Combinatorial Entanglement

TL;DR

This paper introduces grid-labelled graphs to represent quantum states in faulty emitter scenarios, providing new tools for entanglement detection, state characterization, and exploring the complexity of related combinatorial problems.

Contribution

It develops a novel combinatorial framework using grid-labelled graphs to analyze quantum entanglement and introduces new criteria and states, expanding the understanding of entanglement detection.

Findings

Constructed new bound entangled states.

Reformulated entanglement criteria for better detection.

Proved asymptotic entanglement in sparse states.

Abstract

We present new combinatorial objects, which we call grid-labelled graphs, and show how these can be used to represent the quantum states arising in a scenario which we refer to as the faulty emitter scenario: we have a machine designed to emit a particular quantum state on demand, but which can make an error and emit a different one. The device is able to produce a list of candidate states which can be used as a kind of debugging information for testing entanglement. By reformulating the Peres-Horodecki and matrix realignment criteria we are able to capture some characteristic features of entanglement: we construct new bound entangled states, and demonstrate the limitations of matrix realignment. We show how the notion of LOCC is related to a generalisation of the graph isomorphism problem. We give a simple proof that asymptotically almost surely, grid-labelled graphs associated to very sparse density matrices are entangled. We develop tools for enumerating grid-labelled graphs that satisfy the Peres-Horodecki criterion up to a fixed number of vertices, and propose various computational problems for these objects, whose complexity remains an open problem. The proposed mathematical framework also suggests new combinatorial and algebraic ways for describing the structure of graphs.