Recovering Einstein's equation from local correlations with quantum reference frames

The observable spacetime can be viewed as worldline coincidences (events) between a particle system and the observers of an extended (material) reference frame (ERF). Particle positions are then operationally well defined with respect to that frame. In the ideal regime where the ERF contributes negligibly to the stress--energy tensor, the metric field $g_{ab}$ is indifferent to its physical presence. Accordingly, $g_{ab}$ may be viewed as encoding spacetime intervals relative to any ideal ERF placed in the region of interest. In quantum theory, by contrast, the localization events defining such intervals are naturally accompanied by correlations with local observers of the ERF. Motivated by this complementarity, we propose that the metric encodes, in geometric form, the relational information carried by correlations with a local reference frame, thereby dispensing with its explicit presence. Under a suitable constraint on the corresponding conditional entropy, this framework yields the full nonlinear Einstein equation with a reference spacetime whose scalar curvature equals the cosmological constant.