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

Julia Mathé; Ayaka Usui; Otfried Gühne; Giuseppe Vitagliano

doi:10.22331/q-2026-04-21-2078

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

Julia Mathé¹, Ayaka Usui², Otfried Gühne³, and Giuseppe Vitagliano¹

¹Vienna Center for Quantum Science and Technology, Atominstitut, TU Wien, 1020 Vienna, Austria
²Departament de Física, Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain
³Naturwissenschaftlich-Technische Fakultät, Universität Siegen, Walter-Flex-Straße 3, D-57068 Siegen, Germany

Published:	2026-04-21, volume 10, page 2078
Editor:	Aaron Goldberg
Eprint:	arXiv:2504.07814v2
Doi:	https://doi.org/10.22331/q-2026-04-21-2078
Citation:	Quantum 10, 2078 (2026).

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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.

Featured image: We estimate the best separable approximation for an $N$-qubit state (top right), by computing lower and upper bounds (left). Lower bounds are computed from the negative expectation value of an entanglement witness W that is illustrated as blue lines not crossing the set of separable states (shaded ellipse). Upper bounds can be computed from the distance to any given separable state (red/purple lines), minimized over as many separable states as possible. Optimal bounds are illustrated as straight or dotted lines, non-optimal bounds as dashed lines. For states featuring symmetries, the problem can be formulated in a symmetric subspace (bottom right: example matrix for $N = 4$, blocks of different color are a result of permutational invariance, while diagonal matrices have an additional rotation symmetry).

Popular summary

Entanglement in many-body systems is usually analyzed for pure ground states, but realistic systems are often mixed because of temperature, noise, or nonequilibrium dynamics. In such cases, even deciding whether a state is entangled can be difficult, let alone quantifying how much entanglement it contains.

In this work, we study this problem for collective spin states by asking how far a given mixed state is from the set of fully separable states. This distance is quantified by the best separable approximation, which tells us how well the state can still be described by a classical-like mixture of unentangled particles.

We derive a lower bound on this quantity from spin-squeezing inequalities built from standard collective observables, and an upper bound from an iterative algorithm that searches for the closest separable state while exploiting the symmetries of the system.

Applying these tools to thermal states of fully connected spin models, we show that this method can provide insightful quantitative information about mixed-state entanglement across different phases. In particular, we find that entanglement may appear at nonzero temperature even in regimes where the ground state itself is separable, highlighting that relevant quantum correlations can emerge beyond the usual ground-state picture. Our results also suggest that entanglement quantification may refine the usual phase diagram by revealing a finer structure within conventional phases, where states with different amounts of entanglement define distinct regimes.

► BibTeX data

@article{Mathe2026estimatingbest,
  doi = {10.22331/q-2026-04-21-2078},
  url = {https://doi.org/10.22331/q-2026-04-21-2078},
  title = {Estimating the best separable approximation of non-pure spin-squeezed states},
  author = {Math{\'{e}}, Julia and Usui, Ayaka and G{\"{u}}hne, Otfried and Vitagliano, Giuseppe},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {10},
  pages = {2078},
  month = apr,
  year = {2026}
}

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