Characterization of quantum entanglement via a hypercube of Segre embeddings

TL;DR

This paper introduces a hypercube-based algebraic framework for characterizing quantum entanglement in pure states, linking geometric separability with observable measures related to reduced density matrices.

Contribution

It presents a novel hypercube structure to describe the Segre embedding for quantum states and introduces new observables for measuring entanglement.

Findings

The hypercube approach provides a new perspective on entanglement measurement.

The introduced observables correlate well with known entanglement measures.

The method effectively distinguishes different entangled states.

Abstract

A particularly simple description of separability of quantum states arises naturally in the setting of complex algebraic geometry, via the Segre embedding. This is a map describing how to take products of projective Hilbert spaces. In this paper, we show that for pure states of n particles, the corresponding Segre embedding may be described by means of a directed hypercube of dimension n-1, where all edges are bipartite-type Segre maps. Moreover, we describe the image of the original Segre map via the intersections of images of the n-1 edges whose target is the last vertex of the hypercube. This purely algebraic result is then transferred to physics. For each of the last edges of the Segre hypercube, we introduce an observable which measures geometric separability and is related to the trace of the squared reduced density matrix. As a consequence, the hypercube approach gives a novel viewpoint on measuring entanglement, naturally relating bipartitions with $q$-partitions for q>1. We test our observables against well-known states, showing that these provide well-behaved and fine measures of entanglement.