A link between symmetries of critical states and the structure of SLOCC classes in multipartite systems

Oskar Słowik1, Martin Hebenstreit2, Barbara Kraus2, and Adam Sawicki1

1Center for Theoretical Physics, Polish Academy of Sciences, 02-668 Warsaw, Poland
2Institute for Theoretical Physics, University of Innsbruck, A–6020 Innsbruck, Austria

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.

Abstract

Central in entanglement theory is the characterization of local transformations among pure multipartite states. As a first step towards such a characterization, one needs to identify those states which can be transformed into each other via local operations with a non-vanishing probability. The classes obtained in this way are called SLOCC classes. They can be categorized into three disjoint types: the null-cone, the polystable states and strictly semistable states. Whereas the former two are well characterized, not much is known about strictly semistable states. We derive a criterion for the existence of the latter. In particular, we show that there exists a strictly semistable state if and only if there exist two polystable states whose orbits have different dimensions. We illustrate the usefulness of this criterion by applying it to tripartite states where one of the systems is a qubit. Moreover, we scrutinize all SLOCC classes of these systems and derive a complete characterization of the corresponding orbit types. We present representatives of strictly semistable classes and show to which polystable state they converge via local regular operators.

► BibTeX data

► References

[1] M. A. Nielsen and I. L. Chuang, Quantum computation and quantum information (Cambridge University Press, Cambridge, 2009).

[2] R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Rev. Mod. Phys. 81, 865 (2009).
https:/​/​doi.org/​10.1103/​RevModPhys.81.865

[3] M. Hillery, V. Bužek, and A. Berthiaume, Phys. Rev. A 59, 1829 (1999).
https:/​/​doi.org/​10.1103/​PhysRevA.59.1829

[4] D. Gottesman, Phys. Rev. A 61, 042311 (2000).
https:/​/​doi.org/​10.1103/​PhysRevA.61.042311

[5] R. Raussendorf and H. J. Briegel, Phys. Rev. Lett. 86, 5188 (2001).
https:/​/​doi.org/​10.1103/​PhysRevLett.86.5188

[6] V. Giovannetti, S. Lloyd, and L. Maccone, Nature Photonics 5, 222 (2011).
https:/​/​doi.org/​10.1038/​nphoton.2011.35

[7] L. Amico, R. Fazio, A. Osterloh, and V. Vedral, Rev. Mod. Phys. 80, 517 (2008).
https:/​/​doi.org/​10.1103/​RevModPhys.80.517

[8] M. Fannes, B. Nachtergaele, and R. F. Werner, Communications in Mathematical Physics 144, 443 (1992).
https:/​/​doi.org/​10.1007/​BF02099178

[9] D. Perez-Garcia, F. Verstraete, M. M. Wolf, and J. I. Cirac, Quantum Info. Comput. 7, 401 (2007).

[10] G. Vidal, Phys. Rev. Lett. 91, 147902 (2003).
https:/​/​doi.org/​10.1103/​PhysRevLett.91.147902

[11] F. Verstraete and J. I. Cirac, Renormalization algorithms for Quantum-Many Body Systems in two and higher dimensions, arXiv:cond-mat/​0407066 [cond-mat.str-el] (2004).
arXiv:cond-mat/0407066

[12] F. Verstraete, M. M. Wolf, D. Perez-Garcia, and J. I. Cirac, Phys. Rev. Lett. 96, 220601 (2006).
https:/​/​doi.org/​10.1103/​PhysRevLett.96.220601

[13] R. Orús, Annals of Physics 349, 117 (2014).
https:/​/​doi.org/​10.1016/​j.aop.2014.06.013

[14] W. Dür, G. Vidal, and J. I. Cirac, Phys. Rev. A 62, 062314 (2000).
https:/​/​doi.org/​10.1103/​PhysRevA.62.062314

[15] B. Kraus, Phys. Rev. Lett. 104, 020504 (2010a).
https:/​/​doi.org/​10.1103/​PhysRevLett.104.020504

[16] B. Kraus, Phys. Rev. A 82, 032121 (2010b).
https:/​/​doi.org/​10.1103/​PhysRevA.82.032121

[17] M. A. Nielsen, Phys. Rev. Lett. 83, 436 (1999).
https:/​/​doi.org/​10.1103/​PhysRevLett.83.436

[18] E. Chitambar, D. Leung, L. Mančinska, M. Ozols, and A. Winter, Communications in Mathematical Physics 328, 303 (2014).
https:/​/​doi.org/​10.1007/​s00220-014-1953-9

