=0$ is still of the order $\sqrt{|\delta|}$, the distance between their spectral edges behaves like $|\delta|^\nu$ with $\nu\in[1/2,1)$ depending on the rate of decay of the second derivatives of $F$ at infinity.
- [10] arXiv:2411.14891 (cross-list from astro-ph.HE) [pdf, html, other]
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Title: The non-equilibrium Marshak wave problem in non-homogeneous mediaComments: Accepted for publication in Physics of Fluids. arXiv admin note: text overlap with arXiv:2401.05138Subjects: High Energy Astrophysical Phenomena (astro-ph.HE); Mathematical Physics (math-ph); Computational Physics (physics.comp-ph); Plasma Physics (physics.plasm-ph)
We derive a family of similarity solutions to the nonlinear non-equilibrium Marshak wave problem for an inhomogeneous planar medium which is coupled to a time dependent radiation driving source. We employ the non-equilibrium gray diffusion approximation in the supersonic regime. The solutions constitute a generalization of the non-equilibrium nonlinear solutions that were developed recently for homogeneous media. Self-similar solutions are constructed for a power law time dependent surface temperature, a spatial power law density profile and a material model with power law temperature and density dependent opacities and specific energy density. The extension of the problem to non-homogeneous media enables the existence of similarity solutions for a general power law specific material energy. It is shown that the solutions exist for specific values of the temporal temperature drive and spatial density exponents, which depend on the material exponents. We also illustrate how the similarity solutions take various qualitatively different forms which are analyzed with respect to various parameters. Based on the solutions, we define a set of non-trivial benchmarks for supersonic non-equilibrium radiative heat transfer. The similarity solutions are compared to gray diffusion simulations as well as to detailed implicit Monte-Carlo and discrete-ordinate transport simulations in the optically-thick regime, showing a great agreement, which highlights the benefit of these solutions as a code verification test problem.
- [11] arXiv:2411.14909 (cross-list from gr-qc) [pdf, html, other]
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Title: Matter-antimatter (a)symmetry in de Sitter UniverseComments: 5 pages, 1 figure, version accepted for publication in Europhysics Letters (EPL)Subjects: General Relativity and Quantum Cosmology (gr-qc); High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph)
We investigate the matter-antimatter properties of elementary systems, modeled as free quantum fields, within the global structure of de Sitter spacetime. By leveraging the distinctive causal and analytic properties of de Sitter spacetime, we propose that matter-antimatter asymmetry could emerge as an observer-dependent effect shaped by time orientation within a local causal patch, rather than as a fundamental property of de Sitter Universe itself. This kinematic perspective complements, rather than replaces, standard dynamical processes (such as baryon number violation, $\texttt{CP}$ violation, and nonequilibrium processes) that fulfill Sakharov's criteria. Within this framework, the limited presence of antimatter in our predominantly matter-filled Universe, specifically within the causal patch of de Sitter spacetime under consideration, may arise from these mechanisms, though through pathways distinct from conventional interpretations.
- [12] arXiv:2411.14941 (cross-list from quant-ph) [pdf, html, other]
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Title: Completeness of Energy Eigenfunctions for the Reflectionless Potential in Quantum MechanicsComments: This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. This article appeared in American Journalof Physics 92, 950-956 (2024), and may be found at this https URL. This article is distributed under a Creative Commons Attribution-NonCommercial 4.0 International (CC BY-NC) LicenseJournal-ref: American Journal of Physics 92, 950-956 (2024)Subjects: Quantum Physics (quant-ph); Mathematical Physics (math-ph)
There are few exactly solvable potentials in quantum mechanics for which the completeness relation of the energy eigenstates can be explicitly verified. In this article, we give an elementary proof that the set of bound (discrete) states together with the scattering (continuum) states of the reflectionless potential form a complete set. We also review a direct and elegant derivation of the energy eigenstates with proper normalization by introducing an analog of the creation and annihilation operators of the harmonic oscillator problem. We further show that, in the case of a single bound state, the corresponding wave function can be found from the knowledge of continuum eigenstates of the system. Finally, completeness is shown by using the even/odd parity eigenstates of the Hamiltonian, which provides another explicit demonstration of a fundamental property of quantum mechanical Hamiltonians.
