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

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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.

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► References

[1] M. A. Nielsen, I. L. Chuang. Quantum computation and quantum information. Cambridge University Press (2010). DOI: 10.1017/​CBO9780511976667.
https:/​/​doi.org/​10.1017/​CBO9780511976667

[2] Ryszard Horodecki, Paweł Horodecki, Michał Horodecki, and Karol Horodecki. Quantum entanglement. Rev. Mod. Phys. 81, 865 (2009). DOI: 10.1103/​RevModPhys.81.865.
https:/​/​doi.org/​10.1103/​RevModPhys.81.865

[3] Robert Raussendorf, Hans J. Briegel. A one-way quantum computer. Phys. Rev. Lett. 86, 5188 (2001). DOI: 10.1103/​PhysRevLett.86.5188.
https:/​/​doi.org/​10.1103/​PhysRevLett.86.5188

[4] Mark Hillery, Vladimír Bužek, and André Berthiaume. Quantum secret sharing. Phys. Rev. A 59 1829 (1999). DOI: 10.1103/​PhysRevA.59.1829.
https:/​/​doi.org/​10.1103/​PhysRevA.59.1829

[5] Daniel Gottesman. Theory of quantum secret sharing. Phys. Rev. A 61, 042311 (2000). DOI: 10.1103/​PhysRevA.61.042311.
https:/​/​doi.org/​10.1103/​PhysRevA.61.042311

[6] Vittorio Giovannetti, Seth Lloyd, and Lorenzo Maccone. Advances in quantum metrology. Nature Photonics 5, 222–229 (2011). DOI: 10.1038/​nphoton.2011.35.
https:/​/​doi.org/​10.1038/​nphoton.2011.35

[7] Luigi Amico, Rosario Fazio, Andreas Osterloh, and Vlatko Vedral. Entanglement in many-body systems. Rev. Mod. Phys. 80, 517 (2008). DOI: 10.1103/​RevModPhys.80.517.
https:/​/​doi.org/​10.1103/​RevModPhys.80.517

[8] Eric Chitambar, Gilad Gour. Quantum resource theories. Rev. Mod. Phys. 91, 025001 (2019). DOI: 10.1103/​RevModPhys.91.025001.
https:/​/​doi.org/​10.1103/​RevModPhys.91.025001

[9] Gilad Gour, Nolan R. Wallach. Necessary and sufficient conditions for local manipulation of multipartite pure quantum states. New J. Phys. 13, 073013 (2011). DOI: 10.1088/​1367-2630/​13/​7/​073013.
https:/​/​doi.org/​10.1088/​1367-2630/​13/​7/​073013

[10] W. Dür, H. Aschauer, and H.-J. Briegel. Multiparticle Entanglement Purification for Graph States. Phys. Rev. Lett. 91, 107903 (2003). DOI: 10.1103/​PhysRevLett.91.107903.
https:/​/​doi.org/​10.1103/​PhysRevLett.91.107903

[11] M. Hein, J. Eisert, and H.J. Briegel. Multi-party entanglement in graph states. Phys. Rev. A 69, 062311 (2004). DOI: 10.1103/​PhysRevA.69.062311.
https:/​/​doi.org/​10.1103/​PhysRevA.69.062311

[12] D. Gottesman. Stabilizer Codes and Quantum Error Correction. PhD thesis, CalTech, Pasadena (1997). arXiv: quant-ph/​9705052.
arXiv:quant-ph/9705052

[13] Matthias Englbrecht, Barbara Kraus. Symmetries and entanglement of stabilizer states. Phys. Rev. A 101, 062302 (2020). DOI: 10.1103/​PhysRevA.101.062302.
https:/​/​doi.org/​10.1103/​PhysRevA.101.062302

[14] Martin Hebenstreit, Matthias Englbrecht, Cornelia Spee, Julio I. de Vicente, and Barbara Kraus. Measurement outcomes that do not occur and their role in entanglement transformations. New J. Phys 23, 033046 (2021). DOI: 10.1088/​1367-2630/​abe60c.
https:/​/​doi.org/​10.1088/​1367-2630/​abe60c

[15] David Sauerwein, Nolan R. Wallach, Gilad Gour, and Barbara Kraus. Transformations among Pure Multipartite Entangled States via Local Operations are Almost Never Possible. Phys. Rev. X 8, 031020 (2018). DOI: 10.1103/​PhysRevX.8.031020.
https:/​/​doi.org/​10.1103/​PhysRevX.8.031020

[16] Linda Ness, David Mumford. A stratification of the null cone via the moment map. American Journal of Mathematics, 106(6) (1984). DOI: 10.2307/​2374395.
https:/​/​doi.org/​10.2307/​2374395

