Transformations of Stabilizer States in Quantum Networks

Matthias Englbrecht, Tristan Kraft, and Barbara Kraus

Institute for Theoretical Physics, University of Innsbruck, Technikerstraße 21A, 6020 Innsbruck, Austria

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Abstract

Stabilizer states and graph states find application in quantum error correction, measurement-based quantum computation and various other concepts in quantum information theory. In this work, we study party-local Clifford (PLC) transformations among stabilizer states. These transformations arise as a physically motivated extension of local operations in quantum networks with access to bipartite entanglement between some of the nodes of the network. First, we show that PLC transformations among graph states are equivalent to a generalization of the well-known local complementation, which describes local Clifford transformations among graph states. Then, we introduce a mathematical framework to study PLC equivalence of stabilizer states, relating it to the classification of tuples of bilinear forms. This framework allows us to study decompositions of stabilizer states into tensor products of indecomposable ones, that is, decompositions into states from the entanglement generating set (EGS). While the EGS is finite up to $3$ parties [Bravyi et al., J. Math. Phys. $\bf{47}$, 062106 (2006)], we show that for $4$ and more parties it is an infinite set, even when considering party-local unitary transformations. Moreover, we explicitly compute the EGS for $4$ parties up to $10$ qubits. Finally, we generalize the framework to qudit stabilizer states in prime dimensions not equal to $2$, which allows us to show that the decomposition of qudit stabilizer states into states from the EGS is unique.

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► References

[1] H. J. Kimble, "The quantum internet.", Nature 453, 1023 (2008).
https:/​/​doi.org/​10.1038/​nature07127

[2] S. Wehner, D. Elkouss, and R. Hanson, "Quantum internet: A vision for the road ahead.", Science 362, 9288 (2018).
https:/​/​doi.org/​10.1126/​science.aam9288

[3] J. I. Cirac, P. Zoller, H. J. Kimble, and H. Mabuchi, "Quantum State Transfer and Entanglement Distribution among Distant Nodes in a Quantum Network.", Phys. Rev. Lett. 78, 3221 (1997).
https:/​/​doi.org/​10.1103/​PhysRevLett.78.3221

[4] S. Muralidharan, L. Li, J. Kim, N. Lütkenhaus, M. D. Lukin, and L. Jiang, "Efficient long distance quantum communication.", Sci. Rep. 6, 20463 (2016).
https:/​/​doi.org/​10.1038/​srep20463

[5] S. Perseguers, G. J. Lapeyre Jr, D. Cavalcanti, M. Lewenstein, and A. Acín, "Distribution of entanglement in large-scale quantum networks.", Rep. Prog. Phys. 76, 096001 (2013).
https:/​/​doi.org/​10.1088/​0034-4885/​76/​9/​096001

[6] L.-M. Duan, and C. Monroe, " Colloquium: Quantum networks with trapped ions.", Rev. Mod. Phys. 82, 1209 (2010).
https:/​/​doi.org/​10.1103/​RevModPhys.82.1209

[7] A. Reiserer, and G. Rempe, "Cavity-based quantum networks with single atoms and optical photons.", Rev. Mod. Phys. 87,1379 (2015).
https:/​/​doi.org/​10.1103/​RevModPhys.87.1379

[8] J. I. Cirac, A. K. Ekert, S. F. Huelga, and C. Macchiavello, "Distributed quantum computation over noisy channels.", Phys. Rev. A 59, 4249 (1999).
https:/​/​doi.org/​10.1103/​PhysRevA.59.4249

[9] T. J. Proctor, P. A. Knott, and J. A. Dunningham, "Multiparameter Estimation in Networked Quantum Sensors.", Phys. Rev. Lett. 120, 080501 (2018).
https:/​/​doi.org/​10.1103/​PhysRevLett.120.080501

[10] L.-M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, "Long-distance quantum communication with atomic ensembles and linear optics.", Nature 414, 413 (2001).
https:/​/​doi.org/​10.1038/​35106500

