Orbit-injective covariant quantum channels

Abstract

The purpose of this paper is to investigate the quantum channels that preserve and also separate the orbits of pure states under the action of a group unitary representation π. Such a quantum channel will be called π-orbit injective. We prove that for finite group and complex Hilbert space cases, such a channel necessarily separates all the pure states. However, this is no longer true for quantum channels acting on real Hilbert spaces, or quantum channels acting on complex Hilbert spaces with (infinite) compact group representations. In both cases, we obtain necessary and/or sufficient conditions under which the quantum channel is orbit injective. These conditions are given in terms of the so called property (H) of characters (more generally, irreducible representations) of the group, and characterizations of property (H) are presented for real and complex valued multiplicative characters.