Maximally entangled mixed states for qubit-qutrit systems

TL;DR

This paper investigates the maximum entanglement negativity achievable in qubit-qutrit systems with specific spectral and purity constraints, identifying candidate states that optimize entanglement.

Contribution

It provides a complete solution for maximizing negativity with fixed spectrum and partial solutions for fixed purity, introducing a semidefinite programming approach.

Findings

Three-fold degenerate X states of fixed purity reach maximal negativity.

Numerical evidence supports these states as likely maximally entangled.

The study characterizes a sparse family of maximally entangled mixed states.

Abstract

We consider the problems of maximizing the entanglement negativity of X-form qubit-qutrit density matrices with (i) a fixed spectrum and (ii) a fixed purity. In the first case, the problem is solved in full generality whereas, in the latter, partial solutions are obtained by imposing extra spectral constraints such as rank-deficiency and degeneracy, which enable a semidefinite programming treatment for the optimization problem at hand. Despite the technically-motivated assumptions, we provide strong numerical evidence that three-fold degenerate X states of purity $P$ reach the highest entanglement negativity accessible to arbitrary qubit-qutrit density matrices of the same purity, hence characterizing a sparse family of likely qubit-qutrit maximally entangled mixed states.