Non-monotonicity of trace distance under tensor products

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.

References in corpus (11)

• Necessary and sufficient condition for non-zero quantum discord
• Observable measure of quantum coherence in finite dimensional systems
• The Quantum Chernoff Bound
• Quantum speed limit for physical processes
• Quantum speed limits in open system dynamics
• The Chernoff lower bound for symmetric quantum hypothesis testing
• Geometric derivation of the quantum speed limit
• Quantum discord and classical correlation can tighten the uncertainty principle in the presence of quantum memory
• Entanglement and the Lower Bounds on the Speed of Quantum Evolution
• Quantum Discord and its Role in Quantum Information Theory
• Generating pseudo-random discrete probability distributions

Cited by in corpus (4)

• Trace-distance correlations for X states and emergence of the pointer basis in Markovian and non-Markovian regimes
• Hilbert-Schmidt quantum coherence in multi-qudit systems
• Fortran code for generating random probability vectors, unitaries, and quantum states
• The sudden change phenomenon of quantum discord