  A typical quantum experiment has a bunch of apparatuses placed so that
quantum systems can pass between them. We regard each use of an apparatus,
along with some given outcome on the apparatus (a certain detector click or a
certain meter reading for example), as an operation. An operation can have zero
or more quantum systems inputted into it and zero or more quantum systems
outputted from it. We can wire together operations to form circuits. In the
standard framework of quantum theory we must foliate the circuit then calculate
the probability by evolving a state through it. This approach has three
problems. First, we must introduce an arbitrary foliation of the circuit (such
foliations are not unique). Second, we have to pad our expressions with
identities every time two or more foliation hypersurfaces intersect a given
wire. And third, we treat operations corresponding to preparations,
transformations, and results in different ways. In this paper we present the
operator tensor formulation of quantum theory which solves all these problems.
Corresponding to every operation is an operator tensor. The probability for a
circuit is given by simply replacing the operations in the circuit with the
corresponding operator tensors. Wires between operator tensors correspond to
multiplying the tensors in the associated subspace and then taking the partial
trace over that subspace. Operator tensors must be physical (namely, they must
have positive input transpose and satisfy a certain normalization condition).
