Minimax optimization of entanglement witness operator for the quantification of three-qubit mixed-state entanglement

TL;DR

This paper introduces a numerical method using optimal entanglement witness operators and minimax optimization to quantify entanglement in mixed quantum states, offering a verifiable and symmetry-exploiting alternative to traditional approaches.

Contribution

It presents a novel, efficient numerical approach for quantifying multipartite entanglement in mixed states, applicable to general measures and states, with verifiable global optimality.

Findings

Successfully quantified GHZ entanglement in complex three-qubit mixed states.

Demonstrated the method's efficiency and symmetry advantages over conventional techniques.

Provided insights into the structure of optimal witness operators and state convexity.

Abstract

We develop a numerical approach for quantifying entanglement in mixed quantum states by convex-roof entanglement measures, based on the optimal entanglement witness operator and the minimax optimization method. Our approach is applicable to general entanglement measures and states and is an efficient alternative to the conventional approach based on the optimal pure-state decomposition. Compared with the conventional one, it has two important merits: (i) that the global optimality of the solution is quantitatively verifiable, and (ii) that the optimization is considerably simplified by exploiting the common symmetry of the target state and measure. To demonstrate the merits, we quantify Greenberger-Horne-Zeilinger (GHZ) entanglement in a class of three-qubit full-rank mixed states composed of the GHZ state, the W state, and the white noise, the simplest mixtures of states with different genuine multipartite entanglement, which have not been quantified before this work. We discuss some general properties of the form of the optimal witness operator and of the convex structure of mixed states, which are related to the symmetry and the rank of states.