Imaginarity-generating power of unitaries: A resource-theoretic approach

TL;DR

This paper introduces a measure for the ability of unitary operations to generate imaginarity from real quantum states, analyzing its properties, bounds, and typical behavior in high-dimensional systems.

Contribution

It defines the imaginarity-generating power (IGP) of unitaries within a resource-theoretic framework and derives exact formulas, bounds, and typical behaviors for high-dimensional quantum dynamics.

Findings

IGP is monotone under real unital operations.

Pure real states' IGP depends only on intrinsic properties of the unitary.

Haar-random unitaries in high dimensions have IGP near its maximum.

Abstract

Imaginarity, stemming from the complex structure of quantum mechanics, has recently emerged as a fundamental resource, yet its dynamical generation remains largely unexplored. In this work, we introduce the notion of imaginarity-generating power (IGP) of unitary dynamics, which quantifies the ability of unitary operations to produce imaginarity from initially real quantum states. To quantify imaginarity, we employ a measure based on the Hilbert--Schmidt norm, which we show to be monotone under real unital operations. Within the framework of dynamical resource theories, we derive an exact expression for the purity-constrained IGP in arbitrary dimensions and show that, for pure real input states, it depends solely on intrinsic and experimentally accessible properties of the unitary. We further analyze its average behavior over ensembles of states with varying purity under both uniform and Hilbert--Schmidt distributions. We prove that it satisfies the essential properties of a valid resource monotone within the dynamical resource theory of imaginarity. We also characterize the unitaries that maximize the IGP and determine the corresponding bounds. Moreover, for Haar-random unitaries, we show that the IGP concentrates near its maximal value in high dimensions with small fluctuations, indicating that typical high-dimensional quantum dynamics are highly effective at generating imaginarity.