# The Critical Points of Coherent Information on the Manifold of Positive Definite Matrices

 Title: The Critical Points of Coherent Information on the Manifold of Positive Definite Matrices Tehrani, Alireza Department of Mathematics and Statistics Mathematics and Statistics Zeng, BeiPereira, Rajesh The coherent information of quantum channels plays a important role in quantum information theory as it can be used to calculate the quantum capacity of a channel. However, it is a non-linear, non-differentiable optimization problem. 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. In addition, the classification of the maximally mixed state as a local maxima/minima and saddle-point will be solved for the one shot coherent information. The hope of this work is to provide a new avenue to explore the quantum capacity problem. http://hdl.handle.net/10214/17742 2020-01 Attribution 4.0 International All items in the Atrium are protected by copyright with all rights reserved unless otherwise indicated.
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