Gaussian Page Law in Quantum Entanglement

• Gaussian Page law is a framework defining typical entanglement in Gaussian state ensembles, where the volume law is suppressed compared to Haar-random states.
• It offers exact finite-size formulas and asymptotic expressions for average entropy, variance, and symmetry-resolved entanglement in fermionic and bosonic settings.
• The law extends to dynamical contexts in monitored Gaussian circuits, demonstrating that even volume-law entanglement never reaches maximal Page values.

Searching arXiv for papers on Gaussian Page law and closely related work. Gaussian Page law denotes the analogue of Page’s typical-entanglement result when the ensemble of pure states is restricted to Gaussian states rather than the full Haar-random manifold. In the fermionic setting, it gives the average bipartite entanglement entropy of pure random fermionic Gaussian states and exhibits a volume law with a coefficient that is systematically below the Haar-random value, together with qualitatively different finite-size corrections and fluctuations (Bianchi et al., 2021). The term is also used more broadly for related typical-entanglement structures in Gaussian ensembles, including symmetry-resolved extensions for number-conserving fermionic Gaussian states (Murciano et al., 2022), capacity-of-entanglement analogues (Bhattacharjee et al., 2021), bosonic Gaussian Page curves in random linear optics (Iosue et al., 2022), and dynamical contexts where Gaussian circuits realize volume-law phases whose entropy density remains below the maximal Page value (Granet et al., 2022).

1. Definition and conceptual scope

The original Page law concerns the average entanglement entropy of a subsystem of a random pure state. The Gaussian Page law addresses the same question after replacing Haar-random pure states by Gaussian states, which are ground states of quadratic Hamiltonians and are fully characterized by their two-point correlation functions (Bianchi et al., 2021). In this restricted ensemble, typical states remain volume-law entangled, but the Gaussian constraint suppresses entanglement relative to the full Haar case (Bianchi et al., 2021).

In the formulation for pure random fermionic Gaussian states, the full system has $N$ fermionic modes, the subsystem $A$ has $N_A$ modes, and the subsystem fraction is $f=N_A/N$ (Bianchi et al., 2021). The central object is the ensemble average of the bipartite von Neumann entropy $S_A$. The resulting law is “Page-like” because it characterizes typical subsystem entanglement under a natural random-state ensemble, but it differs sharply from the classical Page result in both magnitude and fluctuation structure (Bianchi et al., 2021).

The scope of the notion has expanded. In one direction, symmetry-resolved Page curves ask how the average entanglement decomposes across $U(1)$ charge sectors for random fermionic Gaussian states (Murciano et al., 2022). In another, the same random-Gaussian framework has been used to compute the average capacity of entanglement, defined as the variance of the modular Hamiltonian (Bhattacharjee et al., 2021). A bosonic counterpart has also been developed for random passive Gaussian unitaries acting on squeezed inputs in linear optics, with asymptotically exact Rényi-2 Page curves and bounds on the corresponding von Neumann Page curve (Iosue et al., 2022). Finally, monitored or non-unitary Gaussian dynamics motivate a dynamical usage of the phrase: volume-law entanglement in Gaussian circuits can be “ergodic” in scaling while still remaining below Haar-random maximality, which the cited work explicitly identifies as an instance of a Gaussian Page law (Granet et al., 2022).

2. Fermionic Gaussian Page law: exact formula and thermodynamic limit

The most explicit formulation appears for pure random fermionic Gaussian states (Bianchi et al., 2021). The exact finite-$N$ average entanglement entropy is

$$\langle S_A\rangle_\mathrm{G}\!=\!(N\!-\!\tfrac{1}{2})\Psi(2N)\!+\!(\tfrac{1}{4}\!-\!N_A)\Psi(N)\!+\!(\tfrac{1}{2}\!+\!N_A\!-\!N)\Psi(2N\!-\!2N_A)\!-\!\tfrac{1}{4}\Psi(N\!-\!N_A)\!-\!N_A,$$

where $\Psi$ is the digamma function (Bianchi et al., 2021). This gives a closed finite-size expression for the Gaussian ensemble average.

