Real quantum operations and state transformations

TL;DR

This paper explores the properties and capabilities of real quantum operations within resource theory, providing conditions for state transformations, entanglement measures, and optimal fidelity calculations for pure and mixed states.

Contribution

It offers a comprehensive analysis of real quantum operations, including necessary and sufficient conditions for state transformations and a semidefinite program for optimal fidelity.

Findings

Necessary and sufficient conditions for state transformations under real operations.

Existence of real entanglement monotones.

Analytical expression for optimal fidelity in pure state transformations.

Abstract

Resource theory of imaginarity provides a useful framework to understand the role of complex numbers, which are essential in the formulation of quantum mechanics, in a mathematically rigorous way. In the first part of this article, we study the properties of ``real'' (quantum) operations both in single-party and bipartite settings. As a consequence, we provide necessary and sufficient conditions for state transformations under real operations and show the existence of ``real entanglement'' monotones. In the second part of this article, we focus on the problem of single copy state transformation via real quantum operations. When starting from pure initial states, we completely solve this problem by finding an analytical expression for the optimal fidelity of transformation, for a given probability of transformation and vice versa. Moreover, for state transformations involving arbitrary initial states and pure final states, we provide a semidefinite program to compute the optimal achievable fidelity, for a given probability of transformation.