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arXiv:2605.29732

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

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

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

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.

Comments
11 pages, 1 figure. This is a companion paper to our simultaneous submission with a title "Non-Perturbative Closed Form for the Typical Bipartite Mutual Information of Haar-Random States"

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

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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.

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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.

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We analyze the properties of entangled random pure states of a quantum system partitioned into two smaller subsystems of dimensions N and M. Framing the problem in terms of random matrices with a fixed-trace constraint, we establish, for arbitrary N≤M, a general relation between the n-point densities and the cross moments of the eigenvalues of the reduced density matrix, i.e., the so-called Schmidt eigenvalues, and the analogous functionals of the eigenvalues of the Wishart-Laguerre ensemble of the random matrix theory. This allows us to derive explicit expressions for two-level densities, and also an exact expression for the variance of von Neumann entropy at finite N,M. Then, we focus on the moments E{K^{a}} of the Schmidt number K, the reciprocal of the purity. This is a random variable supported on [1,N], which quantifies the number of degrees of freedom effectively contributing to the entanglement. We derive a wealth of analytical results for E{K^{a}} for N=2 and 3 and arbitrary M, and also for square N=M systems by spotting for the latter a connection with the probability P(x_{min}^{GUE}≥sqrt[2N]ξ) that the smallest eigenvalue x_{min}^{GUE} of an N×N matrix belonging to the Gaussian unitary ensemble is larger than sqrt[2N]ξ. As a by-product, we present an exact asymptotic expansion for P(x_{min}^{GUE}≥sqrt[2N]ξ) for finite N as ξ→∞. Our results are corroborated by numerical simulations whenever possible, with excellent agreement.

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Large NNN factorization of families of tensor trace-invariants

Sylvain Carrozza, Johann Chevrier, Luca Lionni

It was recently proven that, in contrast to their matrix analogues, the moments of a real Gaussian tensor of size N do not in general factorize over their connected components in the asymptotic large N limit. While the original proof of this rather surprising result was not constructive, explicit examples of non-factorizing moments, which are expectation values of trace-invariants, have since then been discovered. We explore further aspects of this problem, with a focus on Haar-distributed (or Gaussian) complex random tensors, which are more directly relevant to quantum information. We start out by exhibiting an explicit example of non-factorizing trace-invariant, thereby filling a gap in the recent literature. We then turn to the opposite question: that of finding interesting families of trace-invariants that do in fact factorize at large N. We establish three main theorems in this regard. The first one provides a sufficient combinatorial bound ensuring large N factorization, that is also simple enough to be applicable to various cases of practical relevance. Our second main result shows that the expectation value of any compatible trace-invariant is dominated by certain tree-like combinatorial structures at large N, which we refer to as tree-like dominant pairings. Our third main theorem establishes that any trace-invariant admitting tree-like dominant pairings does actually factorize at large N. In this way, we are able to prove that various families of trace-invariants that have been previously studied in the literature do factorize at large N. We apply our findings to the theory of multipartite quantum entanglement: to any trace-invariant is associated a multipartite generalization of Rényi entanglement entropy, whose typical expectation value in the uniform random quantum state can be explicitly computed assuming large N factorization.

math-phCross-listedarXiv:2605.12468•May 13, 2026•0 citations

Entanglement Entropy of Eigenstates of Quadratic Fermionic Hamiltonians

Lev Vidmar, Lucas Hackl, Eugenio Bianchi A +1 more

In a seminal paper [D. N. Page, Phys. Rev. Lett. 71, 1291 (1993)], Page proved that the average entanglement entropy of subsystems of random pure states is ${S}_{\text{ave}}\ensuremath{\simeq}\mathrm{ln}{\mathcal{D}}_{\mathrm{A}}\ensuremath{-}(1/2){\mathcal{D}}_{\mathrm{A}}^{2}/\mathcal{D}$ for $1\ensuremath{\ll}{\mathcal{D}}_{\mathrm{A}}\ensuremath{\le}\sqrt{\mathcal{D}}$, where ${\mathcal{D}}_{\mathrm{A}}$ and $\mathcal{D}$ are the Hilbert space dimensions of the subsystem and the system, respectively. Hence, typical pure states are (nearly) maximally entangled. We develop tools to compute the average entanglement entropy $⟨S⟩$ of all eigenstates of quadratic fermionic Hamiltonians. In particular, we derive exact bounds for the most general translationally invariant models $\mathrm{ln}{\mathcal{D}}_{\mathrm{A}}\ensuremath{-}(\mathrm{ln}{\mathcal{D}}_{\mathrm{A}}{)}^{2}/\mathrm{ln}\mathcal{D}\ensuremath{\le}⟨S⟩\ensuremath{\le}\mathrm{ln}{\mathcal{D}}_{\mathrm{A}}\ensuremath{-}[1/(2\mathrm{ln}2)](\mathrm{ln}{\mathcal{D}}_{\mathrm{A}}{)}^{2}/\mathrm{ln}\mathcal{D}$. Consequently, we prove that (i) if the subsystem size is a finite fraction of the system size, then $⟨S⟩<\mathrm{ln}{\mathcal{D}}_{\mathrm{A}}$ in the thermodynamic limit; i.e., the average over eigenstates of the Hamiltonian departs from the result for typical pure states, and (ii) in the limit in which the subsystem size is a vanishing fraction of the system size, the average entanglement entropy is maximal; i.e., typical eigenstates of such Hamiltonians exhibit eigenstate thermalization.

