Annales Henri Poincaré最新文献

Quantum information processing and computing tasks can be understood as quantum networks, comprising quantum states and channels and possible physical transformations on them. It is hence pertinent to estimate the change in informational content of quantum processes due to physical transformations they undergo. The physical transformations of quantum states are described by quantum channels, while the transformations of quantum channels are described by quantum superchannels. In this work, we determine fundamental limitations on how well the physical transformation on quantum channels can be undone or reversed, which are of crucial interest to design and benchmark quantum information and computation devices. In particular, we refine (strengthen) the quantum data processing inequality for quantum channels under the action of quantum superchannels. We identify a class of quantum superchannels, which appears to be the superchannel analog of subunital quantum channels, under the action of which the entropy of an arbitrary quantum channel is nondecreasing. We also provide a refined inequality for the entropy change of quantum channels under the action of an arbitrary quantum superchannel.

A separable version of Ky Fan’s majorization relation is proven for a sum of two operators that are each a tensor product of two positive semi-definite operators. In order to prove it, upper bounds are established on the relevant largest eigenvalue sums in terms of the optimal values of certain linear programs. The objective function of these linear programs is the dual of the direct sum of the spectra of the summands. The feasible sets are bounded polyhedra determined by positive numbers, called alignment terms, that quantify the overlaps between pairs of largest eigenvalue spaces of the summands. By appealing to geometric considerations, tight upper bounds are established on the alignment terms of tensor products of positive semi-definite operators. As an application, the spin alignment conjecture in quantum information theory is affirmatively resolved to the 2-letter level. Consequently, the coherent information of platypus channels is additive to the 2-letter level.

We perform a rigorous study of the Gibbs statistics of high-density hard-core random configurations on a unit triangular lattice (mathbb {A}_2) and a unit honeycomb graph (mathbb {H}_2), for any value of the (Euclidean) repulsion diameter (D>0). Only attainable values of D are relevant, for which (D^2=a^2+b^2+ab), (a, b in mathbb {Z}) (Löschian numbers). Depending on arithmetic properties of (D^2), we identify, for large fugacities, the pure phases (extreme Gibbs measures) and specify their symmetries. The answers depend on the way(s) an equilateral triangle of side-length D can be inscribed in (mathbb {A}_2) or (mathbb {H}_2). On (mathbb {A}_2), our approach works for all attainable (D^2); on (mathbb {H}_2) we have to exclude (D^2 = 4, 7, 31, 133), where a sliding phenomenon occurs, similar to that on a unit square lattice (mathbb {Z}^2). For all values (D^2) apart from the excluded ones, we prove the coexistence of multiple high-density pure phases. Their number grows at least as (O(D^2)); this establishes the existence of a phase transition. The proof is based on the Pirogov–Sinai theory which, in its original form, requires the verification of key assumptions: finiteness of the set of periodic ground states and the Peierls bound. To establish the Peierls bound, we develop a general method based on the concept of a redistributed area for Delaunay triangles. Some of the presented proofs are computer-assisted. As a by-product of the ground state identification, we solve the disk-packing problem on (mathbb {A}_2) and (mathbb {H}_2) for any value of the disk diameter D.

We study the top Lyapunov exponent of a product of random (2 times 2) matrices appearing in the analysis of several statistical mechanical models with disorder, extending a previous treatment of the critical case (Giacomin and Greenblatt, ALEA 19 (2022), 701-728) by significantly weakening the assumptions on the disorder distribution. The argument we give completely revisits and improves the previous proof. As a key novelty we build a probability that is close to the Furstenberg probability, i.e., the invariant probability of the Markov chain corresponding to the evolution of the direction of a vector in (mathbb {R}^2) under the action of the random matrices, in terms of the ladder times of a centered random walk which is directly related to the random matrix sequence. We then show that sharp estimates on the ladder times (renewal) process lead to a sharp control on the probability measure we build and, in turn, to the control of its distance from the Furstenberg probability.