- [13] arXiv:2411.14965 (cross-list from math.SP) [pdf, other]
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Title: The curious spectra and dynamics of non-locally finite crystalsSubjects: Spectral Theory (math.SP); Mathematical Physics (math-ph)
This paper is devoted to the investigation of the spectral theory and dynamical properties of periodic graphs which are not locally finite but carry non-negative, symmetric and summable edge weights. These graphs are shown to exhibit rather intriguing behaviour: for example, we construct a periodic graph whose Laplacian has purely singular continuous spectrum. Regarding point spectrum, and different to the locally finite case, we construct a graph with a partly flat band whose eigenvectors must have infinite support. Concerning dynamical aspects, under some assumptions we prove that motion remains ballistic along at least one layer. We also construct a graph whose Laplacian has purely absolutely continuous spectrum, exhibits ballistic transport, yet fails to satisfy a dispersive estimate. This provides a negative answer to an open question in this context. Furthermore, we include a discussion of the fractional Laplacian for which we prove a phase transition in its dynamical behaviour. Generally speaking, many questions still remain open, and we believe that the studied class of graphs can serve as a playground to better understand exotic spectra and dynamics.
- [14] arXiv:2411.15021 (cross-list from physics.flu-dyn) [pdf, html, other]
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Title: Determination of the Young's angle using static friction in capillary bridgesComments: 15 pages and 7 figures. Comments are welcomeSubjects: Fluid Dynamics (physics.flu-dyn); Mathematical Physics (math-ph)
Recently contact angle hysteresis in two-dimensional droplets lying on a solid surface has been studied extensively in terms of static friction due to pinning forces at contact points. Here we propose a method to determine the coefficient of static friction using two-dimensional horizontal capillary bridges. This method requires only the measurement of capillary force and separation of plates, dispensing with the need for direct measurement of critical contact angles which is notoriously difficult. Based on this determination of friction coefficient, it is possible to determine the Young's angle from its relation to critical contact angles (advancing or receding). The Young's angle determined with our method is different either from the value estimated by Adam and Jessop a hundred years ago or the value argued by Drelich recently, though it is much closer to Adam and Jessop's numerically. The relation between energy and capillary force shows a capillary bridge behaves like a spring. Solving the Young-Laplace's equation, we can also locate the precise positions of neck or bulge and identify the exact moment when a pinch-off occurs.
- [15] arXiv:2411.15038 (cross-list from math.DG) [pdf, html, other]
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Title: Geometric phase and holonomy in the space of 2-by-2 symmetric operatorsSubjects: Differential Geometry (math.DG); Mathematical Physics (math-ph)
We present a non-trivial metric tensor field on the space of 2-by-2 real-valued, symmetric matrices whose Levi-Civita connection renders frames of eigenvectors parallel. This results in fundamental reimagining of the space of symmetric matrices as a curved manifold (rather than a flat vector space) and reduces the computation of eigenvectors of one-parameter-families of matrices to a single computation of eigenvectors at an initial point, while the rest are obtained by the parallel transport ODE. Our work has important applications to vibrations of physical systems whose topology is directly explained by the non-trivial holonomy of the spaces of symmetric matrices.
- [16] arXiv:2411.15077 (cross-list from quant-ph) [pdf, html, other]
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Title: Constructing Multipartite Planar Maximally Entangled States from Phase States and Quantum Secret Sharing ProtocolSubjects: Quantum Physics (quant-ph); Mathematical Physics (math-ph)
In this paper, we explore the construction of Planar Maximally Entangled (PME) states from phase states. PME states form a class of $n$-partite states in which any subset of adjacent particles whose size is less than or equal to half the total number of particles is in a fully entangled state. This property is essential to ensuring the robustness and stability of PME states in various quantum information applications. We introduce phase states for a set of so-called noninteracting $n$ particles and describe their corresponding separable density matrices. These phase states, although individually separable, serve as a starting point for the generation of entangled states when subjected to unitary dynamics. Using this method, we suggest a way to make complex multi-qubit states by watching how unconnected phase states change over time with a certain unitary interaction operator. In addition, we show how to derive PME states from these intricate phase states for two-, three-, four-, and K-qubit systems. This method of constructing PME states is particularly relevant for applications in fields such as quantum teleportation, quantum secret sharing, and quantum error correction, where multiparty entanglement plays a central role in the efficiency of the protocols.
