Minimum Hilbert-Schmidt distance for Schmidt rank 2 states

TL;DR

This paper proves that the minimum Hilbert-Schmidt distance from Schmidt rank 2 states to the set of separable states is non-increasing under LOCC, providing a new analytical expression and proof of its monotonicity.

Contribution

It introduces an analytical formula for the minimum Hilbert-Schmidt distance for Schmidt rank 2 states and demonstrates its non-increasing property under LOCC operations.

Findings

Closed-form expression for the minimum Hilbert-Schmidt distance

Proof of non-increasing behavior under LOCC

Analytical and numerical validation of results

Abstract

The Hilbert-Schmidt distance between two states is proven to be non-contractive under CPTP maps, and therefore is not considered as an entanglement measure. However, that alone does not imply that the minimum Hilbert-Schmidt distance from the set of separable states is not contractive as well. To the contrary, not only do we provide a closed-form expression, we also provide analytical and numerical proof that minimum Hilbert-Schmidt distance for a given bipartite quantum state of Schmidt rank 2 is non-increasing under LOCC. The minimisation is taken to be over the set of separable states. We apply the algorithm by Verstraete et al [Journal of Modern Optics, 49(8), 2002] for the derivation of the analytical expression and Nielsen's theorem for the proof of monotonicity of the distance under LOCC.