Annales Henri Poincaré最新文献

The asymptotic restriction problem for tensors can be reduced to finding all parameters that are normalized, monotone under restrictions, additive under direct sums and multiplicative under tensor products, the simplest of which are the flattening ranks. Over the complex numbers, a refinement of this problem, originating in the theory of quantum entanglement, is to find the optimal rate of entanglement transformations as a function of the error exponent. This trade-off can also be characterized in terms of the set of normalized, additive, multiplicative functionals that are monotone in a suitable sense, which includes the restriction-monotones as well. For example, the flattening ranks generalize to the (exponentiated) Rényi entanglement entropies of order (alpha in [0,1]). More complicated parameters of this type are known, which interpolate between the flattening ranks or Rényi entropies for special bipartitions, with one of the parts being a single tensor factor. We introduce a new construction of subadditive and submultiplicative monotones in terms of a regularized Rényi divergence between many copies of the pure state represented by the tensor and a suitable sequence of positive operators. We give explicit families of operators that correspond to the flattening-based functionals, and show that they can be combined in a nontrivial way using weighted operator geometric means. This leads to a new characterization of the previously known additive and multiplicative monotones, and gives new submultiplicative and subadditive monotones that interpolate between the Rényi entropies for all bipartitions. We show that for each such monotone there exist pointwise smaller multiplicative and additive ones as well. In addition, we find lower bounds on the new functionals that are superadditive and supermultiplicative.

We study k-point correlators of characteristic polynomials in non-Hermitian ensembles of random matrices, focusing on the Ginibre and truncated unitary random matrices. Our approach is based on the technique of character expansions, which expresses the correlator as a sum over partitions involving Schur functions. We show how to sum the expansions in terms of representations which interchange the role of k with the matrix size N. We also provide a probabilistic interpretation of the character expansion analogous to the Schur measure, linking the correlators to the distribution of the top row in certain Young diagrams. In more specific examples, we evaluate these expressions in terms of (k times k) determinants or Pfaffians.

We introduce a meta logarithmic-Sobolev (log-Sobolev) inequality for the Lindbladian of all single-mode phase-covariant Gaussian channels of bosonic quantum systems and prove that this inequality is saturated by thermal states. We show that our inequality provides a general framework to derive information theoretic results regarding phase-covariant Gaussian channels. Specifically, by using the optimality of thermal states, we explicitly compute the optimal constant (alpha _p), for (1le ple 2), of the p-log-Sobolev inequality associated with the quantum Ornstein–Uhlenbeck semigroup. Prior to our work, the optimal constant was only determined for (p=1). Our meta log-Sobolev inequality also enables us to provide an alternative proof for the constrained minimum output entropy conjecture in the single-mode case. Specifically, we show that for any single-mode phase-covariant Gaussian channel (Phi ), the minimum of the von Neumann entropy (Sbig (Phi (rho )big )) over all single-mode states (rho ) with a given lower bound on (S(rho )) is achieved at a thermal state.

We give a rigorous derivation of the free energy of (i) the classical Ising model on the triangular lattice with translation-invariant coupling constants and (ii) the one-dimensional quantum Ising model. We use the method of Kac and Ward. The novel aspect is that the coupling constants may have negative signs. We describe the logarithmic singularity of the specific heat of the classical model and the validity of the Cimasoni–Duminil-Copin–Li formula for the critical temperature. We also discuss the quantum phase transition of the quantum model.

We prove tunneling estimates for two-dimensional Dirac systems which are localized in space due to the presence of a magnetic field. The Hamiltonian driving the motion admits the decomposition ( H = H_0 + W), where (H_0 ) is a rotationally symmetric magnetic Dirac operator and W is a position-dependent matrix-valued potential satisfying certain smoothness condition in the angular variable. A consequence of our results are upper bounds for the growth in time of the expected size of the system and its total angular momentum.

The Aharonov–Casher theorem is a result on the number of the so-called zero modes of a system described by the magnetic Pauli operator in (mathbb {R}^2). In this paper we address the same question for the Dirac operator on a flat two-dimensional manifold with boundary and Atiyah–Patodi–Singer boundary condition. More concretely we are interested in the plane and a disc with a finite number of circular holes cut out. We consider a smooth compactly supported magnetic field on the manifold and an arbitrary magnetic field inside the holes.