Bounding Polynomial Entanglement Measures for Mixed States

TL;DR

This paper introduces a generalized approach to bounding polynomial entanglement measures for mixed states, providing an algorithm to optimize decompositions and improve bounds on entanglement quantification.

Contribution

It generalizes the best separable and W-class approximations to arbitrary polynomial entanglement measures and develops an efficient algorithm for bounding entanglement in mixed states.

Findings

The algorithm's cost scales linearly with the density matrix rank.

The average difference between upper and lower bounds for three-tangle is 0.14.

Three-tangle for random full-rank three-qubit states is less than 0.023 on average.

Abstract

We generalize the notion of the best separable approximation (BSA) and best W-class approximation (BWA) to arbitrary pure state entanglement measures, defining the best zero-$E$ approximation (BEA). We show that for any polynomial entanglement measure $E$, any mixed state $\rho$ admits at least one "$S$-decomposition," i.e., a decomposition in terms of a mixed state on which $E$ is equal to zero, and a single additional pure state with (possibly) non-zero $E$. We show that the BEA is not in general the optimal $S$-decomposition from the point of view of bounding the entanglement of $\rho$, and describe an algorithm to construct the entanglement-minimizing $S$-decomposition for $\rho$ and place an upper bound on $E(\rho)$. When applied to the three-tangle, the cost of the algorithm is linear in the rank $d$ of the density matrix and has accuracy comparable to a steepest descent algorithm whose cost scales as $d^8 \log d$. We compare the upper bound to a lower bound algorithm given by Eltschka and Siewert for the three-tangle, and find that on random rank-two three-qubit density matrices, the difference between the upper and lower bounds is $0.14$ on average. We also find that the three-tangle of random full-rank three qubit density matrices is less than $0.023$ on average.