Quantifying imaginarity of quantum operations

TL;DR

This paper develops a framework to quantify the imaginarity of quantum operations, focusing on their ability to create or detect imaginarity, and provides analytical formulas for qubit unitaries.

Contribution

It introduces new measures of quantum operation imaginarity based on norm and weight, expanding the resource theory of imaginarity.

Findings

Derived analytical formulas for qubit unitary operations.

Established properties and relations of the new imaginarity measures.

Provided insights into the dynamical aspects of quantum imaginarity.

Abstract

Complex numbers are theoretically proved and experimentally confirmed as necessary in quantum mechanics and quantum information, and a resource theory of imaginarity of quantum states has been established. In this work, we establish a framework to quantify the imaginarity of quantum operations from the perspective of the ability to create or detect imaginarity, following the idea by Theurer {\it et al.} [Phys. Rev. Lett. \textbf{122}, 190405 (2019)] used in coherence theory. We present two types of imaginarity measures of quantum operations based on the norm and the weight, investigate their properties and relations, and derive the analytical formulas of the measure under the trace norm for qubit unitary operations. The results provide new insights into imaginarity of operations and deepen our understanding of dynamical imaginarity.