Towards a functorial description of quantum relative entropy. (English) Zbl 1485.94038
Nielsen, Frank (ed.) et al., Geometric science of information. 5th international conference, GSI 2021, Paris, France, July 21–23, 2021. Proceedings. Cham: Springer. Lect. Notes Comput. Sci. 12829, 557-564 (2021).
Summary: A Bayesian functorial characterization of the classical relative entropy (KL divergence) of finite probabilities was recently obtained by J. C. Baez and T. Fritz [Theory Appl. Categ. 29, 422–456 (2014; Zbl 1321.94023)]. This was then generalized to standard Borel spaces by N. Gagné and P. Panangaden [Electron. Notes Theor. Comput. Sci. 336, 135–153 (2018; Zbl 1525.60011)]. Here, we provide preliminary calculations suggesting that the finite-dimensional quantum (Umegaki) relative entropy might be characterized in a similar way. Namely, we explicitly prove that it defines an affine functor in the special case where the relative entropy is finite. A recent non-commutative disintegration theorem provides a key ingredient in this proof.
For the entire collection see [Zbl 1482.94007].
For the entire collection see [Zbl 1482.94007].
MSC:
| 94A17 | Measures of information, entropy |
| 81P45 | Quantum information, communication, networks (quantum-theoretic aspects) |
| 62B10 | Statistical aspects of information-theoretic topics |
References:
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