Average Bipartite Entanglement Entropy

• Average bipartite entanglement entropy is a measure of the mean von Neumann entropy of a subsystem from a bipartition of a pure state, capturing key quantum correlations and typicality.
• Analytic formulas from Haar and Bures–Hall ensembles establish universal scaling laws, including the Page curve and volume law behavior in many-body systems.
• This measure underpins studies in quantum information, statistical mechanics, and non-equilibrium dynamics, enabling insights into thermalization and critical phase transitions.

Average bipartite entanglement entropy quantifies the mean von Neumann entropy of a subsystem resulting from a bipartition of a pure quantum state, serving as a fundamental measure of quantum correlations under random-state or dynamical generation protocols. It is central to quantum information theory, statistical mechanics, and condensed matter physics, providing a rigorous framework to characterize typicality, thermalization, and the structure of quantum phases in complex quantum systems. The analytics predominantly rely on ensemble averages over the Haar measure, random matrix ensembles, or symmetry-constrained sectors, with universal formulas established for diverse regimes.

1. Mathematical Framework for Bipartite Entanglement Entropy

For a pure quantum state $|\psi\rangle$ in a composite Hilbert space $\mathcal{H}_A \otimes \mathcal{H}_B$, the reduced density matrix of subsystem $A$ is $\rho_A = \mathrm{Tr}_B |\psi\rangle\langle\psi|$. The bipartite von Neumann entropy is defined as

$$S_{vN}(\rho_A) = -\mathrm{Tr}[\rho_A \ln \rho_A]\,,$$

and quantifies the entanglement between $A$ and $B$.

The average bipartite entropy is computed as an expectation over the relevant ensemble—most commonly the Haar measure (uniform distribution) over pure states. For $\mathrm{dim}(\mathcal{H}_A)=m$ and $\mathrm{dim}(\mathcal{H}_B)=n$ with $m \leq n$, the average entropy is

$$\langle S_{vN} \rangle = \psi_0(mn+1) - \psi_0(n) - \frac{m+1}{2n}$$

where $\psi_0(x) = \Gamma'(x)/\Gamma(x)$ denotes the digamma function (Wei, 2019, Alonso-Serrano et al., 2017, Nadal et al., 2010).

Under the Bures–Hall ensemble, which is relevant for certain random mixed-state protocols, the closed form is (Wei, 2020)

$$E_f[S_{vN}] = \psi_0(mn - m^2/2 + 1) - \psi_0(n + 1/2)$$

with the parameterization specified by the Cauchy–Laguerre biorthogonal ensemble.

2. Large-Dimension Asymptotics and Universal Laws

In the limit $m,n \to \infty$ with ratio $c=m/n$, the mean entropy admits an asymptotic expansion (Alonso-Serrano et al., 2017, Nadal et al., 2010)

$$\langle S_{vN} \rangle \approx \ln m - \frac{m}{2n} + o(1)$$

For balanced bipartition ($m=n=N$), this specializes to

$\langle S_{vN} \rangle \approx \ln N - \frac{1}{2}$

Random pure states in high dimensions thus yield an "almost maximally mixed" $A$, with subleading $O(1)$ deficit—the "Page curve"—and variance $O(1/N^2)$ (Nadal et al., 2010).

For Rényi entropies $S_q = \frac{1}{1-q}\ln\mathrm{Tr}\rho_A^q$, the analogous formula at leading order reads (Facchi et al., 2019): $$\langle S_q \rangle \simeq \ln N_A - \frac{1}{q-1}\ln\left[\frac{2^{2q}\Gamma(q+1/2)}{\sqrt{\pi}\,\Gamma(q+2)}\right]$$ These results have been extended and confirmed across random matrix models (Facchi et al., 2019), Coulomb gas techniques (Nadal et al., 2010), and free-fermion/harmonic oscillator ensembles (Jafarizadeh et al., 2019).

3. Entropy Scaling Laws in Many-Body Quantum Systems

• Volume Law: In quantum-chaotic many-body systems and generic eigenstates (random pure states), bipartite entanglement entropy scales as the subsystem size

  $S_A \propto L_A$

where $L_A$ is the number of sites or degrees of freedom in $A$ (Vidmar et al., 2017, Łydżba et al., 2021, Jafarizadeh et al., 2019). Subleading corrections typically scale as $\sqrt{L_A}$ for number-conserving sectors away from half filling (Vidmar et al., 2017), or as $O(1)$ for unconstrained random states.

• Universal average in free quadratic models: Translationally invariant free quadratic Hamiltonians (including spin chains via Jordan–Wigner) yield a universal average over eigenstates (Jafarizadeh et al., 2019)

  $\langle S_\ell \rangle_{\rm univ} = N\,s(x) + o(N)$

  where $x = \ell/N$, and $s(x) = -x\ln x - (1-x)\ln(1-x)$ is the binary entropy.

• Quantum-chaotic quadratic Hamiltonians: In random quadratic models with RMT-type single-particle statistics, the leading entropy is independent of particle-number conservation (Łydżba et al., 2021):

  $$\overline{S} = \left(1 - \frac{1 + \frac{1-f}{f}\ln(1-f)}{\ln2}\right) L_A\ln2$$

  with $f=L_A/L$ the subsystem fraction.

