MSc Math Defence: The critical points of coherent information on the manifold of positive definite matrices

CANDIDATE:   ALIREZA TEHRANI

ABSTRACT:

The coherent information of quantum channels plays an important role in quantum information theory. It is the quantum analogue of mutual information and can be used to calculate the quantum capacity of a channel. However, it is a non-linear, non-differentiable optimization problem with very little structure known. This thesis discusses that by restricting to the space of positive definite density matrices and restricting the class of quantum channels to be strictly positive, the coherent information becomes differentiable. This allows the computation of the Riemannian gradient and hessian of the coherent information.  It will be shown that the maximally mixed state is a critical point for the n-shot coherent information of the Pauli, dephrasure and Pauli-erasure channels. Further, the classification of the maximally mixed state as a local maxima/minima and saddle-point will be solved for the one shot coherent information.  Additional topics are explored relating product states to the critical points and the gradient to coherent information. The hope of this work is to provide a new avenue to explore the quantum capacity problem.