The Operator Tensor Formulation of Quantum Theory

TL;DR

The paper introduces an operator tensor framework for quantum theory that simplifies circuit probability calculations, removes the need for arbitrary foliation, and unifies the treatment of different operations.

Contribution

It presents a novel operator tensor formulation that addresses key issues in standard quantum circuit analysis, providing a more consistent and unified approach.

Findings

Operator tensors correspond to each operation in a quantum circuit.

Circuit probabilities are computed by replacing operations with operator tensors and performing tensor contractions.

Operator tensors must satisfy positivity and normalization conditions.

Abstract

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).