Quantum process in probability representation of quantum mechanics

TL;DR

This paper extends the operator-sum representation of quantum processes to the probability (tomographic) framework, introducing kernels that describe system evolution and applying this to various quantum operations.

Contribution

It introduces a kernel-based approach for representing quantum processes in the probability (tomographic) framework, expanding the operator-sum formalism.

Findings

Kernel representation for quantum processes in probability space

Decomposition of kernels for specific quantum operations

Application to qubit flipping and amplitude damping

Abstract

In this work, the operator-sum representation of a quantum process is extended to the probability representation of quantum mechanics. It is shown that each process admitting the operator-sum representation is assigned a kernel, convolving of which with the initial tomogram set characterizing the system state gives the tomographic state of the transformed system. This kernel, in turn, is broken into the kernels of partial operations, each of them incorporating the symbol of the evolution operator related to the joint evolution of the system and an ancillary environment. Such a kernel decomposition for the projection to a certain basis state and a Gaussian-type projection is demonstrated as well as qubit flipping and amplitude damping processes.