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.

Lucretius, On the Nature of the Universe, translated by R. E. Latham (Penguin, Toronto, 1951), p. 66.
L. D.
Z. Physik
or J. Von Neuman, Mathematical Foundations of Quantum Mechanics, translated by R. Beyer (Princeton U.P., Princeton, N.J., 1955), Sec. VI. 2.
Adapted from I.S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965), p. 369, or induction through integration by parts.
This extreme, pure‐case answer σ = n−1(n−1)1/2 coincides with the specialization to k = 1 of Eq. (1), because k = 1 1 forces purity by eliminating the reservoir. This is another check of (l).
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