  We formulate a quantum generalization of maximin surfaces and show that a
quantum maximin surface is identical to the minimal quantum extremal surface,
introduced in the EW prescription. We discuss various subtleties and
complications associated to a maximinimization of the bulk von Neumann entropy
due to corners and unboundedness and present arguments that nonetheless a
maximinimization of the UV-finite generalized entropy should be well-defined.
We give the first general proof that the EW prescription satisfies entanglement
wedge nesting and the strong subadditivity inequality. In addition, we apply
the quantum maximin technology to prove that recently proposed generalizations
of the EW prescription to nonholographic subsystems (including the so-called
"quantum extremal islands") also satisfy entanglement wedge nesting and strong
subadditivity. Our results hold in the regime where backreaction of bulk
quantum fields can be treated perturbatively in $G_{N}\hbar$, but we emphasize
that they are valid even when gradients of the bulk entropy are of the same
order as variations in the area, a regime recently investigated in new models
of black hole evaporation in AdS/CFT.
