  The holographic transformation, belief propagation and loop calculus are
generalized to problems in generalized probabilistic theories including quantum
mechanics. In this work, the partition function of classical factor graph is
represented by an inner product of two high-dimensional vectors both of which
can be decomposed to tensor products of low-dimensional vectors. On the
representation, the holographic transformation is clearly understood by using
adjoint linear maps. Furthermore, on the formulation using inner product, the
belief propagation is naturally defined from the derivation of the loop
calculus formula. As a consequence, the holographic transformation, the belief
propagation and the loop calculus are generalized to measurement problems in
quantum mechanics and generalized probabilistic theories.
