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

• URL: http://arxiv.org/abs/2106.00034v5
• Date: Mon, 6 Nov 2023 17:41:48 GMT
• Title: The quantum switch is uniquely defined by its action on unitary operations
• Authors: Qingxiuxiong Dong, Marco T\'ulio Quintino, Akihito Soeda, Mio Murao
• Abstract summary: The action of the quantum switch on non-unitary operations is chosen to be a natural'' extension of its action on unitary operations. We prove, however, that the natural extension is the only possibility for the quantum switch for the 2-slot case.
• Score: 0.8192907805418581
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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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