Locally Maximally Entangled States of Multipart Quantum Systems

Jim Bryan1, Samuel Leutheusser2,3, Zinovy Reichstein1, and Mark Van Raamsdonk2

1Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, B.C., V6T 1Z1, Canada
2Department of Physics and Astronomy, University of British Columbia, 6224 Agricultural Road, Vancouver, B.C., V6T 1Z2, Canada
3Center for Theoretical Physics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA

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Abstract

For a multipart quantum system, a locally maximally entangled (LME) state is one where each elementary subsystem is maximally entangled with its complement. This paper is a sequel to~[J. Bryan, Z. Reichstein and M. Van Raamsdonk, Existence of Locally Maximally Entangled Quantum States via Geometric Invariant Theory, Ann. Henri Poincaré 19 (2018), no. 8, 2491-2511. MR3830220], which gives necessary and sufficient conditions for a system to admit LME states in terms of its subsystem dimensions $(d_1, d_2, \dots, d_n)$, and computes the dimension of the space ${\cal S}_{LME}/K$ of LME states up to local unitary transformations for all non-empty cases. Here we provide a pedagogical overview and physical interpretation of the underlying mathematics that leads to these results and give a large class of explicit constructions for LME states. In particular, we construct all LME states for tripartite systems with subsystem dimensions $(2,A,B)$ and give a general representation-theoretic construction for a special class of stabilizer LME states. The latter construction provides a common framework for many known LME states. Our results have direct implications for the problem of characterizing SLOCC equivalence classes of quantum states, since points in ${\cal S}_{LME}/K$ correspond to natural families of SLOCC classes. Finally, we give the dimension of the stabilizer subgroup $S \subset \operatorname{SL}(d_1, \mathbb{C}) \times \cdots \times \operatorname{SL}(d_n, \mathbb{C})$ for a generic state in an arbitrary multipart system and identify all cases where this stabilizer is trivial.

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► References

[1] I. Bengtsson, K. Zyczkowski, Geometry of quantum states, second edition, Cambridge University Press, Cambridge, 2017. MR3752196.

[2] P. Bürgisser, A. Garg, R. Oliveira, M. Walter, A. Wigderson, Alternating minimization, scaling algorithms, and the null-cone problem from invariant theory, in 9th Innovations in Theoretical Computer Science, Art. 24, 20 pp, LIPIcs. Leibniz Int. Proc. Inform., 94, Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern. MR3761760.

[3] J. Bryan, Z. Reichstein and M. Van Raamsdonk, Existence of Locally Maximally Entangled Quantum States via Geometric Invariant Theory, Ann. Henri Poincaré 19 (2018), no. 8, 2491–2511. MR3830220 https:/​/​doi.org/​10.1007/​s00023-018-0682-6.
https:/​/​doi.org/​10.1007/​s00023-018-0682-6

[4] J.-L. Brylinski, Algebraic measures of entanglement, in Mathematics of quantum computation, 3–23, Comput. Math. Ser, Chapman & Hall/​CRC, Boca Raton, FL, 2002. MR2007941.

[5] J.-L. Brylinski and R. Brylinski, Universal quantum gates, in Mathematics of quantum computation, 101–116, Comput. Math. Ser, Chapman & Hall/​CRC, Boca Raton, FL, 2002. MR2007944.

[6] J. A. Dieudonné and J. B. Carrell, Invariant theory, old and new, Academic Press, New York, 1971. MR0279102.

[7] W. Dür, G. Vidal and J. I. Cirac, Three qubits can be entangled in two inequivalent ways, Phys. Rev. A (3) 62 (2000), no. 6, 062314, 12 pp. MR1804183, https:/​/​doi.org/​10.1103/​PhysRevA.62.062314.
https:/​/​doi.org/​10.1103/​PhysRevA.62.062314

[8] A. G. Elashvili, Stationary subalgebras of points of general position for irreducible linear Lie groups, Funkcional. Anal. i Priložen. 6 (1972), no. 2, 65–78. MR0304555. English translation in Functional Anal. Appl. 6 (1972), 139–148, https:/​/​doi.org/​10.1007%2FBF01077518.

[9] C. Eltschka and J. Siewert, Quantifying entanglement resources, J. Phys. A 47 (2014), no. 42, 424005, 54 pp. MR3270518, https:/​/​doi.org/​10.1088/​1751-8113/​47/​42/​424005.
https:/​/​doi.org/​10.1088/​1751-8113/​47/​42/​424005

[10] P. Facchi, G. Florio, G. Parisi, and S. Pascazio, Maximally multipartite entangled states, Phys. Rev. A (3) 77 (2008), no. 6, 060304, 4 pp. MR2491404, https:/​/​doi.org/​10.1103/​PhysRevA.77.060304.
https:/​/​doi.org/​10.1103/​PhysRevA.77.060304

[11] W. Fulton and J. Harris, Representation theory, Graduate Texts in Mathematics, 129, Springer-Verlag, New York, 1991. MR1153249.

