Typical entanglement

Let a pure state |ψ〉 be chosen randomly in an NM-dimensional Hilbert space, and consider the reduced density matrix ρA of an N-dimensional subsystem. The bipartite entanglement properties of |ψ〉 are encoded in the spectrum of ρA. By means of a saddle point method and using a “Coulomb gas” model for the eigenvalues, we obtain the typical spectrum of reduced density matrices. We consider the cases of an unbiased ensemble of pure states and of a fixed value of the purity. We finally obtain the eigenvalue distribution by using a statistical mechanics approach based on the introduction of a partition function.