Tensor 2-sums and entanglement

TL;DR

This paper introduces a generalized tensor product of graphs inspired by density matrices to analyze quantum entanglement, providing a combinatorial criterion for separability.

Contribution

It proposes a novel graph-based tensor product framework that captures key properties of quantum entanglement and offers a combinatorial separability test.

Findings

Established a graph-theoretic analogue of the Peres-Horodecki criterion.

Showed that graphs can be expressed as sums of tensor products of adjacency matrices.

Provided a minimal mathematical framework for studying quantum entanglement properties.

Abstract

To define a minimal mathematical framework for isolating some of the characteristic properties of quantum entanglement, we introduce a generalization of the tensor product of graphs. Inspired by the notion of a density matrix, the generalization is a simple one: every graph can be obtained by addition modulo two, possibly with many summands, of tensor products of adjacency matrices. In this picture, we are still able to prove a combinatorial analogue of the Peres-Horodecki criterion for testing separability.