@article{9036, abstract = {In this short note, we prove that the square root of the quantum Jensen-Shannon divergence is a true metric on the cone of positive matrices, and hence in particular on the quantum state space.}, author = {Virosztek, Daniel}, issn = {0001-8708}, journal = {Advances in Mathematics}, keywords = {General Mathematics}, number = {3}, publisher = {Elsevier}, title = {{The metric property of the quantum Jensen-Shannon divergence}}, doi = {10.1016/j.aim.2021.107595}, volume = {380}, year = {2021}, } @article{9412, abstract = {We extend our recent result [22] on the central limit theorem for the linear eigenvalue statistics of non-Hermitian matrices X with independent, identically distributed complex entries to the real symmetry class. We find that the expectation and variance substantially differ from their complex counterparts, reflecting (i) the special spectral symmetry of real matrices onto the real axis; and (ii) the fact that real i.i.d. matrices have many real eigenvalues. Our result generalizes the previously known special cases where either the test function is analytic [49] or the first four moments of the matrix elements match the real Gaussian [59, 44]. The key element of the proof is the analysis of several weakly dependent Dyson Brownian motions (DBMs). The conceptual novelty of the real case compared with [22] is that the correlation structure of the stochastic differentials in each individual DBM is non-trivial, potentially even jeopardising its well-posedness.}, author = {Cipolloni, Giorgio and Erdös, László and Schröder, Dominik J}, issn = {10836489}, journal = {Electronic Journal of Probability}, publisher = {Institute of Mathematical Statistics}, title = {{Fluctuation around the circular law for random matrices with real entries}}, doi = {10.1214/21-EJP591}, volume = {26}, year = {2021}, } @article{9550, abstract = {We prove that the energy of any eigenvector of a sum of several independent large Wigner matrices is equally distributed among these matrices with very high precision. This shows a particularly strong microcanonical form of the equipartition principle for quantum systems whose components are modelled by Wigner matrices. }, author = {Bao, Zhigang and Erdös, László and Schnelli, Kevin}, issn = {20505094}, journal = {Forum of Mathematics, Sigma}, publisher = {Cambridge University Press}, title = {{Equipartition principle for Wigner matrices}}, doi = {10.1017/fms.2021.38}, volume = {9}, year = {2021}, } @article{9912, abstract = {In the customary random matrix model for transport in quantum dots with M internal degrees of freedom coupled to a chaotic environment via 𝑁≪𝑀 channels, the density 𝜌 of transmission eigenvalues is computed from a specific invariant ensemble for which explicit formula for the joint probability density of all eigenvalues is available. We revisit this problem in the large N regime allowing for (i) arbitrary ratio 𝜙:=𝑁/𝑀≤1; and (ii) general distributions for the matrix elements of the Hamiltonian of the quantum dot. In the limit 𝜙→0, we recover the formula for the density 𝜌 that Beenakker (Rev Mod Phys 69:731–808, 1997) has derived for a special matrix ensemble. We also prove that the inverse square root singularity of the density at zero and full transmission in Beenakker’s formula persists for any 𝜙<1 but in the borderline case 𝜙=1 an anomalous 𝜆−2/3 singularity arises at zero. To access this level of generality, we develop the theory of global and local laws on the spectral density of a large class of noncommutative rational expressions in large random matrices with i.i.d. entries.}, author = {Erdös, László and Krüger, Torben H and Nemish, Yuriy}, issn = {1424-0661}, journal = {Annales Henri Poincaré }, pages = {4205–4269}, publisher = {Springer Nature}, title = {{Scattering in quantum dots via noncommutative rational functions}}, doi = {10.1007/s00023-021-01085-6}, volume = {22}, year = {2021}, } @article{10221, abstract = {We prove that any deterministic matrix is approximately the identity in the eigenbasis of a large random Wigner matrix with very high probability and with an optimal error inversely proportional to the square root of the dimension. Our theorem thus rigorously verifies the Eigenstate Thermalisation Hypothesis by Deutsch (Phys Rev A 43:2046–2049, 1991) for the simplest chaotic quantum system, the Wigner ensemble. In mathematical terms, we prove the strong form of Quantum Unique Ergodicity (QUE) with an optimal convergence rate for all eigenvectors simultaneously, generalizing previous probabilistic QUE results in Bourgade and Yau (Commun Math Phys 350:231–278, 2017) and Bourgade et al. (Commun Pure Appl Math 73:1526–1596, 2020).}, author = {Cipolloni, Giorgio and Erdös, László and Schröder, Dominik J}, issn = {1432-0916}, journal = {Communications in Mathematical Physics}, number = {2}, pages = {1005–1048}, publisher = {Springer Nature}, title = {{Eigenstate thermalization hypothesis for Wigner matrices}}, doi = {10.1007/s00220-021-04239-z}, volume = {388}, year = {2021}, }