Cornelia Vogel

OS Mathematical Physics July 20 14:30

Canonical Typicality For Other Ensembles Than Micro-Canonical

Canonical typicality is the known fact in quantum statistical mechanics that for most wave functions ψ from the unit sphere in a high-dimensional subspace H_R (such as a micro-canonical subspace) of the Hilbert space H_S of a macroscopic quantum system S consisting of two subsystems a and b, the reduced density matrix tr_b(|ψ><ψ|) is close to tr_b(ρ^R), where ρ^R is the projection to H_R normalized to trace 1, provided that b is sufficiently large. Here, the word ''most'' refers to the uniform distribution over the unit sphere in H_R, which for a micro-canonical subspace can be regarded as an analog of the micro-canonical ensemble in statistical mechanics. In this talk, we generalize canonical typicality to other ensembles, in particular to an analog of the canonical ensembel, so it expresses a kind of equivalence of ensembles. For a general density matrix ρ, the measure over the unit sphere that forms the analog of the uniform measure but has density matrix ρ is known as GAP(ρ). We show that for any density matrix ρ on H_S with small eigenvalues, most wave functions according to GAP(ρ) are such that tr_b(|ψ><ψ|) is close to tr_b(ρ). We also prove a variant of Lévy's lemma (concentration of measure) for GAP(ρ), and a variant of dynamical typicality for tr_b(|ψ_t><ψ_t|)with GAP(ρ)-typical ψ_0.