Entanglement-symmetries of covariant channels

Dominic Verdon

School of Mathematics, University of Bristol

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Abstract

Let $G$ and $G'$ be monoidally equivalent compact quantum groups, and let $H$ be a Hopf-Galois object realising a monoidal equivalence between these groups' representation categories. This monoidal equivalence induces an equivalence Chan($G$) $\rightarrow$ Chan($G'$), where Chan($G$) is the category whose objects are finite-dimensional $C*$-algebras with an action of G and whose morphisms are covariant channels. We show that, if the Hopf-Galois object $H$ has a finite-dimensional *-representation, then channels related by this equivalence can simulate each other using a finite-dimensional entangled resource. We use this result to calculate the entanglement-assisted capacities of certain quantum channels.

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► References

[1] Samson Abramsky and Bob Coecke. A categorical semantics of quantum protocols. In Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science, 2004., pages 415–425. IEEE, 2004. arXiv:quant-ph/​0402130, doi:10.1109/​LICS.2004.1319636.
https:/​/​doi.org/​10.1109/​LICS.2004.1319636
arXiv:quant-ph/0402130

[2] Albert Atserias, Laura Mančinska, David E Roberson, Robert Šámal, Simone Severini, and Antonios Varvitsiotis. Quantum and non-signalling graph isomorphisms. Journal of Combinatorial Theory, Series B, 136:289–328, 2019. arXiv:1611.09837, doi:10.1016/​j.jctb.2018.11.002.
https:/​/​doi.org/​10.1016/​j.jctb.2018.11.002
arXiv:1611.09837

[3] Michael Brannan, Alexandru Chirvasitu, Kari Eifler, Samuel Harris, Vern Paulsen, Xiaoyu Su, and Mateusz Wasilewski. Bigalois extensions and the graph isomorphism game. Communications in Mathematical Physics, pages 1–33, 2019. arXiv:1812.11474, doi:10.1007/​s00220-019-03563-9.
https:/​/​doi.org/​10.1007/​s00220-019-03563-9
arXiv:1812.11474

[4] Michael Brannan, Priyanga Ganesan, and Samuel J Harris. The quantum-to-classical graph homomorphism game. 2020. arXiv:2009.07229, doi:10.1063/​5.0072288.
https:/​/​doi.org/​10.1063/​5.0072288
arXiv:2009.07229

[5] Julien Bichon. Galois extension for a compact quantum group. 1999. arXiv:math/​9902031.
arXiv:math/9902031

[6] M. Bischoff, Y. Kawahigashi, R. Longo, and K.H. Rehren. Tensor Categories and Endomorphisms of von Neumann Algebras: with Applications to Quantum Field Theory. Springer Briefs in Mathematical Physics. Springer International Publishing, 2015. arXiv:1407.4793.
arXiv:1407.4793

[7] Charles H Bennett, Peter W Shor, John A Smolin, and Ashish V Thapliyal. Entanglement-assisted classical capacity of noisy quantum channels. Physical Review Letters, 83(15):3081, 1999. arXiv:quant-ph/​9904023, doi:10.1103/​PhysRevLett.83.3081.
https:/​/​doi.org/​10.1103/​PhysRevLett.83.3081
arXiv:quant-ph/9904023

[8] Bob Coecke, Chris Heunen, and Aleks Kissinger. Categories of quantum and classical channels. Quantum Information Processing, 15(12):5179–5209, 2016. arXiv:1305.3821, doi:10.1007/​s11128-014-0837-4.
https:/​/​doi.org/​10.1007/​s11128-014-0837-4
arXiv:1305.3821

[9] Bob Coecke, Dusko Pavlovic, and Jamie Vicary. A new description of orthogonal bases. Mathematical Structures in Computer Science, 23(3):555–567, 2013. arXiv:0810.0812, doi:10.1017/​S0960129512000047.
https:/​/​doi.org/​10.1017/​S0960129512000047
arXiv:0810.0812

[10] P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik. Tensor Categories. Mathematical Surveys and Monographs. American Mathematical Society, 2016. URL: http:/​/​www-math.mit.edu/​ etingof/​egnobookfinal.pdf.
http:/​/​www-math.mit.edu/​~etingof/​egnobookfinal.pdf

[11] Chris Heunen, Ivan Contreras, and Alberto S Cattaneo. Relative Frobenius algebras are groupoids. Journal of Pure and Applied Algebra, 217(1):114–124, 2013. arXiv:1112.1284, doi:10.1016/​j.jpaa.2012.04.002.
https:/​/​doi.org/​10.1016/​j.jpaa.2012.04.002
arXiv:1112.1284

[12] Chris Heunen and Jamie Vicary. Categories for Quantum Theory: An Introduction. Oxford Graduate Texts in Mathematics Series. Oxford University Press, 2019. doi:10.1093/​oso/​9780198739623.001.0001.
https:/​/​doi.org/​10.1093/​oso/​9780198739623.001.0001

[13] Emanuel Knill. Non-binary unitary error bases and quantum codes. Technical Report LAUR-96-2717, LANL, 1996. arXiv:quant-ph/​9608048.
arXiv:quant-ph/9608048

