  Vector representations have become a central element in semantic language
modelling, leading to mathematical overlaps with many fields including quantum
theory. Compositionality is a core goal for such representations: given
representations for 'wet' and 'fish', how should the concept 'wet fish' be
represented?
  This position paper surveys this question from two points of view. The first
considers the question of whether an explicit mathematical representation can
be successful using only tools from within linear algebra, or whether other
mathematical tools are needed. The second considers whether semantic vector
composition should be explicitly described mathematically, or whether it can be
a model-internal side-effect of training a neural network.
  A third and newer question is whether a compositional model can be
implemented on a quantum computer. Given the fundamentally linear nature of
quantum mechanics, we propose that these questions are related, and that this
survey may help to highlight candidate operations for future quantum
implementation.