[11] A. Tavakoli, A. Pozas-Kerstjens, M. Luo, and M.-O. Renou, "Bell nonlocality in networks.", Rep. Prog. Phys. (2021).
https:/​/​doi.org/​10.1088/​1361-6633/​ac41bb

[12] O. Gühne, and G. Tóth, "Entanglement detection.", Phys. Rep. 474, 1 (2009).
https:/​/​doi.org/​10.1016/​j.physrep.2009.02.004

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

[14] I. Bengtsson, and K. Zyczkowski, "Geometry of Quantum States.", (Cambridge University Press, 2006).
https:/​/​doi.org/​10.1017/​CBO9780511535048

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

[16] T. Kraft, S. Designolle, C. Ritz, N. Brunner, O. Gühne, and M. Huber, "Quantum entanglement in the triangle network.", Phys. Rev. A 103, L060401 (2021).
https:/​/​doi.org/​10.1103/​PhysRevA.103.L060401

[17] M. Navascués, E. Wolfe, D. Rosset, and A. Pozas-Kerstjens, "Genuine Network Multipartite Entanglement.", Phys. Rev. Lett. 125, 240505 (2020).
https:/​/​doi.org/​10.1103/​PhysRevLett.125.240505

[18] K. Hansenne, Z.-P. Xu, T. Kraft, and O. Gühne, "Symmetries in quantum networks lead to no-go theorems for entanglement distribution and to verification techniques.", Nat. Commun. 13, 496 (2022).
https:/​/​doi.org/​10.1038/​s41467-022-28006-3

[19] M. Hein, W. Dür, J. Eisert, R. Raussendorf, M. Van den Nest, and H.-J. Briegel, "Entanglement in Graph States and its Applications.", in Quantum Computers, Algorithms and Chaos, edited by G. Casati, D. L. Shepelyansky, P. Zoller, and G. Benenti (IOS Press, Amsterdam, 2006).
arXiv:quant-ph/0602096

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

[21] D. Gottesman, "Stabilizer Codes and Quantum Error Correction.", PhD thesis, California Institute of Technology Pasadena, California (1997).
arXiv:quant-ph/9705052

[22] R. Raussendorf and H.J. Briegel, "A One-Way Quantum Computer.", Phys. Rev. Lett. 86, 5188 (2001).
https:/​/​doi.org/​10.1103/​PhysRevLett.86.5188

[23] D. Gottesman, "The Heisenberg Representation of Quantum Computers.", in Proceedings of the XXII International Colloquium on Group Theoretical Methods in Physics, edited by S. P. Corneyet al. (International Press, Cambridge, MA 1999).
arXiv:quant-ph/9807006

[24] D. Schlingemann and R. F. Werner, "Quantum error-correcting codes associated with graphs.", Phys. Rev. A 65, 012308 (2001).
https:/​/​doi.org/​10.1103/​PhysRevA.65.012308

[25] D. Gottesman, and I. L. Chuang, "Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations.", Nature 402, 390 (1999).
https:/​/​doi.org/​10.1038/​46503

[26] M. Van den Nest, J. Dehaene, and B. De Moor, "Graphical description of the action of local Clifford transformations on graph states.", Phys. Rev. A 69, 022316 (2004).
https:/​/​doi.org/​10.1103/​PhysRevA.69.022316

[27] D. Fattal, T. S. Cubitt, Y. Yamamoto, S. Bravyi, & I. L. Chuang, "Entanglement in the stabilizer formalism.", arXiv:quant-ph/​0406168 (2004).
arXiv:quant-ph/0406168

[28] S. Bravyi, D. Fattal, and D. Gottesman, "GHZ extraction yield for multipartite stabilizer states.", J. Math. Phys. 47, 062106 (2006).
https:/​/​doi.org/​10.1063/​1.2203431

[29] S. Y. Looi, and R. B. Griffiths, "Tripartite entanglement in qudit stabilizer states and application in quantum error correction.", Phys. Rev. A 84, 052306 (2011).
https:/​/​doi.org/​10.1103/​PhysRevA.84.052306

