Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2605.29725

Non-Perturbative Closed Form for the Typical Bipartite Mutual Information of Haar-Random States

Zhi-Wei Wang, Pei-Wen Li, Samuel L. Braunstein

quant-phhep-thmath-phstat.APSubmitted May 28, 2026• Announced May 29, 2026• 0 citations• Revised Jun 4, 2026 (v2)
v1 · May 28, 2026v2 · Jun 4, 2026
arXivHTMLPDF

Abstract

The average bipartite quantum mutual information $\langle I(A{:}B)\rangle$ of Haar-random pure states can be expressed exactly through Page's formula in terms of digamma functions. We show that this quantity admits a single non-perturbative closed form: $\langle I(A{:}B)\rangle = (d_A^2-1)(d_B^2-1)\,\mathcal{G}(d_A,d_B,d_E)$, where $\mathcal{G}$ is given by an explicit convergent integral over a Bose--Einstein kernel. The overall factor $(d_A^2-1)(d_B^2-1)=\dim[\mathfrak{su}(d_A)]\cdot\dim[\mathfrak{su}(d_B)]$ is exact, not merely asymptotic. The asymptotic expansion of $\mathcal{G}$ in $1/N$ yields a Bernoulli-factorised series whose coefficients involve $ζ(1{-}2k)$; this series diverges, and our integral is its exact Borel sum. The integral representation also makes $\langle I\rangle < (d_A^2{-}1)(d_B^2{-}1)/(2N)$ manifest via a scale-inversion symmetry of the kernel. Our derivation traces the mutual information's structure to an exact decomposition of Page's entropy into a diagonal (Dirichlet) contribution and a Schur-majorisation eigenvalue correction, whose assembly into the mutual information cleanly separates classical from quantum correlations.

Comments
5 pages. This is a companion paper to our simultaneous submission with a title "Exact Geometric Typicality and Bipartite Entanglement from the Projected Central Limit Theorem on Hyperspheres"

Related papers

Exact Geometric Typicality and Bipartite Entanglement from the Projected Central Limit Theorem on Hyperspheres

Zhi-Wei Wang, Pei-Wen Li, Samuel L. Braunstein

Starting from the exact Projected Central Limit Theorem on hyperspheres, we rederive the Beta distribution for subsystem occupation probabilities and Lubkin's purity formula from elementary hyperspherical moments, quantifying the finite-size ``platykurtic'' suppression of tails relative to the Gaussian approximation used in standard eigenstate-thermalization and typicality treatments. Our main new result concerns the bipartite quantum mutual information $\langle I(A{:}B)\rangle$ for Haar-random pure states. We show that its full asymptotic expansion in $1/N$ admits a Bernoulli-factorized form in which every order $k \ge 1$ carries the symmetric factor $(d_A^{2k}-1)(d_B^{2k}-1)$ and all higher odd-order corrections vanish identically. Through an exact algebraic reorganization of Page's formula (conjectured in Ref.~\cite{Page1993} and subsequently proven~\cite{Foong1994, SanchezRuiz1995, Sen1996}), we establish that the leading finite-size correction separates into a dominant $\mathfrak{su}(d_A) \otimes \mathfrak{su}(d_B)$ bipartite quantum coherence contribution $(d_A^2 - 1)(d_B^2 - 1)/(2N)$ and a subtracted classical-probability (Cartan $\otimes$ Cartan) contribution $(d_A - 1)(d_B - 1)/(2N)$, and we trace this separation to the difference between diagonal and eigenvalue entropies via Schur's majorisation theorem, with the dimensional counts $(d-1)$ and $(d^2-1)$ acquiring meaning through the Cartan structure of the generalised Bloch decomposition. These results admit a single non-perturbative closed form: the exact typical mutual information factors as $\langle I(A{:}B)\rangle = (d_A^2-1)(d_B^2-1)\,\mathcal{G}(d_A,d_B,d_E)$, with $\mathcal{G}$ given by an explicit Bose--Einstein integral whose asymptotic expansion in $1/N$ reproduces the Bernoulli series.

quant-pharXiv:2605.29732•May 29, 2026•0 citations

Random matrix prediction of average entanglement entropy in non-Abelian symmetry sectors

Anwesha Chakraborty, Lucas Hackl, Mario Kieburg

We study the average bipartite entanglement entropy of Haar-random pure states in quantum many-body systems with global $\mathrm{SU}(2)$ symmetry, constrained to fixed total spin $J$ and magnetization $J_z = 0$. Focusing on spin-$\tfrac12$ lattices and subsystem fractions $f < \frac{1}{2}$, we derive a asymptotic expression for the average entanglement entropy up to constant order in the system volume $V$. In addition to the expected leading volume law term, we prove the existence of a $\frac{1}{2}\log V$ finite-size correction resulting from the scaling of the Clebsch-Gordon coefficients and compute explicitly the $O(1)$ contribution reflecting angular-momentum coupling within magnetization blocks. Our analysis uses features of random matrix ensembles and provides a fully analytical treatment for arbitrary spin densities, thereby extending Page type results to non-Abelian sectors and clarifying how $\mathrm{SU}(2)$ symmetry shapes average entanglement.

