Speaker
Michael Brannan, University of Waterloo
Abstract
The set of unital quantum channels on $M_n$ is the natural quantum information theoretic analogue of the convex set of bistochastic matrices in $M_n$. For bistochastic matrices, we have the well-known Birkhoff-von Neumann theorem (B-vN theorem) which says that any bistochastic matrix is a convex combination of permutation matrices. For unital quantum channels, the situation is much more complicated. For instance, several attempts at a quantum version of Birkhoff’s theorem are known to fail (e.g., based on mixed unitary channels or more generally factorizable maps). In this talk, we will look at unital quantum channels from the perspective of noncommutative (a.k.a. matrix) convexity – viewing unital channels as the first level of an enriched matricial convex structure. This naturally leads to a new nc convex set of unital channels we call “mixed quantum unitary channels”, whose nc extreme points are given by quantum automorphisms of $M_n$. For some time, we conjectured that mixed quantum unitary channels would provide the correct framework for a quantum B-vN theorem, but our hopes have been dashed once again! I will present some examples of unital channels which fail to be mixed quantum unitary. Time permitting, I will also explain some consequences this result has on the structure of the operator space spanned by the generators of the free unitary quantum groups. This is joint work with Jason Crann and Alec Gow.
Michael Brannan, University of Waterloo
Abstract
The set of unital quantum channels on $M_n$ is the natural quantum information theoretic analogue of the convex set of bistochastic matrices in $M_n$. For bistochastic matrices, we have the well-known Birkhoff-von Neumann theorem (B-vN theorem) which says that any bistochastic matrix is a convex combination of permutation matrices. For unital quantum channels, the situation is much more complicated. For instance, several attempts at a quantum version of Birkhoff’s theorem are known to fail (e.g., based on mixed unitary channels or more generally factorizable maps). In this talk, we will look at unital quantum channels from the perspective of noncommutative (a.k.a. matrix) convexity – viewing unital channels as the first level of an enriched matricial convex structure. This naturally leads to a new nc convex set of unital channels we call “mixed quantum unitary channels”, whose nc extreme points are given by quantum automorphisms of $M_n$. For some time, we conjectured that mixed quantum unitary channels would provide the correct framework for a quantum B-vN theorem, but our hopes have been dashed once again! I will present some examples of unital channels which fail to be mixed quantum unitary. Time permitting, I will also explain some consequences this result has on the structure of the operator space spanned by the generators of the free unitary quantum groups. This is joint work with Jason Crann and Alec Gow.