Eur. Phys. J. Plus (2013) 128: 48
https://doi.org/10.1140/epjp/i2013-13048-6

Review

Typical entanglement

¹³³⁹ Dipartimento di Matematica, Università di Bari, I-70125, Bari, Italy
²³³⁹ Dipartimento di Fisica and MECENAS, Università di Bari, I-70126, Bari, Italy
³³³⁹ Sezione di Bari, INFN, I-70126, Bari, Italy
⁴³³⁹ Museo Storico della Fisica e Centro Studi e Ricerche “Enrico Fermi”, I-00184, Roma, Italy

^* e-mail: saverio.pascazio@ba.infn.it

Received: 19 March 2013
Accepted: 30 April 2013
Published online: 17 May 2013

Abstract

Let a pure state |ψ〉 be chosen randomly in an NM-dimensional Hilbert space, and consider the reduced density matrix ρ _A of an N-dimensional subsystem. The bipartite entanglement properties of |ψ〉 are encoded in the spectrum of ρ _A . 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.