The quantum switch is uniquely defined by its action on unitary operations

Qingxiuxiong Dong1, Marco Túlio Quintino2,3,4,1, Akihito Soeda5,6,1, and Mio Murao1,7

1Department of Physics, Graduate School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
2Sorbonne Université, CNRS, LIP6, F-75005 Paris, France
3Institute for Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences, Boltzmanngasse 3, A-1090 Vienna, Austria
4Vienna Center for Quantum Science and Technology (VCQ), Faculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090 Vienna, Austria
5Principles of Informatics Research Division, National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku,Tokyo 101-8430, Japan
6Department of Informatics, School of Multidisciplinary Sciences, SOKENDAI (The Graduate University for Advanced Studies), 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan
7Trans-scale Quantum Science Institute, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan

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The quantum switch is a quantum process that creates a coherent control between different unitary operations, which is often described as a quantum process which transforms a pair of unitary operations $(U_1 , U_2)$ into a controlled unitary operation that coherently applies them in different orders as $\vert {0} \rangle\langle {0} \vert \otimes U_1 U_2 + \vert {1} \rangle\langle {1} \vert \otimes U_2 U_1$. This description, however, does not directly define its action on non-unitary operations. The action of the quantum switch on non-unitary operations is then chosen to be a ``natural'' extension of its action on unitary operations. In general, the action of a process on non-unitary operations is not uniquely determined by its action on unitary operations. It may be that there could be a set of inequivalent extensions of the quantum switch for non-unitary operations. We prove, however, that the natural extension is the only possibility for the quantum switch for the 2-slot case. In other words, contrary to the general case, the action of the quantum switch on non-unitary operations (as a linear and completely CP preserving supermap) is completely determined by its action on unitary operations. We also discuss the general problem of when the complete description of a quantum process is uniquely determined by its action on unitary operations and identify a set of single-slot processes which are completely defined by their action on unitary operations.

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[1] Michael Antesberger, Marco Túlio Quintino, Philip Walther, and Lee A. Rozema, "Higher-Order Process Matrix Tomography of a Passively-Stable Quantum Switch", PRX Quantum 5 1, 010325 (2024).

[2] Lee A. Rozema, Teodor Strömberg, Huan Cao, Yu Guo, Bi-Heng Liu, and Philip Walther, "Experimental Aspects of Indefinite Causal Order in Quantum Mechanics", arXiv:2405.00767, (2024).

[3] Giulio Chiribella and Zixuan Liu, "Quantum operations with indefinite time direction", Communications Physics 5 1, 190 (2022).

[4] Jessica Bavaresco, Mio Murao, and Marco Túlio Quintino, "Unitary channel discrimination beyond group structures: Advantages of sequential and indefinite-causal-order strategies", Journal of Mathematical Physics 63 4, 042203 (2022).

[5] Satoshi Yoshida, Akihito Soeda, and Mio Murao, "Universal construction of decoders from encoding black boxes", arXiv:2110.00258, (2021).

[6] Matt Wilson and Giulio Chiribella, "Causality in Higher Order Process Theories", arXiv:2107.14581, (2021).

[7] Satoshi Yoshida, Akihito Soeda, and Mio Murao, "Universal construction of decoders from encoding black boxes", Quantum 7, 957 (2023).

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