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arxiv: 2504.07814 · v2 · submitted 2025-04-10 · 🪐 quant-ph

Estimating the best separable approximation of non-pure spin-squeezed states

Julia Math\'e , Ayaka Usui , Otfried G\"uhne , Giuseppe Vitagliano This is my paper

Pith reviewed 2026-05-22 20:23 UTC · model grok-4.3

classification 🪐 quant-ph
keywords spin-squeezing inequalitiesbest separable approximationcollective spin statesentanglement monotonesthermal statesXXZ modeldistance to separability
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The pith

Spin-squeezing inequalities yield a lower bound on the best separable approximation distance for collective spin states without witness optimization.

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

The paper establishes a method to lower-bound the distance of mixed collective spin states to fully separable states using the complete set of spin-squeezing inequalities. This produces a bound on the best separable approximation that bypasses numerical optimization over linear entanglement witnesses. An improved iterative algorithm finds the closest separable state by exploiting the symmetry of collective states of N spin-1/2 particles. The approach is applied to thermal states of the fully-connected XXZ model, showing that the bound is often tight at zero temperature and at the temperature where entanglement vanishes, and that entanglement can appear at finite temperature even in phases where the ground state is separable.

Core claim

From the complete set of spin-squeezing inequalities, which form nonlinear entanglement criteria based on collective spin variances for states of N spin-1/2 particles, one obtains a lower bound to the best separable approximation, a distance-based entanglement monotone. Combined with symmetry-exploiting optimization for upper bounds, this enables quantitative entanglement analysis for thermal states on fully-connected graphs, including across phases of the XXZ model at nonzero temperature.

What carries the argument

The complete set of spin-squeezing inequalities as nonlinear criteria based on collective spin variances that directly lower-bound the best separable approximation distance.

If this is right

  • Lower bounds to the best separable approximation are obtained directly from the spin-squeezing inequalities without optimizing over witnesses.
  • Symmetry-exploiting iteration finds the closest separable state for collective spin states.
  • Entanglement is quantified in thermal states of the fully-connected XXZ model, including at nonzero temperature in the ordered phase.
  • The lower bound is often tight at zero temperature and at the temperature where entanglement disappears.

Where Pith is reading between the lines

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

  • The method could apply to other symmetric many-body states beyond fully-connected graphs.
  • Quantitative bounds may help detect entanglement in out-of-equilibrium spin systems.
  • Tightness at entanglement disappearance temperatures indicates the inequalities capture the transition precisely in these models.

Load-bearing premise

The spin-squeezing inequalities provide a complete set of nonlinear entanglement criteria for collective states of N spin-1/2 particles.

What would settle it

A thermal state of the XXZ model for which the actual best separable approximation distance computed by other means falls below the lower bound obtained from the spin-squeezing inequalities would disprove the claimed bound.

Figures

Figures reproduced from arXiv: 2504.07814 by Ayaka Usui, Giuseppe Vitagliano, Julia Math\'e, Otfried G\"uhne.

Figure 1
Figure 1. Figure 1: FIG. 1: Overview figure [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Optimization algorithm for symmetric states [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: BSA for di [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: LMG model [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
read the original abstract

We discuss the estimation of the distance of a given mixed many-body quantum state to the set of fully separable states, applied to the concrete scenario of collective spin states. Concretely, we discuss lower bounds to distances from the set of fully separable states based on entanglement criteria and upper bounds to those distances using an iterative algorithm to find the optimal separable state closest to the target. Focusing on collective states of $N$ spin-$1/2$ particles, we consider spin-squeezing inequalities (SSIs), which provide a complete set of nonlinear entanglement criteria based on collective spin variances. First, we find a lower bound to distance-based entanglement monotones, specifically the so-called best separable approximation (BSA) from the complete set of SSIs, thereby bypassing entirely a numerical optimization over a (potentially very large) set of linear entanglement witnesses. Then, we improve current algorithms to iteratively find the closest separable state to a given target state, exploiting the symmetry of the system. These results allow us to study entanglement quantitatively on thermal states of spin systems on fully-connected graphs at nonzero temperature, as well as potentially similar states arising in out-of-equilibrium situations. We thus apply our methods to investigate entanglement across different phases of a fully-connected XXZ model. We observe that our lower bound becomes often tight for zero temperature as well as for the temperature at which entanglement disappears, both of which are thus precisely captured by the SSIs. We further observe, among other things, that entanglement can arise at nonzero temperature even in the ordered phase, where the ground state is separable, revealing the potential usefulness of entanglement quantification also beyond the ground state paradigm.

