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Average relative entropy of random states

T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The paper derives exact closed-form formulas for the average relative entropy of pairs of independent random states from the Hilbert-Schmidt and Bures-Hall ensembles.

desk verdict Hilbert-Schmidt half is solid and new; the Bures-Hall half rests on an impossible entropy formula and a sign error—worth a major-revision round, not a desk reject. read the letter →

arxiv 2509.21846 v1 pith:TWPO4O5Z submitted 2025-09-26 math-ph cs.ITmath.ITmath.MPquant-ph

Lu Wei This is my paper · ORCID
classification math-phcs.ITmath.ITmath.MPquant-ph MSC 60B2081P4594A17
keywords relativeentropyrandomquantumstatesHilbert-SchmidtensembleBures-Hallentanglementunitaryintegrationzonalpolynomialsdigammafunction
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Relative entropy measures how well one quantum state can stand in for another, so its average over random states quantifies typical distinguishability. This paper establishes exact, finite-size formulas for that average when the two states are drawn independently from the Hilbert-Schmidt ensemble, the Bures-Hall ensemble, or one from each. The core step is a factorization: after averaging over the unitary relating the two eigenbases, the cross term $E[\operatorname{tr}(\rho\ln\sigma)]$ collapses to $(1/m)E[\operatorname{tr}(\ln\sigma)]$, reducing a two-state problem to known single-state entropy averages plus a log-determinant average. That produces the closed forms in Propositions 1–3, valid for arbitrary subsystem dimension $m$ and parameters $n_1=m+\alpha_1$, $n_2=m+\alpha_2$, with large-dimension limits that complement the earlier asymptotic replica-method result. Exact formulas of this kind give finite-size predictions for quantum hypothesis testing and for studies of eigenstate thermalization.

What carries the argument

The load-bearing identity is the unitary-integral factorization $$\int_{U(m)}\operatorname{tr}\bigl(\Lambda_\rho U\ln\Lambda_\$\sigma$ U^\dagger\bigr)\,dU=\frac{1}{m}\operatorname{tr}(\ln\$\sigma$),$$ which follows from the zonal-polynomial integral $\int_{U(m)}C_\kappa(XUYU^\dagger)\,dU=C_\kappa(X)C_\kappa(Y)/C_\kappa(I_m)$ and, for this first moment, can also be obtained by Weingarten calculus. It reduces the cross term in relative entropy to a single-state log-determinant average. The remaining task is to compute $E[\operatorname{tr}(\ln\sigma)]$ for each ensemble by differentiating the normalization constants of the Hilbert-Schmidt and Bures-Hall densities with respect to $\alpha$, which produces the digamma-function combinations in (11), (15), and (17).

What would settle it

Draw many independent pairs of random density matrices for a small explicit case, such as $m=2$, $n_1=3$, $n_2=4$, from the Hilbert-Schmidt and Bures-Hall constructions, and compare the sample average of $D(\rho\|\sigma)$ with (11), (15), and (17); a disagreement beyond Monte Carlo error would refute the formulas. A sharper test isolates the factorization by checking whether the sample average of $\operatorname{tr}(\rho\ln\sigma)$ equals $(1/m)$ times the sample average of $\operatorname{tr}(\ln\sigma)$ for independent pairs, and that this equality breaks when the eigenbasis of $\sigma$ is forced to coincide with that of $\rho$.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is that the average relative entropy of two independent random density matrices is controlled by a factorization identity rather than by the joint spectrum of the pair. Writing $\rho=V\Lambda_\rho V^\dagger$ and $\sigma=W\Lambda_\sigma W^\dagger$, the unitary invariance of the Hilbert-Schmidt and Bures-Hall ensembles lets the average over $U=V^\dagger W$ be performed first, giving $E[\operatorname{tr}(\rho\ln\sigma)]=(1/m)E[\operatorname{tr}(\ln\sigma)]$. Combined with the exact mean entanglement entropy of a single random state from each ensemble, this yields explicit digamma-function expressions: Proposition 1 for Hilbert-Schmidt versus Hilbert-Schmidt, Proposition 2 for Bures-Hall versus Bures-Hall, and Proposition 3 for Bures-Hall versus Hilbert-Schmidt, with the complementary Hilbert-Schmidt versus Bures-Hall case recorded as well. The formulas are exact for arbitrary $m$, $n_1=m+\alpha_1$, and $n_2=m+\alpha_2$, and their large-dimension limits recover the previously known asymptotic behavior to leading order.

Load-bearing premise

The chain of proofs depends on the two random states being independent and on each state's eigenvector basis being uniformly random according to the invariant unitary measure and statistically independent of its eigenvalues; if either ensemble lost its unitary invariance, or if the states were not independent, the identity $E[\operatorname{tr}(\rho\ln\sigma)]=(1/m)E[\operatorname{tr}(\ln\sigma)]$ would fail and all three propositions would fall.

