---
title: Lubkin-Page Bounds in Quantum Systems
url: https://www.emergentmind.com/topics/lubkin-page-bounds
type: topic
---

# Lubkin-Page Bounds in Quantum Systems

Searching arXiv for relevant papers on Lubkin-Page bounds and related generalizations.
Lubkin-Page bounds are typicality statements for random pure states in composite quantum systems. In the standard finite-dimensional setting, they show that a subsystem of a Haar-random pure state is almost maximally mixed when its complement is much larger, and that in a tripartite decomposition \(A\otimes B\otimes E\) the mutual information between \(A\) and \(B\) is parametrically suppressed when \(E\) is large. Recent work extends this paradigm from Type I von Neumann algebras to all Type II von Neumann factors, thereby addressing the fact that the observable algebras of quantum field theory and quantum gravity are generically Type II or Type III rather than finite-dimensional tensor products [2607.01873].

## 1. Finite-dimensional Lubkin-Page theorem

In the familiar Type I setting, one considers a pure state drawn uniformly with respect to Haar measure from a bipartite Hilbert space
\[
\mathcal H=\mathcal H_A\otimes\mathcal H_B,
\qquad
\dim\mathcal H_A=d_A,\ \dim\mathcal H_B=d_B.
\]
For \(d_B\ge d_A\), Page’s formula gives the average entanglement entropy of subsystem \(A\) as
\[
\bigl\langle S(A)\bigr\rangle
= \sum_{k=d_B+1}^{d_A d_B}\frac1k-\frac{d_A-1}{2d_B}
= \ln d_A-\frac{d_A}{2d_B}+O\!\bigl((d_A/d_B)^2\bigr).
\]
Equivalently, the average purity is
\[
\bigl\langle \Tr\,\rho_A^2\bigr\rangle
=\frac{d_A+d_B}{d_A d_B+1}
=\frac1{d_A}+\frac1{d_B}+O\!\bigl((d_A/d_B)^2\bigr),
\]
so the reduced state is close to maximally mixed when \(d_B\gg d_A\) [2607.01873].

The tripartite form of the result is central for typicality arguments. If
\[
\mathcal H=\mathcal H_A\otimes\mathcal H_B\otimes\mathcal H_E
\]
with \(d_E\gg d_A d_B\), then the mutual information
\[
I(A:B)=S(A)+S(B)-S(AB)
\]
obeys
\[
\bigl\langle I(A:B)\bigr\rangle
=O\!\Bigl(\frac{(d_A d_B)^2}{d_E^2}\Bigr).
\]
In particular, when the environment \(E\) is large, generic pure states have almost no correlations between \(A\) and \(B\) [2607.01873].

A complementary exact formulation writes the mean entropy for an \(m\)-dimensional subsystem in a total Hilbert space of dimension \(mn\) as
\[
\overline S(m,n)=H_{mn}-H_n-\frac{m-1}{2n},
\]
where \(H_r=\sum_{j=1}^r 1/j\) is the \(r\)-th harmonic number. In the unbalanced regime \(n\gg m\), one has \(\overline S(m,n)=\ln m+O(1/n^2)\), while in the balanced case \(m=n\),
\[
\overline S(n,n)\sim \ln n-\frac12+O\!\bigl(\tfrac1{n^2}\bigr),
\]
so the average entropy lies below the maximum by a half-nat [2606.30941].

## 2. Proof techniques and moment formulas

One standard derivation expands a Haar-random vector as
\[
\ket\Psi=\sum_{i=1}^{d_A}\sum_{\alpha=1}^{d_B}
U_{i\alpha}\,\ket i_A\otimes\ket\alpha_B,
\]
with \(U\in U(d_A d_B)\) uniformly distributed, and then computes moments of the reduced density matrix from the second and fourth Haar moments of the matrix elements. This yields
\[
\bigl\langle \Tr\,\rho_A^2\bigr\rangle=\frac{d_A+d_B}{d_A d_B+1},
\]
from which the entropy can be reconstructed through replica methods or related integral identities [1505.05870].

