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Variance of relative entropy over random states
Linfeng Wei, Lu Wei
TL;DR
Relative entropy between random quantum states has received less attention than entropic quantities for individual states, and its variance characterizes fluctuations that averages alone do not capture. The paper derives exact variances for HS and BH states, including cross-ensemble pairs, and finds that these relative entropies are typical in the considered asymptotic regime.
Problem
Relative entropy between pairs of random states is less studied than entropic quantities for individual states, despite its role as a measure of distinguishability.
Source: §1 · p. 1Method
The paper derives exact variances for same-ensemble and cross-ensemble relative entropies, using zonal polynomials to evaluate required terms.
Source: §1 · p. 3, §2 · pp. 3–4Results
The variances for HS, BH, and cross-ensemble relative entropies vanish, and all three are typical under the considered asymptotic regime.
Source: §2 · p. 6Takeaways & Limitations
In the stated asymptotic regime, the average relative entropy is representative for the HS, BH, and cross-ensemble cases.
Source: §1 · p. 3, §2 · p. 6The bipartite random-state setup assumes m ≤ n, with the system dimension m no larger than the environment dimension n.
Source: §3.1 · p. 8
Abstract
from arXiv · showhide
Average relative entropy over major generic random states is recently studied. In this work, the investigation is extended by deriving the variances of relative entropy over the Hilbert--Schmidt and Bures--Hall ensembles, which characterize its fluctuations around the average value. Exact results for the variances of relative entropy between independent states drawn from the same ensemble and from two different ensembles are obtained. These results further indicate that the average relative entropy is typical in the considered limiting regimes. In addition to the matrix-level calculations similar to the average relative entropy, eigenvalue correlation functions of the ensembles are involved in the variance computations through the use of zonal polynomials.
1 Introduction
The introduction motivates relative entropy as a measure of distinguishability between random quantum states and identifies variance as the missing second-order statistic. It presents exact variance calculations and asymptotic typicality as the paper’s response.
1 Introduction
- Relative entropy between random states is less studied than entropic quantities for individual states, although exact average-relative-entropy formulas for HS, BH, and cross-ensemble cases were recently obtained.The paper builds on those averages by extending the investigation to second-order statistics.Source: §1 · pp. 1–2
- The paper derives exact relative-entropy variances for states drawn from the same HS or BH ensemble and for states drawn from different ensembles.These variances characterize fluctuations around the mean, which can be misleading at finite dimensions when variance is large.Source: §1 · p. 2
- The variance calculation requires the cross moment E[tr(ρ ln ρ) tr(ρ ln σ)] and V[tr(ρ ln σ)], treated using zonal polynomials.The variance contains terms beyond simple combinations of entanglement-entropy cumulants.Source: §1 · pp. 2–3
- The paper uses the variance formulas’ limiting behavior to assess typicality, defined by a vanishing variance with the same dimension scaling as the average.An entropic metric is typical when its average is constant up to a dimension scaling in the limiting regime.Source: §1 · p. 3
2 Main results on variances of relative entropy
The section gives exact relative-entropy variance results for HS, BH, and mixed-ensemble states, then shows that these quantities become typical in a specified large-dimension regime. It also compares their fluctuation patterns and reports numerical agreement with the formulas.