[30] K. Wirthmüller, "Homological invariants of stabilizer states." Quantum Inf. Comput. 8, 595 (2008).
arXiv:quant-ph/0611131

[31] G. Smith, and D. Leung, "Typical entanglement of stabilizer states.", Phys. Rev. A 74, 062314 (2006).
https:/​/​doi.org/​10.1103/​PhysRevA.74.062314

[32] K. Wirthmüller, "Homology of Generic Stabilizer States.", arXiv:0809.3346 (2008).
arXiv:0809.3346

[33] S. Nezami, and M. Walter, "Multipartite Entanglement in Stabilizer Tensor Networks.", Phys. Rev. Lett. 125, 241602 (2020).
https:/​/​doi.org/​10.1103/​PhysRevLett.125.241602

[34] V. V. Sergeĭchuk, "Classification problems for systems of forms and linear mappings.", Math. USSR Izv. 31, 481 (1988).
https:/​/​doi.org/​10.1070/​im1988v031n03abeh001086

[35] A. Dahlberg, J. Helsen, and S. Wehner, "Counting single-qubit Clifford equivalent graph states is #P-Complete.", J. Math. Phys. 61, 022202 (2020).
https:/​/​doi.org/​10.48550/​arXiv.1907.08024

[36] F. E. S. Steinhoff, C. Ritz, N. Miklin, and O. Gühne, "Qudit Hypergraph States.", Phys. Rev. A 95, 052340 (2017).
https:/​/​doi.org/​10.1103/​PhysRevA.95.052340

[37] C. Ritz, "Characterizing the structure of multiparticle entanglement in high-dimensional systems.", PhD thesis, University of Siegen 2018.
https:/​/​nbn-resolving.org/​urn:nbn:de:hbz:467-14299

[38] M. Englbrecht, and B. Kraus, "Symmetries and entanglement of stabilizer states.", Phys. Rev. A 101, 062302 (2020).
https:/​/​doi.org/​10.1103/​PhysRevA.101.062302

[39] D. H. Zhang, H. Fan, and D. L. Zhou, "Stabilizer dimension of graph states.", Phys. Rev. A 79, 042318 (2009).
https:/​/​doi.org/​10.1103/​PhysRevA.79.042318

[40] A. Bouchet, "An efficient algorithm to recognize locally equivalent graphs.", Combinatorica 11, 315 (1991).
https:/​/​doi.org/​10.1007/​BF01275668

[41] A. Bouchet, "Recognizing locally equivalent graphs." Discrete Math. 114, 75 (1993).
https:/​/​doi.org/​10.1016/​0012-365X(93)90357-Y

[42] S.-K. Liao et al., "Satellite-Relayed Intercontinental Quantum Network." Phys. Rev. Lett. 120, 030501 (2018).
https:/​/​doi.org/​10.1103/​PhysRevLett.120.030501

[43] J. I. Cirac, W. Dür, B. Kraus, and M Lewenstein, "Entangling operations and their implementation using a small amount of entanglement." Phys. Rev. Lett. 86, 544 (2001).
https:/​/​doi.org/​10.1103/​PhysRevLett.86.544

[44] G. Gour, and 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

[45] M. Van den Nest, J. Dehaene, and B. De Moor, "Finite set of invariants to characterize local Clifford equivalence of stabilizer states.", Phys. Rev. A 72, 014307 (2005).
https:/​/​doi.org/​10.1103/​PhysRevA.72.014307

[46] Z. Ji, J. Chen, Z. Wei, and M. Ying, "The LU-LC conjecture is false.", Quantum Inf. Comput. 10, 97-108 (2010).
arXiv:0709.1266

[47] R. Scharlau, "Paare alternierender Formen.", Math. Z. 147, 13-19 (1976).
http:/​/​eudml.org/​doc/​172315

[48] G. Belitskii, R. Lipyanski, and V. V. Sergeichuk, "Problems of classifying associative or Lie algebras and triples of symmetric or skew-symmetric matrices are wild.", Linear Algebra Its Appl. 407, 249-262 (2005).
https:/​/​doi.org/​10.1016/​j.laa.2005.05.007

