On the Relative Distance of Entangled Systems in Emergent Spacetime Scenarios

Spacetime emergence from entanglement proposes an alternative to quantizing gravity and typically derives a notion of distance based on the amount of mutual information shared across sub-systems. Albeit promising, this program still faces challenges to describe simple physical systems, such as a maximally entangled Bell pair that is taken apart while preserving its entanglement. We propose a solution to this problem: a reminder that quantum systems can have multiple sectors of independent degrees of freedom, and that each sector can be entangled. Thus, while one sector can decohere, and decrease the amount of total mutual information within the system, another sector, e.g. spin, can remain entangled. We illustrate this with a toy model, showing that only within the particles' momentum uncertainty there can be considerably more entanglement than in the spin sector for a single Bell pair. We finish by introducing some considerations about how spacetime could be tested in the lab in the future.