Relative Entropy for Quantum Channels
Zishuo Zhao
TL;DR
This work extends quantum information concepts to bimodule quantum channels between finite von Neumann algebras. It defines a Pimsner-Popa entropy $H(\Phi|\Psi)$ for completely positive bimodule maps and relates it to a relative entropy for Fourier multipliers $D(\hat{\Phi}\|\hat{\Psi})$, establishing $H(\Phi|\Psi)\le D(\hat{\Phi}\|\hat{\Psi})$, with equality under downward Jones basic construction. An Araki-style relative entropy $S_{\tau}(\Phi,\Psi)$ is introduced for bimodule channels, shown to be monotone and convex, and found to bound $H(\Phi|\Psi)$ from above; Rényi entropies $S_p(\Phi|\Psi)$ interpolate between the log Pimsner-Popa index $\lambda(\Phi,\Psi)$ and $H(\Phi|\Psi)$, yielding $-\log\lambda(\Phi,\Psi)\ge S_p(\Phi|\Psi)\ge H(\Phi|\Psi)$ and a criterion via $S_{1/2}$ for downward Jones basic constructions. Collectively, these results provide a cohesive entropic framework for bimodule quantum channels, connecting finite-index subfactor theory, Fourier-analysis-based entropies, and Araki’s relative entropy in a unified operator-algebraic setting.
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 Rényi 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 Rényi entropy at $1/2$ serves a criterion for the existence of a downward Jones basic construction.