Classical Tensors and Quantum Entanglement II: Mixed States

TL;DR

This paper explores invariant tensor fields on Lie groups to identify entanglement measures in quantum states, recovering known functions like purity and concurrence, and proposing new entanglement monotone candidates.

Contribution

It introduces a method using invariant operator-valued tensor fields on Lie groups to find entanglement monotones, including a new candidate based on a non-linear Lie algebra realization.

Findings

Recovered purity and concurrence functions for two-qubit states.

Identified a new entanglement monotone candidate.

Illustrated the relation between the new candidate and concurrence for specific states.

Abstract

Invariant operator-valued tensor fields on Lie groups are considered. These define classical tensor fields on Lie groups by evaluating them on a quantum state. This particular construction, applied on the local unitary group U(n)xU(n), may establish a method for the identification of entanglement monotone candidates by deriving invariant functions from tensors being by construction invariant under local unitary transformations. In particular, for n=2, we recover the purity and a concurrence related function (Wootters 1998) as a sum of inner products of symmetric and anti-symmetric parts of the considered tensor fields. Moreover, we identify a distinguished entanglement monotone candidate by using a non-linear realization of the Lie algebra of SU(2)xSU(2). The functional dependence between the latter quantity and the concurrence is illustrated for a subclass of mixed states parametrized by two variables.