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Von Neumann entropy and majorization. (English) Zbl 1308.81050
Summary: We consider the properties of the Shannon entropy for two probability distributions which stand in the relationship of majorization. Then we give a generalization of a theorem due to Uhlmann, extending it to infinite dimensional Hilbert spaces. Finally we show that for any quantum channel $$\varPhi$$, one has $$S(\varPhi(\rho))=S(\rho)$$ for all quantum states $$\rho$$ if and only if there exists an isometric operator $$V$$ such that $$\varPhi(\rho)=V\rho V^\ast$$.

##### MSC:
 81P45 Quantum information, communication, networks (quantum-theoretic aspects) 94A40 Channel models (including quantum) in information and communication theory 47H07 Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces 94A17 Measures of information, entropy 81P15 Quantum measurement theory, state operations, state preparations
##### Keywords:
von Neumann entropy; majorization; quantum operation
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##### References:
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