A chain rule for the quantum relative entropy

David Sutter
IBM Zürich Research Laboratory

The chain rule for the conditional entropy allows us to view the conditional entropy of a large composite system as a sum of conditional entropies over small subsystems. This property is one of the main reason why the conditional entropy is used as a mathematical tool within different areas of physics. A more general entropy measure, which contains the conditional entropy as a special case is called relative entropy. Unlike for the conditional entropy no chain rule for the quantum relative entropy is known.
In this talk, I will discuss a chain rule for the quantum relative entropy. The chain rule involves the regularized relative entropy between quantum channels and we show that this regularization is unavoidable. The new chain rule allows us to solve an open problem in the field of quantum statistics. It implies that adaptive strategies are asymptotically no more powerful than non-adaptive strategies for the task of asymmetric discrimination between two different quantum channels. If time permits I will conclude with some open questions related to the chain rule.
(Joint work with Kun Fang, Omar Fawzi, and Renato Renner based on https://arxiv.org/abs/1909.05826)

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