Page's Formula in Quantum Entanglement

• Page's Formula is the exact average von Neumann entropy of the smaller subsystem in Haar-random bipartite pure states.
• It employs harmonic sum and digamma representations to yield precise entanglement and mutual information calculations with clear non-planar corrections.
• The formula underpins advanced studies including free-probabilistic Page curves, tripartite entropy analyses, and comparisons with gravitational matrix models.

Searching arXiv for papers on Page's formula, Haar-random states, PSSY model, and related formulations. Page’s formula, in the quantum-information sense developed in recent arXiv treatments, is the exact average von Neumann entropy of the smaller subsystem of a Haar-random pure state on a bipartite Hilbert space. For subsystem dimensions $m$ and $n$ with $n\ge m$, it is quoted as

$S_R=\sum_{k=n+1}^{mn}\frac{1}{k}-\frac{m-1}{2n},$

and equivalently in digamma form as

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

where $\psi$ is the digamma function (Iizuka et al., 2024, Wang et al., 28 May 2026). In the same literature, this formula functions both as an exact statement about typical entanglement in random pure states and as the seed for later constructions of typical mutual information, Page curves, free-probabilistic generalizations, and non-planar corrections (Wang, 2022).

1. Exact bipartite statement

The formula is introduced as the standard Page result for the entanglement entropy of a random pure state on a bipartite Hilbert space of dimensions $m$ and $n$, assuming $n\ge m$ (Iizuka et al., 2024). In that setting, the entropy being averaged is the von Neumann entropy of the smaller subsystem. The harmonic-sum form and the digamma form are equivalent: $S_R=\sum_{k=n+1}^{mn}\frac{1}{k}-\frac{m-1}{2n},$

$n$0

The term $n$1 is not an asymptotic correction appended after the fact; it is part of the exact formula as quoted in these papers. In the tripartite applications discussed later, this exactness is what permits Page entropies for several subsystems to be assembled into exact formulas for mutual information rather than only leading-order estimates (Wang et al., 28 May 2026).

A structural interpretation is also supplied in later work. The average entropy is decomposed into a diagonal entropy / Dirichlet term,

$n$2

and a Schur-majorisation eigenvalue correction,

$n$3

In that reorganization, the first term is the Shannon entropy of the diagonal probabilities $n$4, while the second is the average gap between the diagonal entropy and the von Neumann entropy (Wang et al., 28 May 2026).

2. Asymptotic regime and symmetric bipartitions

A large-$n$5 expansion is given as

$n$6

(Iizuka et al., 2024). This expansion isolates the near-maximality of the smaller subsystem entropy and makes the finite-size structure explicit.

In the symmetric case $n$7, the same paper writes

$n$8

The value

$n$9

is therefore the leading Page-point behavior at equal subsystem sizes, while the $n\ge m$0 coefficient resolves the first non-planar correction scale studied in matrix-model comparisons.

A later mutual-information analysis develops a different asymptotic organization. There the $n\ge m$1 expansion is Bernoulli-factorised, and the first terms are written

$n\ge m$2

(Wang et al., 28 May 2026). That paper stresses that the associated asymptotic series is divergent because the Bernoulli numbers grow factorially.

3. Tripartite Page entropies and typical mutual information

A recent extension takes Page’s formula as the starting point for a tripartite Hilbert space

$n\ge m$3

with subsystem dimensions $n\ge m$4, $n\ge m$5, $n\ge m$6, total dimension

$n\ge m$7

and the assumption $n\ge m$8 (Wang et al., 28 May 2026). The three relevant entropies are obtained by applying the bipartite Page formula with

$n\ge m$9

for $S_R=\sum_{k=n+1}^{mn}\frac{1}{k}-\frac{m-1}{2n},$0, $S_R=\sum_{k=n+1}^{mn}\frac{1}{k}-\frac{m-1}{2n},$1, and $S_R=\sum_{k=n+1}^{mn}\frac{1}{k}-\frac{m-1}{2n},$2, respectively.

From these, the average mutual information is

$S_R=\sum_{k=n+1}^{mn}\frac{1}{k}-\frac{m-1}{2n},$3

The same work rewrites this as

$S_R=\sum_{k=n+1}^{mn}\frac{1}{k}-\frac{m-1}{2n},$4

with

$S_R=\sum_{k=n+1}^{mn}\frac{1}{k}-\frac{m-1}{2n},$5

and

$S_R=\sum_{k=n+1}^{mn}\frac{1}{k}-\frac{m-1}{2n},$6

The principal new result is an exact non-perturbative closed form,

$S_R=\sum_{k=n+1}^{mn}\frac{1}{k}-\frac{m-1}{2n},$7

where

$S_R=\sum_{k=n+1}^{mn}\frac{1}{k}-\frac{m-1}{2n},$8

and

$S_R=\sum_{k=n+1}^{mn}\frac{1}{k}-\frac{m-1}{2n},$9

The factorization by $$\langle S_{\mathrm{vN}}\rangle=\psi(mn+1)-\psi(n+1)-\frac{m-1}{2n},$$0 is emphasized as exact, not merely asymptotic, and is read as

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

A direct consequence of the integral representation is the manifest bound

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

This reformulation changes the role of Page’s formula. It is no longer only a closed expression for average entropy; it becomes the input for a decomposition into a classical-looking diagonal correlation term and a genuinely quantum eigenvalue correction. The same paper states that the exact integral is the Borel sum of the divergent Bernoulli-factorised asymptotic expansion (Wang et al., 28 May 2026).