In the thermodynamic limit $N\to\infty$ at fixed $A$0, the asymptotic expansion is

$A$1

The leading term is extensive, so the Gaussian Page law is still a volume law (Bianchi et al., 2021). However, its coefficient differs from the Haar-random result, and the subleading structure is explicit rather than exponentially small. The same work emphasizes that the leading-order term agrees with the average over eigenstates of random quadratic Hamiltonians with number conservation, as found by Lydzba, Rigol and Vidmar (Bianchi et al., 2021). This establishes that the Gaussian Page law is not merely an abstract ensemble property but also describes typical eigenstate entanglement in random free-fermion settings.

A closely related quantity is the capacity of entanglement, $A$2, with $A$3 (Bhattacharjee et al., 2021). For fermionic Gaussian states, an exact finite-size expression was derived as a finite series valid for arbitrary bipartition, and in the thermodynamic limit for the complex SYK$A$4 model the average CoE admits a closed-form integral and series representation (Bhattacharjee et al., 2021). At half partition, $A$5, the leading volume-law coefficient becomes $A$6 (Bhattacharjee et al., 2021). This identifies a second Gaussian Page-type benchmark beyond the von Neumann entropy itself.

3. Comparison with Haar-random Page law

The defining distinction between the Gaussian Page law and the Haar-random Page law is entanglement suppression due to the Gaussian manifold (Bianchi et al., 2021). For Haar-random pure states, the average subsystem entropy approaches the maximum possible value $A$7 exponentially fast in $A$8 for $A$9 (Bianchi et al., 2021). For fermionic Gaussian states, by contrast, the average entropy is lower in the thermodynamic limit and approaches that limit algebraically rather than exponentially (Bianchi et al., 2021).

The fluctuation structure is equally different. For random fermionic Gaussian states, the variance satisfies

$N_A$0

so the variance remains finite in the thermodynamic limit (Bianchi et al., 2021). This means Gaussian typicality is weaker than Haar typicality: relative fluctuations shrink, but the standard deviation itself stays $N_A$1 rather than decaying exponentially (Bianchi et al., 2021).

The same contrast appears for the capacity of entanglement. In the Gaussian setting, the variance of the average CoE becomes independent of system size in the thermodynamic limit, whereas the cited comparison with Page’s original random-pure-state result highlights vanishing variance in the chaotic/Haar case (Bhattacharjee et al., 2021). At half system size, the nonzero coefficient $N_A$2 is identified as a distinguishing feature between integrable and quantum-chaotic systems (Bhattacharjee et al., 2021).

A dynamical version of the same comparison arises in non-unitary periodic Gaussian circuits. There, the volume-law coefficient $N_A$3 is always lower than in Haar-random systems: “free circuits never reach maximal (Page) entanglement, even at minimal measurement strength,” and this is explicitly described as an instance of a Gaussian Page law (Granet et al., 2022). This statement sharpens a common misconception: volume-law entanglement does not imply Page-maximality. In Gaussian settings, extensivity and typicality do not erase the algebraic constraints inherited from free-fermion structure.

4. Symmetry-resolved Gaussian Page curves

A major extension concerns systems with a conservation law. For a $N_A$4-symmetric ensemble, symmetry-resolved entanglement asks how bipartite entropy is distributed across sectors of the conserved charge (Murciano et al., 2022). The reduced density matrix decomposes into charge sectors $N_A$5, with sector entropy

$N_A$6

and total entropy splitting into configurational and number entropy contributions (Murciano et al., 2022).

For Haar-random pure states, the symmetry-resolved Page curves can be obtained in an elementary way from the standard Page law (Murciano et al., 2022). This is not true for random fermionic Gaussian states. In the Gaussian case, projections onto fixed charge sectors produce non-Gaussian states, and the analysis therefore requires a combination of random-matrix and large-deviation techniques (Murciano et al., 2022).

