Highly symmetric POVMs and their informational power

We discuss the dependence of the Shannon entropy of normalized finite rank-1 POVMs on the choice of the input state, looking for the states that minimize this quantity. To distinguish the class of measurements where the problem can be solved analytically, we introduce the notion of highly symmetric...

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Bibliographic Details
Published in:Quantum information processing 2016-01, Vol.15 (1), p.565-606
Main Authors: Słomczyński, Wojciech, Szymusiak, Anna
Format: Article
Language:English
Subjects:
Data Structures and Information Theory
Mathematical Physics
Physics
Physics and Astronomy
Quantum Computing
Quantum Information Technology
Quantum Physics
Spintronics
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Summary:We discuss the dependence of the Shannon entropy of normalized finite rank-1 POVMs on the choice of the input state, looking for the states that minimize this quantity. To distinguish the class of measurements where the problem can be solved analytically, we introduce the notion of highly symmetric POVMs and classify them in dimension 2 (for qubits). In this case, we prove that the entropy is minimal, and hence, the relative entropy (informational power) is maximal, if and only if the input state is orthogonal to one of the states constituting a POVM. The method used in the proof, employing the Michel theory of critical points for group action, the Hermite interpolation, and the structure of invariant polynomials for unitary–antiunitary groups, can also be applied in higher dimensions and for other entropy-like functions. The links between entropy minimization and entropic uncertainty relations, the Wehrl entropy, and the quantum dynamical entropy are described.
ISSN:1570-0755
1573-1332
DOI:10.1007/s11128-015-1157-z