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arXiv:2007.15025 · Journal of High Energy Physics

A Rényi quantum null energy condition: proof for free field theories

Mudassir Moosa, Pratik Rath, Vincent Paul Su

Published Jan 13, 2021• 15 citations
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Abstract

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.

DOI
10.1007/jhep01(2021)064

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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 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 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.
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.
The quantum null energy condition (QNEC) is a quantum generalization of the null energy condition, which gives a lower bound on the null energy in terms of the second derivative of the von Neumann entropy or entanglement entropy of some region with respect to a null direction. The QNEC states that $⟨{T}_{kk}{⟩}_{p}\ensuremath{\ge}{\mathrm{lim}}_{A\ensuremath{\rightarrow}0}(\frac{\ensuremath{\hbar}}{2\ensuremath{\pi}A}{S}_{\mathrm{out}}^{\ensuremath{'}\ensuremath{'}})$, where Sout{S}_{\mathrm{out}}Sout​ is the entanglement entropy restricted to one side of a codimension-2 surface $\mathrm{\ensuremath{\Sigma}}$, which is deformed in the null direction about a neighborhood of point ppp with area AAA. A proof of QNEC has been given before, which applies to free and super-renormalizable bosonic field theories, and to any points that lie on a stationary null surface. Using similar assumptions and methods, we prove the QNEC for fermionic field theories.
The Quantum Null Energy Condition (QNEC) relates energy to the second variation of entropy in relativistic quantum field theory. We use the QNEC inequality to bound entanglement entropy in quenches. At early times the entanglement entropy grows quadratically in time, and the QNEC provides an upper bound on the prefactor. We demonstrate that the bound is tight, by showing that it is saturated in certain quench protocols: boundary state quenches in conformal field theories in any dimensions. In higher than two dimensions we compute entanglement entropy using AdS/CFT. Our results are the first purely field theoretic applications of the QNEC.
The quantum focusing conjecture (QFC) is a mathematical expression of the idea that semiclassical gravity remains universally attractive. In its non-gravitational limit, the diagonal inequality is the quantum null energy condition (QNEC), which requires convexity of relative entropy with respect to the vacuum under null deformations. Rényi divergences generalize relative entropy, and positive results for the diagonal Rényi QNEC raise the question of whether they also satisfy the off-diagonal inequalities required by a full focusing principle. We show that classical correlations between distinct null generators obstruct this extension for any non-affine Rényi-type divergence satisfying data processing, tensor additivity, and matched classical-quantum conditioning.