# Coherence cost for violating conservation laws

## TL;DR

This paper establishes a fundamental equality linking the quantum coherence resource to the implementation of unitary operations under conservation laws, resolving a long-standing open problem in quantum resource theory.

## Contribution

It derives an asymptotically exact equality that determines the coherence cost for implementing arbitrary unitaries respecting conservation laws.

## Key findings

- Provides a formula relating coherence cost to implementation error and conservation law violation.
- Clarifies the resource requirements for unitary operations under conservation constraints.
- Offers insights into the resource theory of quantum channels and asymmetry.

## 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.