Antilinear superoperator, quantum geometric invariance, and antilinear symmetry for higher-dimensional quantum systems

TL;DR

This paper systematically studies antilinear superoperators in higher-dimensional quantum systems, exploring their geometric invariance, symmetries, and implications for entanglement and conserved quantities in open quantum systems.

Contribution

It introduces a comprehensive framework for antilinear superoperators, including new classes and their geometric and symmetry properties, advancing understanding of quantum invariance and entanglement.

Findings

Different metrics for Bloch space-time vectors are obtained.

Antilinear superoperator symmetries influence entanglement distribution.

Kramers' degeneracy and conserved quantities are linked to antilinear symmetries.

Abstract

We present a systematic investigation of antilinear superoperators and their applications in studying open quantum systems, particularly focusing on quantum geometric invariance, entanglement distribution, and symmetry. We study several crucial classes of antilinear superoperators, including antilinear quantum channels, antilinearly unital superoperators, antiunitary superoperators, and generalized $\Theta$-conjugation. Using the Bloch representation, we present a systematic investigation of quantum geometric transformations in higher-dimensional quantum systems. By choosing different generalized $\Theta$-conjugations, we obtain various metrics for the space of Bloch space-time vectors, including the Euclidean and Minkowskian metrics. Utilizing these geometric structures, we then investigate the entanglement distribution over a multipartite system constrained by quantum geometric invariance. The strong and weak antilinear superoperator symmetries of the open quantum system are also discussed. Additionally, Kramers' degeneracy and conserved quantities are examined in detail.