arXiv:1703.02979•Physical Review Letters•Jul 11, 2017•134 citations

General Entanglement Diagnostics

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This paper develops a single bipartite entanglement diagnostic and follows it from elementary quantum information into holographic gravity. For a pure state of any Schmidt rank, a natural "balance" quantity built from the elementary symmetric polynomials of the Schmidt spectrum turns out to be exactly half the I-concurrence, its mean hierarchy is Maclaurin's and Newton's inequalities, and—via Newton's identities—it is interchangeable with the replica moments that compute entanglement entropy. Exactly solvable qudit and oscillator thermofield-double models, a closed-form distinction between the maximal-entanglement and first-law (linearized-Einstein) points on the probability simplex, and a Gaussian continuous-variable lift are given. The diagnostic is then lifted to the eternal BTZ black hole, where the concurrence layer is shown to be intrinsically finite-dimensional (it saturates in the continuum) while the moment/relative-entropy layer becomes genuinely geometric. The relative entropy of a single interval is obtained in closed form, S=c3[u26−ln⁡(sinh⁡u/u)]S=\frac{c}{3}[\frac{u^2}{6}-\ln(\sinh u/u)] S=3c[6u2−ln(sinhu/u)]: its first law, an elementary positivity proof, and a quantum-Fisher/bulk-canonical-energy identification follow, and a full reconstruction from the gravitational symplectic flux yields a rigid series in even zeta values resumming to the Ryu–Takayanagi area. The two-interval case is worked out both holographically (mutual information with a phase transition) and for the free fermion via the resolvent (smooth, theory-dependent), making explicit where the universal single-interval law gives way to operator-content dependence. All novel results are stated as propositions/theorems with proofs and verified numerically.