Editorial extensions

If this is right

  • For any fixed finite subsystem dimension $m$ and bipartite parameters $n_1,n_2$, the average relative entropy can now be evaluated exactly instead of approximated, removing finite-size error in regimes where only the asymptotic replica formula was available.
  • In the limit $m\to\infty$ with $n_i/m=c_i$ fixed, the limiting formulas (13), (16), and (18) show that the average relative entropy depends only on the ratios $c_1,c_2$ and decreases monotonically as either parameter grows, the maximum occurring at $c_1=c_2=1$.
  • Comparing the two same-ensemble formulas shows that the Bures-Hall average relative entropy exceeds the Hilbert-Schmidt value for the same parameters, which the paper attributes to the wider spectral width of the Bures-Hall ensemble.
  • The cross-ensemble formula quantifies how much a Bures-Hall state typically differs from a Hilbert-Schmidt model, and its asymmetry under exchanging parameters indicates different robustness of the two ensembles as reference models.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because the factorization step uses only unitary invariance and independence, the same reduction $E[\operatorname{tr}(\rho\ln\sigma)]=(1/m)E[\operatorname{tr}(\ln\sigma)]$ should hold for any two independent states from other unitarily invariant ensembles, so analogous exact formulas could be derived for fermionic Gaussian states once the relevant log-determinant averages are known.
  • The zonal-polynomial machinery used here is suited to higher powers of $\operatorname{tr}(\rho\ln\sigma)$, which suggests the method can be pushed to exact variance or higher-cumulant formulas for relative entropy, not just the mean.
  • If two states are drawn with correlated eigenbases or from a non-unitarily-invariant ensemble, the factorization fails; measuring that failure could serve as a quantitative probe of how eigenvector alignment changes typical distinguishability, a testable extension of the independent-state setting.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper proposes exact closed-form formulas for the average quantum relative entropy E[D(ρ||σ)] for two independent random density matrices drawn from the Hilbert–Schmidt (HS) and Bures–Hall (BH) ensembles, including the mixed HS-vs-BH case. The main mathematical step is a unitary-integral factorization E[tr(ρ ln σ)] = (1/m)E[tr(ln σ)] (Eq. 53), which reduces the problem to known single-state entropy formulas and to derivatives of the ensemble normalization constants. The results are stated as Propositions 1–3 with limiting corollaries, and are compared with numerical simulations.

Significance. The Hilbert–Schmidt part appears correct and useful: the factorization argument via zonal polynomials and Weingarten calculus is clean, Proposition 1 passes the m=1 consistency check, and its large-dimension limit reproduces the known asymptotic result of Kudler-Flam. The Bures–Hall part, however, is not currently reliable: two independent checks (the maximum-entropy bound and the deterministic m=1 limit) fail, and the normalization derivative in Eq. (64) has a sign error. If the BH computations are corrected, the factorization method could still be a valuable contribution, but the paper as it stands does not establish the BH formulas.

major comments (3)
  1. [Proof of Proposition 2, Eq. (64)] The derivative of ln C_BH printed in Eq. (64) has the wrong sign on the term mψ0(m(m+2α)/2). Starting from Eq. (26), d ln C_BH/dα = −2m ln2 + mψ0(m(m+2α)/2) + 2Σψ0(i+2α) − Σψ0(i+α), whereas Eq. (64) uses −mψ0(...). This sign error propagates into Eq. (15), and through Propositions 2 and 3 into Eqs. (16), (17), (18), and (69).
  2. [Section 1, Eq. (7)] The Bures–Hall mean entropy quoted in Eq. (7) cannot be correct: for m=2, n=2 it gives ψ0(3)−ψ0(3/2) ≈ 0.8863, which exceeds the absolute upper bound ln2 ≈ 0.6931 for any 2×2 density matrix. Since Propositions 2 and 3 are built on Eq. (7), their claimed exact formulas and all BH limiting formulas are unsubstantiated as they stand.
  3. [Proposition 2 and Proposition 3, m=1 special case] At m=1, n1=n2=1 both ensembles give the deterministic state ρ=σ=1, so D(ρ||σ)=0. Eq. (15) evaluates to approximately 0.386 and Eq. (17) to approximately −0.614, contradicting both this limiting value and the nonnegativity (3). This internal inconsistency confirms that the Bures–Hall computation of E[tr(ln σ)] or the entropy input is wrong, independent of the maximum-entropy bound.
minor comments (4)
  1. [Section 1, text after Eq. (10)] The symbol "ρ_BS" appears where "σ_BH" or "ρ_BH" seems intended; please correct this notation.
  2. [Figures 1 and 2] The captions say the plots compare exact formulas with simulations, but they do not state the values of n1, n2 (or c1, c2) used for each curve; please add this information for reproducibility.
  3. [Reference [13]] Reference [13] is cited as arXiv:2502.05371; if it has been published, please provide the journal reference.
  4. [Around Eq. (11)] The notation n_i = m + α_i is first used in the propositions without an explicit definition in the introduction; please define it before presenting Proposition 1.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the cross-term factorization is derived from unitary invariance, and the single-state entropy inputs are external results, not fitted to the target quantity.

