Integral representations of quantum Rényi divergences on von Neumann algebras
Abstract
We prove an integral representation of the α-z Rényi divergences for normal states on arbitrary von Neumann algebras. Along each ray z=cα, the divergence is a positive average of relative entropies evaluated in a canonical family of escort states. For orders α 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 Lp 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 α-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.
Related
- Sufficient positive maps between von Neumann algebras: Rényi divergences
- Quantum channels on duals of von Neumann algebras in the Schrödinger picture
- Conditioning on Subalgebras - Entropy Duality and Generalized Quantum Stein's Lemma in von Neumann Algebras
- Exact exponents for smoothing the max-relative entropy and quantum information decoupling in von Neumann algebras
- Thoughts on Quantum Events, Quantizing Gravity and von Neumann Algebras
BibTeX
@misc{kibe2026integral,
title = {{Integral representations of quantum Rényi divergences on von Neumann algebras}},
author = {Tanay Kibe and Pratik Roy},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.21214},
eprint = {2608.21214},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.21214},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Integral representations of quantum Rényi divergences on von Neumann algebras
AU - Tanay Kibe
AU - Pratik Roy
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2608.21214
UR - https://arxiv.org/abs/2608.21214
AB - We prove an integral representation of the α-z Rényi divergences for normal states on arbitrary von Neumann algebras. Along each ray z=cα, the divergence is a positive average of relative entropies evaluated in a canonical family of escort states. For orders α 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 α-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.
ER -