Most states in the Hilbert space are maximally entangled. This fact has proven useful to investigate—among other things—the foundations of statistical mechanics. Unfortunately, most states in the Hilbert space of a quantum many-body system are not physically accessible. We define physical ensembles of states acting on random factorized states by a circuit of length $k$ of random and independent unitaries with local support. We study the typicality of entanglement by means of the purity of the reduced state. We find that for a time $k=O(1)$, the typical purity obeys the area law. Thus, the upper bounds for area law are actually saturated, on average, with a variance that goes to zero for large systems. Similarly, we prove that by means of local evolution a subsystem of linear dimensions $L$ is typically entangled with a volume law when the time scales with the size of the subsystem. Moreover, we show that for large values of $k$ the reduced state becomes very close to the completely mixed state.

• Received 30 September 2011
• Corrected 27 July 2012

DOI:https://doi.org/10.1103/PhysRevLett.109.040502