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arXiv:2609.04016

Quasi-local form for $α$--$z$ Rényi QNEC from fixed-ray escorts

Tanay Kibe, Pratik Roy

hep-thquant-phSubmitted Sep 3, 2026• Announced Sep 4, 2026• 0 citations
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Abstract

The fixed-ray escort integral representation expresses $α$--$z$ Rényi divergence as an average over ordinary relative entropy of a family of escort states. Working in the standard UV-regulated density-matrix description of QFT subregions, we use this representation to derive an escort-averaged entanglement first law, an escort-averaged representation of the $α$--$z$ information kernel, and an escort-averaged Bekenstein-type bound for ball-shaped regions in conformal field theories. For the conjectural $α$--$z$ quantum null energy condition (QNEC), we obtain a quasi-local form in which the null energy is evaluated in an escort-averaged state and is corrected by an escort-transport term encoding the failure of escort formation to commute with restriction to a null-deformed region. The $z=α$ specialization gives a similar quasi-local form of the Rényi QNEC for sandwiched Rényi divergence. We explicitly compute the Rényi QNEC, including the explicit escort transport term, for coherent-state excitations in a free scalar field theory. For the same coherent family, we obtain a positive $α$--$z$ null Hessian, verifying the conjectured diagonal $α$--$z$ QNEC for this family.

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45 pages

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Quasi-local form for ααα--zzz Rényi QNEC from fixed-ray escorts
The fixed-ray escort integral representation expresses ααα--zzz Rényi divergence as an average over ordinary relative entropy of a family of escort states. Working in the standard UV-regulated density-matrix description of QFT subregions, we use this representation to derive an escort-averaged entanglement first law, an escort-averaged representation of the ααα--zzz information kernel, and an escort-averaged Bekenstein-type bound for ball-shaped regions in conformal field theories. For the conjectural ααα--zzz quantum null energy condition (QNEC), we obtain a quasi-local form in which the null energy is evaluated in an escort-averaged state and is corrected by an escort-transport term encoding the failure of escort formation to commute with restriction to a null-deformed region. The z=αz=αz=α specialization gives a similar quasi-local form of the Rényi QNEC for sandwiched Rényi divergence. We explicitly compute the Rényi QNEC, including the explicit escort transport term, for coherent-state excitations in a free scalar field theory. For the same coherent family, we obtain a positive ααα--zzz null Hessian, verifying the conjectured diagonal ααα--zzz QNEC for this family.
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.
The quantum null energy condition (QNEC) is usually stated as a bound on the expectation value of null components of the stress-energy tensor at a point in terms of second null shape variations of the entanglement entropy at the same point. It can be recast as the statement that the sign of the second null shape variation of the relative entropy of any state with respect to the vacuum is positive. Using instead a R\'enyi generalization of relative entropy, called sandwiched R\'enyi divergence (SRD), leads to what is termed the R\'enyi QNEC: The second null shape variation of SRD of any state with respect to the vacuum is positive. In this work, we prove the R\'enyi QNEC for free and superrenormalizable fermionic quantum field theories in spacetime dimensions greater than 2 using null quantization, for the case where the R\'enyi parameter n>1n>1n>1. We end with comments on multiple possible generalizations.
The Quantum Null Energy Condition (QNEC) is a lower bound on the stress-energy tensor in quantum field theory that has been proved quite generally. It can equivalently be phrased as a positivity condition on the second null shape derivative of the relative entropy Srel(ρ||σ) of an arbitrary state ρ with respect to the vacuum σ. The relative entropy has a natural one-parameter family generalization, the Sandwiched Rényi divergence Sn(ρ||σ), which also measures the distinguishability of two states for arbitrary n ∈ [1/2, ∞). A Rényi QNEC, a positivity condition on the second null shape derivative of Sn(ρ||σ), was conjectured in previous work. In this work, we study the Rényi QNEC for free and superrenormalizable field theories in spacetime dimension d > 2 using the technique of null quantization. In the above setting, we prove the Rényi QNEC in the case n > 1 for arbitrary states. We also provide counterexamples to the Rényi QNEC for n < 1.
We prove the Rényi quantum null energy condition for every integer Rényi order n≥2n\geq2n≥2. In local Poincaré-invariant quantum field theory, this states that the sandwiched Rényi divergence (SRD) of an excited state relative to the vacuum is convex along a future-directed null deformation. We assume that the deformation is realized by the canonical translations of a half-sided modular inclusion and that the divergence is finite for the initial, undeformed algebra. The proof establishes convexity of the SRD along the induced positive contraction semigroup on Haagerup Ln(M)L^n(M)Ln(M), for every non-zero positive initial element. We also establish a Rényi ant formula at every even integer order and, under an additional regularity assumption, at odd integer orders. The Rényi ant formula identifies the negative first null derivative of SRD with a multiple of the infimum of the canonical null-translation energy over vector purifications of an escort state constructed from the excited state and the vacuum.
Abstract The Quantum Null Energy Condition (QNEC) is a lower bound on the stress-energy tensor in quantum field theory that has been proved quite generally. It can equivalently be phrased as a positivity condition on the second null shape derivative of the relative entropy Srel(ρ||σ) of an arbitrary state ρ with respect to the vacuum σ. The relative entropy has a natural one-parameter family generalization, the Sandwiched Rényi divergence Sn(ρ||σ), which also measures the distinguishability of two states for arbitrary n ∈ [1/2, ∞). A Rényi QNEC, a positivity condition on the second null shape derivative of Sn(ρ||σ), was conjectured in previous work. In this work, we study the Rényi QNEC for free and superrenormalizable field theories in spacetime dimension d > 2 using the technique of null quantization. In the above setting, we prove the Rényi QNEC in the case n > 1 for arbitrary states. We also provide counterexamples to the Rényi QNEC for n < 1.
The trade-off relations between the two types of error probabilities in binary i.i.d. quantum state discrimination can be expressed by single-copy formulas in terms of the Petz-type and the sandwiched Rényi divergences of the two states representing the two hypotheses. In the non-i.i.d. setting, the error exponents can usually be expressed in terms of regularized Rényi divergences, which do not admit explicit formulas in general. Here, we consider a class of states, translation-invariant and gauge-invariant quasifree states on doubly infinite fermionic chains, and give explicit formulas for a wide range of regularized Rényi divergences between such states, including (α,z)(α,z)(α,z), log-Euclidean, geometric, measured, and the recently introduced integral Rényi divergences. We show that the case where there is a single mode at each lattice site becomes asymptotically classical, with all the different types of regularized Rényi divergences being equal, while in the case of multiple modes per site, non-commutativity persists under regularization, and for any fixed ααα, the regularized Rényi (α,z)(α,z)(α,z)-divergences give different regularized values for different zzz parameters in general. We also generalize a previous construction from [Bunth, Maróti, Mosonyi, Zimborás, Lett.~Math.~Phys.~113:(7), 2023] to the case of multiple modes per lattice site to obtain a large class of states exhibiting super-exponential decay of the discrimination error probabilities.