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arxiv: 2605.18986 · v1 · pith:WE7W33OQnew · submitted 2026-05-18 · ❄️ cond-mat.stat-mech · quant-ph

Non-Gaussianity of random quantum states

Filiberto Ares , Sara Murciano , Pasquale Calabrese This is my paper

Pith reviewed 2026-05-20 08:02 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech quant-ph
keywords non-Gaussianityrandom quantum statesHaar measureU(1) symmetryreduced density matrixWeingarten calculusfermionic systemstypicality
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The pith

In Haar-random quantum states, fermionic non-Gaussianity vanishes for subsystems smaller than half the total size without symmetries but stays small and finite with U(1) symmetry, becoming extensive for larger subsystems.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines fermionic non-Gaussianity in typical quantum states using Haar-random qubit states, both with and without a global U(1) symmetry. Non-Gaussianity is measured via the relative entropy between a subsystem's reduced density matrix and a Gaussian state that matches its two-point correlations. Analytical results from Weingarten calculus identify two regimes set by the subsystem-to-system size ratio ℓ/L. Without symmetries, the measure drops to zero for ℓ/L below 1/2 because reduced states approach the maximally mixed state exponentially closely, while a U(1) symmetry keeps a small finite value; above ℓ/L = 1/2 the non-Gaussianity grows linearly with system size.

Core claim

The central claim is that the non-Gaussianity of reduced density matrices in Haar-random states is controlled by the ratio ℓ/L: it vanishes exponentially for ℓ/L < 1/2 in the absence of symmetries because typical reduced density matrices are exponentially close to the maximally mixed state, remains small but finite in the presence of a global U(1) symmetry, and becomes extensive for ℓ/L > 1/2.

What carries the argument

The relative entropy between the reduced density matrix and its Gaussianized counterpart, averaged over Haar-random states via Weingarten calculus.

If this is right

  • For ℓ/L < 1/2 without symmetries the non-Gaussianity vanishes exponentially.
  • A global U(1) symmetry leaves a small but nonzero non-Gaussianity in the same regime.
  • For ℓ/L > 1/2 the non-Gaussianity scales extensively with total system size.
  • The results fix the typical scaling of fermionic non-Gaussianity in random states under symmetry constraints.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Symmetry constraints appear to protect a residual non-Gaussianity even in otherwise typical small-subsystem states.
  • The sharp transition at ℓ/L = 1/2 may link to other subsystem-size thresholds in quantum information and many-body typicality.
  • Sampling random circuits on quantum hardware could directly test the predicted regime change by measuring the same relative entropy.

Load-bearing premise

That Haar-random states represent typical quantum states and that relative entropy to a Gaussianized counterpart is the right measure of fermionic non-Gaussianity.

What would settle it

Exact numerical evaluation of the relative entropy for a system of size L=20 at ℓ/L=0.4 without symmetry (expecting exponentially small value) and at ℓ/L=0.6 (expecting linear growth in L).

Figures

Figures reproduced from arXiv: 2605.18986 by Filiberto Ares, Pasquale Calabrese, Sara Murciano.

Figure 1
Figure 1. Figure 1: Average entanglement entropy of the Gaussianized re [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Average non-Gaussianity for Haar random states of a [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: Average entanglement entropy of the Gaussianized re [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: Average non-Gaussianity (33) for U(1) symmetric Haar￾random states at different fillings ν in the thermodynamic limit L → ∞, with ℓ/L finite. For ℓ < L/2, the non-Gaussianity is not extensive in L and, unlike in the absence of any symmetry, is a small but finite function of ℓ/L that is independent of ν. The inset shows the dependence on ν for ℓ/L = 3/4 (dashed curve) and ℓ = L (solid curve). complexity req… view at source ↗
read the original abstract

We study the fermionic non-Gaussianity in typical quantum states, focusing on Haar random states of qubits with or without a global $U(1)$ symmetry. Using the Weingarten calculus, we derive analytical predictions for the non-Gaussianity, defined as the relative entropy between the reduced density matrix and its Gaussianized counterpart. We identify two regimes controlled by the ratio between the subsystem and the system size, $\ell/L$. For $\ell/L < 1/2$, the non-Gaussianity vanishes in the absence of symmetries, because typical reduced density matrices are exponentially close to the maximally mixed state. In the presence of a global $U(1)$ symmetry, instead, it remains small but finite. By contrast, in the regime $\ell/L > 1/2$, the non-Gaussianity becomes extensive. These results establish the typical scaling of fermionic non-Gaussianity in random states and analyze how this is modified by the presence of global symmetries.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper studies fermionic non-Gaussianity in typical Haar-random pure states of L qubits (with and without global U(1) symmetry), defining it as the relative entropy between the reduced density matrix on ℓ sites and its Gaussianized counterpart obtained by matching two-point functions. Using Weingarten calculus, it derives analytical predictions showing two regimes controlled by ℓ/L: for ℓ/L < 1/2 the non-Gaussianity vanishes exponentially without symmetry (as reduced states approach the maximally mixed state) but remains small yet finite with symmetry; for ℓ/L > 1/2 it becomes extensive.

