Weak Typicality of von Neumann Entanglement Entropy in Gaussian Boson Sampling

• URL: http://arxiv.org/abs/2608.17274v1
• Date: Tue, 18 Aug 2026 02:04:28 GMT
• Title: Weak Typicality of von Neumann Entanglement Entropy in Gaussian Boson Sampling
• Abstract summary: 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 $s$.<n>For subsystem of $k_n$ modes satisfying $k_n/nto rin(0,1)$, we prove that, for every $varepsilon>0$ and all sufficiently large $n$, $mathbbPleft(left|fracS_1,nmathbbES_1,n$ to $1$, a typical
• Score: 2.8224889996383395
• License: http://creativecommons.org/licenses/by/4.0/
• 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 $\varepsilon>0$ and all sufficiently large $n$, $\mathbb{P}\left(\left|\frac{S_{1,n}}{\mathbb{E}S_{1,n}}-1\right|\geq\varepsilon\right)\leq2\exp\left[-\frac{c_{s,r}\varepsilon^2n^2}{\log^2(en)}\right].$ 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 $\mathrm{Var}(S_{1,n})=O_s(\log^2 n)$. An accompanying Lean 4 development verifies the proof chain.

Related papers

• Log-Sobolev inequality, von Neumann entropy and Entanglement of Formation [0.0]
  We present two results derived from the log-Sobolev inequality for the uniform measure on a complete graph.<n>The results concern the von Neumann entropy and the Entanglement of Formation of a state of finite and infinite-dimensional quantum systems.
  arXiv  Detail & Related papers  (2026-09-11T10:12:16Z)
• Exact Moments of Gaussian Gram Hafnians Reveal an $n^2/\log n$ Threshold for Weak Anticoncentration [2.8224889996383395]
  Anticoncentration is central to hardness arguments for approximate sampling.<n> finite dimensional Haar moment transfer and high probability small ball anticoncentration remain separate problems.
  arXiv  Detail & Related papers  (2026-08-17T19:12:22Z)
• Rényi exponent landscape of multipartite entanglement in free-fermion systems [51.56484100374058]
  We show that the Rényi tripartite information $I_3() exhibits a qualitatively $exclusion-dependent scaling at small Fermi momentum.<n>A direct consequence is a replica obstruction: $I_m(n)/I_m(1) sim zm-1 to 0$ for all integer $n geq 2$, so the leading von Neumann signal cannot be reconstructed from integer Rényi data.
  arXiv  Detail & Related papers  (2026-03-09T22:27:00Z)
• Dimension-free Private Mean Estimation for Anisotropic Distributions [55.86374912608193]
  Previous private estimators on distributions over $mathRd suffer from a curse of dimensionality. We present an algorithm whose sample complexity has improved dependence on dimension.
  arXiv  Detail & Related papers  (2024-11-01T17:59:53Z)
• Entanglement entropy in type II$_1$ von Neumann algebra: examples in Double-Scaled SYK [6.990954253986022]
  In this paper, we study the entanglement entropy $S_n$ of the fixed length state $|nrangle$ in Double-Scaled Sachdev-Ye-Kitaev model.
  arXiv  Detail & Related papers  (2024-04-03T04:27:07Z)
• A Unified Framework for Uniform Signal Recovery in Nonlinear Generative Compressed Sensing [68.80803866919123]
  Under nonlinear measurements, most prior results are non-uniform, i.e., they hold with high probability for a fixed $mathbfx*$ rather than for all $mathbfx*$ simultaneously. Our framework accommodates GCS with 1-bit/uniformly quantized observations and single index models as canonical examples. We also develop a concentration inequality that produces tighter bounds for product processes whose index sets have low metric entropy.
  arXiv  Detail & Related papers  (2023-09-25T17:54:19Z)
• Learning a Single Neuron with Adversarial Label Noise via Gradient Descent [50.659479930171585]
  We study a function of the form $mathbfxmapstosigma(mathbfwcdotmathbfx)$ for monotone activations. The goal of the learner is to output a hypothesis vector $mathbfw$ that $F(mathbbw)=C, epsilon$ with high probability.
  arXiv  Detail & Related papers  (2022-06-17T17:55:43Z)
• Low-degree learning and the metric entropy of polynomials [44.99833362998488]
  We prove that any (deterministic or randomized) algorithm which learns $mathscrF_nd$ with $L$-accuracy $varepsilon$ requires at least $Omega(sqrtvarepsilon)2dlog n leq log mathsfM(mathscrF_n,d,|cdot|_L,varepsilon) satisfies the two-sided estimate $$c (1-varepsilon)2dlog
  arXiv  Detail & Related papers  (2022-03-17T23:52:08Z)
• Random quantum circuits transform local noise into global white noise [118.18170052022323]
  We study the distribution over measurement outcomes of noisy random quantum circuits in the low-fidelity regime. For local noise that is sufficiently weak and unital, correlations (measured by the linear cross-entropy benchmark) between the output distribution $p_textnoisy$ of a generic noisy circuit instance shrink exponentially. If the noise is incoherent, the output distribution approaches the uniform distribution $p_textunif$ at precisely the same rate.
  arXiv  Detail & Related papers  (2021-11-29T19:26:28Z)
• Sublinear quantum algorithms for estimating von Neumann entropy [18.30551855632791]
  We study the problem of obtaining estimates to within a multiplicative factor $gamma>1$ of the Shannon entropy of probability distributions and the von Neumann entropy of mixed quantum states. We work with the quantum purified query access model, which can handle both classical probability distributions and mixed quantum states, and is the most general input model considered in the literature.
  arXiv  Detail & Related papers  (2021-11-22T12:00:45Z)
• Random matrices in service of ML footprint: ternary random features with no performance loss [55.30329197651178]
  We show that the eigenspectrum of $bf K$ is independent of the distribution of the i.i.d. entries of $bf w$. We propose a novel random technique, called Ternary Random Feature (TRF) The computation of the proposed random features requires no multiplication and a factor of $b$ less bits for storage compared to classical random features.
  arXiv  Detail & Related papers  (2021-10-05T09:33:49Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.