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.
Highlights
The quintessential result that arose from this connection is the Quantum Null Energy Condition (QNEC), which follows from the Quantum Focusing Conjecture [1]
We show that the second sandwiched Rényi divergence (SRD) variation is positive for n > 1, proving the Rényi QNEC in this case
By considering states that are perturbatively close to the vacuum and computing the second SRD variation in a perturbative expansion, we provide evidence that the Rényi QNEC could be saturated, just like the QNEC [12]
Summary
The Sandwiched Rényi Divergence (SRD), SnM(Ψ||Φ), is a measure of distinguishability of two quantum states |Ψ and |Φ given an algebra of operators M [15, 16]. Since the Rényi QNEC is formulated in terms of the SRD, we first review its definition and properties. This definition is not directly applicable to QFT where reduced density matrices do not exist in the continuum limit. Having defined SRD, our main focus will be constructing a physically reasonable set of states using the Euclidean path integral for which SRD with respect to the vacuum state is finite
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