Distance to boundary and minimum-error discrimination

Abstract

We introduce the concept of boundariness capturing the most efficient way of expressing a given element of a convex set as a probability mixture of its boundary elements. In other words, this number measures (without the need of any explicit topology) how far the given element is from the boundary. It is shown that one of the elements from the boundary can be always chosen to be an extremal element. We focus on evaluation of this quantity for quantum sets of states, channels and observables. We show that boundariness is intimately related to (semi)norms that provide an operational interpretation of this quantity. In particular, the minimum error probability for discrimination of a pair of quantum devices is lower bounded by the boundariness of each of them. We proved that for states and observables this bound is saturated and conjectured this feature for channels. The boundariness is zero for infinite-dimensional quantum objects as in this case all the elements are boundary elements.

14 pages, 6 figures (final version)

References in corpus (2)

• Effects as functions on projective Hilbert space
• Radon-Nikodym derivatives of quantum operations

Cited by in corpus (8)

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• Probabilistic transformations of quantum resources
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• Continuous-variable supraquantum nonlocality
• Convex decomposition of dimension-altering quantum channels
• Exploring boundaries of quantum convex structures: special role of unitary processes
• Coherent information of a quantum channel or its complement is generically positive