Communications in Mathematical Physics最新文献

Inspired by Etingof–Varchenko’s dynamical Weyl group for Lie algebras and Reshetikhin–Stokman’s boundary fusion operators for split symmetric pairs, we introduce a dynamical Weyl group for split symmetric pairs. We then turn to the study of ((mathfrak {so}_{2n},textrm{O}_m))-duality and prove that the standard Knizhnik–Zamolodchikov and dynamical operators (both differential and difference) on the (mathfrak {so}_{2n})-side are exchanged with the symmetric pair analogs, for (textrm{O}_msubset textrm{GL}_m), on the (textrm{O}_m)-side.

We construct random walks on simple Lie groups that quickly converge to the Haar measure for all moments up to order t. Specifically, a step of the walk on the unitary or orthogonal group of dimension (2^{{textsf{n}}}) is a random Pauli rotation (e^{mathrm i theta P /2}). The spectral gap of this random walk is shown to be (Omega (1/t)), which coincides with the best previously known bound for a random walk on the permutation group on ({0,1}^{{textsf{n}}}). This implies that the walk gives an (varepsilon )-approximate unitary t-design in depth (mathcal O({textsf{n}} t^2 + t log frac{1}{varepsilon })d) where (d=mathcal {O}(log {textsf{n}})) is the circuit depth to implement (e^{mathrm i theta P /2}). Our simple proof uses quadratic Casimir operators of Lie algebras.

The quantum data processing inequality asserts that two quantum states become harder to distinguish when a noisy channel is applied. On the other hand, a reverse quantum data processing inequality characterizes whether distinguishability is preserved after the application of a noisy channel. In this work, we explore these concepts through contraction and expansion coefficients of the relative entropy of quantum channels. Our first result is that quantum channels with an input dimension greater than or equal to the output dimension do not have a non-zero expansion coefficient, which means that they cannot admit a reverse data-processing inequality. We propose a comparative approach by introducing a relative expansion coefficient, to assess how one channel expands relative entropy compared to another. We show that this relative expansion coefficient is positive for three important classes of quantum channels: depolarizing channels, generalized dephasing channels, and amplitude damping channels. As an application, we give the first rigorous construction of level-1 less noisy quantum channels that are non-degradable.

The purpose of this article is to show a geometric version of Zabrodin–Wiegmann conjecture for an integer quantum Hall state. Given an effective reduced divisor on a compact connected Riemann surface, using the canonical holomorphic section of the associated canonical line bundle as well as certain initial data and local normalisation data, we construct a canonical non-zero element in the determinant line of the cohomology of the p-tensor power of the line bundle. When endowed with proper metric data, the square of the (L^{2})-norm of our canonical element is the partition function associated to an integer quantum Hall state. We establish an asymptotic expansion for the logarithm of the partition function when (prightarrow +infty ). The constant term of this expansion includes the holomorphic analytic torsion and matches a geometric version of Zabrodin–Wiegmann’s prediction. Our proof relies on Bismut–Lebeau’s embedding formula for the Quillen metrics, Bismut–Vasserot and Finski’s asymptotic expansion for the analytic torsion associated to the higher tensor product of a positive Hermitian holomorphic line bundle.

We investigate periodic trajectories in a classical system consisting of multiple mutually repelling electrons constrained to move along a half-line, with an attractive nucleus fixed at the origin. Adopting a variational framework, we seek critical points of the associated Lagrangian action functional using a modified Lusternik-Schnirelmann theory for manifolds with boundary. Furthermore, in the limit where the electron charges vanish, we demonstrate that frozen planet orbits converge to segments of a brake orbit in a Kepler-type problem, thereby drawing a strong analogy with Schubart orbits in the gravitational N-body problem.

