Weak Typicality of von Neumann Entanglement Entropy in Gaussian Boson Sampling

Abstract: We study the von Neumann entanglement entropy generated by a Haar distributed passive interferometer acting on $n$ equally squeezed input modes with fixed nonzero squeezing strength $s$. Previous work established proportional weak typicality for integer R'enyi orders $α\geq 2$ and stated a sublinear von Neumann result, while the proportional von Neumann case remained open. For a subsystem of $k_n$ modes satisfying $k_n/n\to r\in(0,1)$, we prove that, for every $\varepsilon&gt;0$ and all sufficiently large $n$, $$\mathbb{P}\left(\left|\frac{S_{1,n}}{\mathbb{E}S_{1,n}}-1\right|\geq\varepsilon\right)\leq2\exp\left[-\frac{c_{s,r}\varepsilon<sup>2n<sup>2}{\log<sup>2(en)}\right].$$ The proof represents the entropy as a singular value statistic of a principal block of $UU^{\mathsf T}$, where $U$ denotes the unitary interferometer. It regularizes the logarithmic singularity at the endpoint corresponding to a pure Gaussian mode and applies concentration on the unitary group. The result establishes proportional von Neumann weak typicality and further implies almost sure convergence of $S_{1,n}/\mathbb{E}S_{1,n}$ to $1$, a typical volume law, and the variance bound $\mathrm{Var}(S_{1,n})=O_s(\log² n)$. An accompanying Lean 4 development verifies the proof chain.

• The paper proves proportional weak typicality for von Neumann entanglement entropy, with relative deviations bounded by 2 exp[-c ε²n²/log²(en)] for Haar-random interferometers and proportional subsystems.
• The proof combines an exact singular-value representation, endpoint regularization, an extensive mean lower bound, and Haar concentration to establish a typical volume law, Page-curve convergence, and Var(S₁,ₙ) = Oₛ(log²(en)).
• The paper shows strong typicality fails for every proportional cut except that the balanced case remains unresolved, while unequal squeezing, optical loss, and nonintegrable endpoint singularities remain important open challenges.

Setting and problem

Gaussian boson sampling (GBS) prepares squeezed vacuum inputs, mixes them through a passive linear optical interferometer, and measures photon counts. Before measurement, the output is a pure bosonic Gaussian state whose entanglement across a bipartition of the modes is the von Neumann entropy $S_{1,n}$ of either reduced state. The paper studies this entropy for a Haar distributed interferometer $U \in U(n)$ acting on $n$ equally squeezed inputs with fixed nonzero squeezing strength $s$, and a subsystem of $k_n$ modes with $k_n/n \to r \in (0,1)$.

Prior work left a specific gap. Iosue et al. proved proportional weak typicality for Rényi order 2 and stated von Neumann weak typicality only for sublinear subsystems; Youm et al. extended proportional weak typicality to every integer Rényi order $\alpha \ge 2$ and derived the asymptotically exact mean Page curve, but explicitly left proportional von Neumann typicality open. The obstruction is analytic: the one-mode entropy profile has a derivative that diverges logarithmically as a reduced symplectic eigenvalue approaches its pure value $\nu = 1$. The paper closes this gap with a quantitative relative tail bound and an accompanying Lean 4 formalization.

Main result

The central theorem establishes weak typicality in the proportional regime. For fixed $s \neq 0$ and any sequence $k_n/n \to r \in (0,1)$, there exists $U \in U(n)$0 such that for all sufficiently large $U \in U(n)$1,

$U \in U(n)$2

The right-hand side is summable in $U \in U(n)$3, so by Borel–Cantelli the convergence holds almost surely for every coupling whose size-$U \in U(n)$4 marginal is Haar on $U \in U(n)$5; no independence across dimensions is required. Three corollaries follow directly:

• Typical volume law: for any $U \in U(n)$6, the probability that $U \in U(n)$7 tends to one, so $U \in U(n)$8. Purity sharpens the deterministic upper bound to $U \in U(n)$9.
• Typical Page curve value: combined with the published mean limit $n$0 of Youm et al., the entropy density converges in probability (and almost surely) to the Page curve density at each fixed ratio $n$1. This is pointwise in $n$2; it does not assert simultaneous concentration over all ratios or mode subsets.
• Variance bound: $n$3, with deviations below every polynomial scale almost surely, i.e. $n$4.