- [17] arXiv:2411.15132 (cross-list from quant-ph) [pdf, html, other]
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Title: Efficient Eigenstate Preparation in an Integrable Model with Hilbert Space FragmentationComments: 42 pages, 19 figuresSubjects: Quantum Physics (quant-ph); Statistical Mechanics (cond-mat.stat-mech); Mathematical Physics (math-ph); Exactly Solvable and Integrable Systems (nlin.SI)
We consider the preparation of all the eigenstates of spin chains using quantum circuits. It is known that generic eigenstates of free-fermionic spin chains can be prepared with circuits whose depth grows only polynomially with the length of the chain and the number of particles. We show that the polynomial growth is also achievable for selected interacting models where the interaction between the particles is sufficiently simple. Our working example is the folded XXZ model, an integrable spin chain that exhibits Hilbert space fragmentation. We present the explicit quantum circuits that prepare arbitrary eigenstates of this model on an open chain efficiently. We perform error-mitigated noisy simulations with circuits of up to 13 qubits and different connectivities between qubits, achieving a relative error below 5%. As a byproduct, we extend a recent reformulation of the Bethe ansatz as a quantum circuit from closed to open boundary conditions.
Cross submissions (showing 14 of 14 entries)
- [18] arXiv:2403.06441 (replaced) [pdf, html, other]
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Title: On the group-theoretical approach to energy quantization of a perturbed vortex ring: spectrum calculating in the pipe-type domainComments: 13 pages, 1 figureSubjects: Mathematical Physics (math-ph); Quantum Physics (quant-ph)
In this study, the problem of the energy spectrum of a quantum vortex loop moving in a thin long pipe is solved for the first time. We quantize this dynamic system using a new method, which leads to non-trivial results for circulation $\Gamma$ and energy values $E$. It is shown that the spectrum has a quasi-continuous fractal structure. In the final form, we present the spectrum of the vortex loop in the form of a ''Regge trajectory'' $E = E(\Gamma)$. The vortex quantization problem is considered outside of two-fluid hydrodynamics and other conventional approaches. We also discuss ways to improve the model, which could allow us to apply the results we've obtained to describe a quantum turbulent flow.
- [19] arXiv:2411.01389 (replaced) [pdf, html, other]
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Title: Fluid dynamics duality and solution of decaying turbulenceComments: 22 pages, six figures, added discussion, and reformatted for the AIP journal submissionSubjects: Mathematical Physics (math-ph); Exactly Solvable and Integrable Systems (nlin.SI); Fluid Dynamics (physics.flu-dyn)
We describe the duality of incompressible Navier-Stokes fluid dynamics in three dimensions, leading to its reformulation in terms of a one-dimensional momentum loop equation. \textbf{The momentum loop equation does not have finite-time blow-up solutions.} The decaying turbulence is a solution of this equation equivalent to a string theory with discrete target space made of regular star polygons and Ising degrees of freedom on the sides. This string theory is solvable in the turbulent limit, equivalent to the quasiclassical approximation in a nontrivial calculable background. \textbf{As a result, the spectrum of decay indexes is analytically computed, and it agrees very well with real and numerical experiments. } Among the decay indexes, there are complex conjugate pairs related to zeros of the Riemann zeta function. The Kolmogorov scaling laws are replaced by certain number theory functions, which are nonlinear in the log-log scale.