4. Page curve, islands, and free multiplicative convolution

In the Penington-Shenker-Stanford-Yang model, Page’s formula appears as the planar entropy law for the radiation subsystem in the maximally entangled case. The flat-spectrum result is

$$\langle S_{\mathrm{vN}}\rangle=\psi(mn+1)-\psi(n+1)-\frac{m-1}{2n},$$3

which is identified as the PSSY version of Page’s formula (Wang, 2022). In that regime, for $$\langle S_{\mathrm{vN}}\rangle=\psi(mn+1)-\psi(n+1)-\frac{m-1}{2n},$$4 the entropy grows like $$\langle S_{\mathrm{vN}}\rangle=\psi(mn+1)-\psi(n+1)-\frac{m-1}{2n},$$5, while for $$\langle S_{\mathrm{vN}}\rangle=\psi(mn+1)-\psi(n+1)-\frac{m-1}{2n},$$6 it saturates at $$\langle S_{\mathrm{vN}}\rangle=\psi(mn+1)-\psi(n+1)-\frac{m-1}{2n},$$7.

The free-probabilistic generalization replaces this piecewise minimum by an exact spectral formula. The fundamental radiation density operator $$\langle S_{\mathrm{vN}}\rangle=\psi(mn+1)-\psi(n+1)-\frac{m-1}{2n},$$8 has spectral distribution

$$\langle S_{\mathrm{vN}}\rangle=\psi(mn+1)-\psi(n+1)-\frac{m-1}{2n},$$9

and therefore

$\psi$0

The same framework yields the Rényi entropies

$\psi$1

(Wang, 2022).

The derivation is combinatorial. The replica-trick gravitational path integral produces a noncrossing-partition expansion,

$\psi$2

which is matched to the standard moment formula for free multiplicative convolution,

$\psi$3

The paper concludes that replica wormhole combinatorics implements

$\psi$4

This exact entropy formula supersedes the island formula when the bulk radiation spectrum is non-flat and multiple saddles contribute comparably. The island formula is recovered only when the convolution factorizes, which the paper characterizes by one-shot entropy conditions. In the young-black-hole regime,

$\psi$5

one has $\psi$6, while in the old-black-hole regime,

$\psi$7

one has $\psi$8 (Wang, 2022). The same work therefore treats Page’s formula not as the generic answer but as the flat-spectrum factorized limit of a more general free-convolution law.

5. Non-planar corrections and matrix-model comparison

A further development studies non-planar corrections to the Page curve in the PSSY model through a correspondence with the IOP matrix model (Iizuka et al., 2024). The planar PSSY result gives

$\psi$9

which matches the large-dimension limit of Page’s formula. At the Page point $m$0, this becomes

$m$1

The IOP/PSSY correspondence is built by matching Schwinger-Dyson equations, Feynman diagrams, and parameters. At equal subsystem sizes, the PSSY spectral density is the Marchenko–Pastur law

$m$2

and the entropy is computed by

$m$3

The non-planar analysis resums a specific class of crossing diagrams. Mapping the IOP $m$4 correction back to PSSY yields the entropy expansion

$m$5

This reproduces the $m$6 part of the equal-dimension Page expansion,

$m$7

but not the full coefficient (Iizuka et al., 2024).

The discrepancy is attributed to extra-handle-in-bulk geometries, which have no direct counterpart in the IOP matrix model. The paper therefore interprets the crossing resummation as a partial non-planar reconstruction of Page’s formula. A plausible implication is that the full finite-$m$8 Page expansion in the gravitational model requires resummation of all non-planar bulk topologies, not only crossings.

6. Homonymous and thematic usages

The name “Page’s formula” is not unique to quantum typical entanglement in the supplied literature. One mathematically distinct usage appears in the study of Lucas-Lehmer polynomials and Gray-code-ordered nested radicals. There the term refers to the classical nested-radical limit for $m$9,

$n$0

which is tied to the smallest positive zero of $n$1; the same paper derives exact $n$2-identities by equating Gray-code nested radicals with trigonometric expressions for the zeros of $n$3 (Vellucci et al., 2016).

A related elliptic-integral paper does not explicitly derive Page’s formula in that classical guise, but presents Ramanujan-type series for the complete elliptic integrals $n$4 and $n$5,

$n$6

$n$7

and describes these as closely analogous to Page-like closed forms (Bagis, 2011).

By contrast, one number-theoretic paper on the prime counting function states explicitly that Page’s Formula is not mentioned and that any connection is only thematic (Gaitanas, 2013). Another record concerning monochrome-layer compositing cannot support a reliable reconstruction of a theorem because the supplied content reports that no PDF is available (Bali, 2017).

In this sense, the technically standard modern usage in the supplied arXiv corpus is the entanglement-entropy formula for Haar-random bipartite pure states. Around that core, the literature develops three distinct directions: exact mutual-information reorganizations, free-probabilistic Page-curve generalizations, and non-planar corrections in gravitational matrix models (Wang et al., 28 May 2026, Wang, 2022, Iizuka et al., 2024).