A notable negative companion result: strong typicality ($n$5 in probability) fails at every proportional ratio $n$6, proved via the Jacobi linear statistics CLT applied to the squared singular values, which form a real $n$7-Jacobi ensemble with nondegenerate limiting Gaussian fluctuations.

Proof structure

The argument proceeds in four steps.

Exact spectral reduction. For equal squeezing, the reduced covariance matrix is $n$8, where $n$9, $s$0, and $s$1 is built from the leading $s$2 principal block $s$3 of $s$4, which is complex symmetric and COE distributed. Anticommutation with the symplectic form yields the exact finite-$s$5 identity

$s$6

where $s$7 are the singular values of $s$8 and $s$9 is the one-mode entropy function. This reduces the entropy to a singular value statistic of a principal COE block.

Endpoint regularization. The function $k_n$0 is continuous but its derivative diverges logarithmically at $k_n$1: $k_n$2. Truncating within $k_n$3 of the endpoint defines $k_n$4 with Lipschitz constant $k_n$5 while the uniform approximation error satisfies $k_n$6. The cost–benefit ratio is favorable because the cutoff error scales as $k_n$7 while the Lipschitz penalty grows only as $k_n$8; choosing $k_n$9 makes the former vanish without degrading concentration. A direct variance calculation is thereby avoided entirely.

Extensive mean lower bound. Concentration must be compared against the mean scale. Using $k_n/n \to r \in (0,1)$0, a chord bound on the logarithm, and an elementary self-contained derivation of the COE second moment $k_n/n \to r \in (0,1)$1 — obtained by congruence symmetries rather than Weingarten calculus — the paper proves

$k_n/n \to r \in (0,1)$2

which is $k_n/n \to r \in (0,1)$3 in the proportional regime.

Haar concentration. Applying the Meckes–Meckes log-Sobolev concentration inequality on $k_n/n \to r \in (0,1)$4 to the Lipschitz statistic $k_n/n \to r \in (0,1)$5, and transferring back through the uniform approximation, yields the main tail. The variance bound follows by integrating the same tail.

Formal verification

A Lean 4 development (pinned toolchain and Mathlib commit, archived on Zenodo) kernel-checks the core proof chain. Its scope is explicitly bounded: full unitary Haar concentration, Mirsky's singular value inequality, and the Gaussian covariance/normal-form results are represented as explicit theorem assumptions rather than re-proved; the published hypergeometric Page curve computation is likewise assumed for the transfer corollary. The formalization transcribes but does not construct Fock space, trace-class operators, or partial traces, identifying them with the abstract spectral model only through the stated Gaussian interface. The strong-typicality failure result lies outside the formalization scope.

Limitations and open questions

The paper states its boundaries plainly. Strong typicality at the balanced cut $k_n/n \to r \in (0,1)$6 remains undetermined: there the limiting Jacobi support reaches the logarithmically singular endpoint, so the $k_n/n \to r \in (0,1)$7 CLT used elsewhere does not apply, though weak typicality still holds. Equal squeezing is essential to the reduction to a single principal COE block; with unequal squeezing profiles the reduction is lost, and extending weak typicality to that experimentally relevant case is identified as the natural next problem. The result concerns the ideal lossless premeasurement state — with loss the global state is mixed and $k_n/n \to r \in (0,1)$8 no longer isolates entanglement — and it makes no claim about sampling hardness or anticoncentration of output probabilities. The method's reliance on integrable endpoint singularities also means it would not transfer unchanged to observables with power-law singularities, where cutoff error and Lipschitz cost compete at the extensive scale.

Conclusion

The paper proves proportional weak typicality of the von Neumann entanglement entropy for Haar random passive interferometers acting on equally squeezed vacuum inputs, closing the cell left open in prior work. The proof combines an exact singular-value representation, a vanishing logarithmic endpoint cutoff, a self-contained extensive mean lower bound, and Haar concentration, yielding a summable relative tail, a typical volume law, logarithmic tightness, and $k_n/n \to r \in (0,1)$9. Together with the known mean Page curve, it promotes the average Page curve value at each fixed proportional bipartition to a statement about typical samples, while proving that additive (strong) typicality fails away from the balanced cut.