•Jun 6, 2026•0 citations
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.
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.
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 analyze the properties of entangled random pure states of a quantum system partitioned into two smaller subsystems of dimensions N and M. Framing the problem in terms of random matrices with a fixed-trace constraint, we establish, for arbitrary N≤M, a general relation between the n-point densities and the cross moments of the eigenvalues of the reduced density matrix, i.e., the so-called Schmidt eigenvalues, and the analogous functionals of the eigenvalues of the Wishart-Laguerre ensemble of the random matrix theory. This allows us to derive explicit expressions for two-level densities, and also an exact expression for the variance of von Neumann entropy at finite N,M. Then, we focus on the moments E{K^{a}} of the Schmidt number K, the reciprocal of the purity. This is a random variable supported on [1,N], which quantifies the number of degrees of freedom effectively contributing to the entanglement. We derive a wealth of analytical results for E{K^{a}} for N=2 and 3 and arbitrary M, and also for square N=M systems by spotting for the latter a connection with the probability P(x_{min}^{GUE}≥sqrt[2N]ξ) that the smallest eigenvalue x_{min}^{GUE} of an N×N matrix belonging to the Gaussian unitary ensemble is larger than sqrt[2N]ξ. As a by-product, we present an exact asymptotic expansion for P(x_{min}^{GUE}≥sqrt[2N]ξ) for finite N as ξ→∞. Our results are corroborated by numerical simulations whenever possible, with excellent agreement.
It was recently proven that, in contrast to their matrix analogues, the moments of a real Gaussian tensor of size N do not in general factorize over their connected components in the asymptotic large N limit. While the original proof of this rather surprising result was not constructive, explicit examples of non-factorizing moments, which are expectation values of trace-invariants, have since then been discovered. We explore further aspects of this problem, with a focus on Haar-distributed (or Gaussian) complex random tensors, which are more directly relevant to quantum information. We start out by exhibiting an explicit example of non-factorizing trace-invariant, thereby filling a gap in the recent literature. We then turn to the opposite question: that of finding interesting families of trace-invariants that do in fact factorize at large N. We establish three main theorems in this regard. The first one provides a sufficient combinatorial bound ensuring large N factorization, that is also simple enough to be applicable to various cases of practical relevance. Our second main result shows that the expectation value of any compatible trace-invariant is dominated by certain tree-like combinatorial structures at large N, which we refer to as tree-like dominant pairings. Our third main theorem establishes that any trace-invariant admitting tree-like dominant pairings does actually factorize at large N. In this way, we are able to prove that various families of trace-invariants that have been previously studied in the literature do factorize at large N. We apply our findings to the theory of multipartite quantum entanglement: to any trace-invariant is associated a multipartite generalization of Rényi entanglement entropy, whose typical expectation value in the uniform random quantum state can be explicitly computed assuming large N factorization.
In a seminal paper [D. N. Page, Phys. Rev. Lett. 71, 1291 (1993)], Page proved that the average entanglement entropy of subsystems of random pure states is ${S}_{\text{ave}}\ensuremath{\simeq}\mathrm{ln}{\mathcal{D}}_{\mathrm{A}}\ensuremath{-}(1/2){\mathcal{D}}_{\mathrm{A}}^{2}/\mathcal{D}$ for $1\ensuremath{\ll}{\mathcal{D}}_{\mathrm{A}}\ensuremath{\le}\sqrt{\mathcal{D}}$, where DA{\mathcal{D}}_{\mathrm{A}}DA​ and D\mathcal{D}D are the Hilbert space dimensions of the subsystem and the system, respectively. Hence, typical pure states are (nearly) maximally entangled. We develop tools to compute the average entanglement entropy ⟨S⟩⟨S⟩⟨S⟩ of all eigenstates of quadratic fermionic Hamiltonians. In particular, we derive exact bounds for the most general translationally invariant models $\mathrm{ln}{\mathcal{D}}_{\mathrm{A}}\ensuremath{-}(\mathrm{ln}{\mathcal{D}}_{\mathrm{A}}{)}^{2}/\mathrm{ln}\mathcal{D}\ensuremath{\le}⟨S⟩\ensuremath{\le}\mathrm{ln}{\mathcal{D}}_{\mathrm{A}}\ensuremath{-}[1/(2\mathrm{ln}2)](\mathrm{ln}{\mathcal{D}}_{\mathrm{A}}{)}^{2}/\mathrm{ln}\mathcal{D}$. Consequently, we prove that (i) if the subsystem size is a finite fraction of the system size, then ⟨S⟩<lnDA⟨S⟩<\mathrm{ln}{\mathcal{D}}_{\mathrm{A}}⟨S⟩<lnDA​ in the thermodynamic limit; i.e., the average over eigenstates of the Hamiltonian departs from the result for typical pure states, and (ii) in the limit in which the subsystem size is a vanishing fraction of the system size, the average entanglement entropy is maximal; i.e., typical eigenstates of such Hamiltonians exhibit eigenstate thermalization.
This paper develops a single bipartite entanglement diagnostic and follows it from elementary quantum information into holographic gravity. For a pure state of any Schmidt rank, a natural "balance" quantity built from the elementary symmetric polynomials of the Schmidt spectrum turns out to be exactly half the I-concurrence, its mean hierarchy is Maclaurin's and Newton's inequalities, and—via Newton's identities—it is interchangeable with the replica moments that compute entanglement entropy. Exactly solvable qudit and oscillator thermofield-double models, a closed-form distinction between the maximal-entanglement and first-law (linearized-Einstein) points on the probability simplex, and a Gaussian continuous-variable lift are given. The diagnostic is then lifted to the eternal BTZ black hole, where the concurrence layer is shown to be intrinsically finite-dimensional (it saturates in the continuum) while the moment/relative-entropy layer becomes genuinely geometric. The relative entropy of a single interval is obtained in closed form, S=c3[u26−ln⁡(sinh⁡u/u)]S=\frac{c}{3}[\frac{u^2}{6}-\ln(\sinh u/u)] S=3c[6u2−ln(sinhu/u)]: its first law, an elementary positivity proof, and a quantum-Fisher/bulk-canonical-energy identification follow, and a full reconstruction from the gravitational symplectic flux yields a rigid series in even zeta values resumming to the Ryu–Takayanagi area. The two-interval case is worked out both holographically (mutual information with a phase transition) and for the free fermion via the resolvent (smooth, theory-dependent), making explicit where the universal single-interval law gives way to operator-content dependence. All novel results are stated as propositions/theorems with proofs and verified numerically.