Real quantum operations and state transformations

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.

Automated summary

“The resource theory of imaginarity provides a useful framework for understanding the role of complex numbers in mathematical rigor, particularly in quantum mechanics. This article explores the properties of 'real' (quantum) operations in both single-party and bipartite scenarios. It also discusses the necessary and sufficient conditions for state transformations under real operations and introduces the concept of 'real entanglement' monotones. Furthermore, the article focuses on single copy state transformation through real quantum operations, providing analytical expressions for optimal fidelity and presenting a semidefinite program for computing optimal achievable fidelity in various scenarios.”