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The entanglement content of high-dimensional random pure states is almostmaximal, nevertheless, we show that, due to the complexity of such states, thedetection of their entanglement using witness operators is rather difficult. Wediscuss the case of unknown random states, and the case of known random statesfor which we can optimize the entanglement witness. Moreover, we show thatcoarse graining, modeled by considering mixtures of m random states instead ofpure ones, leads to a decay in the entanglement detection probabilityexponential with m. Our results also allow to explain the emergence ofclassicality in coarse grained quantum chaotic dynamics.

Detecting entanglement of random states with an entanglement witness