  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
$S_{\text{rel}}(\rho||\sigma)$ of an arbitrary state $\rho$ with respect to the
vacuum $\sigma$. The relative entropy has a natural one-parameter family
generalization, the Sandwiched Renyi divergence $S_n(\rho||\sigma)$, which also
measures the distinguishability of two states for arbitrary $n\in[1/2,\infty)$.
A Renyi QNEC, a positivity condition on the second null shape derivative of
$S_n(\rho||\sigma)$, was conjectured in previous work. In this work, we study
the Renyi 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 Renyi QNEC in the case $n>1$ for arbitrary states. We also provide
counterexamples to the Renyi QNEC for $n<1$.
