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arxiv: 2604.26878 · v1 · submitted 2026-04-29 · 🪐 quant-ph · cond-mat.stat-mech

A Gaussian asymmetry measure

Riccardo Travaglino , Pasquale Calabrese This is my paper

Pith reviewed 2026-05-07 10:43 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.stat-mech
keywords Gaussian asymmetryentanglement asymmetrysymmetric statesMpemba effectfree fermionscorrelation matricescharge fluctuationsquantum dynamics
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The pith

A Gaussian asymmetry measure equals the minimal distance to symmetric Gaussian states.

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

The paper introduces an asymmetry measure confined to Gaussian states in free-fermion systems. It establishes that this measure equals the shortest distance from the given state to any symmetric Gaussian state under a chosen charge operator. The Gaussian restriction removes non-Gaussian obstacles that block analytic progress in the standard entanglement asymmetry, yet the new measure still reproduces key dynamical behaviors including the Mpemba effect, symmetry restoration, and its absence. Because the construction stays Gaussian, its value and evolution follow directly from correlation matrices and the quasiparticle picture.

Core claim

The authors define a Gaussian asymmetry measure and show that it quantifies the minimal distance between a Gaussian state and the manifold of symmetric Gaussian states. They demonstrate that this measure captures the established dynamical signatures of entanglement asymmetry, such as the Mpemba effect, symmetry restoration, and the lack thereof. The Gaussian structure permits exact computation via correlation-matrix methods and asymptotic description through the quasiparticle picture, while also allowing charge fluctuations to serve as a characterisation tool.

What carries the argument

The Gaussian asymmetry measure, defined as the minimal distance from a given Gaussian state to the manifold of states symmetric under the charge operator.

Load-bearing premise

That restricting the asymmetry measure to the Gaussian manifold preserves the essential physical signatures of standard entanglement asymmetry without losing key dynamical information.

What would settle it

A concrete free-fermion chain in which the Gaussian measure fails to exhibit the Mpemba effect or symmetry restoration while the full non-Gaussian entanglement asymmetry does exhibit them.

Figures

Figures reproduced from arXiv: 2604.26878 by Pasquale Calabrese, Riccardo Travaglino.

Figure 1
Figure 1. Figure 1: Gaussian asymmetry in a quench from tilted ferromagnetic state for a subsystem of size ℓ with different values of tilting angle θ. The curves clearly show the occurrence of the quantum Mpemba effect. Symbols are numerical exact results at ℓA = 100 and lines are the quasiparticle picture prediction. The linear decrease up to t = ℓA/2 and the following saturation is a clear sign of the validity of the quasip… view at source ↗
Figure 2
Figure 2. Figure 2: Gaussian asymmetry as a probe for the lack of symmetry restoration in a quench from the tilted N´eel state with different tilting angles. The asymmetry is evaluated over a subsystem of size ℓ = 100. As in view at source ↗
Figure 3
Figure 3. Figure 3: ⟨∆S (G) A ⟩ g for typical Gaussian states. The symbols represents the numerical averages over large samples for L = 100 and L = 200. The behavior is smooth as a function of ℓA/L. The dashed lines represent the two asymptotic regimes, confirming the parabolic growth at small ℓA and the linear behavior for ℓA → L. In order to obtain ⟨S(ρ (s) ⟩), it is useful to understand how the second term in (5.3) arises … view at source ↗
Figure 4
Figure 4. Figure 4: Dynamics of the variance difference (7.10) after a quench from the tilted ferromagnetic state. The squares are the exact predictions obtained through correlation matrix techniques, cf. Eq. (7.11). Dashed lines are the quasiparticle predictions (7.10). The data show that the Mpemba effect can be diagnosed through the variance difference of the charge distribution. where the pair contribution is given by [12… view at source ↗
read the original abstract

The study of Entanglement Asymmetry has emerged in recent years as a powerful tool to characterise the symmetry properties of quantum states in relation to a given charge operator through the lens of entanglement. While extremely powerful and general, the standard definition of asymmetry introduces significant non-Gaussian features in free-fermionic systems, leading to certain analytical limitations. In this work, we introduce an asymmetry measure that remains strictly within the Gaussian manifold and analyse its properties. In particular, we show that it quantifies the minimal distance between a Gaussian state and the manifold of symmetric Gaussian states. We further demonstrate that this measure captures the established dynamical signatures of entanglement asymmetry, such as the Mpemba effect, symmetry restoration, and the lack thereof. The Gaussian structure allows these novel asymmetry measures to be computed exactly using correlation matrix techniques, and to be described asymptotically through the quasiparticle picture. We also comment on the possibility of using charge fluctuations to characterise the asymmetry of a Gaussian state.

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 introduces a Gaussian-restricted asymmetry measure for free-fermionic systems that avoids the non-Gaussian features of the standard entanglement asymmetry. It claims this measure equals the minimal distance from a given Gaussian state to the manifold of symmetric Gaussian states, and demonstrates that it reproduces key dynamical signatures of entanglement asymmetry including the Mpemba effect, symmetry restoration, and the absence of restoration. The Gaussian structure permits exact computation via correlation matrices and asymptotic analysis via the quasiparticle picture; the paper also comments on characterizing asymmetry via charge fluctuations.

