Quantum Imaginarity-Mixedness Trade-off: Characterizing Maximally Imaginary Mixed States

TL;DR

This paper characterizes the maximum imaginarity states in quantum systems, explores their trade-offs with mixedness, and analyzes their evolution under specific quantum channels.

Contribution

It introduces the concept of maximally imaginary mixed states (MIMS) and identifies them for qubit, qutrit, and high-dimensional systems, revealing new trade-off relations.

Findings

Identified MIMS for qubit and qutrit systems.

Established trade-off relations between imaginarity and mixedness.

Analyzed the evolution of these states under four Markovian channels.

Abstract

We investigate the trade-off relations between imaginarity and mixedness in arbitrary $d$-dimensional quantum systems. For given mixedness, a quantum state with maximum imaginarity is defined to be a "maximally imaginary mixed state" (MIMS). By using the $l_{1}$ norm of imaginarity and the normalized linear entropy, we conclusively identify the MIMSs for both qubit and qutrit systems. For high-dimensional quantum systems, we present a comprehensive class of MIMSs, which also gives rise to complementarity relations between the $1$-norm of imaginarity and the $1$-norm of mixedness, as well as between the relative entropy of imaginarity and the von Neumann entropy. Furthermore, we examine the evolution of the trade-off relation for single-qubit states under four specific Markovian channels: bit flip channel, phase damping channel, depolarizing channel and amplitude damping channel.