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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 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