The operator tensor formulation of quantum theory

Abstract

In this paper, we provide what might be regarded as a manifestly covariant presentation of discrete quantum theory. 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 anoperation. An operation (e.g.Bb2a3a1) can have zero or more quantum systems inputted into it and zero or more quantum systems outputted from it. The operationBb2a3a1has one system of typeainputted, and one system of typeband one system of typeaoutputted. We can wire together operations to form circuits, for example,. Each repeated integer label here denotes a wire connecting an output to an input of the same type. As each operation in a circuit has an outcome associated with it, a circuit represents a set of outcomes that can happen in a run of the experiment. In the operator tensor formulation of quantum theory, each operation corresponds to anoperator tensor. For example, the operationBb2a3a1corresponds to the operator tensor. Further, the probability for a general circuit is given by replacing operations with corresponding operator tensors as in1Repeated integer labels indicate that we multiply in the associated subspace and then take the partial trace over that subspace. Operator tensors must be physical (namely, they must have positive input transpose and satisfy a certain normalization condition).