Exact variance results
- Exact variance formulas are derived for relative entropy between two HS states and between two BH states, with results expressed using polygamma functions and finite rational-function sums.When n1 = n2 = m, the results reduce to closed-form expressions.Source: §2 · pp. 3–4
Mixed ensembles
- The analysis also gives an exact variance for relative entropy with a BH state as ρ and an HS state as σ, a mixed-ensemble case.The reverse ordering, with an HS state as ρ and a BH state as σ, is also derived for completeness.Source: §2 · pp. 4–5
Asymptotic behavior
- Under regime F, the variances for HS, BH, and inter-ensemble relative entropy vanish, and all three relative entropies are typical.Regime F takes m, n1, and n2 to infinity while keeping the differences between n1 and m and between n2 and m fixed.Source: §2 · pp. 5–7
Numerical studies
- Numerical simulations over 10^5 density-matrix realizations match the exact formulas for same-ensemble and inter-ensemble variances.The simulations cover selected parameter instances; Figure 1 shows the same-ensemble comparisons.Source: §2 · p. 6, Figure 1 · p. 7
Fluctuation comparisons
- At fixed system dimension, increasing either environment dimension reduces fluctuations, with the same increase in n2 reducing them more than an increase in n1.The reported comparison is for fixed m and with the other environment parameter held fixed.Source: §2 · p. 6
- At the same system dimension and fixed parameters, HS has the smallest variance, BH the largest, and the inter-ensemble case lies between them near HS.The inter-ensemble behavior is described as close to HS, with slightly larger fluctuation.Source: §2 · p. 6
3 Derivation of variance formulas
This section derives exact variance formulas for relative entropy by reducing its terms to matrix moments and cumulants, evaluated using zonal polynomials and ensemble-specific eigenvalue correlations. It also treats the reverse HS-to-BH and degenerate cases, and derives vanishing-variance limits establishing typicality when limiting average relative entropies are constant.
3.2 Calculation of moment integrals via zonal polynomials
- A change of variables and zonal-polynomial expansion of trace powers turn the required joint moments into single-statistic cumulant integrals.Source: §3 · p. 7, §3.2 · pp. 9–10
3.1 Random states ensembles
- HS states arise as fixed-trace Wishart reductions of bipartite Gaussian states, while BH states extend the construction through random-unitary rotation and Bures-prior weighting.Source: §3.1 · pp. 7–9
3 Derivation of variance formulas
- The derivation targets E[tr(ρ ln σ)], E[tr(ρ ln ρ) tr(ρ ln σ)], and V[tr(ρ ln σ)] as constituent terms in the variance formulas.Source: §3.2 · p. 9
3.2 Calculation of moment integrals via zonal polynomials
- Unitary invariance separates eigenvalue and eigenvector integrations, while zonal-polynomial identities handle the eigenvector integrals.Source: §3.2 · p. 11
3.3 Proof of Proposition 1
- For HS, the remaining H_k moments are linear statistics of the Laguerre ensemble and are evaluated with its correlation kernel to obtain variance result (12).Source: §3.3 · pp. 13–14
3.4 Proof of Proposition 2
- For BH, the analogous calculation uses Cauchy-Laguerre biorthogonal correlation kernels, Mellin and Meijer G integral identities, and finite sums to obtain result (16).Source: §3.4 · pp. 15–16
- Resolving the indeterminacy in (16) yields result (17).Source: §3.4 · p. 17
3.5 Proof of Proposition 3 and the other variance results
- For BH-to-HS variance, required trace expectations are assembled from individual ensemble results before resolving indeterminacy to obtain (21).Source: §3.5 · p. 17
- The variance of D(ρHS||σBH) has an exact formula, with a degenerate case when n1 = n2 = m.Source: §3.5 · p. 17
- The D(ρHS||σBH) expression uses finite summation B_BH and closed-form C_F4 terms, and its degenerate case follows by resolving indeterminacy.Source: §3.5 · p. 17
3.6 Proof of Corollary 1
- Polygamma-function limiting approximations applied to exact variances (12), (16), and (20) yield their asymptotic behavior.Source: §3.6 · p. 17
- In regime (26), variance vanishes, so relative entropy is typical when its average approaches a constant.Source: §3.6 · p. 18
- Exact results in give constant limiting average relative entropies in regime (26); with vanishing variance, relative entropy is typical for HS, BH, and cross-ensemble cases.Source: §3.6 · p. 18
A Closed-form expressions in Proposition 1–3 and (121)
The appendix lists closed-form expressions involving polygamma functions and finite summations used in the variance results. It also specifies how removable singularities in the integral sums are interpreted.
A Closed-form expressions in Proposition 1–3 and (121)
- This section lists closed-form expressions involving polygamma functions in the propositions, but the cited passage does not identify particular proposition or equation numbers.Source: A Closed-form expressions in… · p. 19
B Finite summation of integrals (115)–(118)
- Appendix B presents finite summations of integrals from Cauchy-Laguerre two-point kernels using Pochhammer’s symbol.Removable singularities are interpreted by taking limits as ǫ → 0.Source: B Finite summation of integrals… · p. 20