Coherence cost for violating conservation laws

Abstract

In isolated quantum systems that have conserved quantities, such as energy and total magnetization, the conserved quantities cannot vary their values with their own dynamics. To violate conservation laws of the system keeping the unitary nature in the system's dynamics, quantum coherence must be supplied from the outer environment. This provides a general and abstract interpretation of the underlying mechanism of unitary operations on systems of interest. In this paper, we consider the amount of quantum coherence cost to realize the desired unitary operation. We derive upper and lower bounds for the coherence cost to implement arbitrary unitary operation within the desired error. These two bounds asymptotically match each other in a small error regime, and give an asymptotic equality. We present applications of our theories to several physical situations such as quantum heat engines and entanglement erasure.