- [20] arXiv:2012.01453 (replaced) [pdf, html, other]
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Title: Constructing quantum codes from any classical code and their embedding in ground space of local HamiltoniansComments: 21 pages + references ; 6 figures, accepted versionSubjects: Quantum Physics (quant-ph); Strongly Correlated Electrons (cond-mat.str-el); Mathematical Physics (math-ph)
Implementing robust quantum error correction (QEC) is imperative for harnessing the promise of quantum technologies. We introduce a framework that takes {\it any} classical code and explicitly constructs the corresponding QEC code. Our framework can be seen to generalize the CSS codes, and goes beyond the stabilizer formalism~(Fig.~1). A concrete advantage is that the desirable properties of a classical code are automatically incorporated in the design of the resulting quantum code. We reify the theory by various illustrations some of which outperform the best previous constructions. We then introduce a local quantum spin-chain Hamiltonian whose ground space we analytically completely characterize. We utilize our framework to demonstrate that the ground space contains explicit quantum codes with linear distance. This side-steps the Bravyi-Terhal no-go theorem.
- [21] arXiv:2203.09677 (replaced) [pdf, html, other]
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Title: Geodesics and dynamical information projections on the manifold of H\"older equilibrium probabilitiesComments: Keywords: Geodesics; infinite-dimensional Riemannian manifold; equilibrium probabilities; KL-divergence; information projections; Pythagorean inequalities; Fourier-like basisSubjects: Dynamical Systems (math.DS); Information Theory (cs.IT); Mathematical Physics (math-ph); Differential Geometry (math.DG); Probability (math.PR)
We consider here the discrete time dynamics described by a transformation $T:M \to M$, where $T$ is either the action of shift $T=\sigma$ on the symbolic space $M=\{1,2,...,d\}^\mathbb{N}$, or, $T$ describes the action of a $d$ to $1$ expanding transformation $T:S^1 \to S^1$ of class $C^{1+\alpha}$ (\,for example $x \to T(x) =d\, x $ (mod $1) $\,), where $M=S^1$ is the unit circle. It is known that the infinite-dimensional manifold $\mathcal{N}$ of equilibrium probabilities for Hölder potentials $A:M \to \mathbb{R}$ is an analytical manifold and carries a natural Riemannian metric associated with the asymptotic variance. We show here that under the assumption of the existence of a Fourier-like Hilbert basis for the kernel of the Ruelle operator there exists geodesics paths. When $T=\sigma$ and $M=\{0,1\}^\mathbb{N}$ such basis exists.
In a different direction, we also consider the KL-divergence $D_{KL}(\mu_1,\mu_2)$ for a pair of equilibrium probabilities. If $D_{KL}(\mu_1,\mu_2)=0$, then $\mu_1=\mu_2$. Although $D_{KL}$ is not a metric in $\mathcal{N}$, it describes the proximity between $\mu_1$ and $\mu_2$. A natural problem is: for a fixed probability $\mu_1\in \mathcal{N}$ consider the probability $\mu_2$ in a convex set of probabilities in $\mathcal{N}$ which minimizes $D_{KL}(\mu_1,\mu_2)$. This minimization problem is a dynamical version of the main issues considered in information projections. We consider this problem in $\mathcal{N}$, a case where all probabilities are dynamically invariant, getting explicit equations for the solution sought. Triangle and Pythagorean inequalities will be investigated. - [22] arXiv:2307.11008 (replaced) [pdf, html, other]
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Title: Distillable entanglement under dually non-entangling operationsComments: 10+24 pages; v2 is close to the published versionJournal-ref: Nature Communications 15, 10120 (2024)Subjects: Quantum Physics (quant-ph); Mathematical Physics (math-ph)
Computing the exact rate at which entanglement can be distilled from noisy quantum states is one of the longest-standing questions in quantum information. We give an exact solution for entanglement distillation under the set of dually non-entangling (DNE) operations -- a relaxation of the typically considered local operations and classical communication, comprising all channels which preserve the sets of separable states and measurements. We show that the DNE distillable entanglement coincides with a modified version of the regularised relative entropy of entanglement in which the arguments are measured with a separable measurement. Ours is only the second known regularised formula for the distillable entanglement under any class of free operations in entanglement theory, after that given by Devetak and Winter for (one-way) local operations and classical communication. An immediate consequence of our finding is that, under DNE, entanglement can be distilled from any entangled state. As our second main result, we construct a general upper bound on the DNE distillable entanglement, using which we prove that the separably measured relative entropy of entanglement can be strictly smaller than the regularisation of the standard relative entropy of entanglement, solving an open problem posed by Li and Winter [CMP 326, 63 (2014)]. Finally, we study also the reverse task of entanglement dilution and show that the restriction to DNE operations does not change the entanglement cost when compared with the larger class of non-entangling operations. This implies a strong form of irreversiblility of entanglement theory under DNE operations: even when asymptotically vanishing amounts of entanglement may be generated, entangled states cannot be converted reversibly.
