2026-10-23 · arXiv Quantum Physics
arXiv:2608.21214v2 Announce Type: replace
Abstract: We prove an integral representation of the $\alpha$-$z$ R\'enyi 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 R\'enyi 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.
原文链接：https://arxiv.org/abs/2608.21214
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