quant-pharXiv:2512.22942•Dec 30, 2025•0 citations

Relative entropy of entanglement of Haar random states

Takato Mori, Beni Yoshida

We determine the relative entropy of entanglement of a bipartite mixed state $ρ_{AB}$ obtained by tracing out one subsystem of a tripartite Haar-random pure state $|ψ\rangle_{ABC}$, finding $E_R(ρ_{AB})=\log\frac{d_Ad_B}{\max(d_A,d_B,d_C)}+O(1)$. Equivalently, the relative entropy of entanglement nearly saturates the smaller of the entanglement of formation $E_F(ρ_{AB})$ and the mutual information $I(A:B)$. The upper bound is achieved by an explicit separable state obtained through one-sided Schmidt dephasing, which is therefore approximately closest.

quant-pharXiv:2608.05274•Aug 7, 2026•0 citations

Symmetry-resolved Page curves

Sara Murciano, Pasquale Calabrese, Lorenzo Piroli

Given a statistical ensemble of quantum states, the corresponding Page curve quantifies the average entanglement entropy associated with each possible spatial bipartition of the system. In this work, we study a natural extension in the presence of a conservation law and introduce the symmetry-resolved Page curves, characterizing average bipartite symmetry-resolved entanglement entropies. We derive explicit analytic formulas for two important statistical ensembles with a $U(1)$-symmetry: Haar-random pure states and random fermionic Gaussian states. In the former case, the symmetry-resolved Page curves can be obtained in an elementary way from the knowledge of the standard one. This is not true for random fermionic Gaussian states. In this case, we derive an analytic result in the thermodynamic limit based on a combination of techniques from random-matrix and large-deviation theories. We test our predictions against numerical calculations and discuss the subleading finite-size corrections.

arXiv:2206.05083•Physical review. D/Physical review. D.•Aug 30, 2022•52 citations

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

Charlie Shakeshaft, Christopher David White, Harry J. D. Miller

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.

quant-pharXiv:2610.39712•Oct 30, 2026•0 citations

Page curve for fermionic Gaussian states

Eugenio Bianchi A, Lucas Hackl, Mario Kieburg

In a seminal paper, Page found the exact formula for the average entanglement entropy for a pure random state. We consider the analogous problem for the ensemble of pure fermionic Gaussian states, which plays a crucial role in the context of random free Hamiltonians. Using recent results from random matrix theory, we show that the average entanglement entropy of pure random fermionic Gaussian states in a subsystem of ${N}_{A}$ out of $N$ degrees of freedom is given by ${\ensuremath{\langle}{S}_{A}\ensuremath{\rangle}}_{\mathrm{G}}=(N\ensuremath{-}\frac{1}{2})\mathrm{\ensuremath{\Psi}}(2N)+(\frac{1}{4}\ensuremath{-}{N}_{A})\mathrm{\ensuremath{\Psi}}(N)+(\frac{1}{2}+{N}_{A}\ensuremath{-}N)\mathrm{\ensuremath{\Psi}}(2N\ensuremath{-}2{N}_{A})\ensuremath{-}\frac{1}{4}\mathrm{\ensuremath{\Psi}}(N\ensuremath{-}{N}_{A})\ensuremath{-}{N}_{A}$, where $\mathrm{\ensuremath{\Psi}}$ is the digamma function. Its asymptotic behavior in the thermodynamic limit is given by ${\ensuremath{\langle}{S}_{A}\ensuremath{\rangle}}_{\mathrm{G}}=N(log2\ensuremath{-}1)f+N(f\ensuremath{-}1)log(1\ensuremath{-}f)+\frac{1}{2}f+\frac{1}{4}log(1\ensuremath{-}f)\phantom{\rule{0.16em}{0ex}}+\phantom{\rule{0.16em}{0ex}}O(1/N)$, where $f={N}_{A}/N\ensuremath{\le}1/2$. Remarkably, its leading order agrees with the average over eigenstates of random quadratic Hamiltonians with number conservation, as found by \L{}yd\ifmmode \dot{z}\else \.{z}\fi{}ba, Rigol, and Vidmar. Finally, we compute the variance in the thermodynamic limit, given by the constant ${lim}_{N\ensuremath{\rightarrow}\ensuremath{\infty}}{(\mathrm{\ensuremath{\Delta}}{S}_{A})}_{\mathrm{G}}^{2}=\frac{1}{2}[f+{f}^{2}+log(1\ensuremath{-}f)]$.

