Maximum-Entropy Ensemble of Quantum Subsystems with a Fixed Average State

Abstract

The reduced density matrices of Haar-random bipartite states form a fixed-trace version of the Wishart ensemble of random matrices, with an asymptotic eigenvalue density given by the Marchenko--Pastur (MP) law. We establish a similar universal form for the maximum-entropy distribution of mixed states subject to a fixed ensemble average, arguing that its eigenvalue density converges asymptotically to that of the correlated Wishart ensemble (CWE). The density is thus obtained by a free multiplicative convolution formula, which we apply to the transverse-field Ising model (TFIM) with a thermal average state to derive a high-temperature leading-order correction to the MP law. We then consider measurement by a random POVM on the purification of the mixed reference state. We show that the resulting mixed-state ensemble is asymptotically equivalent to the CWE, and provide confirming numerical evidence. We return to the TFIM example to compute the entanglement of the system with the ancilla after measurement, and give finite-temperature Page curves for the average entanglement entropies.

The paper asks how the induced ensemble of random mixed states changes when its ensemble average is fixed to a reference state $\rho_0$. Starting from the measure obtained by tracing a Haar-random state on $\mathbb C^d\otimes\mathbb C^K$, it maximises relative entropy subject to $\langle\rho\rangle=\rho_0$, obtaining $P_{\mathrm{ME}}(\rho)\propto e^{-\operatorname{Tr}M\rho}(\det\rho)^{K-d}$ on unit-trace positive matrices. It derives a partition function through an inverse Laplace transform, giving access to constraints and observable cumulants. The main result connects this ensemble asymptotically to normalised correlated Wishart matrices with covariance $\rho_0$. The proofs couple both ensembles to a distribution with fixed weights in the reference-state eigenbasis, giving Wasserstein bounds of order $K^{-1/2}$. Trace-concentration bounds then connect normalised and unnormalised Wishart spectra. The resulting limiting spectral density is expressed as the free multiplicative convolution of the spectral measure of $d\rho_0$ with the Marchenko–Pastur law. A continuous Gaussian-indexed POVM on a purification produces a Born-weighted version of the normalised Wishart ensemble, whose total-variation distance from the latter is bounded by $1/(2\sqrt K)$. For thermal transverse-field Ising reference states, numerical inversion of the S-transform gives spectral densities and finite-temperature Page curves, with reduced entanglement and asymmetric subsystem-size dependence. A high-temperature expansion gives the leading correction to Marchenko–Pastur statistics. The entropy-maximisation construction assumes $K\ge d$; the multiplier identification $M\approx K\rho_0^{-1}$ is argued non-rigorously. The Ising calculations use an approximately Gaussian density of states, and simultaneous constraints on both subsystems remain an extension.