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Quantum Channels with Quantum Group Symmetry

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Abstract

In this paper we will demonstrate that any compact quantum group can be used as symmetries for quantum channels, which leads us to the concept of covariant channels. We then unearth the structure of the convex set of covariant channels by identifying all extreme points under the assumption of multiplicity-free condition for the associated fusion rule, which provides a wide generalization of the results of Mozrzymas et al. (J Math Phys 58(5):052204, 2017). The presence of quantum group symmetry in contrast to the group symmetry will be highlighted in the examples of quantum permutation groups and \(SU_q(2)\). In the latter example, we will see the necessity of the Heisenberg picture coming from the non-Kac type condition. This paper ends with the covariance with respect to projective representations, which leads us back to Weyl covariant channels and its fermionic analogue.

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Acknowledgements

S.-G. Youn was funded by the New Faculty Startup Fund from Seoul National University, by Samsung Science and Technology Foundation under Project Number SSTF-BA2002-01 and by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2020R1C1C1A01009681). H.H. Lee was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) Grant NRF-2017R1E1A1A03070510 and the National Research Foundation of Korea (NRF) Grant funded by the Korean Government (MSIT) (Grant No. 2017R1A5A1015626).

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Authors and Affiliations

  1. Department of Mathematical Sciences and the Research Institute of Mathematics, Seoul National University, Gwanak-ro 1, Gwanak-gu, Seoul, 08826, Republic of Korea

    Hun Hee Lee

  2. Department of Mathematics Education, Seoul National University, Gwanak-ro 1, Gwanak-gu, Seoul, 08826, Republic of Korea

    Sang-Gyun Youn

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  1. Hun Hee Lee
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Correspondence to Sang-Gyun Youn.

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Communicated by Y. Kawahigashi.

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Lee, H.H., Youn, SG. Quantum Channels with Quantum Group Symmetry. Commun. Math. Phys. 389, 1303–1329 (2022). https://doi.org/10.1007/s00220-021-04283-9

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