Non-monotonicity of trace distance under tensor products

TL;DR

This paper investigates the non-monotonicity of trace distance under tensor products in quantum states, revealing it occurs in a significant fraction of mixed states and should be considered when using trace distance as a measure.

Contribution

The study provides the first detailed analytical and numerical analysis of trace distance non-monotonicity across different quantum state classes.

Findings

Non-monotonicity occurs in a non-negligible fraction of mixed states.

The occurrence of non-monotonicity decreases with increasing system dimension.

Pure states and some mixed states do not exhibit this property.

Abstract

The trace distance (TD) possesses several of the good properties required for a faithful distance measure in the quantum state space. Despite its importance and ubiquitous use in quantum information science, one of its questionable features, its possible non-monotonicity under taking tensor products of its arguments (NMuTP), has been hitherto unexplored. In this article we advance analytical and numerical investigations of this issue considering different classes of states living in a discrete and finite dimensional Hilbert space. Our results reveal that although this property of TD does not shows up for pure states and for some particular classes of mixed states, it is present in a non-negligible fraction of the regarded density operators. Hence, even though the percentage of quartets of states leading to the NMuTP drawback of TD and its strength decrease as the system's dimension grows, this property of TD must be taken into account before using it as a figure of merit for distinguishing mixed quantum states.