$su(d)$-squeezing and many-body entanglement geometry in finite-dimensional systems

Giuseppe Vitagliano; Otfried Gühne; Géza Tóth

doi:10.22331/q-2025-09-03-1844

$su(d)$-squeezing and many-body entanglement geometry in finite-dimensional systems

Giuseppe Vitagliano¹, Otfried Gühne², and Géza Tóth^3,4,5,6,7

¹Vienna Center for Quantum Science and Technology, Atominstitut, TU Wien, A-1020 Vienna, Austria
²Naturwissenschaftlich-Technische Fakultät, Universität Siegen, Walter-Flex-Straße 3, D-57068 Siegen, Germany
³Department of Theoretical Physics, University of the Basque Country UPV/EHU, P.O. Box 644, E-48080 Bilbao, Spain
⁴EHU Quantum Center, University of the Basque Country UPV/EHU, Barrio Sarriena s/n, E-48940 Leioa, Biscay, Spain
⁵Donostia International Physics Center DIPC, Paseo Manuel de Lardizabal 4, E-20018 San Sebastián, Spain
⁶IKERBASQUE, Basque Foundation for Science, E-48009 Bilbao, Spain
⁷HUN-REN Wigner Research Centre for Physics, P.O. Box 49, H-1525 Budapest, Hungary

Published:	2025-09-03, volume 9, page 1844
Editor:	Himadri Shekhar Dhar
Eprint:	arXiv:2406.13338v2
Doi:	https://doi.org/10.22331/q-2025-09-03-1844
Citation:	Quantum 9, 1844 (2025).

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Abstract

Generalizing the well-known spin-squeezing inequalities, we study the relation between squeezing of collective $N$-particle $su(d)$ operators and many-body entanglement geometry in multi-particle systems. For that aim, we define the set of pseudo-separable states, which are mixtures of products of single-particle states that lie in the $(d^2-1)$-dimensional Bloch sphere but are not necessarily positive semidefinite. We obtain a set of necessary conditions for states of $N$ qudits to be of the above form. Any state that violates these conditions is entangled. We also define a corresponding $su(d)$-squeezing parameter that can be used to detect entanglement in large particle ensembles. Geometrically, this set of conditions defines a convex set of points in the space of first and second moments of the collective $N$-particle $su(d)$ operators. We prove that, in the limit $N\gg 1$, such set is filled by pseudo-separable states, while any state corresponding to a point outside of this set is necessarily entangled. We also study states that are detected by these inequalities: We show that states with a bosonic symmetry are detected if and only if the two-body reduced state violates the positive partial transpose (PPT) criterion. On the other hand, highly mixed states states close to the $su(d)$ singlet are detected which have a separable two-body reduced state and are also PPT with respect to all possible bipartitions. We also provide numerical examples of thermal equilibrium states that are detected by our set of inequalities, comparing the spin-squeezing inequalities with the $su(3)$-squeezing inequalities.

Featured image: Top: Representation of pseudo-separable states as a product of (sections of) single-particle Bloch-spheres. Bottom left: representation of sets of quantum states, separable states and pseudo-separable states; bottom right: A section of the su(d)-squeezing polytope.

Popular summary

Entanglement is one of the central features of quantum physics and a key resource for emerging quantum technologies. Yet, as the number of particles in a system grows, identifying whether they are entangled becomes increasingly difficult. A successful strategy for large ensembles of spin-1/2 particles relies on spin squeezing, where quantum fluctuations in one spin direction are reduced at the expense of increased fluctuations in another. Spin-squeezing inequalities not only reveal entanglement but also connect directly to enhanced precision in quantum metrology.

In our work, we extend this approach to particles with more than two internal levels—so-called qudits, which can be thought of as spins larger than 1/2. These higher-dimensional systems naturally arise in cold atoms, ions, and other platforms, and offer new opportunities for both fundamental studies and practical applications. To analyze them, we introduce a family of inequalities based on su(d) operators, the natural generalization of spin operators to qudits.

Geometrically, these inequalities carve out a well-defined region—the su(d) squeezing polytope—that contains all states which can be described without entanglement. Any state found outside this region must be entangled. Remarkably, in the limit of many particles, this picture becomes in a certain sense complete: the polytope is fully filled by certain states that we call “pseudo-separable”, and resemble the usual separable states, but with single-particle states that are not necessarily positive but lie inside a qudit Bloch sphere. Thus, any point outside of the su(d) polytopes unambiguously signals non-pseudo-separability, which also implies many-body entanglement.

We further connect our framework to two-particle properties, showing that for symmetric states (such as those realized in bosonic systems), our conditions are equivalent to the well-known positive partial transpose (PPT) test of entanglement. At the same time, we identify highly mixed states close to special “singlet” states that our inequalities detect as entangled, even though they have separable two-particle density matrix.

Finally, we illustrate the power of our method by analyzing thermal states in realistic spin-1 and qutrit models, showing that su(d)-squeezing can reveal entanglement beyond the reach of traditional spin-squeezing inequalities. In particular, unlike the original spin squeezing criterion, our method can detect the entanglement of quantum states without a large spin in some direction, such as singlet states appearing in condensed matter physics and Dicke states appearing in quantum optics.

Our results open new avenues for characterizing and harnessing entanglement in higher-spin ensembles, with direct relevance for experiments in cold atoms and other multi-level quantum systems.

► BibTeX data

@article{Vitagliano2025sudsqueezingmany,
  doi = {10.22331/q-2025-09-03-1844},
  url = {https://doi.org/10.22331/q-2025-09-03-1844},
  title = {{$su(d)$}-squeezing and many-body entanglement geometry in finite-dimensional systems},
  author = {Vitagliano, Giuseppe and G{\"{u}}hne, Otfried and T{\'{o}}th, G{\'{e}}za},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1844},
  month = sep,
  year = {2025}
}

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[1] Julia Mathé, Ayaka Usui, Otfried Gühne, and Giuseppe Vitagliano, "Estimating the best separable approximation of non-pure spin-squeezed states", Quantum 10, 2078 (2026).

[2] Samuel R Hedemann, "Multi-operator quantum uncertainty relations from new Cauchy–Schwarz inequalities", Journal of Physics A: Mathematical and Theoretical 59 6, 065302 (2026).

[3] Christopher Wilson, John Drew Wilson, Luke Coffman, Shah Saad Alam, and Murray J. Holland, "Geometric invariants of quantum metrology", Physical Review A 113 6, 063725 (2026).

[4] Szilárd Szalay and Géza Tóth, "Alternatives of entanglement depth and metrological entanglement criteria", Quantum 9, 1718 (2025).

[5] Allan Tameshtit, "Emergence of Symmetry in a System of Distinguishable but Identical Particles", arXiv:2509.11779, (2025).

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