Bulk-edge correspondence is a wide-ranging principle that applies to topological matter, as well as a precise result established in a large and growing number of cases. According to the principle, the distinctive topological properties of matter, thought of as extending indefinitely in space, are equivalently reflected in the excitations running along its boundary, when one is present. Indices encode those properties, and their values, when differing, are witness to a violation of that correspondence. We address such violations, as they are encountered in a hydrodynamic context. The model concerns a shallow layer of fluid in a rotating frame and provides a local description of waves propagating either across the oceans or along a coastline; it becomes topological when suitably modified at short distances. The edge index is sensitive to boundary conditions, as exemplified in earlier work, hence exhibiting a violation. Here we present classification of all (local, self-adjoint) boundary conditions and a parameterization of their manifold. They come in four families, distinguished in part by the degree of their underlying differential operators. Essentially, that degree counts the degrees of freedom of the hydrodynamic field that are constrained at the boundary by way of their normal derivatives. Generally, both the correspondence and its violation are typical. Within families though, the maximally possible amount of violation is increasing with its degree. Several indices of interest are charted for all boundary conditions. A single spectral mechanism for the onset of violations is furthermore identified. The role of a symmetry is investigated.

In this paper, we consider the Hölder continuity of the integrated density of states (IDS). Applying Avila’s almost reducible result and KAM technique, we proved that there exists a dense subset of Liouvillean frequencies (alpha ), for which the IDS of the analytic quasi-periodic Schrödinger operator is ((chi )-(log ))-Hölder continuous for any (chi >1), provided that the subcritical strip of the operator satisfies (h_0 > 2beta (alpha ) ). We also proved the (chi )-Hölder continuity of the IDS for a dense subset of Liouvillean frequencies for operators with (0< chi < frac{1}{2}), if the subcritical strip satisfies (h_0 > frac{8beta }{ 1-2chi }).

The group symmetries inherent in quantum channels often make them tractable and applicable to various problems in quantum information theory. In this paper, we introduce natural probability distributions for covariant quantum channels. Specifically, this is achieved through the application of “twirling operations” on random quantum channels derived from the Stinespring representation that uses Haar-distributed random isometries. We explore various types of group symmetries, including unitary and orthogonal covariance, hyperoctahedral covariance, and diagonal orthogonal covariance (DOC), and analyze their properties related to quantum entanglement based on the model parameters. In particular, we discuss the threshold phenomenon for positive partial transpose and entanglement breaking properties, comparing thresholds among different classes of random covariant channels. Finally, we contribute to the (hbox {PPT}^2) conjecture by showing that the composition between two random DOC channels is generically entanglement breaking.

We study the mean diagonal overlap of left and right eigenvectors associated with complex eigenvalues in (Ntimes N) non-Hermitian random Gaussian matrices. In a well-known work by Chalker and Mehlig the expectation of this (self-)overlap was computed for the complex Ginibre ensemble as (Nrightarrow infty ) (Chalker and Mehlig in Phys Rev Lett 81(16):3367–3370, 1998). In the present work, we consider the same quantity in the real and complex elliptic Ginibre ensembles, which are characterised by correlations between off-diagonal entries controlled by a parameter (tau in [0,1]), with (tau =1) corresponding to the Hermitian limit. We derive exact expressions for the mean diagonal overlap in both ensembles at any finite N, for any eigenvalue off the real axis. We further investigate several scaling regimes as (Nrightarrow infty ), both in the limit of strong non-Hermiticity keeping a fixed (tau in [0,1)) and in the weak non-Hermiticity limit, with (tau ) approaching unity in such a way that (N(1-tau )) remains finite.

In this paper, we study the distribution of temperature of a body due to the transfer of radiation. Specifically, the boundary value problem for the stationary radiative transfer equation is considered. In all the analysis, we assume the so-called local thermal equilibrium (LTE), i.e., there is a well-defined temperature of the body at each point. We consider the limit in which the mean free path of the photons is much smaller than the characteristic length of the domain. In this case, we can approximate the solution by means of the so-called diffusion approximation. The analysis of this paper is restricted to the case in which the absorption coefficient is independent of the frequency ( nu ) (the so-called Grey approximation). We ignore also scattering effects. Under these assumptions, we show that the density of radiative energy u, which is proportional to the fourth power of the temperature, solves in the limit an elliptic equation. The boundary values for that limit equation can be determined uniquely analyzing a suitable boundary layer problem. The method developed here allows to prove all the results using maximum principle arguments for a class of non-local elliptic equations.

A local time decay estimate of fractional Schrödinger operators with slowly decaying positive potentials is studied. It is shown that the resolvent is smooth near zero, and the time propagator exhibits fast local time decay, which is very different from very short-range cases. The key element of the proof is to establish a weaker Agmon estimate for a classically forbidden region using exotic symbol calculus. As a byproduct, we prove that the Riesz operator is a pseudodifferential operator with an exotic symbol.