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Temperley–Lieb Quantum Channels

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Abstract

We study a class of quantum channels arising from the representation theory of compact quantum groups that we call Temperley–Lieb quantum channels. These channels simultaneously extend those introduced by Brannan and Collins (Commun Math Phys 358(3):1007–1025, 2018), Nuwairan (Int J Math 25(6):1450048, 2014) and Lieb and Solovej (Acta Math 212(2):379–398, 2014). (Quantum) Symmetries in quantum information theory arise naturally from many points of view, providing an important source of new examples of quantum phenomena, and also serve as useful tools to simplify or solve important problems. This work provides new applications of quantum symmetries in quantum information theory. Among others, we study entropies and capacitites of Temperley–Lieb channels, their (anti-) degradability, PPT and entanglement breaking properties, as well as the behaviour of their tensor products with respect to entangled inpurs. Finally we compare the Tempereley–Lieb channels with the (modified) TRO-channels recently introduced by Gao et al. (Commun Math Phys 364(1):83–121, 2018)).

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Notes

  1. The term “trace” comes from the fact that under the fiber functor \(\text {TL}(d) \rightarrow \text {Rep}(O^+_F)\), \(\tau _k\) corresponds to the well-known Markov trace \(\tau _k:\text {TL}_{k,k}(d) \rightarrow \mathbb {C}\) obtained by tracial closure of Temperley–Lieb diagrams [KL94].

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Acknowledgements

MB’s research was supported by NSF Grant DMS-1700267. BC’s research was supported by JSPS KAKENHI 17K18734, 17H04823, 15KK0162. HHL and SY’s research 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). SY’s research was also supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2020R1C1C1A01009681). The authors are grateful to Marius Junge for useful comments and discussons during various stages of preparation of this manuscript.

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Authors and Affiliations

  1. Department of Mathematics, Texas A&M University, Mailstop 3368, College Station, TX, 77843-3368, USA

    Michael Brannan

  2. Department of Mathematics, Kyoto University, Kyoto, 606-8502, Japan

    Benoît Collins

  3. 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

  4. 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. Michael Brannan
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  2. Benoît Collins
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  3. Hun Hee Lee
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  4. Sang-Gyun Youn
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Correspondence to Sang-Gyun Youn.

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Communicated by Y. Kawahigashi

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Brannan, M., Collins, B., Lee, H.H. et al. Temperley–Lieb Quantum Channels. Commun. Math. Phys. 376, 795–839 (2020). https://doi.org/10.1007/s00220-020-03731-2

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