[12] N. Gisin and H. Bechmann-Pasquinucci, Bell inequality, Bell states and maximally entangled states for $n$ qubits, Phys. Lett. A 246 (1998), no. 1-2, 1–6. MR1643978, https:/​/​doi.org/​10.1016/​S0375-9601(98)00516-7.
https:/​/​doi.org/​10.1016/​S0375-9601(98)00516-7

[13] G. Gour and N. R. Wallach, All maximally entangled four-qubit states, J. Math. Phys. 51 (2010), no. 11, 112201, 24 pp. MR2759471, https:/​/​doi.org/​10.1063/​1.3511477.
https:/​/​doi.org/​10.1063/​1.3511477

[14] G. Gour, N. R. Wallach, Classification of Multipartite Entanglement of All Finite Dimensionality, Phys. Rev. Lett. 111, 060502 (2013), https:/​/​doi.org/​10.1103/​PhysRevLett.111.060502.
https:/​/​doi.org/​10.1103/​PhysRevLett.111.060502

[15] G. Gour, B. Kraus and N. R. Wallach, Almost all multipartite qubit quantum states have trivial stabilizer, J. Math. Phys. 58 (2017), no. 9, 092204, 14 pp. MR3706585, https:/​/​doi.org/​10.1063/​1.5003015.
https:/​/​doi.org/​10.1063/​1.5003015

[16] D. Goyeneche, K. Zyczkowski, Genuinely multipartite entangled states and orthogonal arrays , PRA 90, 022316 (2014), https:/​/​doi.org/​10.1103/​PhysRevA.90.022316.
https:/​/​doi.org/​10.1103/​PhysRevA.90.022316

[17] W. Helwig, W. Cui, A. Riera, J. I. Latorre and H. K. Lo, ``Absolute Maximal Entanglement and Quantum Secret Sharing,'' Phys. Rev. A 86 (2012) 052335 ,https:/​/​doi.org/​10.1103/​PhysRevA.86.052335.
https:/​/​doi.org/​10.1103/​PhysRevA.86.052335

[18] A. Higuchi and A. Sudbery, How entangled can two couples get?, Phys. Lett. A 273 (2000), no. 4, 213–217. MR1781603, https:/​/​doi.org/​10.1016/​S0375-9601(00)00480-1.
https:/​/​doi.org/​10.1016/​S0375-9601(00)00480-1

[19] V. Hoskins, Geometric Invariant Theory and Symplectic Quotients, lecture notes, 2012, http:/​/​userpage.fu-berlin.de/​hoskins/​GITnotes.pdf.
http:/​/​userpage.fu-berlin.de/​hoskins/​GITnotes.pdf

[20] F. Huber, O. Guhne, and J. Siewert, Absolutely maximally entangled states of seven qubits do not exist, Phys. Rev. Lett. 118, 200502 (2017), https:/​/​doi.org/​10.1103/​PhysRevLett.118.200502.
https:/​/​doi.org/​10.1103/​PhysRevLett.118.200502

[21] A. A. Klyachko, Coherent states, entanglement, and geometric invariant theory, 2002, https:/​/​arxiv.org/​abs/​quant-ph/​0206012.
arXiv:quant-ph/0206012

[22] A. A. Klyachko, Quantum marginal problem and representations of the symmetric group, 2004, https:/​/​arxiv.org/​abs/​quant-ph/​0409113.
arXiv:quant-ph/0409113

[23] A. A. Klyachko. Dynamical symmetry approach to entanglement., In Physics and theoretical computer science, volume 7 of NATO Secur. Sci. Ser. D Inf. Commun. Secur., pages 25–54. IOS, Amsterdam, 2007, https:/​/​arxiv.org/​abs/​0802.4008.
arXiv:0802.4008

[24] T. Maciazek, M. Oszmaniec and A. Sawicki, How many invariant polynomials are needed to decide local unitary equivalence of qubit states?, J. Math. Phys. 54 (2013), no. 9, 092201, 15 pp. MR3135592, https:/​/​doi.org/​10.1063/​1.4819499.
https:/​/​doi.org/​10.1063/​1.4819499

[25] T. Maciazek and V. Tsanov, Quantum marginals from pure doubly excited states, J. Phys. A: Math. Theor. 50 465304, 2017, https:/​/​doi.org/​10.1088/​1751-8121/​aa8c5f.
https:/​/​doi.org/​10.1088/​1751-8121/​aa8c5f

[26] D. Mumford, J. Fogarty and F. Kirwan, Geometric invariant theory, third edition, Ergebnisse der Mathematik und ihrer Grenzgebiete (2), 34, Springer-Verlag, Berlin, 1994. MR1304906.