[14] Joachim Kock. Frobenius Algebras and 2-D Topological Quantum Field Theories. London Mathematical Society Student Texts. Cambridge University Press, 2003. doi:10.1017/​CBO9780511615443.
https:/​/​doi.org/​10.1017/​CBO9780511615443

[15] Paul-André Melliès. Functorial boxes in string diagrams. In International Workshop on Computer Science Logic, pages 1–30. Springer, 2006. URL: https:/​/​www.irif.fr/​ mellies/​mpri/​mpri-ens/​articles/​mellies-functorial-boxes.pdf, doi:10.1007/​11874683_1.
https:/​/​doi.org/​10.1007/​11874683_1
https:/​/​www.irif.fr/​~mellies/​mpri/​mpri-ens/​articles/​mellies-functorial-boxes.pdf

[16] Benjamin Musto, David Reutter, and Dominic Verdon. A compositional approach to quantum functions. Journal of Mathematical Physics, 59(8):081706, 2018. arXiv:1711.07945, doi:10.1063/​1.5020566.
https:/​/​doi.org/​10.1063/​1.5020566
arXiv:1711.07945

[17] Benjamin Musto, David Reutter, and Dominic Verdon. The Morita theory of quantum graph isomorphisms. Communications in Mathematical Physics, 365(2):797–845, 2019. arXiv:1801.09705, doi:10.1007/​s00220-018-3225-6.
https:/​/​doi.org/​10.1007/​s00220-018-3225-6
arXiv:1801.09705

[18] Sergey Neshveyev and Lars Tuset. Compact Quantum Groups and Their Representation Categories. Collection SMF.: Cours spécialisés. Société Mathématique de France, 2013.

[19] Sergey Neshveyev and Makoto Yamashita. Categorically Morita equivalent compact quantum groups. Documenta Mathematica, 23:2165–2216, 2018. arXiv:1704.04729, doi:10.25537/​dm.2018v23.2165-2216.
https:/​/​doi.org/​10.25537/​dm.2018v23.2165-2216
arXiv:1704.04729

[20] Viktor Ostrik. Module categories over the Drinfeld double of a finite group. International Mathematics Research Notices, 2003(27):1507–1520, 01 2003. arXiv:math/​0202130, doi:10.1155/​S1073792803205079.
https:/​/​doi.org/​10.1155/​S1073792803205079
arXiv:math/0202130

[21] Peter Selinger. A survey of graphical languages for monoidal categories. In New Structures for Physics, pages 289–355. Springer, 2010. arXiv:0908.3347, doi:10.1007/​978-3-642-12821-9_4.
https:/​/​doi.org/​10.1007/​978-3-642-12821-9_4
arXiv:0908.3347

[22] Thomas Timmerman. An invitation to quantum groups and duality. EMS Textbooks in Mathematics. European Mathematical Society Publishing House, 2008. doi:10.4171/​043.
https:/​/​doi.org/​10.4171/​043

[23] Ivan G Todorov and Lyudmila Turowska. Quantum no-signalling correlations and non-local games. 2020. arXiv:2009.07016.
arXiv:2009.07016

[24] Dominic Verdon. Unitary pseudonatural transformations. 2020. arXiv:2004.12760.
arXiv:2004.12760

[25] Dominic Verdon. A covariant Stinespring theorem. Journal of Mathematical Physics, 63(9):091705, 2022. arXiv:2108.09872, doi:10.1063/​5.0071215.
https:/​/​doi.org/​10.1063/​5.0071215
arXiv:2108.09872

[26] Dominic Verdon. Entanglement-invertible channels. 2022. arXiv:2204.04493.
arXiv:2204.04493

[27] Dominic Verdon. Unitary transformations of fibre functors. Journal of Pure and Applied Algebra, 226(7), July 2022. arXiv:2004.12761, doi:10.1016/​j.jpaa.2021.106989.
https:/​/​doi.org/​10.1016/​j.jpaa.2021.106989
arXiv:2004.12761

[28] Jamie Vicary. Categorical formulation of finite-dimensional quantum algebras. Communications in Mathematical Physics, 304(3):765–796, 2011. arXiv:0805.0432, doi:10.1007/​s00220-010-1138-0.
https:/​/​doi.org/​10.1007/​s00220-010-1138-0
arXiv:0805.0432

[29] Shuzhou Wang. Quantum symmetry groups of finite spaces. Communications in Mathematical Physics, 195:195–211, 1998. arXiv:math/​9807091, doi:10.1007/​s002200050385.
https:/​/​doi.org/​10.1007/​s002200050385
arXiv:math/9807091

Cited by

[1] Dominic Verdon, "A covariant Stinespring theorem", Journal of Mathematical Physics 63 9, 091705 (2022).

[2] Dominic Verdon, "Entanglement-invertible channels", arXiv:2204.04493, (2022).

[3] Dominic Verdon, "Unitary transformations of fibre functors", arXiv:2004.12761, (2020).

[4] Dominic Verdon, "Covariant Quantum Combinatorics with Applications to Zero-Error Communication", Communications in Mathematical Physics 405 2, 51 (2024).

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