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Books > Science & Mathematics > Physics > Quantum physics (quantum mechanics)
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Quantum f-Divergences in von Neumann Algebras - Reversibility of Quantum Operations (Paperback, 1st ed. 2021)
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Quantum f-Divergences in von Neumann Algebras - Reversibility of Quantum Operations (Paperback, 1st ed. 2021)
Series: Mathematical Physics Studies
Expected to ship within 10 - 15 working days
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Relative entropy has played a significant role in various fields of
mathematics and physics as the quantum version of the
Kullback-Leibler divergence in classical theory. Many variations of
relative entropy have been introduced so far with applications to
quantum information and related subjects. Typical examples are
three different classes, called the standard, the maximal, and the
measured f-divergences, all of which are defined in terms of
(operator) convex functions f on (0, ) and have respective
mathematical and information theoretical backgrounds. The -Renyi
relative entropy and its new version called the sandwiched -Renyi
relative entropy have also been useful in recent developments of
quantum information. In the first half of this monograph, the
different types of quantum f-divergences and the Renyi-type
divergences mentioned above in the general von Neumann algebra
setting are presented for study. While quantum information has been
developing mostly in the finite-dimensional setting, it is widely
believed that von Neumann algebras provide the most suitable
framework in studying quantum information and related subjects.
Thus, the advance of quantum divergences in von Neumann algebras
will be beneficial for further development of quantum information.
Quantum divergences are functions of two states (or more generally,
two positive linear functionals) on a quantum system and measure
the difference between the two states. They are often utilized to
address such problems as state discrimination, error correction,
and reversibility of quantum operations. In the second half of the
monograph, the reversibility/sufficiency theory for quantum
operations (quantum channels) between von Neumann algebras via
quantum f-divergences is explained, thus extending and
strengthening Petz' previous work. For the convenience of the
reader, an appendix including concise accounts of von Neumann
algebras is provided.
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