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Almost surely convergence of the quantum entropy of random graph states and the area law

Zhi Yin, Liang Zhao

TL;DR

The paper proves that the von Neumann entropy of the reduced state of random graph states satisfies an area law almost surely as the local dimension $N\to\infty$. By analyzing fluctuations of the empirical eigenvalue distribution through Weingarten calculus and a flow-network–based minimization of combinatorial functionals $F_{n,n}$ and $F_n$, the authors show the empirical measure $\mu_N$ converges in moments to a compactly supported measure $\mu$ and that $H(\rho_{\mathcal{S}})=|\partial\mathcal{S}|\log N-\int t\log t\, d\mu + o(1)$ almost surely. The maximum-flow value in the associated network equals the boundary size, i.e., $|\partial\mathcal{S}|$, which links geometry directly to spectral properties. The approach yields explicit examples on a chain and a lattice, where $H(\rho_{\mathcal{S}})=3\log N-\tfrac12+o(1)$ and $6\log N+o(1)$, respectively, and connects the spectral limit to a free Poisson distribution. These results extend prior average-area-law findings to almost-sure statements, with potential applicability to non-Hermitian ensembles via Wick-type analyses.

Abstract

In [1], Collins et al. showed that the quantum entropy of random graph states satisfies the so-called area law as the local dimension tends to be large. In this paper, we continue to study the fluctuation of the convergence and thus prove the area law holds almost surely.

Almost surely convergence of the quantum entropy of random graph states and the area law

TL;DR

The paper proves that the von Neumann entropy of the reduced state of random graph states satisfies an area law almost surely as the local dimension . By analyzing fluctuations of the empirical eigenvalue distribution through Weingarten calculus and a flow-network–based minimization of combinatorial functionals and , the authors show the empirical measure converges in moments to a compactly supported measure and that almost surely. The maximum-flow value in the associated network equals the boundary size, i.e., , which links geometry directly to spectral properties. The approach yields explicit examples on a chain and a lattice, where and , respectively, and connects the spectral limit to a free Poisson distribution. These results extend prior average-area-law findings to almost-sure statements, with potential applicability to non-Hermitian ensembles via Wick-type analyses.

Abstract

In [1], Collins et al. showed that the quantum entropy of random graph states satisfies the so-called area law as the local dimension tends to be large. In this paper, we continue to study the fluctuation of the convergence and thus prove the area law holds almost surely.
Paper Structure (13 sections, 11 theorems, 107 equations, 6 figures)

This paper contains 13 sections, 11 theorems, 107 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Non-crossing partitions and permutations
  4. Weingarten's formula
  5. Random graph states
  6. Flow network and maximum flow problem
  7. Main results
  8. Minimization problem of $F_{n,n} (\alpha, \beta)$ and $F_{n} ({\tilde{\alpha}}, {\tilde{\beta}})$
  9. Proof of Theorem \ref{['thm:almost-sure']}
  10. Examples
  11. One dimensional chain
  12. Two-dimensional lattice
  13. Conclusion

Key Result

Theorem 2.1

Let $U=(u_{i,j})_{i,j=1}^N \in \mathbb{U}_N(\mathbb{C})$, we have for all $N \geq n$ that where we have used that And $\mathop{\mathrm{Wg}}\nolimits(\cdot, \cdot)$ is called the Weingarten function.

Figures (6)

  • Figure 1: One dimensional chain.
  • Figure 2: Chain Case (Fattened Graph).
  • Figure 3: Chain Case (Flow Network).
  • Figure 4: Two-dimensional lattice.
  • Figure 5: Lattice Case (Fattened graph).
  • ...and 1 more figures

Theorems & Definitions (30)

  • Definition 2.1: Non-crossing partition NS2006
  • Definition 2.2: Non-crossing permutation NS2006
  • Definition 2.3: MS17
  • Theorem 2.1: Weingarten's formula NS2006
  • Remark 2.1
  • Definition 2.4: BIK3
  • Definition 2.5: Random graph state BIK0
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • ...and 20 more