  We investigate the quantum null energy condition (QNEC) in holographic CFTs,
focusing on half-spaces and particular classes of states. We present direct,
and in certain cases nonperturbative, calculations for both the diagonal and
off- diagonal variational derivatives of entanglement entropy. In d > 2, we
find that the QNEC is saturated. We compute relations between the off-diagonal
variation of entanglement, boundary relative entropy, and the bulk stress
tensor. Strong subadditivity then leads to energy conditions in the bulk. In d
= 2, we find that the QNEC is in general not saturated when the Ryu-Takayanagi
surface intersects bulk matter. Moreover, when bulk matter is present the QNEC
can imply new bulk energy conditions. For a simple class of states, we derive
an example that is stronger than the bulk averaged null energy condition and
reduces to it in certain limits.