Significance. If the central claims hold, the work supplies an analytically tractable tool for studying symmetry properties in Gaussian states, leveraging correlation-matrix techniques and the quasiparticle picture to obtain exact and asymptotic results. This could streamline investigations of symmetry breaking and restoration in free-fermion models while retaining the qualitative dynamical features of the broader entanglement-asymmetry framework.

major comments (2)
  1. [dynamical signatures / quasiparticle analysis] The central claim that the Gaussian measure captures the Mpemba effect, symmetry restoration, and lack thereof (stated in the abstract and demonstrated in the dynamical sections) rests on the assumption that restricting to the Gaussian manifold preserves essential signatures without loss from non-Gaussian contributions generated by the charge operator. The demonstrations appear confined to Gaussian states and correlation-matrix evolution; without an explicit comparison or counter-example check against the full non-Gaussian entanglement asymmetry in at least one free-fermion evolution where non-Gaussianity is known to matter, it is unclear whether restoration times or Mpemba crossings are preserved.
  2. [definition / properties section] The statement that the measure 'quantifies the minimal distance between a Gaussian state and the manifold of symmetric Gaussian states' is presented as a key property. The supporting derivation (presumably in the definition or properties section) must be checked for whether the distance is taken with respect to a specific metric on the Gaussian manifold (e.g., Hilbert-Schmidt or Bures) and whether the minimum is achieved uniquely at the symmetrized state; any dependence on the choice of metric would affect the interpretation as a canonical asymmetry quantifier.
minor comments (2)
  1. [final comments] The abstract mentions 'we also comment on the possibility of using charge fluctuations to characterise the asymmetry'; if this is developed only briefly, consider expanding it into a short dedicated subsection with an explicit formula relating fluctuations to the new measure.
  2. [throughout] Notation for the new asymmetry measure and the symmetric manifold should be introduced with a clear symbol (e.g., A_G or similar) and distinguished from the standard entanglement asymmetry throughout the text and figures.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address each major comment below.

read point-by-point responses

  1. Referee: [dynamical signatures / quasiparticle analysis] The central claim that the Gaussian measure captures the Mpemba effect, symmetry restoration, and lack thereof (stated in the abstract and demonstrated in the dynamical sections) rests on the assumption that restricting to the Gaussian manifold preserves essential signatures without loss from non-Gaussian contributions generated by the charge operator. The demonstrations appear confined to Gaussian states and correlation-matrix evolution; without an explicit comparison or counter-example check against the full non-Gaussian entanglement asymmetry in at least one free-fermion evolution where non-Gaussianity is known to matter, it is unclear whether restoration times or Mpemba crossings are preserved.

    Authors: Our work introduces a Gaussian-restricted measure specifically for free-fermionic systems to enable exact correlation-matrix computations and quasiparticle analysis. All dynamical demonstrations, including the Mpemba effect and symmetry restoration (or its absence), are performed within Gaussian states and their exact evolution under quadratic Hamiltonians. We agree that non-Gaussian contributions arising from the charge operator could quantitatively affect results in certain evolutions, and our manuscript does not include a direct numerical comparison to the full entanglement asymmetry in regimes where non-Gaussianity is known to be significant. We will add a clarifying paragraph in the discussion section stating that the Gaussian measure is intended as an analytically tractable proxy that reproduces the key qualitative signatures, while noting the absence of such a benchmark as a limitation. revision: partial

  2. Referee: [definition / properties section] The statement that the measure 'quantifies the minimal distance between a Gaussian state and the manifold of symmetric Gaussian states' is presented as a key property. The supporting derivation (presumably in the definition or properties section) must be checked for whether the distance is taken with respect to a specific metric on the Gaussian manifold (e.g., Hilbert-Schmidt or Bures) and whether the minimum is achieved uniquely at the symmetrized state; any dependence on the choice of metric would affect the interpretation as a canonical asymmetry quantifier.

    Authors: In the definition and properties section we derive the measure explicitly as the minimal distance to the manifold of symmetric Gaussian states, where the distance is the Hilbert-Schmidt distance between density operators (restricted to the Gaussian submanifold). We prove that the minimum is achieved uniquely at the Gaussian state obtained by symmetrizing the original state with respect to the charge operator. The Hilbert-Schmidt metric is chosen because it is compatible with the correlation-matrix representation used throughout the paper. We will revise the text to state the metric explicitly and to include the uniqueness argument in a dedicated lemma. revision: yes

standing simulated objections not resolved
  • Lack of explicit comparison between the Gaussian measure and the full non-Gaussian entanglement asymmetry in at least one free-fermion evolution where non-Gaussianity is known to matter.

Circularity Check

0 steps flagged

No circularity: Gaussian asymmetry measure defined independently with derived properties

full rationale

The paper introduces a new asymmetry measure restricted to the Gaussian manifold, motivated by analytical limitations of the standard (non-Gaussian) definition. It then shows via explicit construction that this measure equals the minimal distance to the manifold of symmetric Gaussian states, and demonstrates capture of dynamical signatures (Mpemba effect, symmetry restoration) through correlation-matrix and quasiparticle computations. No step reduces by construction to a fitted parameter, self-citation chain, or tautological renaming; the distance identification and dynamical demonstrations are presented as independent results rather than presupposed inputs. The derivation remains self-contained against external benchmarks without load-bearing self-referential elements.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the standard domain assumption that free-fermionic states are Gaussian and on the new definition of the asymmetry measure itself; no free parameters or additional invented entities beyond the measure are indicated in the abstract.

axioms (1)
  • domain assumption Free-fermionic systems are described by Gaussian states fully characterized by two-point correlation matrices.
    Standard assumption invoked when restricting to the Gaussian manifold.
invented entities (1)
  • Gaussian asymmetry measure no independent evidence
    purpose: Quantify asymmetry while remaining strictly Gaussian and equal to minimal distance to symmetric Gaussian states
    Newly introduced definition in the work.

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