[19] E. Chitambar, Phys. Rev. Lett. 107, 190502 (2011).
https:/​/​doi.org/​10.1103/​PhysRevLett.107.190502

[20] E. Chitambar and M.-H. Hsieh, Nature Communications 8, 2086 (2017).
https:/​/​doi.org/​10.1038/​s41467-017-01887-5

[21] F. Verstraete, J. Dehaene, B. De Moor, and H. Verschelde, Phys. Rev. A 65, 052112 (2002a).
https:/​/​doi.org/​10.1103/​PhysRevA.65.052112

[22] P. Mathonet, S. Krins, M. Godefroid, L. Lamata, E. Solano, and T. Bastin, Phys. Rev. A 81, 052315 (2010).
https:/​/​doi.org/​10.1103/​PhysRevA.81.052315

[23] T. Bastin, S. Krins, P. Mathonet, M. Godefroid, L. Lamata, and E. Solano, Phys. Rev. Lett. 103, 070503 (2009).
https:/​/​doi.org/​10.1103/​PhysRevLett.103.070503

[24] P. Migdał, J. Rodriguez-Laguna, and M. Lewenstein, Phys. Rev. A 88, 012335 (2013).
https:/​/​doi.org/​10.1103/​PhysRevA.88.012335

[25] J.-G. Luque and J.-Y. Thibon, Phys. Rev. A 67, 042303 (2003).
https:/​/​doi.org/​10.1103/​PhysRevA.67.042303

[26] O. Viehmann, C. Eltschka, and J. Siewert, Phys. Rev. A 83, 052330 (2011).
https:/​/​doi.org/​10.1103/​PhysRevA.83.052330

[27] C. Eltschka and J. Siewert, Journal of Physics A: Mathematical and Theoretical 47, 424005 (2014).
https:/​/​doi.org/​10.1088/​1751-8113/​47/​42/​424005

[28] G. Gour and N. R. Wallach, Phys. Rev. Lett. 111, 060502 (2013).
https:/​/​doi.org/​10.1103/​PhysRevLett.111.060502

[29] A. Osterloh and J. Siewert, Phys. Rev. A 72, 012337 (2005).
https:/​/​doi.org/​10.1103/​PhysRevA.72.012337

[30] G. Gour, B. Kraus, and N. R. Wallach, Journal of Mathematical Physics 58, 092204 (2017).
https:/​/​doi.org/​10.1063/​1.5003015

[31] D. Sauerwein, N. R. Wallach, G. Gour, and B. Kraus, Phys. Rev. X 8, 031020 (2018a).
https:/​/​doi.org/​10.1103/​PhysRevX.8.031020

[32] J. Bryan, S. Leutheusser, Z. Reichstein, and M. V. Raamsdonk, Quantum 3, 115 (2019).
https:/​/​doi.org/​10.22331/​q-2019-01-06-115

[33] G. Kempf and L. Ness, in Algebraic Geometry, edited by K. Lønsted (Springer Berlin Heidelberg, Berlin, Heidelberg, 1979) pp. 233–243.

[34] A. Osterloh and J. Siewert, New Journal of Physics 12, 075025 (2010).
https:/​/​doi.org/​10.1088/​1367-2630/​12/​7/​075025

[35] A. Klyachko, Coherent states, entanglement, and geometric invariant theory (2002), arXiv:quant-ph/​0206012 [quant-ph].
arXiv:quant-ph/0206012

[36] G. Gour and N. R. Wallach, New Journal of Physics 13, 073013 (2011).
https:/​/​doi.org/​10.1088/​1367-2630/​13/​7/​073013

[37] D. Sauerwein, K. Schwaiger, and B. Kraus, Discrete and differentiable entanglement transformations (2018b), arXiv:1808.02819 [quant-ph].
arXiv:1808.02819

[38] L. Ness and D. Mumford, American Journal of Mathematics 106, 1281 (1984).
https:/​/​doi.org/​10.2307/​2374395

[39] M. Walter, B. Doran, D. Gross, and M. Christandl, Science 340, 1205 (2013).
https:/​/​doi.org/​10.1126/​science.1232957

[40] A. Sawicki, M. Oszmaniec, and M. Kuś, Reviews in Mathematical Physics 26, 1450004 (2014).
https:/​/​doi.org/​10.1142/​S0129055X14500044

[41] T. Maciążek and A. Sawicki, Journal of Physics A: Mathematical and Theoretical 48, 045305 (2015).
https:/​/​doi.org/​10.1088/​1751-8113/​48/​4/​045305

[42] T. Maciążek and A. Sawicki, Journal of Physics A: Mathematical and Theoretical 51, 07LT01 (2018).
https:/​/​doi.org/​10.1088/​1751-8121/​aaa4d7