[27] T. Maciazek, A. Sawicki, Critical points of the linear entropy for pure $L$-qubit states, J. Phys. A 48 (2015), no. 4, 045305, 25 pp. MR3300268, https:/​/​doi.org/​10.1088/​1751-8113/​48/​4/​045305.
https:/​/​doi.org/​10.1088/​1751-8113/​48/​4/​045305

[28] F. Pastawski, B. Yoshida, D. Harlow and J. Preskill, Holographic quantum error-correcting codes: Toy models for the bulk/​boundary correspondence, J. High Energy Phys. 2015, no. 6, 149, front matter+53 pp. MR3370186, https:/​/​doi.org/​10.1007/​JHEP06(2015)149.
https:/​/​doi.org/​10.1007/​JHEP06(2015)149

[29] A. M. Popov, Finite isotropy subgroups in general position of irreducible semisimple linear Lie groups, Trans. Moscow Math. Soc. 1988, 205–249; translated from Trudy Moskov. Mat. Obshch. 50(1987), 209–248. MR0912058, http:/​/​mi.mathnet.ru/​eng/​mmo/​v50/​p209.
http:/​/​mi.mathnet.ru/​eng/​mmo/​v50/​p209

[30] A. J. Scott, Multipartite entanglement, quantum-error-correcting codes, and entangling power of quantum evolutions, PRA 69, 052330 (2004), https:/​/​doi.org/​10.1103/​PhysRevA.69.052330.
https:/​/​doi.org/​10.1103/​PhysRevA.69.052330

[31] A. Sawicki, M. Oszmaniec, M. Kus, Critical sets of the total variance of state detect all SLOCC entanglement classes, Phys. Rev. A 86, 040304(R) (2012), https:/​/​doi.org/​10.1103/​PhysRevA.86.040304.
https:/​/​doi.org/​10.1103/​PhysRevA.86.040304

[32] A. Sawicki, M. Oszmaniec, M. Kus, Convexity of momentum map, Morse index, and quantum entanglement, Reviews in Mathematical Physics, 26, 1450004, (2014), https:/​/​doi.org/​10.1142/​S0129055X14500044.
https:/​/​doi.org/​10.1142/​S0129055X14500044

[33] D. Sauerwein, N.R. Wallach, G. Gour, B. Kraus, Transformations among pure multipartite entangled states via local operations are almost never possible, Phys. Rev. X 8, 031020, https:/​/​doi.org/​10.1103/​PhysRevX.8.031020.
https:/​/​doi.org/​10.1103/​PhysRevX.8.031020

[34] F. Tennie, D. Ebler, V. Vedral, and C. Schilling, Pinning of fermionic occupation numbers: General concepts and one spatial dimension, Phys. Rev. A, 93(4), 042126, 2016, https:/​/​doi.org/​10.1103/​PhysRevA.93.042126.
https:/​/​doi.org/​10.1103/​PhysRevA.93.042126

[35] F. Tennie, V. Vedral, and C. Schilling, Influence of the fermionic exchange symmetry beyond Pauli's exclusion principle, Phys. Rev. A, 95(2), 022336, 2017, https:/​/​doi.org/​10.1103/​PhysRevA.95.022336.
https:/​/​doi.org/​10.1103/​PhysRevA.95.022336

[36] N. R. Wallach, Quantum computing and entanglement for mathematicians, in Representation theory and complex analysis, lectures given at the C.I.M.E. Summer School held in Venice, Italy, June 10-17, 2004, 345–376, Lecture Notes in Math., 1931, Springer, Berlin, 2008. MR2409702.

[37] F. Verstraete, J. Dehaene, and B. De Moor, Normal forms and entanglement measures for multipartite quantum states, Phys. Rev. A, 68 (1), 2003, https:/​/​doi.org/​10.1103/​PhysRevA.68.012103.
https:/​/​doi.org/​10.1103/​PhysRevA.68.012103

[38] Michèle Vergne, Michael Walter, Inequalities for moment cones of finite-dimensional representations, Journal of Symplectic Geometry 15 (2017), no. 4, 1209-1250 https:/​/​doi.org/​10.4310/​JSG.2017.v15.n4.a8.
https:/​/​doi.org/​10.4310/​JSG.2017.v15.n4.a8

[39] M. Walter, Multipartite Quantum States and their Marginals, Ph.D. Thesis, ETH Zurich, 2014, https:/​/​arxiv.org/​abs/​1410.6820.
arXiv:1410.6820

[40] M. Walter, B. Doran, D. Gross, M. Christandl, Entanglement polytopes: multiparticle entanglement from single-particle information, Science 340 (2013), no. 6137, 1205–1208. MR3087706, https:/​/​doi.org/​10.1126/​science.1232957.
https:/​/​doi.org/​10.1126/​science.1232957

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[12] Jim Bryan, Zinovy Reichstein, and Mark Van Raamsdonk, "Existence of Locally Maximally Entangled Quantum States via Geometric Invariant Theory", Annales Henri Poincaré 19 8, 2491 (2018).

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