State Ensembles and Quantum Entropy

Shunlong Luo, Nan Li, Xuelian Cao

Harry J. D. Miller

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.

Sohail, Lu Wei

Random states play a foundational role in different branches of modern quantum science. In this work, we study a recently proposed random state ensemble induced from von Neumann entropy through the Bogoliubov-Kubo-Mori (BKM) metric. In particular, we derive an exact yet explicit formula of average entanglement entropy over BKM ensemble. In obtaining the formula, we only make use of properties of normalization constant of the ensemble in the absence of its correlation kernel, contrary to average entropy computation of other ensembles. This new framework paves the way for calculating higher-order cumulants of BKM ensemble beyond the average.

Olga V. Man’ko

Richard Jozsa, J. Schlienz

Let ${|{\ensuremath{\psi}}_{1}〉,\dots{},|{\ensuremath{\psi}}_{n}〉{;p}_{1},\dots{}{,p}_{n}}$ be an ensemble of pure quantum states. We show that it is possible to increase all of the pairwise overlaps $|〈{\ensuremath{\psi}}_{i}|{\ensuremath{\psi}}_{j}〉|,$ i.e., make each constituent pair of the states more parallel (while keeping the prior probabilities the same), in such a way that the von Neumann entropy S is increased, and dually, make all pairs more orthogonal while decreasing S. We show that this phenomenon cannot occur for ensembles in two dimensions but that it is a feature of almost all ensembles of three states in three dimensions. It is known that the von Neumann entropy characterizes the classical and quantum information capacities of the ensemble and we argue that information capacity, in turn, is a manifestation of the distinguishability of the signal states. Hence, our result shows that the notion of distinguishability within an ensemble is a global property that cannot be reduced to considering distinguishability of each constituent pair of states.

A. P. Balachandran, Amilcar R. de Queiroz, Sachindeo Vaidya