Semiclassical analysis of SU(2)-equivariant quantum channels

Equivariant (or covariant) channels are completely positive, trace preserving maps that intertwine the action of a group G on two (irreducible) representations V and W. An interesting question is to characterise the set of states that minimise the output von Neumann entropy of such channels. For the group SU(2), this question was partially answered by Lieb and Solovej in 2014: for a certain class of SU(2)-equivariant channels, coherent states are minimisers of all concave functions of the output of channels. In this talk, I will present some asymptotic properties of SU(2)-equivariant quantum channels. I will explain how, in the limit of large output dimension, it is possible to approximate any channel by the quantisation of a “symbol” map related to the classical Husimi function. As a consequence, I will give an asymptotic expansion of the output entropy of SU(2)-equivariant quantum channels, which can be used to relate the problem of minimisation of the von Neumann entropy and the classical problem of minimisation of the Wehrl entropy. This is based on joint work with B. Ruba and J. P. Solovej.