A more recent derivation replaces the usual random-matrix-theoretic route by Schur-Weyl duality and the character theory of the symmetric group \(S_k\). For a random pure state in \(\mathbb C^{mn}\),
\[
E\bigl[\tr\rho_A^k\bigr]
=\frac{1}{(mn)_k}\sum_{\mu\vdash k}
\chi_\mu((k))\,\frac{C_\mu(n)\,C_\mu(m)}{H_\mu},
\]
where \((x)_k\) is the rising Pochhammer, \(H_\mu\) is the product of hook lengths, \(C_\mu(n)\) is the content polynomial, and \(\chi_\mu((k))\) is the character of a \(k\)-cycle in the \(S_k\)-irrep \(\mu\). Differentiation at \(k=1\) reproduces the exact Page formula [2606.30941].

These derivations make precise what is meant by “typical.” In the finite-dimensional theorem, typicality is defined by Haar measure on the full Hilbert-space sphere. The resulting concentration around near-maximal entropy is the mechanism behind the suppression of mutual information in the tripartite regime [2607.01873].

## 3. Type II von Neumann factors

The recent extension to Type II von Neumann factors begins by replacing ordinary dimension counting with traces, finite-dimensional approximants, and index theory. A Type II\(_1\) factor \(R\) is an infinite-dimensional von Neumann algebra with trivial center and a unique faithful normal trace
\[
\tau:R\to\mathbb C,\qquad \tau(1)=1,\qquad \tau(xy)=\tau(yx).
\]
The hyperfinite II\(_1\) factor is built by an approximately finite-dimensional chain
\[
M_{d_1}\subset M_{d_2}\subset\cdots\subset R,
\qquad
\tau|_{M_{d_n}}=\frac1{d_n}\Tr_{d_n}.
\]
For a subfactor \(A\subset R\), relative size is measured by the Jones index \([R\!:\!A]\); in the tensor-product case \(R\cong A\otimes B\), one has
\[
[R\!:\!A]=[R\!:\!B]^{-1}=(\dim B)^2,
\]
mirroring the finite-dimensional relation \(M_d\cong M_{d_A}\otimes M_{d_B}\) with index \(d_B^2\) [2607.01873].

A Type II\(_\infty\) factor has the form
\[
M=R\otimes \mathcal B(\mathcal H),
\]
where \(R\) is II\(_1\) and \(\mathcal B(\mathcal H)\) is the algebra of bounded operators on a separable infinite-dimensional \(\mathcal H\). It carries a semifinite trace \(\tau=\tau_R\otimes\Tr\) with \(\tau(1)=\infty\). In both the II\(_1\) and II\(_\infty\) settings, one recovers effective dimensions by choosing increasing sequences of finite-dimensional subalgebras. If
\[
A_n\cong M_{d_n^{(A)}},\qquad
B_n\cong M_{d_n^{(B)}},\qquad
E_n\cong M_{d_n^{(E)}},
\]
then
\[
M_n=A_n\otimes B_n\otimes E_n\cong M_{d_n},
\qquad
d_n=d_n^{(A)}d_n^{(B)}d_n^{(E)},
\]
and the conditional expectation onto \(M_n\) preserves the trace and induces finite-dimensional approximants of states in \(R\) [2607.01873].

This framework is the algebraic substitute for the tensor-product Hilbert-space assumptions used in the original Lubkin-Page theorem. The generalization is therefore not a cosmetic extension: it addresses precisely the mismatch between finite-dimensional typicality arguments and the operator-algebraic structures that arise in QFT and quantum gravity [2607.01873].