arXiv:2103.05416•Physical review. B./Physical review. B•Jun 29, 2021•63 citations
The average bipartite quantum mutual information ⟨I(A:B)⟩\langle I(A{:}B)\rangle⟨I(A:B)⟩ of Haar-random pure states can be expressed exactly through Page's formula in terms of digamma functions. We show that this quantity admits a single non-perturbative closed form: ⟨I(A:B)⟩=(dA2−1)(dB2−1) G(dA,dB,dE)\langle I(A{:}B)\rangle = (d_A^2-1)(d_B^2-1)\,\mathcal{G}(d_A,d_B,d_E)⟨I(A:B)⟩=(dA2​−1)(dB2​−1)G(dA​,dB​,dE​), where G\mathcal{G}G is given by an explicit convergent integral over a Bose--Einstein kernel. The overall factor (dA2−1)(dB2−1)=dim⁡[su(dA)]⋅dim⁡[su(dB)](d_A^2-1)(d_B^2-1)=\dim[\mathfrak{su}(d_A)]\cdot\dim[\mathfrak{su}(d_B)](dA2​−1)(dB2​−1)=dim[su(dA​)]⋅dim[su(dB​)] is exact, not merely asymptotic. The asymptotic expansion of G\mathcal{G}G in 1/N1/N1/N yields a Bernoulli-factorised series whose coefficients involve ζ(1−2k)ζ(1{-}2k)ζ(1−2k); this series diverges, and our integral is its exact Borel sum. The integral representation also makes ⟨I⟩<(dA2−1)(dB2−1)/(2N)\langle I\rangle < (d_A^2{-}1)(d_B^2{-}1)/(2N)⟨I⟩<(dA2​−1)(dB2​−1)/(2N) manifest via a scale-inversion symmetry of the kernel. Our derivation traces the mutual information's structure to an exact decomposition of Page's entropy into a diagonal (Dirichlet) contribution and a Schur-majorisation eigenvalue correction, whose assembly into the mutual information cleanly separates classical from quantum correlations.
Starting from the exact Projected Central Limit Theorem on hyperspheres, we rederive the Beta distribution for subsystem occupation probabilities and Lubkin's purity formula from elementary hyperspherical moments, quantifying the finite-size ``platykurtic'' suppression of tails relative to the Gaussian approximation used in standard eigenstate-thermalization and typicality treatments. Our main new result concerns the bipartite quantum mutual information ⟨I(A:B)⟩\langle I(A{:}B)\rangle⟨I(A:B)⟩ for Haar-random pure states. We show that its full asymptotic expansion in 1/N1/N1/N admits a Bernoulli-factorized form in which every order k≥1k \ge 1k≥1 carries the symmetric factor (dA2k−1)(dB2k−1)(d_A^{2k}-1)(d_B^{2k}-1)(dA2k​−1)(dB2k​−1) and all higher odd-order corrections vanish identically. Through an exact algebraic reorganization of Page's formula (conjectured in Ref.~\cite{Page1993} and subsequently proven~\cite{Foong1994, SanchezRuiz1995, Sen1996}), we establish that the leading finite-size correction separates into a dominant su(dA)⊗su(dB)\mathfrak{su}(d_A) \otimes \mathfrak{su}(d_B)su(dA​)⊗su(dB​) bipartite quantum coherence contribution (dA2−1)(dB2−1)/(2N)(d_A^2 - 1)(d_B^2 - 1)/(2N)(dA2​−1)(dB2​−1)/(2N) and a subtracted classical-probability (Cartan ⊗\otimes⊗ Cartan) contribution (dA−1)(dB−1)/(2N)(d_A - 1)(d_B - 1)/(2N)(dA​−1)(dB​−1)/(2N), and we trace this separation to the difference between diagonal and eigenvalue entropies via Schur's majorisation theorem, with the dimensional counts (d−1)(d-1)(d−1) and (d2−1)(d^2-1)(d2−1) acquiring meaning through the Cartan structure of the generalised Bloch decomposition. These results admit a single non-perturbative closed form: the exact typical mutual information factors as ⟨I(A:B)⟩=(dA2−1)(dB2−1) G(dA,dB,dE)\langle I(A{:}B)\rangle = (d_A^2-1)(d_B^2-1)\,\mathcal{G}(d_A,d_B,d_E)⟨I(A:B)⟩=(dA2​−1)(dB2​−1)G(dA​,dB​,dE​), with G\mathcal{G}G given by an explicit Bose--Einstein integral whose asymptotic expansion in 1/N1/N1/N reproduces the Bernoulli series.
