Quantum concentration inequalities

Abstract

We establish transportation cost inequalities (TCI) with respect to the quantum Wasserstein distance by introducing quantum extensions of well-known classical methods: first, using a non-commutative version of Ollivier's coarse Ricci curvature, we prove that high temperature Gibbs states of commuting Hamiltonians on arbitrary hypergraphs $H=(V,E)$ satisfy a TCI with constant scaling as $O(|V|)$. Second, we argue that the temperature range for which the TCI holds can be enlarged by relating it to recently established modified logarithmic Sobolev inequalities. Third, we prove that the inequality still holds for fixed points of arbitrary reversible local quantum Markov semigroups on regular lattices, albeit with slightly worsened constants, under a seemingly weaker condition of local indistinguishability of the fixed points. Finally, we use our framework to prove Gaussian concentration bounds for the distribution of eigenvalues of quasi-local observables and argue the usefulness of the TCI in proving the equivalence of the canonical and microcanonical ensembles and an exponential improvement over the weak Eigenstate Thermalization Hypothesis.

31 pages, one figure

References in corpus (3)

• Structure of states which satisfy strong subadditivity of quantum entropy with equality
• Complete entropic inequalities for quantum Markov chains
• Equivalence of Statistical Mechanical Ensembles for Non-Critical Quantum Systems

Cited by in corpus (17)

• Classical simulation of short-time quantum dynamics
• Entropy decay for Davies semigroups of a one dimensional quantum lattice
• Learning quantum many-body systems from a few copies
• Improving the speed of variational quantum algorithms for quantum error correction
• Rapid thermalization of dissipative many-body dynamics of commuting Hamiltonians
• Concentration bounds for quantum states and limitations on the QAOA from polynomial approximations
• Deviation bounds and concentration inequalities for quantum noises
• The Wasserstein distance of order 1 for quantum spin systems on infinite lattices
• Quantum Optimal Transport: Quantum Channels and Qubits
• Ricci curvature of quantum channels on non-commutative transportation metric spaces
• Quantum Wasserstein distance of order 1 between channels
• Order $p$ quantum Wasserstein distances from couplings
• Classical shadows meet quantum optimal mass transport
• Wasserstein Complexity of Quantum Circuits
• Efficient learning of ground & thermal states within phases of matter
• Quantum concentration inequalities and equivalence of the thermodynamical ensembles: an optimal mass transport approach
• Benchmarking Quantum Simulators