Designing locally maximally entangled quantum states with arbitrary local symmetries

Oskar Słowik; Adam Sawicki; Tomasz Maciążek

doi:10.22331/q-2021-05-01-450

Designing locally maximally entangled quantum states with arbitrary local symmetries

Oskar Słowik¹, Adam Sawicki¹, and Tomasz Maciążek²

¹Center for Theoretical Physics, Polish Academy of Sciences, Al. Lotników 32/46, 02-668 Warsaw, Poland
²School of Mathematics, University of Bristol, Fry Building, Woodland Road, Bristol BS8 1UG, UK

Published:	2021-05-01, volume 5, page 450
Eprint:	arXiv:2011.04078v5
Doi:	https://doi.org/10.22331/q-2021-05-01-450
Citation:	Quantum 5, 450 (2021).

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Abstract

One of the key ingredients of many LOCC protocols in quantum information is a multiparticle (locally) maximally entangled quantum state, aka a critical state, that possesses local symmetries. We show how to design critical states with arbitrarily large local unitary symmetry. We explain that such states can be realised in a quantum system of distinguishable traps with bosons or fermions occupying a finite number of modes. Then, local symmetries of the designed quantum state are equal to the unitary group of local mode operations acting diagonally on all traps. Therefore, such a group of symmetries is naturally protected against errors that occur in a physical realisation of mode operators. We also link our results with the existence of so-called strictly semistable states with particular asymptotic diagonal symmetries. Our main technical result states that the $N$th tensor power of any irreducible representation of $\mathrm{SU}(N)$ contains a copy of the trivial representation. This is established via a direct combinatorial analysis of Littlewood-Richardson rules utilising certain combinatorial objects which we call telescopes.

Featured image: A quantum system consisting of three distinguishable traps each containing two bosons that can occupy three modes. Such a system contains a Locally Maximally Entangled state (wrt to traps) with diagonal SU(3)-symmetry in local diagonal mode special unitary operations.

► BibTeX data

@article{Slowik2021designinglocally,
  doi = {10.22331/q-2021-05-01-450},
  url = {https://doi.org/10.22331/q-2021-05-01-450},
  title = {Designing locally maximally entangled quantum states with arbitrary local symmetries},
  author = {S{\l{}}owik, Oskar and Sawicki, Adam and Maci{\k{a}}{\.{z}}ek, Tomasz},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {5},
  pages = {450},
  month = may,
  year = {2021}
}

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