- [23] arXiv:2312.15358 (replaced) [pdf, html, other]
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Title: Seat number configuration of the box-ball system, and its relation to the 10-elimination and invariant measuresComments: 31 pages, 2 figuresSubjects: Combinatorics (math.CO); Mathematical Physics (math-ph); Probability (math.PR)
The box-ball system (BBS) is a soliton cellular automaton introduced by [TS], and it is known that the dynamics of the BBS can be linearized by several methods. Recently, a new linearization method, called the seat number configuration, is introduced by [MSSS]. The aim of this paper is fourfold. First, we introduce the $k$-skip map $\Psi_{k} : \Omega \to \Omega$, where $\Omega$ is the state space of the BBS, and show that the $k$-skip map induces a shift operator of the seat number configuration. Second, we show that the $k$-skip map is a natural generalization of the $10$-elimination, which was originally introduced by [MIT] to solve the initial value problem of the periodic BBS. Third, we generalize the notions and results of the seat number configuration and the $k$-skip map for the BBS on the whole-line. Finally, we investigate the distribution of $\Psi_{k}(\eta), \eta \in \Omega$ when the distribution of $\eta$ belongs to a certain class of invariant measures of the BBS introduced by [FG]. As an application of the above results, we obtain the long-time behavior of the integrated current of $\Psi_{k}(\eta)$ with Markov stationary initial distributions.
- [24] arXiv:2403.11719 (replaced) [pdf, html, other]
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Title: Tight concentration inequalities for quantum adversarial setups exploiting permutation symmetryComments: 29 pages, 4 figures, published versionSubjects: Quantum Physics (quant-ph); Mathematical Physics (math-ph)
We developed new concentration inequalities for a quantum state on an $N$-qudit system or measurement outcomes on it that apply to an adversarial setup, where an adversary prepares the quantum state. Our one-sided concentration inequalities for a quantum state require the $N$-qudit system to be permutation invariant and are thus de-Finetti type, but they are tighter than the one previously obtained. We show that the bound can further be tightened if each qudit system has an additional symmetry. Furthermore, our concentration inequality for the outcomes of independent and identical measurements on an $N$-qudit quantum system has no assumption on the adversarial quantum state and is much tighter than the conventional one obtained through Azuma's inequality. We numerically demonstrate the tightness of our bounds in simple quantum information processing tasks.
- [25] arXiv:2406.01498 (replaced) [pdf, other]
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Title: The Hadamard condition on a Cauchy surface and the renormalized stress-energy tensorComments: 65 pages, discussion improvedJournal-ref: JCAP 10 (2024) 002Subjects: General Relativity and Quantum Cosmology (gr-qc); High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph)
Given a Cauchy surface in a curved spacetime and a suitably defined quantum state on the CCR algebra of the Klein-Gordon quantum field on that surface, we show, by expanding the squared spacetime geodesic distance and the `$U$' and `$V$' Hadamard coefficients (and suitable derivatives thereof) in sufficiently accurate covariant Taylor expansions on the surface that the renormalized expectation value of the quantum stress-energy tensor on the surface is determined by the geometry of the surface and the first 4 time derivatives of the metric off the surface, in addition to the Cauchy data for the field's two-point function. This result has been anticipated in and is motivated by a previous investigation by the authors on the initial value problem in semiclassical gravity, for which the geometric initial data corresponds {\it a priori} to the metric on the surface and up to 3 time derivatives off the surface, but where it was argued that the fourth derivative can be obtained with aid of the field equations on the initial surface.
