Scalable Ellipsoidal Classification for Bipartite Quantum States

TL;DR

This paper introduces a scalable ellipsoidal classification method for bipartite quantum states to efficiently determine separability and detect bound entangled states using geometric optimization techniques.

Contribution

It proposes a novel approach using minimum volume ellipsoids and Euclidean distances in vectorized state space for quantum state separability testing.

Findings

Method effectively detects separability and entanglement.

Scalable to high-dimensional quantum systems.

Able to identify bound entangled states.

Abstract

The Separability Problem is approached from the perspective of Ellipsoidal Classification. A Density Operator of dimension N can be represented as a vector in a real vector space of dimension $N^{2}- 1$, whose components are the projections of the matrix onto some selected basis. We suggest a method to test separability, based on successive optimization programs. First, we find the Minimum Volume Covering Ellipsoid that encloses a particular set of properly vectorized bipartite separable states, and then we compute the Euclidean distance of an arbitrary vectorized bipartite Density Operator to this ellipsoid. If the vectorized Density Operator falls inside the ellipsoid, it is regarded as separable, otherwise it will be taken as entangled. Our method is scalable and can be implemented straightforwardly in any desired dimension. Moreover, we show that it allows for detection of Bound Entangled States