Entropy-based random quantum states

TL;DR

This paper introduces an entropy-based random quantum state ensemble, providing a new method for generating mixed states, analyzing their statistical properties, and exploring their connection to chaotic Hamiltonians and quantum chaos phenomena.

Contribution

It develops a novel entropy-based random matrix ensemble for quantum states, establishes a duality with chaotic Hamiltonian models, and demonstrates applications in quantum chaos and Bayesian quantum tomography.

Findings

The entropy-based ensemble exhibits Wigner level repulsion.

Duality with chaotic Hamiltonian models is established.

Application to quantum chaos via survival probability analysis.

Abstract

In quantum information geometry, the curvature of von-Neumann entropy and relative entropy induce a natural metric on the space of mixed quantum states. Here we use this information metric to construct a random matrix ensemble for states and investigate its key statistical properties such as the asymptotic eigenvalue density and mean entropy. We present an algorithm for generating these entropy-based random density matrices, thus providing a new recipe for random state generation that differs from the well established Hilbert-Schmidt and Bures-Hall ensemble approaches. We also prove a duality between the entropy-based state ensemble and a random Hamiltonian model constructed from the thermodynamic length over the set of Gibbs states. This Hamiltonian model is found to display Wigner level repulsion, implying that the dual state ensemble can be realised as a random Gibbs state with respect to a class of chaotic Hamiltonians. As an application we use our model to compute the survival probability of a randomly evolved thermofield double state, predicting a ramp and plateau over time that is characteristic of quantum chaos. For other applications, the entropy-based ensemble can be used as an uninformative prior for Bayesian quantum state or Hamiltonian tomography.