Tests For Complete Positivity of Quantum Channels

TL;DR

This paper reviews various methods for efficiently testing the complete positivity of quantum channels using Choi representations, aiming to improve reliability and computational efficiency in quantum information analysis.

Contribution

It compares different CP tests based on Choi's proof, highlighting approaches that avoid full spectral calculations for faster, more reliable channel analysis.

Findings

Different CP tests can be applied depending on the situation.

Some methods avoid the computational cost of spectral analysis.

Analytical and computational tests are crucial for negative channel detection.

Abstract

A quantum channel will have a Choi representation from which the complete positivity (CP) can be determined in a number of different ways. Every method relies on Choi's proof which relates CP to the positive semi-definiteness of a specially constructed matrix (i.e. the "Choi representation of the channel"). It can be shown that different CP tests can be used in different situations to avoid the computationally expensive brute force method of calculating the full spectrum of the Choi representation, which is the traditional method for testing whether or not a channel is CP. It is important to have quick, reliable analytical and computational tests for CP as negative channels become more prevalent in the study of quantum information.