Certifying optimality for convex quantum channel optimization problems

Abstract

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.

Geometric conditions for saturating the data processing inequality

“…a result previously shown in [25]. So by theorem 3.1, saturation of the DPI for the fidelity implies the equation…”

“…Our goal in this section will be to understand the derivatives of functions f : Pos(H) → R. We develop this machinery so that for any distinguishability measure B : Pos(H) × Pos(H) → R, and any positive operator σ, we will be able to compute the derivative of the restricted map B| σ : Pos(H) → R defined by B| σ (ρ) = B(ρ, σ). Our pedagogy in this section roughly follows section 5 of [25].…”