## 4. The Type II\(_1\) and Type II\(_\infty\) bounds

For the hyperfinite Type II\(_1\) factor, let
\[
R\cong A\otimes B\otimes E
\]
with a compatible AFD tower
\[
M_n=A_n\otimes B_n\otimes E_n\subset R,
\]
and assume for large \(n\) that
\[
d_n^{(A)}\,d_n^{(B)}\le d_n^{(E)}.
\]
If one draws a Haar-random unit vector in the GNS space \(L^2(M_n,\tau)\cong \mathbb C^{d_n^2}\), traces out \(E_n\), and denotes the resulting state on \(A_n\otimes B_n\) by \(\varphi_n\), then
\[
\bigl\langle I(A_n\!:\!B_n)_{\varphi_n}\bigr\rangle
\le
\frac{(d_n^{(A)})^2(d_n^{(B)})^2}{2\,\ln2\,(d_n^{(E)})^2}.
\]
Hence, as \(d_n^{(E)}/(d_n^{(A)}d_n^{(B)})\to\infty\), the bound tends to zero, and any weak\(^*\) limit point \(\varphi\) satisfies \(I(A:B)_\varphi=0\) [2607.01873].

The proof uses three ingredients. First, at each finite level \(n\), the problem reduces to an honest finite-dimensional Haar-random state on
\[
\mathbb C^{(d_n^{(A)})^2}\otimes
\mathbb C^{(d_n^{(B)})^2}\otimes
\mathbb C^{(d_n^{(E)})^2},
\]
so the standard Lubkin-Page bound applies. Second, lower semicontinuity of the Araki relative entropy under weak\(^*\) limits transfers the estimate to the infinite-dimensional limit. Third, Markov’s inequality, or concentration of measure, shows that with high probability the mutual information is close to its mean [2607.01873].

For Type II\(_\infty\), one writes
\[
M=R\otimes\mathcal B(\mathcal H)
\cong
(A_0\otimes B_0\otimes E_0)\otimes
\bigl(\mathcal B(\mathcal H_A)\otimes\mathcal B(\mathcal H_B)\otimes\mathcal B(\mathcal H_E)\bigr),
\]
and truncates the Type I\(_\infty\) factors to dimensions \(d_A,d_B,d_E\). At level \((n,D)\),
\[
M_{n,D}
=\bigl(A_{0,n}\otimes M_{d_A}\bigr)\otimes
\bigl(B_{0,n}\otimes M_{d_B}\bigr)\otimes
\bigl(E_{0,n}\otimes M_{d_E}\bigr),
\]
with total GNS-dimension squared
\[
(d_n^{(A)})^2d_A^2\times (d_n^{(B)})^2d_B^2\times (d_n^{(E)})^2d_E^2.
\]
Assuming
\[
d_n^{(A)}d_A\times d_n^{(B)}d_B\le d_n^{(E)}d_E,
\]
the Haar-random state \(\varphi_{n,D}\) on \(\mathcal A_{n,D}\otimes\mathcal B_{n,D}\) obtained by tracing out \(\mathcal E_{n,D}\) satisfies
\[
\bigl\langle I(\mathcal A_{n,D}\!:\!\mathcal B_{n,D})\bigr\rangle
\le
\frac{(d_n^{(A)})^2d_A^2\times (d_n^{(B)})^2d_B^2}
{2\,\ln2\times (d_n^{(E)})^2d_E^2}.
\]
This is the Type II\(_\infty\) analogue of the finite-dimensional Lubkin-Page suppression [2607.01873].

## 5. Black holes, Page curves, and Hamiltonian refinements

In the gravitational crossed-product construction, the effective environment dimension satisfies
\[
d_E\sim e^{S_{\rm BH}},
\qquad
S_{\rm BH}=\frac{A}{4G\hbar},
\]
so the Type II\(_\infty\) bound becomes
\[
\bigl\langle I\bigr\rangle\lesssim
\frac{d_A^2 d_B^2}{2\ln2}\,e^{-2S_{\rm BH}}.
\]
For macroscopic horizons this is an exponentially tiny correlation. The construction uses the modular flow, or equivalently the horizon area operator, to extend a QFT Type III algebra to Type II\(_\infty\); the Bekenstein-Hawking bound cuts off the spectrum of the modular Hamiltonian and produces a finite effective dimension \(e^{S_{\rm BH}}\) [2607.01873].

