Abstract
Nature imposes many restrictions on the operations that we perform. Many of these restrictions can be interpreted in terms of {\it resource} required to realize the operations. Classifying required resource for different types of operations and determining the amount of resource are the crucial subjects in physics. Among many types of operations, a unitary operation is one of the most fundamental operation that has been studied for long time in terms of the resource implicitly and explicitly. Yet, it is a long standing open problem to identify the resource and to clarify the necessary and sufficient amount of resource for implementing a general unitary operation under conservation laws. In this paper, we provide a solution to this open problem. We derive an asymptotically exact equality that clarifies the necessary and sufficient amount of quantum coherence as a resource to implement arbitrary unitary operation within a desired error. In this equality, the required coherence cost is asymptotically expressed with the implementation error and the degree of violation of conservation law in the desired unitary operation. We also discuss the underlying physics in several physical situations from the viewpoint of coherence cost based on the equality. This work does not only provide a solution to a long-standing problem on the unitary control, but also clarifies the key question of the resource theory of the quantum channels in the region of resource theory of asymmetry, for the case of unitary channels.
Highlights
Conservation laws indicate that conserved quantities do not change their values with the unitary dynamics of isolated quantum systems
We address the following problem: What is the fundamental limitation to implement unitary dynamics violating the conservation law using the resource stored in the external system? In our formulation, the required resource is measured by the quantum Fisher information, which is a well-known resource measure in the resource theory of asymmetry
We introduce the basic notions of the resource theory of asymmetry
Summary
Conservation laws indicate that conserved quantities do not change their values with the unitary dynamics of isolated quantum systems. The first type is categorized into studies focusing on sufficient conditions for ancillary coherence to overcome conservation laws and to achieve desired and specific unitary dynamics. (See the schematic in Fig. 1.) We derive upper and lower bounds for the coherence cost, the necessary and sufficient amount of quantum coherence in the battery to achieve unitary time evolution in the target system within the desired error.
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