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 show 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.

Hilbert–Schmidt quantum coherence in multi-qudit systems

“…Actually, one can show that this holds true for all p-norm distances [20]. For general qudit states, the probability for random quartets of states to present the NMuTP behavior of HSD decays with increasing the system dimension, also similar to what was seen for the TD [49]. For completeness, we show such probabilities in Fig.…”

“…One can verify that, similarly to what happens with the trace distance (TD) [49], the Hilbert-Schmidt distance (HSD) does not suffer from the non-monotonicity under tensor product (NMuTP) issue for general pure states or for one-qubit collinear states. Actually, one can show that this holds true for all p-norm distances [20].…”

“…We will look at the possibility of controlling the local and/or non-local coherences [45][46][47] of a system by manipulating its local populations. And we will investigate the issue of a possible non-monotonic behavior of quantum distance measures under tensor products [48,49].…”

“…The non-monotonicity under tensor products (NMuTP) of a distance measure, investigated recently for the trace distance [49], is the possibility for an inversion in the dissimilarity relation between pairs of states when we consider their copies. That is to say, for four density matrices ρ, σ, ξ, η, it can happen that…”

Fortran Code for Generating Random Probability Vectors, Unitaries, and Quantum States

“…In Quantum Information Science (QIS), a multidisciplinary field aiming an efficient and far reaching use and manipulation of information, the panorama is not different. The creation of random states and unitaries can be useful for encryption, remote state preparation, data hiding, classical correlation locking, quantum devices and decoherence characterization and tailoring, and for quantumness and correlations quantification [15][16][17][18][19][20][21][22][23][24][25], to name but a few examples.…”