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

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.

Correlation functions and quantum measures of descendant states

“…Notice that for h → 0 we recover the same scaling as in (4.9). In all the examples that we considered, the SRD proved to be a convex function of x, providing further evidence to the validity of the Rényi QNEC in two dimensions [46] for large enough central charge.…”

“…We also checked explicitly if predictions from the generalized version of QNEC [14,46] are true for descendant states, namely that the sandwiched Rényi divergence is a convex function of subsystem size. In the Ising model and Potts model in all the cases we checked, the SRD is a convex function.…”

“…After the QNEC has been proven in free and holographic theories [49][50][51] it has since been shown to hold quite generally in the context of Tomita-Takesaki modular theory [52,53]. Recently a generalized version of QNEC has been suggested in [14] and later proven to be true in free theories in dimensions larger than two [46]. This generalization may be called 'Rényi Quantum Null Energy Condition' and is formulated as the positivity of the second variation of sandwiched Rényi entropies.…”

“…In[46] the interested reader can find arguments why this is not simply an artifact of the CFT construction but holds generally when one assumes that the state is prepared from a Euclidean path integral.…”

Holographic Rényi relative divergence in JT gravity

“…It has also nice properties which are inherited from relative entropy [23]. Recent discussions of applications of RRD include [24][25][26][27][28]. One of the difficulties in computing RRD is that this quantity breaks the U(1) translational symmetry of Euclidean time direction.…”