**In minisymposium: Linear Algebra and Quantum Computation**

On faces of the set of quantum channels

Raphael Loewy (Technion-Israel Institute of Technology)

A linear map $T$ from ${\mathbb{C}}^{n\times n}$ into ${\mathbb{C}}^{n\times n}$ is called a quantum channel if it is completely positive and trace preserving. The set ${\mathcal{L}}_{n}$ of all such quantum channels is known to be a compact convex set. While the extreme points of ${\mathcal{L}}_{n}$ can be characterized, not much is known about its boundary. We obtain several results on the face structure of ${\mathcal{L}}_{n}$. In particular, it is shown that for any $n \ge 3$ there exist faces of dimension one and two. The maximum dimension of a proper face of ${\mathcal{L}}_{n}$ is also computed. The analysis depends on the so called Choi matrix associated with $T$.