Geometric measures of entanglement and the Schmidt decomposition

TL;DR

This paper generalizes geometric measures of entanglement by considering unnormalized product states, linking the closest product state's components to Schmidt decomposition eigenvectors and eigenvalues, with analytical solutions for symmetric states.

Contribution

It introduces a generalized geometric entanglement measure using unnormalized product states and connects it to Schmidt decomposition, providing analytical solutions for symmetric states.

Findings

Analytical solutions for symmetric states.

Connection between closest product states and Schmidt eigenvectors.

Extension of geometric entanglement measure to unnormalized states.

Abstract

In the standard geometric approach, the entanglement of a pure state is $\sin^2\theta$, where $\theta$ is the angle between the entangled state and the closest separable state of products of normalised qubit states. We consider here a generalisation of this notion by considering separable states that consist of products of unnormalised states of different dimension. The distance between the target entangled state and the closest unnormalised product state can be interpreted as a measure of the entanglement of the target state. The components of the closest product state and its norm have an interpretation in terms of, respectively, the eigenvectors and eigenvalues of the reduced density matrices arising in the Schmidt decomposition of the state vector. For several cases where the target state has a large degree of symmetry, we solve the system of equations analytically, and look specifically at the limit where the number of qubits is large.