Related black-hole applications appear in analyses of the Page curve. In a qubit-pair toy model of evaporation, the stepwise entropy change
\[
\Delta S(n)=S_{\rm rad}(n+1)-S_{\rm rad}(n)
\]
obeys
\[
1-4\epsilon_2^2-\sqrt{1-\gamma^2}
\le \frac{\Delta S}{\ln2}
\le \sqrt{1-4\epsilon_2^2},
\]
with
\[
\|\Lambda^{(2)}\|^2=\epsilon^2,\qquad
\langle\Lambda^{(1)}|\Lambda^{(2)}\rangle=\epsilon_2,\qquad
\gamma^2=1-4\bigl[\epsilon^2(1-\epsilon^2)-\epsilon_2^2\bigr].
\]
When \(\epsilon=\epsilon_2=0\), the bound collapses to \(\Delta S_{\rm Page}=\ln2\), recovering the standard Lubkin-Page result for a two-dimensional subsystem. To obtain any negative \(\Delta S\), one must satisfy
\[
\epsilon^2(1-\epsilon^2)+\epsilon_2^2>\tfrac34,
\]
so small corrections do not make the Page curve turn over [1908.09669].

A distinct refinement replaces Haar typicality on the full Hilbert space by canonical typicality in a microcanonical energy shell. For a nonzero Hamiltonian
\[
H=H_A+H_B+V_{AB},
\]
the typical reduced state of \(A\) is
\[
\rho_A\approx \frac{e^{-\beta H_A}}{Z_A(\beta)},
\qquad
Z_A(\beta)=\Tr_A e^{-\beta H_A},
\]
rather than the maximally mixed state. The corresponding entanglement entropy is approximately thermal entropy,
\[
S_A\approx S_{\rm therm}(A,\beta),
\]
which is in general strictly less than \(\ln d_A\). In the black-hole context, one paper argues that this nonmaximal entanglement changes the standard firewall inference that was based on the zero-Hamiltonian Page limit [1505.05870]. This suggests that Lubkin-Page typicality is sensitive to the ensemble and dynamical constraints used to define “typical.”

## 6. Type III obstructions and open questions

The extension from Type II to Type III remains open. Type III factors, as in algebraic QFT on Minkowski space, admit no trace, no density matrices, and no Haar measure on state space. One instead has the Araki relative entropy
\[
S(\varphi\|\omega)
=-\langle\Omega_\varphi,\log\Delta_{\omega,\varphi}\,\Omega_\varphi\rangle
\]
and the mutual information
\[
I(\varphi;\mathcal M_A,\mathcal M_B)
=
S\bigl(\varphi|_{\mathcal M_A\vee\mathcal M_B}\big\|
\varphi|_{\mathcal M_A}\otimes\varphi|_{\mathcal M_B}\bigr).
\]
Three obstacles are identified: there is no trace or maximally mixed state, there is no Haar measure, and there is no ordinary notion of dimension. The Jones index is replaced by the Kosaki-Longo index, but its relation to concentration of measure is not worked out [2607.01873].

Several open directions have been proposed. One is to use Tomita-Takesaki modular theory, including the Bisognano-Wichmann framework, to replace the partial-trace decomposition. Another is to invoke the eigenstate thermalization hypothesis for local algebras as a dynamical notion of typicality. A third is to identify the commutant in holographic settings, such as the black-hole interior, as a natural thermal bath; this may allow one to drop the explicit dimension constraint \(d_A d_B\le d_E\) and obtain tighter one-shot bounds [2607.01873].

A common misconception is that the finite-dimensional Lubkin-Page theorem automatically applies to local algebras in QFT. The Type II results show that a substantial generalization is possible, but they also clarify that Type III algebras require different tools and that the role of the ensemble, the trace, and the effective notion of dimension is structurally nontrivial. In that sense, the modern formulation of Lubkin-Page bounds is less a single theorem than a hierarchy of typicality statements whose validity depends on the operator-algebraic setting and on how randomness is defined.

Source: https://www.emergentmind.com/topics/lubkin-page-bounds