  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.
