Quantifying the imaginarity via different distance measures

TL;DR

This paper introduces new measures of quantum imaginarity using various distance metrics, analyzes their decay under channels, and studies state ordering invariance, advancing the resource theory of imaginarity.

Contribution

It proposes well-defined imaginarity measures based on advanced distance metrics and analyzes their behavior under quantum channels and state transformations.

Findings

Tsallis relative α-entropy shows higher decay rates under quantum channels.

Imaginarity measures maintain state ordering invariance under bit-flip channels for certain parameters.

The study deepens understanding of imaginarity as a quantum resource.

Abstract

The recently introduced resource theory of imaginarity facilitates a systematic investigation into the role of complex numbers in quantum mechanics and quantum information theory. In this work, we propose well-defined measures of imaginarity using various distance metrics, drawing inspiration from recent advancements in quantum entanglement and coherence. Specifically, we focus on quantitatively evaluating imaginarity through measures such as Tsallis relative $\alpha$-entropy, Sandwiched R\'{e}nyi relative entropy, and Tsallis relative operator entropy. Additionally, we analyze the decay rates of these measures. Our findings reveal that the Tsallis relative $\alpha$-entropy of imaginarity exhibits higher decay rate under quantum channels compared to other measures. Finally, we examine the ordering of single-qubit states under these imaginarity measures, demonstrating that the order remains invariant under the bit-flip channel for specific parameter ranges. This study enhances our understanding of imaginarity as a quantum resource and its potential applications in quantum information theory.