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Variance of relative entropy over random states

This paper derives exact analytical formulas for the variances of relative entropy over Hilbert–Schmidt and Bures–Hall ensembles for both independent and cross-ensemble random states, utilizing zonal polynomials and eigenvalue correlation functions to demonstrate the typicality of average relative entropy in limiting regimes.

Original authors: Linfeng Wei, Lu Wei

Published 2026-10-26

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Linfeng Wei, Lu Wei

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the vast landscape of quantum science, researchers often study the properties of single, randomly chosen quantum states to understand the behavior of complex systems. These random states act as a kind of statistical baseline, helping scientists predict how quantum devices perform or how information is stored and lost. A central tool for measuring these states is a concept called relative entropy. Think of this as a ruler for difference; it quantifies how distinguishable one quantum state is from another, or how much information is lost when one state is used to approximate a different one. While scientists have long known the average value of this difference across many random states, knowing the average alone can be misleading. Just as knowing the average temperature of a year tells you little about the risk of a sudden frost or a heatwave, knowing the average difference between quantum states does not reveal how much that difference might fluctuate in a real, physical system. If the difference varies wildly from one instance to the next, the average becomes a poor predictor of what will actually happen.

A team of researchers at Texas Tech University has now filled this gap by calculating exactly how much this difference fluctuates. They focused on two specific ways of generating random quantum states, known as the Hilbert–Schmidt and Bures–Hall ensembles. These are not just abstract mathematical ideas; they represent the most fundamental models for how quantum systems are prepared and how they interact with their surroundings. The researchers derived precise formulas to measure the variance, or the spread, of the relative entropy between pairs of these random states. They looked at cases where both states came from the same type of ensemble, as well as cases where one state came from one type of ensemble and the other from the different type. This last scenario is particularly relevant for practical computing, where a full, complex model might be approximated by a simpler, less detailed one to speed up calculations.

The results of this work provide a clear picture of stability in the quantum world. The researchers found that for both types of ensembles, and even when mixing them, the fluctuations in the relative entropy are not random chaos but follow a strict mathematical pattern. They calculated the exact values for these fluctuations using advanced mathematical tools that break down complex interactions into simpler, manageable parts. Their calculations showed that as the size of the quantum system grows, these fluctuations do not just get smaller; they vanish entirely. In the limit of large systems, the difference between any two random states becomes incredibly predictable, clustering tightly around a single average value. This phenomenon, known as typicality, means that for large quantum systems, one does not need to check every single possible state to know how they will behave; the average is a reliable guide for almost every individual case.

The study also revealed subtle differences between the two types of ensembles. The fluctuations were found to be smallest when both states were drawn from the Hilbert–Schmidt ensemble, which is the most neutral and least informative model. The fluctuations were largest for the Bures–Hall ensemble, which incorporates more specific physical constraints. When mixing the two, the behavior fell somewhere in between, leaning closer to the Hilbert–Schmidt results. This suggests that when a complex quantum model is approximated by a simpler one, the reliability of that approximation is governed by the statistical properties of the simpler model. Furthermore, the researchers discovered that the size of the environment surrounding the quantum system plays a crucial role. Increasing the size of this environment reduces the fluctuations more effectively than changing other parameters, making the distinguishability between states more consistent.

By proving that these fluctuations disappear in large systems, the work confirms that relative entropy is a robust and stable measure for quantum information. This is a significant step forward for fields like quantum hypothesis testing, where scientists must decide which of two possible states a system is in. If the difference between states fluctuated wildly, a test might succeed one moment and fail the next, even if the average performance looked good. The new formulas show that such wild swings are unlikely in large systems, providing a stronger foundation for designing quantum devices and algorithms. The researchers backed their exact formulas with extensive computer simulations, generating millions of random states to verify that the theoretical predictions matched the numerical reality. The agreement was perfect, confirming that the mathematical derivation accurately captures the physical behavior of these quantum systems. This work transforms relative entropy from a theoretical average into a practical, predictable tool for the future of quantum technology.

Technical Summary: Variance of Relative Entropy over Random States

Problem Statement
While the statistical properties of entropic quantities, such as entanglement entropy, have been extensively investigated for random quantum states drawn from the Hilbert–Schmidt (HS) and Bures–Hall (BH) ensembles, the study of distinguishability measures between pairs of random states remains less developed. Specifically, while exact formulas for the average relative entropy between random states from these ensembles (and between different ensembles) have recently been derived, the variance of this quantity—which characterizes fluctuations around the mean and determines the typicality of the metric—had not been established. Understanding these fluctuations is critical for finite-dimensional systems where large variance can render the mean misleading, and for applications like quantum hypothesis testing where performance guarantees depend on the stability of distinguishability.