- [26] arXiv:2406.14857 (replaced) [pdf, html, other]
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Title: Replica bound for Ising spin glass models in one dimensionComments: 18 pages, 0 figureSubjects: Disordered Systems and Neural Networks (cond-mat.dis-nn); Statistical Mechanics (cond-mat.stat-mech); Mathematical Physics (math-ph)
The interpolation method is a powerful tool for rigorous analysis of mean-field spin glass models, both with and without dilution. In this study, we show that the interpolation method can be applied to Ising spin glass models in one dimension, such as a one-dimensional chain and a two-leg ladder. In one dimension, the replica symmetric (RS) cavity method is naturally expected to be rigorous for Ising spin glass models. Using the interpolation method, we rigorously prove that the RS cavity method provides lower bounds on the quenched free energies of Ising spin glass models in one dimension at any finite temperature in the thermodynamic limit.
- [27] arXiv:2407.21605 (replaced) [pdf, html, other]
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Title: Geometry of Integrable Systems Related to the Restricted GrassmannianJournal-ref: SIGMA 20 (2024), 104, 18 pagesSubjects: Exactly Solvable and Integrable Systems (nlin.SI); Mathematical Physics (math-ph)
A hierarchy of differential equations on a Banach Lie-Poisson space related to the restricted Grassmannian is studied. Flows on the groupoid of partial isometries and on the restricted Grassmannian are described, and a momentum map picture is presented.
- [28] arXiv:2408.08181 (replaced) [pdf, html, other]
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Title: On geometric bases for quantum A-polynomials of knotsComments: 16 pages, v2: minor updatesJournal-ref: Phys.Lett.B 860 (2025) 139139Subjects: High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph); Geometric Topology (math.GT); Quantum Algebra (math.QA)
A simple geometric way is suggested to derive the Ward identities in the Chern-Simons theory, also known as quantum $A$- and $C$-polynomials for knots. In quasi-classical limit it is closely related to the well publicized augmentation theory and contact geometry. Quantization allows to present it in much simpler terms, what could make these techniques available to a broader audience. To avoid overloading of the presentation, only the case of the colored Jones polynomial for the trefoil knot is considered, though various generalizations are straightforward. Restriction to solely Jones polynomials (rather than full HOMFLY-PT) is related to a serious simplification, provided by the use of Kauffman calculus. Going beyond looks realistic, however it remains a problem, both challenging and promising.
- [29] arXiv:2409.12389 (replaced) [pdf, html, other]
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Title: Instantaneous tunneling time within the theory of time-of-arrival operatorsSubjects: Quantum Physics (quant-ph); Mathematical Physics (math-ph)
It has been shown in \href{this https URL}{\textit{Phys. Rev. Lett.}, \textbf{108} 170402 (2012)}, that quantum tunneling is instantaneous using a time-of-arrival (TOA) operator constructed by Weyl quantization of the classical TOA. However, there are infinitely many possible quantum images of the classical TOA, leaving it unclear if one is uniquely preferred over the others. This raises the question on whether instantaneous tunneling time is simply an artifact of the chosen ordering rule. Here, we demonstrate that tunneling time vanishes for all possible quantum images of the classical arrival time, irrespective of the ordering rule between the position and momentum observables. The result still holds for TOA-operators that are constructed independent of canonical quantization, while still imposing the correct algebra defined by the time-energy canonical commutation relation.
- [30] arXiv:2410.03175 (replaced) [pdf, html, other]
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Title: CMM formula as superintegrability property of unitary modelComments: 12 pagesSubjects: High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph)
A typical example of superintegrability is provided by expression of the Hopf link hyperpolynomial in an arbitrary representation through a pair of the Macdonald polynomials at special points. In the simpler case of the Hopf link HOMFLY-PT polynomial and a pair of the Schur functions, it is a relation in the unitary matrix model. We explain that the Cherednik-Mehta-Macdonald (CMM) identity for bilinear Macdonald residues with an elliptic weight function is nothing but a reformulation of these same formulas. Their lifting to arbitrary knots and links, even torus ones remains obscure.
