Entanglement entropy, black holes and holography

TL;DR

This paper explores how entanglement entropy can grow faster than area in certain quantum states, but gravitational constraints limit this growth, framing holography as an entropy bound rather than a Hilbert space dimension limit.

Contribution

It demonstrates that pure states with long-range correlations can exhibit volume-law entanglement entropy growth, and shows gravitational collapse conditions restrict this growth, offering a new perspective on holography.

Findings

Entanglement entropy can grow proportionally to volume in certain states.

Gravitational collapse conditions prevent entropy from exceeding area law.

Holography acts as an upper bound on entropy, not Hilbert space size.

Abstract

We observe that the entanglement entropy resulting from tracing over a subregion of an initially pure state can grow faster than the surface area of the subregion (indeed, proportional to the volume), in contrast to examples studied previously. The pure states with this property have long-range correlations between interior and exterior modes and are constructed by purification of the desired density matrix. We show that imposing a no-gravitational collapse condition on the pure state is sufficient to exclude faster than area law entropy scaling. This observation leads to an interpretation of holography as an upper bound on the realizable entropy (entanglement or von Neumann) of a region, rather than on the dimension of its Hilbert space.