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.
Highlights
Several problems and settings that arise in quantum information theory can be expressed as optimization problems in which a real-valued function, defined for a class of quantum channels or measurements, is either minimized or maximized
Other examples arise in the study of quantum cloning [11, 28] and the closely related notion of quantum money [1], where one is generally interested in knowing how well an optimally selected quantum channel can transform a single copy of a given state into multiple copies of the same state, with respect to a number of different figures of merit
Another example can be found in quantum complexity theory, in which two-message quantum interactive proof systems [21] are naturally analyzed as optimization problems in which the objective function describes the probability that a given verifier accepts, and where the optimization is over all quantum channels of a fixed size, which describe the possible actions of a prover
Summary
Several problems and settings that arise in quantum information theory can be expressed as optimization problems in which a real-valued function, defined for a class of quantum channels or measurements, is either minimized or maximized. The problem of minimum error quantum state discrimination [3], in which a quantum state randomly selected from a known ensemble of states is to be identified with the smallest possible probability of error by means of a measurement, provides a well-known example This problem is naturally expressed as the optimization of a real-valued linear function defined on the set of all measurements with a fixed number of outcomes. Described explicitly later in this paper, are relatively easy to check; the problem of finding or approximating an optimal measurement, while efficiently solvable through the use of semidefinite programming [13, 20, 22], is in general a more computationally involved task These optimality conditions can be extended to obtain optimality conditions for realvalued linear functions defined on the set of all quantum channels transforming one quantum system to another. Expositions of this topic can be found in [30] and Chapter 12 of [32]
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