Certifying optimality for convex quantum channel optimization problems

Bryan Coutts1,2, Mark Girard1, and John Watrous1,2,3

1Institute for Quantum Computing, University of Waterloo, Canada
2School of Computer Science, University of Waterloo, Canada
3Canadian Institute for Advanced Research, Toronto, Canada

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We identify necessary and sufficient conditions for a quantum channel to be optimal for any convex optimization problem in which the optimization is taken over the set of all quantum channels of a fixed size. Optimality conditions for convex optimization problems over the set of all quantum measurements of a given system having a fixed number of measurement outcomes are obtained as a special case. In the case of linear objective functions for measurement optimization problems, our conditions reduce to the well-known Holevo-Yuen-Kennedy-Lax measurement optimality conditions. We illustrate how our conditions can be applied to various state transformation problems having non-linear objective functions based on the fidelity, trace distance, and quantum relative entropy.

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