Quantifying the Imaginarity of Quantum Mechanics

TL;DR

This paper develops a resource theory for the use of imaginary numbers in quantum mechanics, characterizing free states and operations, and providing measures and conditions for state transformations.

Contribution

It introduces a basis-dependent resource theory of imaginarity, linking it to coherence, and characterizes free operations and state transformation conditions.

Findings

Resource theory of imaginarity is basis-dependent.

Free operations are identical to certain non-generating operations.

Provides measures and conditions for pure state transformations.

Abstract

The use of imaginary numbers in modelling quantum mechanical systems encompasses the wave-like nature of quantum states. Here we introduce a resource theoretic framework for imaginarity, where the free states are taken to be those with density matrices that are real with respect to a fixed basis. This theory is closely related to the resource theory of coherence, as it is basis dependent, and the imaginary numbers appear in the off-diagonal elements of the density matrix. Unlike coherence however, the set of physically realizable free operations is identical to both completely resource non-generating operations, and stochastically resource non-generating operations. Moreover, the resource theory of imaginarity does not have a self-adjoint resource destroying map. After introducing and characterizing the free operations, we provide several measures of imaginarity, and give necessary and sufficient conditions for pure state transformations via physically consistent free operations in the single shot regime.