For random fermionic Gaussian states with fixed total particle number $N_A$7, subsystem fraction $N_A$8, subsystem charge density $N_A$9, and total density $f=N_A/N$0, the thermodynamic-limit density of symmetry-resolved von Neumann entropy is given explicitly by Eq. (68) of the paper, summarized in the supplied data as

$f=N_A/N$1

The cited work states that, unlike the Haar case, equipartition is explicitly broken in the Gaussian case; entanglement in each charge sector is systematically lower because of the Wick-structured ensemble (Murciano et al., 2022). The leading finite-size correction is reported as $f=N_A/N$2 in both Haar and Gaussian cases, while the number entropy is subleading and has the same leading structure in both ensembles (Murciano et al., 2022).

This suggests that the Gaussian Page law has a layered structure under symmetry resolution: the global entropy already lies below the Haar value, and the sector-wise distribution further departs from Haar-style equipartition once charge conservation and Gaussianity are imposed simultaneously.

5. Bosonic Gaussian Page curves in linear optics

A bosonic counterpart has been developed for pure bosonic Gaussian states generated by applying a random passive Gaussian unitary to a product of squeezed vacuum modes (Iosue et al., 2022). The entanglement measure studied most directly is the Rényi-2 entropy,

$f=N_A/N$3

where $f=N_A/N$4 is the covariance matrix of the subsystem (Iosue et al., 2022). For a subsystem of $f=N_A/N$5 modes in an $f=N_A/N$6-mode system with equal squeezing $f=N_A/N$7, the asymptotic Page curve per mode is

$f=N_A/N$8

with $f=N_A/N$9 the Catalan numbers (Iosue et al., 2022). The curve is symmetric under $S_A$0, as required for bipartite entanglement in a pure state (Iosue et al., 2022). At half partition, the asymptotic value simplifies to $S_A$1 per mode (Iosue et al., 2022).

The same work defines a Page correction as the deficit from the maximum possible Rényi-2 entropy. At $S_A$2,

$S_A$3

This plays the same role as in finite-dimensional Page theory: it quantifies how close typical entanglement comes to the maximal allowed value (Iosue et al., 2022).

Typicality results were also established. For equal squeezing, the Rényi-2 entropy is weakly typical for any subsystem size $S_A$4, and strongly typical when $S_A$5 (Iosue et al., 2022). For the von Neumann entropy, symmetric bounds between $S_A$6 and $S_A$7 imply weak typicality for $S_A$8 (Iosue et al., 2022). These results supply a continuous-variable Gaussian analogue of the finite-dimensional typical-entanglement paradigm and situate the Gaussian Page law within random linear optics and Gaussian Boson Sampling (Iosue et al., 2022).

6. Dynamical and circuit manifestations

The phrase “Gaussian Page law” also appears in dynamical settings where Gaussian states arise from monitored or open-system evolution rather than static random ensembles. In a non-unitary periodic Gaussian circuit with alternating unitary gates and post-selected weak measurements, spatial translation symmetry, and time periodicity, the entanglement phase is controlled by fixed-point properties of momentum-resolved Möbius transformations (Granet et al., 2022). The coherent-state parameter $S_A$9 transforms as

$U(1)$0

and entanglement scaling is classified by whether the corresponding Möbius matrix $U(1)$1 has a unique stable fixed point, isolated critical momenta, or an interval of critical momenta (Granet et al., 2022).

Within the volume-law phase, the entropy has leading form

$U(1)$2

for Hartley, von Neumann, and Rényi entropies (Granet et al., 2022). The same paper identifies explicit log-law to area-law and volume-law to area-law transitions, including a two-cycle protocol with phase boundary

$U(1)$3

As $U(1)$4, the volume-law coefficient vanishes with entropy-dependent critical behavior: $U(1)$5 for Hartley entropy and $U(1)$6 for $U(1)$7, while the von Neumann entropy exhibits $U(1)$8 (Granet et al., 2022).