- [31] arXiv:2410.20610 (replaced) [pdf, html, other]
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Title: Critical Droplets and Replica Symmetry BreakingComments: 12 pages, no figures, new version includes a Note Added in Proof and an additional reference ([38]). arXiv admin note: text overlap with arXiv:2110.11229Journal-ref: Front. Phys. 12:1473378 (2024)Subjects: Disordered Systems and Neural Networks (cond-mat.dis-nn); Mathematical Physics (math-ph)
We show that the notion of critical droplets is central to an understanding of the nature of ground states in the Edwards-Anderson Ising model of a spin glass in arbitrary dimension. Given a specific ground state, suppose the coupling value for a given edge is varied with all other couplings held fixed. Beyond some specific value of the coupling, a droplet will flip leading to a new ground state; we refer to this as the critical droplet for that edge and ground state. We show that the distribution of sizes and energies over all edges for a specific ground state can be used to determine which of the leading scenarios for the spin glass phase is correct. In particular, the existence of low-energy interfaces between incongruent ground states as predicted by replica symmetry breaking is equivalent to the presence of critical droplets whose boundaries comprise a positive fraction of edges in the infinite lattice.
- [32] arXiv:2411.09075 (replaced) [pdf, other]
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Title: Weak Poincar\'e Inequalities, Simulated Annealing, and Sampling from Spherical Spin GlassesComments: 94 pages, removed an incorrect application to the ferromagnetic Potts modelSubjects: Probability (math.PR); Disordered Systems and Neural Networks (cond-mat.dis-nn); Data Structures and Algorithms (cs.DS); Mathematical Physics (math-ph)
There has been a recent surge of powerful tools to show rapid mixing of Markov chains, via functional inequalities such as Poincaré inequalities. In many situations, Markov chains fail to mix rapidly from a worst-case initialization, yet are expected to approximately sample from a random initialization. For example, this occurs if the target distribution has metastable states, small clusters accounting for a vanishing fraction of the mass that are essentially disconnected from the bulk of the measure. Under such conditions, a Poincaré inequality cannot hold, necessitating new tools to prove sampling guarantees.
We develop a framework to analyze simulated annealing, based on establishing so-called weak Poincaré inequalities. These inequalities imply mixing from a suitably warm start, and simulated annealing provides a way to chain such warm starts together into a sampling algorithm. We further identify a local-to-global principle to prove weak Poincaré inequalities, mirroring the spectral independence and localization schemes frameworks for analyzing mixing times of Markov chains.
As our main application, we prove that simulated annealing samples from the Gibbs measure of a spherical spin glass for inverse temperatures up to a natural threshold, matching recent algorithms based on algorithmic stochastic localization. This provides the first Markov chain sampling guarantee that holds beyond the uniqueness threshold for spherical spin glasses, where mixing from a worst-case initialization is provably slow due to the presence of metastable states. As an ingredient in our proof, we prove bounds on the operator norm of the covariance matrix of spherical spin glasses in the full replica-symmetric regime.
Additionally, we resolve a question related to sampling using data-based initializations. - [33] arXiv:2411.14390 (replaced) [pdf, html, other]
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Title: Persistent Homology for Structural Characterization in Disordered SystemsComments: 19 pages, 17 figuresSubjects: Disordered Systems and Neural Networks (cond-mat.dis-nn); Materials Science (cond-mat.mtrl-sci); Machine Learning (cs.LG); Mathematical Physics (math-ph)
We propose a unified framework based on persistent homology (PH) to characterize both local and global structures in disordered systems. It can simultaneously generate local and global descriptors using the same algorithm and data structure, and has shown to be highly effective and interpretable in predicting particle rearrangements and classifying global phases. Based on this framework, we define a non-parametric metric, the Separation Index (SI), which not only outperforms traditional bond-orientational order parameters in phase classification tasks but also establishes a connection between particle environments and the global phase structure. Our methods provide an effective framework for understanding and analyzing the properties of disordered materials, with broad potential applications in materials science and even wider studies of complex systems.