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Typical Entanglement

Let a pure state ψbe chosen randomly in an NM-dimensional Hilbert space, and consider the reduced density matrix ρof an N-dimensional subsystem. The bipartite entanglement properties of ψare encoded in the spectrum of ρ. 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.