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Entropy-based random quantum states

Harry J. D. Miller

TL;DR

This work introduces an entropy-based approach to sampling random quantum states by leveraging the Bogoliubov-Kubo-Mori (BKM) metric, a geometric generalization tied to von Neumann entropy. It derives the metric-induced volume form and the associated eigenvalue density, revealing a sampling procedure linked to the Muttalib–Borodin ensemble and yielding a new random-density ensemble with higher purity than HS or BH ensembles. The paper provides a Page-like entropy estimate for the BKM ensemble, demonstrates concentration of entropy, and offers the X–Y sampling algorithm that reproduces the BKM statistics; these results enable an uninformative prior for Bayesian quantum state tomography, particularly in high-purity regimes, and offer a tool to study typical entanglement in finite-depth quantum circuits. It also outlines future directions, including cost analyses, thermodynamic connections, and extensions to infinite-dimensional systems.

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 the eigenvalue density and probability distribution of entropy. We present an algorithm for generating these entropy-based random density matrices by sampling a class of bipartite pure states, thus providing a new recipe for random state generation that differs from the well established Hilbert-Schmidt and Bures-Hall ensemble approaches. We find that a distinguishing feature of the ensemble is its larger purity and increased volume towards the boundary of full-rank states. The entropy-based ensemble can thus be used as a uninformative prior for Bayesian quantum state tomography in high purity regimes, and as a tool for quantifying typical entanglement in finite depth quantum circuits.

Entropy-based random quantum states

TL;DR

This work introduces an entropy-based approach to sampling random quantum states by leveraging the Bogoliubov-Kubo-Mori (BKM) metric, a geometric generalization tied to von Neumann entropy. It derives the metric-induced volume form and the associated eigenvalue density, revealing a sampling procedure linked to the Muttalib–Borodin ensemble and yielding a new random-density ensemble with higher purity than HS or BH ensembles. The paper provides a Page-like entropy estimate for the BKM ensemble, demonstrates concentration of entropy, and offers the X–Y sampling algorithm that reproduces the BKM statistics; these results enable an uninformative prior for Bayesian quantum state tomography, particularly in high-purity regimes, and offer a tool to study typical entanglement in finite-depth quantum circuits. It also outlines future directions, including cost analyses, thermodynamic connections, and extensions to infinite-dimensional systems.

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 the eigenvalue density and probability distribution of entropy. We present an algorithm for generating these entropy-based random density matrices by sampling a class of bipartite pure states, thus providing a new recipe for random state generation that differs from the well established Hilbert-Schmidt and Bures-Hall ensemble approaches. We find that a distinguishing feature of the ensemble is its larger purity and increased volume towards the boundary of full-rank states. The entropy-based ensemble can thus be used as a uninformative prior for Bayesian quantum state tomography in high purity regimes, and as a tool for quantifying typical entanglement in finite depth quantum circuits.

Paper Structure

This paper contains 8 sections, 38 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Metric-induced random density matrices: known approaches
  3. The entropy-based metric and its volume form
  4. Asymptotic properties of the ensemble and relation to random matrix theory
  5. Generating the states
  6. Summary
  7. Normalisation constant
  8. Generating BKM states

Figures (2)

  • Figure 1: (a) Plot of the asymptotic marginal eigenvalue density for the BKM ensemble \ref{['eq:lambert']}, compared with the equivalent distribution for Hilbert--Schmidt states \ref{['eq:HS']} and Bures--Hall states \ref{['eq:BH']}. (b) Plot of the average von Neumann entropy $\langle S\rangle$ as a function of dimension $N$ for the BKM ensemble (black) and Bures--Hall ensemble (red), along with the maximum entropy $\ln N$ (blue).
  • Figure 2: Probability density function of standardised von-Neumann entropy compared with a Gaussian fitting for dimension $N=70$ and $10^5$ samples. Inset: standard deviation in entropy $\sigma[S]$ as a function of $N$.