Integral representations of quantum Rényi divergences on von Neumann algebras

Abstract

We prove an integral representation of the $\alpha$-$z$ Rényi divergences for normal states on arbitrary von Neumann algebras. Along each ray $z=c\alpha$, the divergence is a positive average of relative entropies evaluated in a canonical family of escort states. For orders $\alpha$ above one, we require the divergence to be finite, whereas for orders below one support inclusion suffices. An explicit boundary correction removes the lower-order support assumption. The proof uses a compatible weighted Haagerup $L^p$ scale and applies outside the data-processing region. For faithful density matrices, we express the derivative of the escort relative entropy with respect to the Rényi order in terms of a nonnegative fluctuation quantity. Differences of order-one derivatives of the divergence give quantum Fisher information and Wigner-Yanase skew information. For faithful bipartite density matrices, we also express optimized conditional $\alpha$-$z$ entropies at orders above one as averages of ordinary conditional entropies. Further applications concern coherent-state divergences in quantum field theory and the optimizing order for the strong converse exponent in hypothesis-testing.

arXiv comment48 pages