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arXiv:1106.4295 · Journal of Physics A Mathematical and Theoretical

Extreme commutative quantum observables are sharp

Teiko Heinosaari, Juha-Pekka Pellonpää

Published Jul 8, 2011• 4 citations
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Abstract

It is well known that, in the description of quantum observables, positive operator valued measures (POVMs) generalize projection valued measures (PVMs) and they also turn out be more optimal in many tasks. We show that a commutative POVM is an extreme point in the convex set of all POVMs if and only if it is a PVM. This results implies that non-commutativity is a necessary ingredient to overcome the limitations of PVMs.

DOI
10.1088/1751-8113/44/31/315303

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It is well known that, in the description of quantum observables, positive operator valued measures (POVMs) generalize projection valued measures (PVMs) and they also turn out be more optimal in many tasks. We show that a commutative POVM is an extreme point in the convex set of all POVMs if and only if it is a PVM. This results implies that non-commutativity is a necessary ingredient to overcome the limitations of PVMs.
Measurements on quantum channels are described by so-called process positive operator valued measures, or process POVMs. We study implementing schemes of extremal process POVMs. As it turns out, the corresponding measurement must satisfy certain extremality property, which is stronger than the usual extremality given by the convex structure. This property motivates the introduction and investigation of the A-convex structure of POVMs, which generalizes both the usual convex and C*-convex structures. We show that extremal points and faces of the set of process POVMs are closely related to A-extremal points and A-faces of POVMs, for a certain subalgebra A. We also give a characterization of A-extremal and A-pure POVMs.
Convex sets of quantum states and processes play a central role in quantum theory and quantum information. Many important examples of convex sets in quantum theory are spectrahedra, that is, sets of positive operators satisfying affine constraints. These examples include sets of quantum states with given expectation values of a set of observables, sets of multipartite quantum states with given marginals, sets of quantum measurements, channels and multitime quantum processes, as well as sets of higher-order quantum maps and quantum causal structures. This contribution provides a characterization of the extreme points of general spectrahedra, and bounds on the ranks of the corresponding operators. The general results are applied to several special cases, and then used to retrieve classic results such as Choi’s characterization of the extreme quantum channels, Parthasarathy’s characterization of the extreme quantum states with given marginals and the quantum version of Birkhoff’s theorem for qubit unital channels. Finally, we propose a notion of positive operator valued measures (POVMs) with general affine constraints for their normalization, and we characterize the extremal POVMs.
Convex sets of quantum states and processes play a central role in quantum theory and quantum information. Many important examples of convex sets in quantum theory are spectrahedra, that is, sets of positive operators subject to affine constraints. These examples include sets of quantum states with given expectation values of some observables of interest, sets of multipartite quantum states with given marginals, sets of quantum measurements, channels, and multitime quantum processes, as well as sets of higher-order quantum maps and quantum causal structures. This contribution provides a characterisation of the extreme points of general spectrahedra, and bounds on the ranks of the corresponding operators. The general results are applied to several special cases, and are used to retrieve classic results such as Choi's characterisation of the extreme quantum channels, Parhasarathy's characterisation of the extreme quantum states with given marginals, and the quantum version of Birkhoff's theorem for qubit unital channels. Finally, we propose a notion of positive operator valued measures (POVMs) with general affine constraints for their normalisation, and we characterise the extremal POVMs.
In many a traditional physics textbook, a quantum measurement is defined as a projective measurement represented by a Hermitian operator. In quantum information theory, however, the concept of a measurement is dealt with in complete generality and we are therefore forced to confront the more general notion of positive-operator valued measures (POVMs), which suffice to describe all measurements that can be implemented in quantum experiments. We study the (in)compatibility of such POVMs and show that quantum theory realizes all possible (in)compatibility relations among sets of POVMs. This is in contrast to the restricted case of projective measurements for which commutativity is essentially equivalent to compatibility. Our result therefore points out a fundamental feature with respect to the (in)compatibility of quantum observables that has no analog in the case of projective measurements.