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arXiv:2608.21214

Integral representations of quantum Rényi divergences on von Neumann algebras

Tanay Kibe, Pratik Roy

quant-phhep-thmath-phmath.OASubmitted Aug 21, 2026• Announced Aug 24, 2026• 0 citations• Revised Oct 22, 2026 (v2)
v1 · Aug 21, 2026v2 · Oct 22, 2026
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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 $L^p$ 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.

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48 pages

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We prove an integral representation of the ααα-zzz Rényi divergences for normal states on arbitrary von Neumann algebras. Along each ray z=cαz=cαz=cα, the divergence is a positive average of relative entropies evaluated in a canonical family of escort states. For orders ααα 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 LpL^pLp 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 ααα-zzz 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.
We prove recovery theorems for ααα-zzz Rényi divergences under normal unital positive maps between von Neumann algebras. Previous results in this setting required 2-positivity, while recent finite dimensional work showed that this assumption can be relaxed to mere positivity. Our proof uses sufficiency for JW*-subalgebras and LpL^pLp-spaces over them to characterize equality in the data processing inequality. As a further application of our methods, we solve a problem discussed by Haagerup and Stormer: Every conditional expectation of a von Neumann algebra onto a JW*-subalgebra factors through a conditional expectation onto the generated von Neumann subalgebra.
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