Significance. If the central derivations hold and the relative entropy remains well-defined, the results establish exact typical scalings of non-Gaussianity in random states and clarify the effects of global symmetries. This provides useful benchmarks for many-body quantum systems, entanglement studies, and numerical simulations in condensed-matter contexts, with the Weingarten-calculus approach offering parameter-free analytical control.

major comments (2)
  1. [regime ℓ/L > 1/2 derivation] § on the ℓ/L > 1/2 regime (central claim of extensivity): when ℓ > L/2 the reduced state ρ satisfies rank(ρ) ≤ 2^{L-ℓ} < 2^ℓ. The Gaussianized state ρ_G is constructed from the covariance matrix; if supp(ρ_G) does not contain supp(ρ), the relative entropy S(ρ || ρ_G) diverges. The manuscript must supply an explicit argument (or regularization) showing the supports are compatible or that the quantity remains finite, as this is load-bearing for the extensivity result.
  2. [Weingarten calculus application] Weingarten-calculus derivations (abstract and main results sections): the reported regimes are stated to follow from Haar averages, yet no explicit intermediate steps, error bounds, or numerical cross-checks for the relative-entropy expressions are visible. This reduces verifiability of the analytic predictions, particularly the transition at ℓ/L = 1/2.
minor comments (2)
  1. [methods/notation] Clarify notation for the Gaussianized state and covariance matching in the methods section; add a reference to standard Weingarten-calculus literature for the specific integrals used.
  2. [figures] Figure captions could explicitly state the system sizes and averaging procedure used in any numerical checks.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address the major comments point by point below and will revise the manuscript to incorporate clarifications and additional details as outlined.

read point-by-point responses

  1. Referee: [regime ℓ/L > 1/2 derivation] § on the ℓ/L > 1/2 regime (central claim of extensivity): when ℓ > L/2 the reduced state ρ satisfies rank(ρ) ≤ 2^{L-ℓ} < 2^ℓ. The Gaussianized state ρ_G is constructed from the covariance matrix; if supp(ρ_G) does not contain supp(ρ), the relative entropy S(ρ || ρ_G) diverges. The manuscript must supply an explicit argument (or regularization) showing the supports are compatible or that the quantity remains finite, as this is load-bearing for the extensivity result.

    Authors: We thank the referee for identifying this technical subtlety in the ℓ/L > 1/2 regime. The reduced density matrix ρ indeed has rank at most 2^{L-ℓ}. The Gaussian state ρ_G is obtained by matching the two-point functions via the covariance matrix. For typical Haar-random states, the averaged correlations ensure that the support of ρ lies within the support of ρ_G, keeping the relative entropy finite. In the revised manuscript we will add an explicit argument establishing this compatibility using the properties of the Weingarten-averaged covariance matrix. As a safeguard we will also introduce a regularization (adding a small εI term to the covariance matrix and taking ε → 0 after the thermodynamic limit), which preserves the extensivity result. This clarification will be inserted in the relevant section. revision: yes

  2. Referee: [Weingarten calculus application] Weingarten-calculus derivations (abstract and main results sections): the reported regimes are stated to follow from Haar averages, yet no explicit intermediate steps, error bounds, or numerical cross-checks for the relative-entropy expressions are visible. This reduces verifiability of the analytic predictions, particularly the transition at ℓ/L = 1/2.

    Authors: We agree that greater transparency in the derivations would strengthen the paper. In the revised version we will expand the Weingarten-calculus sections to include the key intermediate steps for computing the Haar averages of the relative entropy. We will also derive explicit error bounds for the large-L approximations and add numerical benchmarks on small systems (L ≤ 12) that confirm the analytic expressions and the sharp transition at ℓ/L = 1/2. These additions will be placed in an extended methods subsection and an appendix. revision: yes

Circularity Check

0 steps flagged

No circularity: derivations use external Weingarten calculus on Haar measure

full rationale

The paper derives analytical expressions for fermionic non-Gaussianity (relative entropy to Gaussianized reduced density matrix) by applying Weingarten calculus to compute Haar averages over random states, both with and without U(1) symmetry. This is an independent mathematical tool for unitary integrals, not a fit to the target non-Gaussianity quantities, not a self-definition, and not reliant on load-bearing self-citations. The two regimes (ℓ/L < 1/2 vs > 1/2) and the vanishing vs extensive behavior follow directly from the averaged expressions without reducing to the input data or prior author results by construction. The rank-support concern for ℓ/L > 1/2 is an applicability question outside the circularity analysis.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard assumptions in quantum information about random states and the mathematical framework for averaging over them.

axioms (2)
  • domain assumption Quantum states are distributed according to the Haar measure on the unitary group
    This is the definition of typical or random states used throughout the abstract.
  • standard math Weingarten calculus yields exact expressions for the relevant averages over Haar-random unitaries
    Invoked to obtain the analytical predictions for non-Gaussianity in both symmetric and non-symmetric cases.

pith-pipeline@v0.9.0 · 5699 in / 1447 out tokens · 84448 ms · 2026-05-20T08:02:23.484792+00:00 · methodology

discussion (0)

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Lean theorems connected to this paper

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  • IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    We study the fermionic non-Gaussianity in typical quantum states, focusing on Haar random states of qubits with or without a global U(1) symmetry. Using the Weingarten calculus, we derive analytical predictions for the non-Gaussianity, defined as the relative entropy between the reduced density matrix and its Gaussianized counterpart.

  • IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    For ℓ/L < 1/2, the non-Gaussianity vanishes in the absence of symmetries, because typical reduced density matrices are exponentially close to the maximally mixed state. … in the regime ℓ/L > 1/2, the non-Gaussianity becomes extensive.

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matches
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supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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