Upper bound for SL-invariant entanglement measures of mixed states

TL;DR

This paper introduces an algorithm to compute upper bounds for polynomial SL-invariant entanglement measures of mixed states, using a method based on convex-roof calculations for rank-two density matrices, with applications to specific quantum states.

Contribution

It presents a novel algorithm that approximates the convex roof for full-rank density matrices by leveraging rank-two decompositions, providing upper bounds for entanglement measures.

Findings

Algorithm overestimates the threetangle for tested states.

Applied to GHZ-Werner state, providing insights into its optimal decomposition.

Demonstrated the algorithm's effectiveness on the transverse quantum Ising model.

Abstract

An algorithm is proposed that serves to handle full rank density matrices, when coming from a lower rank method to compute the convex-roof. This is in order to calculate an upper bound for any polynomial SL invariant multipartite entanglement measure E. Here, it is exemplifyed how this algorithm works, based on a method for calculating convex-roofs of rank two density matrices. It iteratively considers the decompositions of the density matrix into two states each, exploiting the knowledge for the rank-two case. The algorithm is therefore quasi exact as far as the two rank case is concerned, and it also gives hints where it should include more states in the decomposition of the density matrix. Focusing on the threetangle, I show the results the algorithm gives for two states, one of which being the $GHZ$-Werner state, for which the exact convex roof is known. It overestimates the threetangle in the state, thereby giving insight into the optimal decomposition the $GHZ$-Werner state has. As a proof of principle, I have run the algorithm for the threetangle on the transverse quantum Ising model. I give qualitative and quantitative arguments why the convex roof should be close to the upper bound found here.