Geometry of Generalized Depolarizing Channels

TL;DR

This paper explores the geometric structure of generalized depolarizing channels in quantum systems, proving that for qubits in the Pauli basis, the compression vectors form a simplex, and extends this to other bases.

Contribution

It proves a conjecture that the set of compression vectors forms a simplex for qubits in the Pauli basis and characterizes when this holds in other bases.

Findings

Compression vectors form a simplex for qubits in the Pauli basis

The geometry of compression vectors varies with the basis used

Conditions are identified when the set of compression vectors forms a simplex

Abstract

A generalized depolarizing channel acts on an N-dimensional quantum system to compress the ``Bloch ball'' in N^2-1 directions; it has a corresponding compression vector. We investigate the geometry of these compression vectors and prove a conjecture of Dixit and Sudarshan [1], namely that when N=2^d (i.e. the system consists of d qubits) and we work in the Pauli basis then the set of all compression vectors forms a simplex. We extend this result by investigating the geometry in other bases; in particular we find precisely when the set of all compression vectors forms a simplex.