On some classes of bipartite unitary operators

TL;DR

This paper characterizes classes of bipartite unitary operators that generate specific types of quantum channels, providing conditions, dimensions, and relations among these classes, with implications for quantum information theory.

Contribution

It offers explicit characterizations and conditions for bipartite unitaries generating various quantum channels, and analyzes their algebraic and dimensional properties.

Findings

Explicit characterizations of bipartite unitaries for certain channel types

Necessary and sufficient conditions for membership in these classes

Many classes coincide in low-dimensional cases

Abstract

We investigate unitary operators acting on a tensor product space, with the property that the quantum channels they generate, via the Stinespring dilation theorem, are of a particular type, independently of the state of the ancilla system in the Stinespring relation. The types of quantum channels we consider are those of interest in quantum information theory: unitary conjugations, constant channels, unital channels, mixed unitary channels, PPT channels, and entanglement breaking channels. For some of the classes of bipartite unitary operators corresponding to the above types of channels, we provide explicit characterizations, necessary and/or sufficient conditions for membership, and we compute the dimension of the corresponding algebraic variety. Inclusions between these classes are considered, and we show that for small dimensions, many of these sets are identical.