Combinatorial entanglement

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 physical 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 if the resulting state is entangled. By reformulating the Peres-Horodecki and matrix realignment entanglement 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 local operations and classical communication (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, the computational complexity of which we leave as an open problem. The proposed mathematical framework also suggests new combinatorial and algebraic ways for describing the structure of graphs. (C) 2021 Elsevier Inc. All rights reserved.

Automated summary

The paper introduces new combinatorial objects, grid-labelled graphs, to represent quantum states arising in a specific physical scenario. By reformulating entanglement criteria, new bound entangled states are constructed and the limitations of matrix realignment are demonstrated. The relationship between local operations and classical communication (LOCC) and a generalisation of the graph isomorphism problem is also discussed.