[43] M. Johansson, M. Ericsson, E. Sjöqvist, and A. Osterloh, Phys. Rev. A 89, 012320 (2014).
https:/​/​doi.org/​10.1103/​PhysRevA.89.012320

[44] F. Verstraete, J. Dehaene, and B. De Moor, Phys. Rev. A 68, 012103 (2003).
https:/​/​doi.org/​10.1103/​PhysRevA.68.012103

[45] A. Sawicki, M. Oszmaniec, and M. Kuś, Phys. Rev. A 86, 040304 (2012).
https:/​/​doi.org/​10.1103/​PhysRevA.86.040304

[46] A. Sawicki, T. Maciążek, M. Oszmaniec, K. Karnas, K. Kowalczyk-Murynka, and M. Kuś, Rep. Math. Phys. 82, 81 (2018).
https:/​/​doi.org/​10.1016/​S0034-4877(18)30072-7

[47] E. Chitambar, C. A. Miller, and Y. Shi, Journal of Mathematical Physics 51, 072205 (2010).
https:/​/​doi.org/​10.1063/​1.3459069

[48] M. Hebenstreit, M. Gachechiladze, O. Gühne, and B. Kraus, Phys. Rev. A 97, 032330 (2018).
https:/​/​doi.org/​10.1103/​PhysRevA.97.032330

[49] L. Kronecker, Sitzungsberichte d. Preußischen Adad. d. Wissenschaften , 763 (1890).

[50] N. R. Wallach, Geometric invariant theory over the real and complex numbers, 1st Edition (Springer-Verlag, 2017).
https:/​/​doi.org/​10.1007/​978-3-319-65907-7

[51] M. Brion, Les cours du CIRM 1, 1 (2010).
https:/​/​ccirm.centre-mersenne.org/​item/​?id=CCIRM_2010__1_1_1_0

[52] S. Mukai, An Introduction to Invariants and Moduli (Cambridge University Press, 2003).
https:/​/​doi.org/​10.1017/​CBO9781316257074

[53] I. Dolgachev, Lectures on Invariant Theory, London Mathematical Society Lecture Note Series (296) (Cambridge University Press, 2010).
https:/​/​doi.org/​10.1017/​CBO9780511615436

[54] M. Johansson, M. Ericsson, K. Singh, E. Sjöqvist, and M. S. Williamson, Phys. Rev. A 89, 012320 (2012).
https:/​/​doi.org/​10.1103/​PhysRevA.89.012320

[55] F. Verstraete, J. Dehaene, and B. De Moor, Phys. Rev. A 65, 032308 (2002b).
https:/​/​doi.org/​10.1103/​PhysRevA.65.032308

[56] D. Mumford, The Red Book of Varieties and Schemes (Springer, 1999).
https:/​/​doi.org/​10.1007/​b62130

[57] P. Tauvel and R. Yu, Introduction to actions of algebraic groups (Springer, 2015).

[58] F. Gantmacher, The Theory of Matrices, Vol. 1 and 2 (Chelsea Publishing Company, 1959).

[59] D. Sauerwein, A. Molnar, J. I. Cirac, and B. Kraus, Phys. Rev. Lett. 123, 170504 (2019).
https:/​/​doi.org/​10.1103/​PhysRevLett.123.170504

[60] V. Hoskins, Lecture notes, Zurich (2012).

[61] K. Smith, L. Kahanpää, P. Kekäläinen, and W. Traves, An Invitation to Algebraic Geometry (Springer, 2000).
https:/​/​doi.org/​10.1007/​978-1-4757-4497-2

[62] I. R. Shafarevich, Basic Algebraic Geometry, Third Edition (Springer, 2013).
https:/​/​doi.org/​10.1007/​978-3-642-37956-7

[63] R. Duan and Y. Shi, Quantum Info. Comput. 10, 925 (2010).
https:/​/​doi.org/​10.26421/​QIC10.11-12

Cited by

[1] Adam Burchardt and Zahra Raissi, "Stochastic local operations with classical communication of absolutely maximally entangled states", arXiv:2003.13639, Physical Review A 102 2, 022413 (2020).

[2] Antoine Neven, David Gunn, Martin Hebenstreit, and Barbara Kraus, "Local Transformations of Multiple Multipartite States", arXiv:2007.06256.

The above citations are from Crossref's cited-by service (last updated successfully 2020-09-20 19:20:04) and SAO/NASA ADS (last updated successfully 2020-09-20 21:20:16). The list may be incomplete as not all publishers provide suitable and complete citation data.