The coherent information on the manifold of positive definite density matrices

This paper will explore the restriction of the coherent information to the positive definite density matrices in the special case where the quantum channels are strictly positive linear maps. The space of positive definite density matrices is equipped with an embedded submanifold structure of the real vector space of Hermitian matrices. These ensure that the n-shot coherent information is differentiable and allows for the computation of its gradient and Hessian. We show that any tensor products of critical points preserve being a critical point of the coherent information. Furthermore, we show that for any positive integer n, the maximally mixed state is always a critical point for the class of mixed unitary quantum channels with orthogonal, unitary Kraus operators. We determine when the maximally mixed state is a local maximum/minimum or saddle point, including its eigenvectors, for the class of Pauli-erasure channels when n is equal to 1. This class includes the dephrasure channel and Pauli channel and refines potential regions where super-additivity is thought to occur. These techniques can be used to study other optimization problems over density matrices and allow the use of manifold optimization algorithms and a better understanding of the quantum capacity problem by utilizing the first and second order geometry.