[49] M. Barot, "Representations of quivers.", available at https:/​/​www.matem.unam.mx/​ barot/​articles/​ notes_ictp.pdf.
https:/​/​www.matem.unam.mx/​~barot/​articles/​notes_ictp.pdf

[50] L. E. Danielsen, Database of Entanglement in Graph States (2011), available at http:/​/​www.ii.uib.no/​ larsed/​entanglement/​.
http:/​/​www.ii.uib.no/​~larsed/​entanglement/​

[51] E. Hostens, J. Dehaene, and B. De Moor, "Stabilizer states and Clifford operations for systems of arbitrary dimensions and modular arithmetic.", Phys. Rev. A 71, 042315 (2005).
https:/​/​doi.org/​10.1103/​PhysRevA.71.042315

[52] J. M. Farinholt, "An ideal characterization of the Clifford operators." J. Phys. A 47, 305303 (2014).
https:/​/​doi.org/​10.1088/​1751-8113/​47/​30/​305303

[53] M. Bahramgiri, and S. Beigi, "Graph states under the action of local Clifford group in non-binary case.", arXiv:quant-ph/​0610267 (2006).
arXiv:quant-ph/0610267

[54] T. Kraft, C. Ritz, N. Brunner, M. Huber, and O. Gühne, "Characterizing Genuine Multilevel Entanglement.", Phys. Rev. Lett. 120, 060502 (2018).
https:/​/​doi.org/​10.1103/​PhysRevLett.120.060502

[55] A. Neven, D. K. Gunn, M. Hebenstreit, and B. Kraus, "Local transformations of multiple multipartite states.", SciPost Phys., 11, 042 (2021).
https:/​/​doi.org/​10.21468/​SciPostPhys.11.2.042

[56] C. Kruszynska, and B. Kraus. "Local entanglability and multipartite entanglement." Phys. Rev. A 79, 052304 (2009).
https:/​/​doi.org/​10.1103/​PhysRevA.79.052304

[57] M. Aschbacher, Finite group theory, (Cambridge University Press, Cambridge, MA, 2000).
https:/​/​doi.org/​10.1017/​CBO9781139175319

[58] S. Lang, Algebra, (Springer, New York, NY, 2002).
https:/​/​doi.org/​10.1007/​978-1-4613-0041-0

[59] D. Gross, "Hudson’s theorem for finite-dimensional quantum systems." J. Math. Phys. 47, 122107 (2006).
https:/​/​doi.org/​10.1063/​1.2393152

[60] F. J. MacWilliams, and N. J. Sloane, The Theory of Error-Correcting Codes, North-Holland Publishing Company (1977).

[61] G. Marsaglia, "Bounds for the Rank of the Sum of Two Matrices." Mathematical Note, Boeing Scientific Research Labs, Seattle, WA, (1964).
https:/​/​apps.dtic.mil/​sti/​pdfs/​AD0600471.pdf

[62] C. G. Bartolone, and M. A. Vaccaro. "The action of the symplectic group associated with a quadratic extension of fields." J. Algebra 220, 115 (1999).
https:/​/​doi.org/​10.1006/​jabr.1999.7921

[63] G. Falcone, M. A. Vaccaro, "Kronecker modules and reductions of a pair of bilinear forms.", Acta Univ. Palacki. Olomuc., Fac. rer. nat., Mathematica 43, 55 (2004).
https:/​/​dml.cz/​bitstream/​handle/​10338.dmlcz/​132949/​ActaOlom_43-2004-1_5.pdf

[64] J. Dieudonné, "Sur la réduction canonique des couples de matrices." Bull. Soc. Math. Fr. 74, 130 (1946).
https:/​/​doi.org/​10.24033/​bsmf.1380

[65] Y. A. Drozd, and A. I. Plakosh, "On nilpotent Chernikov 2-groups with elementary tops.", Algebra Discrete Math. 22, 201 (2016).
arXiv:1610.05030

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