Geodesic
Reversibility conditions for quantum channels and their applications
Sbornik. Mathematics, Tome 204 (2013) no. 8, pp. 1215-1237

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Conditions for a quantum channel (noncommutative Markov operator) to be reversible with respect to complete families of quantum states with bounded rank are obtained. A description of all quantum channels reversible with respect to a given (orthogonal or nonorthogonal) complete family of pure states is given. Some applications in quantum information theory are considered. Bibliography: 20 titles.
Keywords: quantum states, quantum channels, relative entropy, $\chi$-quantity of ensemble of quantum states. 
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     title = {Reversibility conditions for quantum channels and their applications},
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M. E. Shirokov. Reversibility conditions for quantum channels and their applications. Sbornik. Mathematics, Tome 204 (2013) no. 8, pp. 1215-1237. http://geodesic.mathdoc.fr/item/SM_2013_204_8_a6/