Computing partial transposes and related entanglement functions

TL;DR

This paper provides explicit formulas and Fortran code for computing the partial transpose and related entanglement measures in bipartite and multipartite quantum systems, including analytical results for two-qudit systems.

Contribution

It introduces new explicit formulas and numerical tools for partial transpose calculations and entanglement quantification, avoiding Lagrange multipliers used in prior work.

Findings

Explicit formulas for partial transpose and entanglement functions

Fortran code for numerical implementation

Analytical expression for Hilbert-Schmidt entanglement of two-qudit systems

Abstract

The partial transpose (PT) is an important function for entanglement testing and quantification and also for the study of geometrical aspects of the quantum state space. In this article, considering general bipartite and multipartite discrete systems, explicit formulas ready for the numerical implementation of the PT and of related entanglement functions are presented and the Fortran code produced for that purpose is described. What is more, we obtain an analytical expression for the Hilbert-Schmidt entanglement of two-qudit systems and for the associated closest separable state. In contrast to previous works on this matter, we only use the properties of the PT, not applying Lagrange multipliers.