  We study entanglement and other correlation properties of random states in
high-dimensional bipartite systems. These correlations are quantified by
parameters that are subject to the "concentration of measure" phenomenon,
meaning that on a large-probability set these parameters are close to their
expectation. For the entropy of entanglement, this has the counterintuitive
consequence that there exist large subspaces in which all pure states are close
to maximally entangled. This, in turn, implies the existence of mixed states
with entanglement of formation near that of a maximally entangled state, but
with negligible quantum mutual information and, therefore, negligible
distillable entanglement, secret key, and common randomness. It also implies a
very strong locking effect for the entanglement of formation: its value can
jump from maximal to near zero by tracing over a number of qubits negligible
compared to the size of total system. Furthermore, such properties are generic.
Similar phenomena are observed for random multiparty states, leading us to
speculate on the possibility that the theory of entanglement is much simplified
when restricted to asymptotically generic states. Further consequences of our
results include a complete derandomization of the protocol for universal
superdense coding of quantum states.
