Conditional Entropy of Bipartite Quantum Processes

What would be a reasonable definition of the conditional entropy of bipartite quantum processes, and what novel insight would it provide? We develop this notion using desirable information-theoretic axioms and define the corresponding quantitative formulas based on the generalized channel divergences, e.g., quantum relative entropy and max-relative entropy. We show that the conditional entropy of a bipartite quantum channel provides insight into the underlying causal structure of the channel. We realize a necessary and sufficient condition for the (von Neumann) conditional entropy of a quantum channel, for which the channel shows no causal influence from the non-conditioning input system to the conditioning output system. Additionally, our definition of conditional entropy establishes the strong subadditivity of the entropy for quantum channels. Furthermore, we extend our approach behind the definition of conditional entropy to quantify the total amount of correlations due to quantum processes by defining the mutual information and conditional mutual information of quantum channels.

The related pre-print is arXiv:2410.01740 [quant-ph].

Short bio:
Siddhartha Das tries to understand physical laws of the nature (universe) from quantum information-theoretic viewpoint. He is an Assistant Professor at the International Institute of Information Technology, Hyderabad, India since January 2022. He obtained his PhD from Louisiana State University, Baton Rouge, USA and was a PostDoc at Université libre de Bruxelles, Brussels, Belgium. His pre-prints and publications can be found at arxiv.org/a/das_s_6.html