[17] T. Maciążek, A. Sawicki. Critical points of the linear entropy for pure L-qubit states. Journal of Physics A: Mathematical and Theoretical 48(4), 045305 (2015). DOI: 10.1088/​1751-8113/​48/​4/​045305.
https:/​/​doi.org/​10.1088/​1751-8113/​48/​4/​045305

[18] T. Maciążek, A. Sawicki. Asymptotic properties of entanglement polytopes for large number of qubits. Journal of Physics A: Mathematical and Theoretical 51, 07LT01 (2018). DOI: 10.1088/​1751-8121/​aaa4d7.
https:/​/​doi.org/​10.1088/​1751-8121/​aaa4d7

[19] Adam Sawicki, Michał Oszmaniec, and Marek Kuś. Critical sets of the total variance can detect all stochastic local operations and classical communication classes of multiparticle entanglement. Phys. Rev. A 86, 040304 (2012). DOI: 10.1103/​PhysRevA.86.040304.
https:/​/​doi.org/​10.1103/​PhysRevA.86.040304

[20] A. Sawicki, T. Maciążek, M. Oszmaniec, K. Karnas, K. Kowalczyk-Murynka, and M. Kuś. Multipartite quantum correlations: symplectic and algebraic geometry approach. Rep. Math. Phys., 82(1):81 – 111 (2018). DOI: 10.1016/​S0034-4877(18)30072-7.
https:/​/​doi.org/​10.1016/​S0034-4877(18)30072-7

[21] Michael Walter, Brent Doran, David Gross, and Matthias Christandl. Entanglement polytopes: Multiparticle entanglement from single-particle information. Science 340, 6137 (2013). DOI: 10.1126/​science.1232957.
https:/​/​doi.org/​10.1126/​science.1232957

[22] Adam Sawicki, Michał Oszmaniec, and Marek Kuś. Convexity of momentum map, morse index, and quantum entanglement. Reviews in Mathematical Physics 26(03), 1450004 (2014). DOI: 10.1142/​S0129055X14500044.
https:/​/​doi.org/​10.1142/​S0129055X14500044

[23] Jim Bryan, Samuel Leutheusser, Zinovy Reichstein, and Mark Van Raamsdonk. Locally Maximally Entangled States of Multipart Quantum Systems. Quantum 3, 115 (2019). DOI: 10.22331/​q-2019-01-06-115.
https:/​/​doi.org/​10.22331/​q-2019-01-06-115

[24] Alex Arne, Matthias Kalus, Alan Huckelberry, and Jan von Delft. A numerical algorithm for the explicit calculation of $\mathrm{SU}({N})$ and $\mathrm{SL}({N}, \mathbb{C})$ Clebsch-Gordan coefficients. J. Math. Phys 52, 023507 (2011). DOI: 10.1063/​1.3521562.
https:/​/​doi.org/​10.1063/​1.3521562

[25] M. Altunbulak, A. Klyachko. The Pauli principle revisited. Commun. Math. Phys. 282 (2008). DOI: 10.1007/​s00220-008-0552-z.
https:/​/​doi.org/​10.1007/​s00220-008-0552-z

[26] William Fulton. Eigenvalues, invariant factors, highest weights, and Schubert calculus. Bull. Amer. Math. Soc. 37, 209-249 (2000). DOI: 10.1090/​S0273-0979-00-00865-X.
https:/​/​doi.org/​10.1090/​S0273-0979-00-00865-X

[27] George Kempf, Linda Ness. The length of vectors in representation spaces. In: Knud Lønsted (eds) Algebraic Geometry. Lecture Notes in Mathematics 732. Springer (1979). DOI: 10.1007/​BFb0066647.
https:/​/​doi.org/​10.1007/​BFb0066647

[28] Oskar Słowik, Martin Hebenstreit, Barbara Kraus, and Adam Sawicki. A link between symmetries of critical states and the structure of SLOCC classes in multipartite systems. Quantum 4, 300 (2020). DOI: 10.22331/​q-2020-07-20-300.
https:/​/​doi.org/​10.22331/​q-2020-07-20-300

[29] J.R. Stembridge, Jean-Yves Thibon, and M.A.A. van Leeuwen. Interaction of Combinatorics and Representation Theory. Part 3. The Littlewood-Richardson Rule, and Related Combinatorics. MSJ Memoirs 11. Cambridge University Press (2001). DOI: 10.2969/​msjmemoirs/​01101C030.
https:/​/​doi.org/​10.2969/​msjmemoirs/​01101C030

[30] William Fulton, Joe Harris. Representation Theory: A First Course. Graduate Texts in Mathematics 129. Springer-Verlag (2004). DOI: 10.1007/​978-1-4612-0979-9.
https:/​/​doi.org/​10.1007/​978-1-4612-0979-9

[31] I.M. Gelfand, M.L. Tsetlin. Matrix elements for the unitary group. Dokl. Akad. Nauk SSSR 71, 825 (1950).

