Conditional entropy and information of quantum processes

TL;DR

This paper introduces a new way to define and analyze the conditional entropy of bipartite quantum processes, revealing insights into their causal structure and correlations using information-theoretic principles.

Contribution

It develops a novel definition of quantum channel conditional entropy based on divergences, linking it to causal influence and establishing properties like strong subadditivity.

Findings

Conditional entropy relates to causal influence in quantum channels.

Channels with certain entropy values exhibit signaling behavior.

The framework establishes strong subadditivity for quantum channel entropy.

Abstract

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 four information-theoretic axioms and define the corresponding quantitative formulas. Our definitions of the conditional entropies of channels are based on the generalized state and channel divergences, for instance, quantum relative entropy. We find that the conditional entropy of quantum channels has potential to reveal insights for quantum processes that aren't already captured by the existing entropic functions, entropy or conditional entropy, of the states and channels. The von Neumann conditional entropy $S[A|B]_{\mathcal{N}}$ of the channel $\mathcal{N}_{A'B'\to AB}$ is based on the quantum relative entropy, with system pairs $A',A$ and $B',B$ being nonconditioning and conditioning systems, respectively. We identify a connection between the underlying causal structure of a bipartite channel and its conditional entropy. In particular, we provide a necessary and sufficient condition for a bipartite quantum channel $\mathcal{N}_{A'B'\to AB}$ in terms of its von Neumann conditional entropy $S[A|B]_{\mathcal{N}}$, to have no causal influence from $A'$ to $B$. As a consequence, if $S[A|B]_{\mathcal{N}}< -\log|A|$ then the channel necessarily has causal influence (signaling) from $A'$ to $B$. Our definition of the conditional entropy establishes the strong subadditivity of the entropy for quantum channels. We also study the total amount of correlations possible due to quantum processes by defining the multipartite mutual information of quantum channels.