Bounding the entanglement of a state from its spectrum

Jofre Abellanet-Vidal; Guillem Müller-Rigat; Albert Rico; Anna Sanpera

doi:10.22331/q-2026-11-06-2260

Bounding the entanglement of a state from its spectrum

Jofre Abellanet-Vidal¹, Guillem Müller-Rigat², Albert Rico^1,3, and Anna Sanpera^1,4

¹Física Teòrica: Informació i Fenòmens Quàntics, Departament de Física, Universitat Autònoma de Barcelona, E-08193 Bellaterra, Spain.
²Faculty of Physics, Astronomy and Applied Computer Science, Jagiellonian University, ul. Łojasiewicza 11, 30-348 Kraków, Poland.
³Naturwissenschaftlich-Technische Fakultät, Universität Siegen, Walter-Flex-Straße 3, 57068 Siegen, Germany.
⁴ICREA – Institució Catalana de Recerca i Estudis Avançats, Lluis Companys 23, 08010 Barcelona, Spain.

Published:	2026-11-06, volume 10, page 2260
Editor:	Remigiusz Augusiak
Eprint:	arXiv:2604.02420v4
Doi:	https://doi.org/10.22331/q-2026-11-06-2260
Citation:	Quantum 10, 2260 (2026).

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Abstract

We introduce a framework to upper bound the entanglement content of a bipartite quantum state from its spectrum alone. Using linear maps and their inverses, we derive rigorous constraints on the maximal entanglement that can be activated under global unitary transformations. We use as entanglement quantifiers the negativity and the Schmidt number; however, our framework is general and applies to any other entanglement measure. Our approach yields compact analytical sufficient criteria for bounding the entanglement of full-rank states in arbitrary dimensions and reveals new spectral constraints on Schmidt number witnesses.

Popular summary

Entanglement is one of the main quantum features, and a key resource for quantum-enhanced technologies. For this reason, quantifying this resource in a state is of crucial importance to assess the advantages they may provide. However, physical systems suffer from noise due to interaction with the environment, which results into mixed quantum states. When this occurs, it is unclear how much entanglement can be recovered by means of unitary dynamics (or quantum circuit gates). Instead, existing works have been only able to answer whether or not a state can be entangled, leaving open the question of how much entanglement it contains.
In this work, we tackle this open problem and find useful criteria to determine the maximal amount of entanglement that can be possibly created in a mixed quantum state with unitary evolutions. Although the problem is in general a difficult optimization, the bounds introduced depend solely on two eigenvalues of the state, and in fact are shown to be tight for certain noise models. Thus they apply effectively at low computational cost, and allow us to address the problem for high-dimensional quantum systems. We bound both continuous and discrete entanglement measures, namely the entanglement negativity and the so-called Schmidt number (which accounts for the amount of entangled degrees of freedom).

► BibTeX data

@article{AbellanetVidal2026bounding,
  doi = {10.22331/q-2026-11-06-2260},
  url = {https://doi.org/10.22331/q-2026-11-06-2260},
  title = {Bounding the entanglement of a state from its spectrum},
  author = {Abellanet-Vidal, Jofre and M{\"{u}}ller-Rigat, Guillem and Rico, Albert and Sanpera, Anna},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {10},
  pages = {2260},
  month = nov,
  year = {2026}
}

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