4. Entanglement Entropy under Symmetry Constraints

Average bipartite entanglement entropy in the presence of non-Abelian constraints (e.g., global SU(2) conservation) exhibits modified scaling (Chakraborty et al., 28 Dec 2025): $$\langle S_A \rangle_j = V f\,\beta(j) + \tfrac12 \log V + C_0(f, j) + o(1)$$ where $j = 2J/V$ is the normalized global spin, $$\beta(j) = -\tfrac{1+j}{2}\ln\tfrac{1+j}{2} - \tfrac{1-j}{2}\ln\tfrac{1-j}{2}$$, and $C_0(f,j)$ encodes symmetry-dependent constants. The subleading $\frac12\log V$ term reflects the scaling of Clebsch–Gordan coefficients, and the O(1) constant encodes angular-momentum block effects.

5. Critical Phenomena and Spectral Phase Diagrams

The entanglement spectrum and average entropy admit sharp phase transitions in the thermodynamic limit (Facchi et al., 2019, Nadal et al., 2010):

• Concentration line ($u_C(q)$): Onset of eigenvalue density at zero; third-order transition for $q \neq 1$, fourth-order at $q=1$.
• Evaporation line ($u_E(q)$): Detachment of the largest eigenvalue ("macroscopic occupation"); first-order for $q \neq 1$, second-order for $q=1$.

States below $u_C$ reside in the entangled (semicircle-law-like) phase; $u_C<u<u_E$ delimit the typical regime; above $u_E$ enters the separable phase (macroscopic eigenvalue). These critical lines organize the "phase diagram" for bipartite entanglement (Facchi et al., 2019, Nadal et al., 2010).

6. Non-equilibrium and Measurement-Induced Entanglement

Average bipartite entropy also figures prominently in non-equilibrium settings:

• Quench dynamics: Time evolution after a sudden parameter change leads to rapid saturation ("plateau") in entanglement entropy, with distinct behavior in integrable vs non-integrable (chaotic) systems (Poilblanc, 2010).
• Diagonal ensemble average: The infinite-time average of entropy (over the trajectory of $\rho_A(t)$) differs markedly from thermal equilibrium values in integrable models, reflecting memory of conserved charges (Poilblanc, 2010).
• Measurement-induced entanglement: In quantum protocols such as gravitational wave detection, bipartite measurement-induced entanglement entropy scales logarithmically with mean occupation number (Jones et al., 2024), distinguishing semi-classical scenarios from genuine quantum signatures.

7. Generalizations, Universality, and Experimental Implications

Multiple generalizations exist for tripartite and multipartite systems, with average subsystem entropies serving as anchors for discussions of multipartite entanglement, average tangle, and concurrence (Alonso-Serrano et al., 2017). The universality of the Page curve and its variants is confirmed by analytic, Coulomb-gas, and random matrix techniques, and has been extended to scenarios with symmetry constraints, integrals of motion, and to both Hermitian and quasi-Hermitian evolutions (Moise et al., 2022).

Experimental access to average bipartite entanglement entropy uses randomized measurements (second Rényi protocols), duality relations in free models, and analysis of multi-interval entropies, especially in platforms such as cold atoms and trapped ions (Jafarizadeh et al., 2019).

Table: Analytic Formulas for $\langle S_{vN} \rangle$ Under Key Ensembles

Ensemble / Regime	Formula for $\langle S_{vN} \rangle$	Scaling Behavior
Haar Random Pure States	$\psi_0(mn+1) - \psi_0(n) - \frac{m+1}{2n}$	$\ln m - m/(2n) + o(1)$
Bures–Hall Ensemble	$\psi_0(mn - m^2/2 + 1) - \psi_0(n + 1/2)$	$O(\ln m)$
Unconstrained Quadratic	$N s(x) + o(N)$, $s(x)=-x\ln x -(1-x)\ln(1-x)$	Volume law
Quantum-Chaotic Quadratic	$(1 - [1+\frac{1-f}{f}\ln(1-f)]/\ln2)L_A\ln2$	Volume law
SU(2) Symmetry Sector	$Vf\,\beta(j) + \frac12\log V + C_0(f,j) + o(1)$	Volume + $\frac12\log V$

All formulas are directly traceable to the cited arXiv literature (Wei, 2020, Wei, 2019, Alonso-Serrano et al., 2017, Facchi et al., 2019, Chakraborty et al., 28 Dec 2025, Łydżba et al., 2021, Jafarizadeh et al., 2019, Nadal et al., 2010).

• (Wei, 2020) Proof of Sarkar-Kumar's Conjectures on Average Entanglement Entropies over the Bures-Hall Ensemble
• (Wei, 2019) Skewness of von Neumann entanglement entropy
• (Alonso-Serrano et al., 2017) Multi-partite analysis of average-subsystem entropies
• (Facchi et al., 2019) Phase diagram of bipartite entanglement
• (Chakraborty et al., 28 Dec 2025) Random matrix prediction of average entanglement entropy in non-Abelian symmetry sectors
• (Łydżba et al., 2021) Entanglement in many-body eigenstates of quantum-chaotic quadratic Hamiltonians
• (Jafarizadeh et al., 2019) Bipartite entanglement entropy of the excited states of the free fermions and the harmonic oscillators
• (Nadal et al., 2010) Statistical distribution of quantum entanglement for a random bipartite state
• (Poilblanc, 2010) Out-of-equilibrium Correlated Systems: Bipartite Entanglement as a Probe of Thermalization
• (Jones et al., 2024) Measurement-induced entanglement entropy of gravitational wave detections
• (Moise et al., 2022) Entropy and entanglement in a bipartite quasi-Hermitian system and its Hermitian counterparts