Extreme quantum states and processes, and extreme points of general
spectrahedra in finite dimensional algebras
- URL: http://arxiv.org/abs/2311.10929v1
- Date: Sat, 18 Nov 2023 01:31:16 GMT
- Title: Extreme quantum states and processes, and extreme points of general
spectrahedra in finite dimensional algebras
- Authors: Giulio Chiribella
- Abstract summary: 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.
This contribution provides a characterisation of the extreme points of general spectrahedra, and bounds on the ranks of the corresponding operators.
- Score: 0.27195102129094995
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: 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.
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