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

1 major / 2 minor

Summary. The manuscript claims that the complete set of spin-squeezing inequalities (SSIs) yields a computable lower bound on the best separable approximation (BSA) distance for collective spin states of N spin-1/2 particles, thereby avoiding optimization over linear entanglement witnesses. It further presents a symmetry-exploiting improvement to iterative algorithms for computing the closest separable state (upper bound on the distance) and applies both to thermal states of the fully-connected XXZ model, reporting that the lower bound is often tight at T=0 and at the entanglement-disappearance temperature.

Significance. If the claimed lower bound construction holds, the work supplies a practical, witness-optimization-free route to quantitative entanglement measures for symmetric collective states at finite temperature. The symmetry reduction in the iterative search is a concrete algorithmic improvement. The observation that SSIs capture the exact disappearance temperature in the XXZ model is a falsifiable prediction that strengthens the utility beyond ground-state studies.

major comments (1)
  1. [Introduction and the section presenting the lower-bound derivation] The central construction of the lower bound on the BSA distance from the maximal SSI violation is stated in the abstract and introduction but the explicit mapping (how the violation quantity is converted into a distance lower bound) is not shown with intermediate steps or an equation; without this, it is impossible to verify that the bound is indeed obtained solely from the SSIs without hidden optimization or additional assumptions.
minor comments (2)
  1. [Algorithm section] The description of the improved iterative algorithm would benefit from an explicit statement of the symmetry reduction (e.g., which variables are fixed or eliminated) and a convergence criterion or iteration count used in the XXZ numerics.
  2. [Results section / figures] Figure captions for the temperature-dependent plots should explicitly mark the ordered and disordered phases of the XXZ model and the entanglement-disappearance temperature to make the tightness observations immediately readable.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive feedback. We address the major comment below.

read point-by-point responses

  1. Referee: [Introduction and the section presenting the lower-bound derivation] The central construction of the lower bound on the BSA distance from the maximal SSI violation is stated in the abstract and introduction but the explicit mapping (how the violation quantity is converted into a distance lower bound) is not shown with intermediate steps or an equation; without this, it is impossible to verify that the bound is indeed obtained solely from the SSIs without hidden optimization or additional assumptions.

    Authors: We agree that the explicit mapping requires clearer presentation. In the revised manuscript we will insert a dedicated paragraph (or short subsection) immediately following the statement of the complete set of SSIs. This paragraph will contain the intermediate algebraic steps that convert the maximal SSI violation into a lower bound on the BSA distance, together with the resulting explicit inequality relating the two quantities. The added material will make transparent that the bound follows directly from the SSIs without any auxiliary optimization or extra assumptions. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation starts from the established completeness of spin-squeezing inequalities (SSIs) as an external input and constructs a lower bound on the best separable approximation distance directly from maximal SSI violation; this step does not reduce by the paper's own equations to a fitted parameter, self-definition, or self-citation chain. The subsequent iterative algorithm improvement rests on symmetry properties of collective spin states and is independent of the bound. No self-definitional, fitted-input, or ansatz-smuggling patterns appear in the load-bearing steps.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the domain assumption that the spin-squeezing inequalities form a complete set for collective spin states; no free parameters or new postulated entities are introduced.

axioms (1)
  • domain assumption Spin-squeezing inequalities provide a complete set of nonlinear entanglement criteria based on collective spin variances for collective states of N spin-1/2 particles.
    This completeness is invoked to obtain the lower bound to the best separable approximation without optimizing over linear witnesses.

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