We consider rigorous consequences of modular invariance for two-dimensional unitary non-rational CFTs with (c > 1). Simple estimates for the torus partition function can lead to remarkably strong results. We show in particular that the spectral density of spin-J operators must grow like (exp left( pi sqrt{frac{2}{3}(c-1) J} right) /sqrt{2J}) in any twist interval at or above ((c-1)/12), with a known twist-dependent prefactor. This proves that the large J spectrum becomes dense even without averaging over spins. For twists below ((c-1)/12) we establish that the growth must be strictly slower. Finally, we estimate how fast the maximal gap between two spin-J operators must go to zero as J becomes large.

We develop the theory of local operations and classical communication (LOCC) for bipartite quantum systems represented by commuting von Neumann algebras. Our central result is the extension of Nielsen’s Theorem, stating that the LOCC ordering of bipartite pure states is equivalent to the majorization of their restrictions to arbitrary factors. As a consequence, we find that in bipartite system modeled by commuting factors in Haag duality, (a) all states have infinite single-shot entanglement if and only if the local factors are not of type I, (b) type III factors are characterized by LOCC transitions of arbitrary precision between any two pure states, and c) the latter holds even without classical communication for type (hbox {III}_{1}) factors. In the case of semifinite factors, the usual construction of pure state entanglement monotones carries over. Together with recent work on embezzlement of entanglement, this gives a one-to-one correspondence between the classification of factors into types and subtypes and operational entanglement properties. In the appendix, we provide a self-contained treatment of majorization on semifinite von Neumann algebras and (sigma )-finite measure spaces.

We show the strong graded locality of all unitary minimal W-algebras, so that they give rise to irreducible graded-local conformal nets. Among these unitary vertex superalgebras, up to taking tensor products with free fermion vertex superalgebras, there are the unitary Virasoro vertex algebras ((N=0)) and the unitary (N=1,2,3,4) super-Virasoro vertex superalgebras. Accordingly, we have a uniform construction that gives, besides the already known (N=0,1,2) super-Virasoro nets, also the new (N=3,4) super-Virasoro nets. All strongly rational unitary minimal W-algebras give rise to previously known completely rational graded-local conformal nets and we conjecture that the converse is also true. We prove this conjecture for all unitary W-algebras corresponding to the (N=0,1,2,3,4) super-Virasoro vertex superalgebras.

We give a classification of the regular soliton solutions of the KP hierarchy, referred to as the KP solitons, under the Gel’fand-Dickey (ell )-reductions in terms of the permutation of the symmetric group. As an example, we show that the regular soliton solutions of the (good) Boussinesq equation as the 3-reduction can have at most one resonant soliton in addition to two sets of solitons propagating in opposite directions. We also give a systematic construction of these soliton solutions for the (ell )-reductions using the vertex operators. In particular, we show that the non-crossing permutation gives the regularity condition for the soliton solutions.

The von Neumann entropy of an n-partite system (A_1^n) given a system B can be written as the sum of the von Neumann entropies of the individual subsystems (A_k) given (A_1^{k-1}) and B. While it is known that such a chain rule does not hold for the smooth min-entropy, we prove a counterpart of this for a variant of the smooth min-entropy, which is equal to the conventional smooth min-entropy up to a constant. This enables us to lower bound the smooth min-entropy of an n-partite system in terms of, roughly speaking, equally strong entropies of the individual subsystems. We call this a universal chain rule for the smooth min-entropy, since it is applicable for all values of n. Using duality, we also derive a similar relation for the smooth max-entropy. Our proof utilises the entropic triangle inequality based technique developed in Marwah and Dupuis (Commun Math Phys 405(9):211, 2024. https://doi.org/10.1007/s00220-024-05074-8) for analysing approximation chains. Additionally, we also prove an approximate version of the entropy accumulation theorem, which significantly relaxes the conditions required on the state to bound its smooth min-entropy. In particular, it does not require the state to be produced through a sequential process like previous entropy accumulation type bounds. In our companion paper Marwah and Dupuis (Security proof for parallel DIQKD, 2025. https://arxiv.org/abs/2507.03991), we use it to prove the security of parallel device independent quantum key distribution.