Methodology
The authors derive exact formulas for the variance of relative entropy, $V[D(\rho||\sigma)]$ , for three distinct cases:

Two states $\rho, \sigma$ drawn from the HS ensemble.
Two states $\rho, \sigma$ drawn from the BH ensemble.
One state from the BH ensemble and the other from the HS ensemble.

The derivation relies on decomposing the variance into terms involving the moments of linear statistics of the density matrices. Specifically, the calculation requires evaluating expectations of the form $E[\text{tr}(\rho \ln \rho) \text{tr}(\rho \ln \sigma)]$ and $V[\text{tr}(\rho \ln \sigma)]$ .

The core technical strategy involves:

Variable Transformation: Introducing a change of variables to separate the trace (radial) components from the eigenvalue components of the density matrices.
Zonal Polynomials: Utilizing the theory of zonal polynomials to decompose integrals over the unitary group. This allows the authors to break down joint cumulants of trace powers into single cumulants. The method leverages the integral identity of zonal polynomials over the unitary group to decouple eigenvector dependencies.
Eigenvalue Correlation Functions: The remaining calculations involve computing moments of linear statistics (specifically involving $\ln x$ ) over the eigenvalue distributions of the ensembles. For the HS ensemble, this involves the Laguerre orthogonal polynomial ensemble. For the BH ensemble, it involves the Cauchy-Laguerre biorthogonal ensemble.
Polygamma Functions: The final exact results are expressed in terms of polygamma functions ( $\psi_k$ ) and finite summations of rational functions.

Key Contributions and Results
The paper presents three main propositions providing exact closed-form expressions (in terms of polygamma functions and finite sums) for the variance of relative entropy:

HS Ensemble (Proposition 1): An exact formula for $V[D(\rho_{HS}||\sigma_{HS})]$ is derived. In a degenerate case where the environment dimensions equal the system dimension ( $n_1=n_2=m$ ), the result simplifies to a specific closed-form expression.
BH Ensemble (Proposition 2): An exact formula for $V[D(\rho_{BH}||\sigma_{BH})]$ is derived, involving a specific finite summation term denoted as $BBH$ .
Inter-Ensemble Case (Proposition 3): An exact formula for $V[D(\rho_{BH}||\sigma_{HS})]$ is derived, addressing the case where the two states originate from different ensembles. This is noted as having information-theoretic interest for comparing full probabilistic models with reduced-order approximations.

Additionally, the paper provides:

Asymptotic Analysis (Corollary 1): Under the asymptotic regime where the system dimension $m$ and environment dimensions $n_1, n_2$ all tend to infinity, with the differences $n_1-m$ and $n_2-m$ remaining fixed, the authors prove that the variance of the relative entropy vanishes ( $V \to 0$ ).
Typicality: Based on the vanishing variance and previously established limiting averages, the authors conclude that the relative entropy is a "typical" quantity in these limiting regimes. This implies that for large systems, the average relative entropy is a robust predictor for individual realizations.
Numerical Validation: The theoretical results are validated against numerical simulations involving random sampling of Wishart matrices (for HS) and random unitary matrices via the Metropolis–Hastings algorithm (for BH). The simulations confirm the analytical formulas and illustrate that increasing the environment dimension of the second state ( $n_2$ ) significantly reduces fluctuations.

Significance
The paper claims that these results extend the study of relative entropy from first-order statistics (mean) to second-order statistics (variance). The significance lies in:

Characterizing Fluctuations: Providing the necessary tools to understand how much individual instances of relative entropy deviate from the mean, which is crucial for finite-dimensional quantum systems.
Establishing Typicality: Demonstrating that relative entropy is typical in the large-dimension limit, thereby justifying the use of ensemble averages to predict the behavior of generic random states in high-dimensional quantum systems.
Differentiating Ensembles: The distinct variance formulas and numerical results highlight the structural differences between the HS and BH ensembles, as well as the behavior of inter-ensemble comparisons.
Operational Relevance: The findings are relevant to quantum hypothesis testing and the assessment of distinguishability, where low variance ensures consistent discrimination performance across random state pairs.

The work relies on exact mathematical derivations using zonal polynomials and special functions, offering a rigorous foundation for the statistical analysis of quantum state distinguishability without proposing new experimental setups or unverified applications.

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