Entanglement-symmetries of covariant channels

Dominic Verdon

doi:10.22331/q-2024-02-29-1272

Entanglement-symmetries of covariant channels

Dominic Verdon

School of Mathematics, University of Bristol

Published:	2024-02-29, volume 8, page 1272
Eprint:	arXiv:2012.05761v9
Doi:	https://doi.org/10.22331/q-2024-02-29-1272
Citation:	Quantum 8, 1272 (2024).

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Updated version: The authors have uploaded version v10 of this work to the arXiv which may contain updates or corrections not contained in the published version v9. The authors left the following comment on the arXiv:

34 pages, many coloured pictures | Rev 10: Included missing funder information

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.

► BibTeX data

@article{Verdon2024entanglement,
  doi = {10.22331/q-2024-02-29-1272},
  url = {https://doi.org/10.22331/q-2024-02-29-1272},
  title = {Entanglement-symmetries of covariant channels},
  author = {Verdon, Dominic},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {8},
  pages = {1272},
  month = feb,
  year = {2024}
}

► 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, "Entanglement-invertible channels", Journal of Mathematical Physics 65 6, 062203 (2024).

[2] Sang-Jun Park, Yeong-Gwang Jung, Jeongeun Park, and Sang-Gyun Youn, "A universal framework for entanglement detection under group symmetry", Journal of Physics A: Mathematical and Theoretical 57 32, 325304 (2024).

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

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

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