Traditional statistical physics studies random states, i.e., probability distributions on the set of states. In quantum physics, such distributions can be described by mixed states \(\rho\), i.e., crudely speaking, states in which several pure states \(\psi_i\) occur with probabilities \(\lambda_i\), \(\sum_{i=1}^N \lambda_i=1\). It is desirable to know what happens in a “generic” mixed state. For that, it is reasonable to introduce a natural probability measure on the set of all mixed states, and to study what happens for almost all mixed states – almost all in the sense of this measure. Mixed states form a manifold; so, to describe a probability measure, it is makes sense to select a Riemannian metric; then, the metric-induced volume defines a probability measure. Several metrics have been proposed; one of the most physically meaningful among them is the Bures metric that describes the mutual information between the mixed states \(D_B(\rho,\sigma)=\sqrt{2-2\mathrm{Tr}\{[\sqrt{\rho}\sigma\sqrt{\rho}]\}}\).
One of the physically relevant questions is how pure the random mixed states are. In a pure state, \(\lambda_i=1\) for some \(i\) and \(\lambda_j=0\) for all \(j\neq i\). A natural measure of corresponding uncertainty is the entropy \(S=-\sum \lambda_i\cdot \ln(\lambda_i)\): it is 0 if and only if the state is pure. However, entropy is difficult to compute, so most quantum researchers use a linear approximation \(S_{\mathrm{lin}}\), in which the term \(\ln(\lambda_i)=\ln(1+x)\) is replaced by the linear part \(x=\lambda_i-1\) of its Taylor expansion. The resulting expression \(S_{\mathrm{lin}}=\sum \lambda_i^2+\sum\lambda_i=1-\sum \lambda_i^2\) is also equal to 0 if and only if the state is pure. The opposite value \(\Sigma_2=1-S_{\mathrm{lin}}\) is equal to 1 if and only if the state is pure; this value \(\Sigma_2\) is thus called purity.
The authors provide explicit formulas for the asymptotic distribution of purity of a random mixed state when \(N\to \infty\). They find three different asymptotic expressions corresponding to different ranges of parameters. Computations are performed not only for the random Bures mixed states, but also for a reasonable 1-parametric generalization of the Bures measure. Estimates are obtained by a reduction to a (non-quantum) statistical Coulomb gas model: a gas in which molecules are interacting according to Coulomb’s law.