Entropy of an n-system from its correlation with a k-reservoir

Let a random pure state vector be chosen in nk-dimensional Hilbert space, and consider an n-dimensional subsystem’s density matrix P. P will usually be close to the totally unpolarized mixed state if k is large. Specifically, the rms deviation of a probability from the mean value 1/n is (1−1/n2)/(kn +1)1/2. ’’Random’’ refers to unitarily invariant Haar measure.