We study the average bipartite entanglement entropy of Haar-random pure states in quantum many-body systems with global SU(2)\mathrm{SU}(2)SU(2) symmetry, constrained to fixed total spin JJJ and magnetization Jz=0J_z = 0Jz​=0. Focusing on spin-12\tfrac1221​ lattices and subsystem fractions f<12f < \frac{1}{2}f<21​, we derive a asymptotic expression for the average entanglement entropy up to constant order in the system volume VVV. In addition to the expected leading volume law term, we prove the existence of a 12log⁡V\frac{1}{2}\log V21​logV finite-size correction resulting from the scaling of the Clebsch-Gordon coefficients and compute explicitly the O(1)O(1)O(1) contribution reflecting angular-momentum coupling within magnetization blocks. Our analysis uses features of random matrix ensembles and provides a fully analytical treatment for arbitrary spin densities, thereby extending Page type results to non-Abelian sectors and clarifying how SU(2)\mathrm{SU}(2)SU(2) symmetry shapes average entanglement.
We determine the relative entropy of entanglement of a bipartite mixed state ρABρ_{AB}ρAB​ obtained by tracing out one subsystem of a tripartite Haar-random pure state ∣ψ⟩ABC|ψ\rangle_{ABC}∣ψ⟩ABC​, finding ER(ρAB)=log⁡dAdBmax⁡(dA,dB,dC)+O(1)E_R(ρ_{AB})=\log\frac{d_Ad_B}{\max(d_A,d_B,d_C)}+O(1)ER​(ρAB​)=logmax(dA​,dB​,dC​)dA​dB​​+O(1). Equivalently, the relative entropy of entanglement nearly saturates the smaller of the entanglement of formation EF(ρAB)E_F(ρ_{AB})EF​(ρAB​) and the mutual information I(A:B)I(A:B)I(A:B). The upper bound is achieved by an explicit separable state obtained through one-sided Schmidt dephasing, which is therefore approximately closest.
Given a statistical ensemble of quantum states, the corresponding Page curve quantifies the average entanglement entropy associated with each possible spatial bipartition of the system. In this work, we study a natural extension in the presence of a conservation law and introduce the symmetry-resolved Page curves, characterizing average bipartite symmetry-resolved entanglement entropies. We derive explicit analytic formulas for two important statistical ensembles with a U(1)U(1)U(1)-symmetry: Haar-random pure states and random fermionic Gaussian states. In the former case, the symmetry-resolved Page curves can be obtained in an elementary way from the knowledge of the standard one. This is not true for random fermionic Gaussian states. In this case, we derive an analytic result in the thermodynamic limit based on a combination of techniques from random-matrix and large-deviation theories. We test our predictions against numerical calculations and discuss the subleading finite-size corrections.
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.
In a seminal paper, Page found the exact formula for the average entanglement entropy for a pure random state. We consider the analogous problem for the ensemble of pure fermionic Gaussian states, which plays a crucial role in the context of random free Hamiltonians. Using recent results from random matrix theory, we show that the average entanglement entropy of pure random fermionic Gaussian states in a subsystem of NA{N}_{A}NA​ out of NNN degrees of freedom is given by ${\ensuremath{\langle}{S}_{A}\ensuremath{\rangle}}_{\mathrm{G}}=(N\ensuremath{-}\frac{1}{2})\mathrm{\ensuremath{\Psi}}(2N)+(\frac{1}{4}\ensuremath{-}{N}_{A})\mathrm{\ensuremath{\Psi}}(N)+(\frac{1}{2}+{N}_{A}\ensuremath{-}N)\mathrm{\ensuremath{\Psi}}(2N\ensuremath{-}2{N}_{A})\ensuremath{-}\frac{1}{4}\mathrm{\ensuremath{\Psi}}(N\ensuremath{-}{N}_{A})\ensuremath{-}{N}_{A}$, where $\mathrm{\ensuremath{\Psi}}$ is the digamma function. Its asymptotic behavior in the thermodynamic limit is given by ${\ensuremath{\langle}{S}_{A}\ensuremath{\rangle}}_{\mathrm{G}}=N(log2\ensuremath{-}1)f+N(f\ensuremath{-}1)log(1\ensuremath{-}f)+\frac{1}{2}f+\frac{1}{4}log(1\ensuremath{-}f)\phantom{\rule{0.16em}{0ex}}+\phantom{\rule{0.16em}{0ex}}O(1/N)$, where $f={N}_{A}/N\ensuremath{\le}1/2$. Remarkably, its leading order agrees with the average over eigenstates of random quadratic Hamiltonians with number conservation, as found by \L{}yd\ifmmode \dot{z}\else \.{z}\fi{}ba, Rigol, and Vidmar. Finally, we compute the variance in the thermodynamic limit, given by the constant ${lim}_{N\ensuremath{\rightarrow}\ensuremath{\infty}}{(\mathrm{\ensuremath{\Delta}}{S}_{A})}_{\mathrm{G}}^{2}=\frac{1}{2}[f+{f}^{2}+log(1\ensuremath{-}f)]$.