When restricted to one side of a spatial partition, the vacuum state of a quantum field has entropy because the two sides are entangled

Under a simultaneous variation of the geometry and the state of the quantum fields, the diamond entanglement entropy variation will consist of two contributions, a state-independent UV part δSUV from the area change induced by δgab, and a state-dependent IR part δSIR from δ|ψ

Given our assumptions, that the semiclassical Einstein equation holds, for first order variations of the vacuum, if and only if the entropy in small causal diamonds is stationary at constant volume, when varied from a maximally symmetric vacuum state of geometry and quantum fields

We assumed the diamond size is much smaller than the local curvature length, the wavelength of any excitations of the vacuum, and the scales in the matter field theory, but much larger than the UV scale at which quantum gravity effects become strong

Not all energy registers as a change of entanglement. This is consistent with the hypothesis of maximal vacuum entanglement, the Einstein equation implies that the entropy has decreased — relative to vacuum — by more than it needs to in order to satisfy the hypothesis

A derivation of Einstein’s equation invoking a quantum limit to measurements of the spacetime geometry of small causal diamonds was given in Ref

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