  Holant problems are a family of counting problems parameterised by sets of
algebraic-complex valued constraint functions, and defined on graphs. They
arise from the theory of holographic algorithms, which was originally inspired
by concepts from quantum computation. Here, we employ quantum information
theory to explain existing results about holant problems in a concise way and
to derive two new dichotomies: one for a new family of problems, which we call
Holant$^+$, and, building on this, a full dichotomy for Holant$^c$. These two
families of holant problems assume the availability of certain unary constraint
functions -- the two pinning functions in the case of Holant$^c$, and four
functions in the case of Holant$^+$ -- and allow arbitrary sets of
algebraic-complex valued constraint functions otherwise. The dichotomy for
Holant$^+$ also applies when inputs are restricted to instances defined on
planar graphs. In proving these complexity classifications, we derive an
original result about entangled quantum states.
