Visualization of all two-qubit states via partial-transpose-moments

TL;DR

This paper visualizes the complete 3D region of two-qubit states using partial-transpose moments, providing an operational criterion for entanglement detection based on measurable quantities.

Contribution

It extends previous work by visualizing the full 3D region of two-qubit states with the fourth PT-moment, enabling complete entanglement detection.

Findings

Identified the 3D region of all two-qubit states using PT-moments.

Derived a dividing surface separating entangled and separable states.

Provided a measurable, operational criterion for entanglement detection.

Abstract

Efficiently detecting entanglement based on measurable quantities is a basic problem for quantum information processing. Recently, the measurable quantities called partial-transpose (PT)-moments have been proposed to detect and characterize entanglement. In the recently published paper [L. Zhang \emph{et al.}, \href{https://doi.org/10.1002/andp.202200289}{Ann. Phys.(Berlin) \textbf{534}, 2200289 (2022)}], we have already identified the 2-dimensional (2D) region, comprised of the second and third PT-moments, corresponding to two-qubit entangled states, and described the whole region for all two-qubit states. In the present paper, we visualize the 3D region corresponding to all two-qubit states by further involving the fourth PT-moment (the last one for two-qubit states). The characterization of this 3D region can finally be achieved by optimizing some polynomials. Furthermore, we identify the dividing surface which separates the two parts of the whole 3D region corresponding to entangled and separable states respectively. Due to the measurability of PT-moments, we obtain a complete and operational criterion for the detection of two-qubit entanglement.