[32] I.M. Gelfand, R.A. Minlos, and Z.Ya. Shapiro. Representations of the Rotation and Lorentz Group. Translated from the Russian edition (Moscow, 1958) by G. Cummins and T. Boddington. Pergamon (1963). DOI: 10.1126/​science.144.3617.402-a.
https:/​/​doi.org/​10.1126/​science.144.3617.402-a

[33] L.C. Biedenharn, J.D. Louck. A pattern calculus for tensor operators in the unitary groups. Commun. Math. Phys 8 (1968). DOI: 10.1007/​BF01645800.
https:/​/​doi.org/​10.1007/​BF01645800

[34] Anthony W. Knapp. Lie Groups Beyond an Introduction. Progress in Mathematics 140. Birkhäuser Boston (1996). DOI: 10.1007/​978-1-4757-2453-0.
https:/​/​doi.org/​10.1007/​978-1-4757-2453-0

[35] William Fulton. Young Tableaux, With Applications to Representation Theory and Geometry. London Mathematical Society Student Texts 35. Cambridge University Press (2012). DOI: 10.1017/​CBO9780511626241.
https:/​/​doi.org/​10.1017/​CBO9780511626241

[36] Katarzyna Górska, Karol A. Penson. Multidimensional Catalan and related numbers as Hausdorff moments. Probability and Mathematical Statistics 33, 2 (2013). URL: math.uni.wroc.pl/​ pms/​publications.php?nr=33.2.
https:/​/​www.math.uni.wroc.pl/​~pms/​publications.php?nr=33.2

[37] Michael W. Kirson. Introductory Algebra for Physicists. Course Notes. Weizmann Institute of Science (2016). URL: webhome.weizmann.ac.il/​home/​fnkirson/​Alg15/​Young_diagrams.pdf.
https:/​/​webhome.weizmann.ac.il/​home/​fnkirson/​Alg15/​Young_diagrams.pdf

[38] N.J.A. Sloane. The On-Line Encyclopedia of Integer Sequences. In: M. Kauers, M. Kerber, R. Miner, W. Windsteiger (eds) Towards Mechanized Mathematical Assistants. MKM 2007, Calculemus 2007. Lecture Notes in Computer Science 4573. Springer (2007). DOI: 10.1007/​978-3-540-73086-6_123.
https:/​/​doi.org/​10.1007/​978-3-540-73086-6_12

[39] Thomas Curtright, Thomas van Kortryk, and Cosmas Zachos. Spin Multiplicities. hal-01345527v2 (2016). URL: https:/​/​hal.archives-ouvertes.fr/​hal-01345527v2.
https:/​/​doi.org/​10.1016/​j.physleta.2016.12.006
https:/​/​hal.archives-ouvertes.fr/​hal-01345527v2

[40] Kazuhiko Koike. On the Decomposition of Tensor Products of the Representations of the Classical Groups: By Means of the Universal Characters. Advances in Mathematics 74 (1989). DOI: 10.1016/​0001-8708(89)90004-2.
https:/​/​doi.org/​10.1016/​0001-8708(89)90004-2

[41] A. U. Klimyk. Decomposition of the direct product of irreducible representations of semisimple Lie algebras into irreducible representations. Ukrain. Mat. Z. 18, 5 (1966).

[42] Jing-Song Huang, Chen-Bo Zhu. Weyl's Construction and Tensor Power Decomposition for $G_2$. Proceedings of the American Mathematical Society 127, 3 (1999). URL: ams.org/​journals/​proc/​1999-127-03/​.
https:/​/​doi.org/​10.1090/​S0002-9939-99-04583-9
https:/​/​www.ams.org/​journals/​proc/​1999-127-03/​

[43] Robert Fegera, Thomas W. Kephartb, and Robert J. Saskowski. LieART 2.0 – A Mathematica Application for Lie Algebras and Representation Theory. Computer Physics Communications 257, 107490 (2020). DOI: 10.1016/​j.cpc.2020.107490.
https:/​/​doi.org/​10.1016/​j.cpc.2020.107490

[44] V. V. Tsanov, Secant Varieties and Degrees of Invariants. Journal of Geometry and Symmetry in Physics 51 (2019). DOI: 10.7546/​jgsp-51-2019-73-85.
https:/​/​doi.org/​10.7546/​jgsp-51-2019-73-85

[45] Brian C. Hall. Lie Groups, Lie Algebras, and Representations: An Elementary Introduction. Graduate Texts in Mathematics 222. Springer (2015). DOI: 10.1007/​978-3-319-13467-3.
https:/​/​doi.org/​10.1007/​978-3-319-13467-3

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