Quantum f-Divergences in von Neumann Algebras

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 α-Rényi relative entropy and its new version called the sandwiched α-Rényi 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 Rényi-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.

“The presentation of this monograph is very friendly, since it is addressed to a large audience made up of mathematicians and physicists, with a generous set of appendices on the basics of von Neumann algebras, positive self-adjoint operators (unbounded) in Hilbert spaces, operator convex functions on (0, 1), and operator connections with normal positive forms. It gathers updated information on f-divergences and it is very useful for specialists, although not exclusively.” (Aurelian Gheondea, Mathematical Reviews, October, 2022)

“The contents of the book presents very interesting topics for quantum information scientists … . I would like to recommend this book for two kinds of special groups of scientists: pure mathematicians, who want to learn the power of quantum computing, and quantum information theorists, who are tired of quantum computing. … this book will be a good guide. For the second group, including people like me, this book stimulates the dormant instincts about quantum information theory.” (Kabgyun Jeong, zbMATH 1476.81004, 2022)

The author is currently Professor Emeritus at Tohoku University.