Relative Entropy for Quantum Channels (2312.16576v2)
Abstract: We introduce an quantum entropy for bimodule quantum channels on finite von Neumann algebras, generalizing the remarkable Pimsner-Popa entropy. The relative entropy for Fourier multipliers of bimodule quantum channels establishes an upper bound of the quantum entropy. Additionally, we present the Araki relative entropy for bimodule quantum channels, revealing its equivalence to the relative entropy for Fourier multipliers and demonstrating its left/right monotonicities and convexity. Notably, the quantum entropy attains its maximum if there is a downward Jones basic construction. By considering R\'{e}nyi entropy for Fourier multipliers, we find a continuous bridge between the logarithm of the Pimsner-Popa index and the Pimsner-Popa entropy. As a consequence, the R\'{e}nyi entropy at $1/2$ serves a criterion for the existence of a downward Jones basic construction.
- A.Ocneanu. Quantized groups, string algebras, and Galois theory for algebras, volume 2 of London Mathematical Society Lecture Note Series, pages 119–172. Cambridge University Press, 1989.
- H. Araki. Relative entropy of states of von Neumann algebras II. Publications of The Research Institute for Mathematical Sciences, 13:173–192, 1977.
- D. Bisch. Bimodules, higher relative commutants and the fusion algebra associated to a subfactor. The Fields Institute for Research in Mathematical Sciences Communications Series, 13, 1997.
- M. Burns. Subfactors, Planar Algebras and Rotations. Phd thesis, University of California, Berkeley, May 2003. Available at https://arxiv.org/abs/1111.1362.
- A. Connes. Noncommutative Geometry. Academic Press, 1994.
- A. Connes and E. Størmer. Entropy for automorphisms of II1 von Neumann algebras. Acta Mathematica, 134:289 – 306, 1975.
- Phase groups of bimodule quantum channels, 2023.
- The quantum perron-frobenius space, 2023.
- Quantum convolution inequalities on frobenius von neumann algebras, 2022.
- Quantum fourier analysis. Proceedings of the National Academy of Sciences, 117(20):10715–10720, 2020.
- Noncommutative uncertainty principles. Journal of Functional Analysis, 270(1):264–311, 2016.
- V. Jones. Index for subfactors. Inventiones mathematicae, 72:1–26, 1983.
- Relative entropy for von neumann subalgebras. International Journal of Mathematics, 31(06):2050046, may 2020.
- R. Longo. Index of subfactors and statistics of quantum fields. I. Communications in Mathematical Physics, 126(2):217 – 247, 1989.
- On quantum Rényi entropies: A new generalization and some properties. Journal of Mathematical Physics, 54(12):122203, 12 2013.
- D. Petz. Sufficiency of channels over von Neumann algebras. Quarterly Journal of Mathematics, 39:97–108, 1988.
- M. Pimsner and S. Popa. Entropy and Index for Subfactors. Annales Scientifiques De L Ecole Normale Superieure, 1986.
- M. Pimsner and S. Popa. Iterating the basic construction. Transactions of the American Mathematical Society, 1988.
- S. Popa. Correspondences. Available at https://www.math.ucla.edu/~popa/popa-correspondences.pdf, 1986.
- S. Popa. The relative Dixmier property for inclusions of von Neumann algebras of finite index. Annales scientifiques de l’École Normale Supérieure, 32(6):743–767, 1999.
- S. Popa and C. Anantharaman. An introduction to II1 factors. Available at https://www.math.ucla.edu/~popa/Books/IIunV15.pdf, 2010.
- M. Takesaki. Theory of Operator Algebras II. Springer Berlin, Heidelberg, 2003.
- H. Umegaki. Conditional expectations in an operator algebra iv (entropy and information). Kodai Mathematical Seminar Reports, 14(2):59–85, 1988.