The specific relevance to Gaussian Page law is the statement that, even in the volume-law regime, the entropy density is always lower than in Haar-random systems and never reaches maximal Page entanglement (Granet et al., 2022). The monitored Gaussian circuit therefore realizes a dynamical analogue of the static Gaussian Page law: extensive entanglement constrained below the random-state maximum by Gaussianity.

Related, but conceptually distinct, are solvable fermionic models exhibiting Page-curve-in-time dynamics rather than subsystem-size Page curves. In a free fermion system coupled to a large environment, the von Neumann entropy grows, peaks at a Page time, and then decays to zero as the system empties (Kehrein, 2023). An interacting extension preserves the Page-like curve while introducing a non-analyticity in the min-entropy at a critical time before the Page time (Jha et al., 5 Feb 2025). These works discuss a connection to Gaussian Page law only in the sense of contrast: their dynamical Page curve does not arise from random-state typicality, and the maximal entropy at the Page time remains below the Haar value (Kehrein, 2023).

7. Significance, limitations, and common confusions

The Gaussian Page law provides a benchmark for typical entanglement in free or quadratic many-body settings. Its importance lies in giving exact or asymptotically exact reference formulas for ensembles that are structurally central in condensed matter theory, quantum information, random quadratic Hamiltonians, linear optics, and monitored free-fermion circuits (Bianchi et al., 2021, Iosue et al., 2022, Granet et al., 2022).

Several points are easily conflated and are best separated.

Topic	Gaussian result	Contrast
Typical scaling	Volume law in many regimes (Bianchi et al., 2021)	Volume law does not imply maximal Page value
Fluctuations	Variance remains finite in thermodynamic limit (Bianchi et al., 2021)	Haar variance decays exponentially
Symmetry resolution	Equipartition is broken (Murciano et al., 2022)	Haar case is simpler and effectively equipartitioned
Dynamics vs ensemble	Page-curve-in-time can occur in solvable fermion models (Kehrein, 2023, Jha et al., 5 Feb 2025)	This differs from the static Page law as a function of subsystem size

A first common confusion is to identify any Page curve with the Gaussian Page law. The static Gaussian Page law concerns average entanglement over Gaussian ensembles as a function of bipartition size (Bianchi et al., 2021, Iosue et al., 2022). By contrast, the Page-curve dynamics in system-environment fermionic chains concerns time evolution of entanglement under a specific initial state and Hamiltonian (Kehrein, 2023, Jha et al., 5 Feb 2025).

A second confusion is to equate Gaussian volume-law entanglement with Haar-random entanglement. The monitored-circuit analysis explicitly rules this out: the leading coefficient in the Gaussian volume-law phase is always suppressed relative to the Haar case (Granet et al., 2022). The fermionic Gaussian ensemble results say the same in static form (Bianchi et al., 2021).

A third confusion is to assume that conservation laws merely add subleading structure. The symmetry-resolved analysis shows that for random fermionic Gaussian states the sector-by-sector entropy is not obtained by a trivial projection of the ordinary Page law; instead, Gaussianity plus $U(1)$9 symmetry produces genuinely different thermodynamic formulas and breaks equipartition across sectors (Murciano et al., 2022).

Taken together, these results position the Gaussian Page law as a precise entanglement-typicality framework for constrained quantum state manifolds. Its central message is not the absence of typical entanglement, but the modification of typicality by Gaussian structure: extensive entanglement survives, maximality does not, and both fluctuations and symmetry resolution retain nontrivial signatures of the underlying free-state geometry (Bianchi et al., 2021, Murciano et al., 2022, Bhattacharjee et al., 2021, Iosue et al., 2022, Granet et al., 2022).