full rationale

The derivation chain is self-contained at every load-bearing step. The new ingredient is Eq. (53), E[tr(rho ln sigma)] = (1/m)E[tr(ln sigma)], which is proved directly from unitary invariance via zonal polynomials (Eqs. (39)-(43)) or equivalently Weingarten calculus (Eqs. (48)-(52)), not assumed or fitted. The remaining quantities E[tr(rho ln rho)] are taken from prior literature: Eq. (6) for the Hilbert-Schmidt ensemble and Eq. (7) for the Bures-Hall ensemble. These are single-state entropies, distinct from the target relative entropy, and they are not adjusted to reproduce the final formulas. The Bures-Hall input is credited jointly to [22] and [26], and although [26] shares an author with the present paper, the cited result concerns a different quantity and is not the conclusion being derived. E[tr(ln sigma)] is then computed by differentiating the ensemble normalization constants with respect to alpha, Eqs. (55)-(62) and (63)-(64), which is a parameter-free calculation. No prediction in Propositions 1-3 is equivalent by construction to an input; no fitted parameter is renamed as a prediction; and no ansatz is smuggled in through self-citation. The possible sign error in Eq. (64) noted by a skeptical reader would be a mathematical correctness issue, not a circularity. Accordingly, the score is 0.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

No free parameters are fitted; m, n1, and n2 are physical Hilbert-space dimensions. The main inputs are the two unitarily invariant ensemble densities and their normalization constants, the known single-state entropy formulas, the zonal polynomial integration identity, and an unproved interchange of derivative and integral. No new entities are postulated.

assumptions (8)
  • domain assumption The Hilbert-Schmidt ensemble density (20) and normalization constant (21) are valid for α ≥ 0.
    Used to compute E[tr(ln σ)] via the derivative of C_HS in Eqs. (56)-(60).
  • domain assumption The Bures-Hall ensemble density (25) and normalization constant (26) are valid for α ≥ 0 and m ≥ 2.
    Used to compute E[tr(ln σ)] via the derivative of C_BH; the m equals one counterexample indicates the stated formulas implicitly exclude m equals one.
  • domain assumption Unitarily invariant ensembles have eigenvector matrices Haar-distributed and independent of the eigenvalues.
    Basis for the factorization in Eq. (53).
  • domain assumption The known single-state average entropy formulas (6) and (7) are correct.
    External prior results used as inputs for E[tr(ρ ln ρ)].
  • standard math The zonal polynomial integral identity (39) is valid.
    Used for unitary averaging; Weingarten calculus gives the same result.
  • ad hoc to paper Differentiation under the integral in Eqs. (56)-(57) and (63)-(64) is permitted.
    Required to convert dC/dα into E[ln det σ]; the necessary integrability conditions are not stated.
  • standard math The digamma summation formulas (61) and (65) are correct.
    Used to simplify the derivatives of C_HS and C_BH.
  • ad hoc to paper The term αψ0(α) is interpreted by its limit minus one as α tends to zero.
    The formulas for α equals zero contain αψ0(α), which is undefined without this limiting convention; the paper does not state it.

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Pith. "Pith review of Average relative entropy of random states." pith.science (2026). https://pith.science/paper/TWPO4O5Z

@misc{pith2026250921846,
  author       = {Pith},
  title        = {Pith review of: Average relative entropy of random states},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TWPO4O5Z}},
  note         = {Machine review of arXiv:2509.21846}
}
read the original abstract

Relative entropy serves as a cornerstone concept in quantum information theory. In this work, we study relative entropy of random states from major generic state models of Hilbert-Schmidt and Bures-Hall ensembles. In particular, we derive exact yet explicit formulas of average relative entropy of two independent states of arbitrary dimensions from the same ensemble as well as from two different ensembles. One ingredient in obtaining the results is the observed factorization of ensemble averages after evaluating the required unitary integral. The derived exact formula in the case of Hilbert-Schmidt ensemble complements the work by Kudler-Flam (2021 Phys Rev Lett 126 171603), where the corresponding asymptotic formula for states of equal dimensions was obtained based on the replica method.

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