Canonical typicality

It is well known that a system, S, weakly coupled to a heat bath, B, is described by the canonical ensemble when the composite, S + B, is described by the microcanonical ensemble corresponding to a suitable energy shell. This is true both for classical distributions on the phase space and for quantum density matrices. Here we show that a much stronger statement holds for quantum systems. Even if the state of the composite corresponds to a single wave function rather than a mixture, the reduced density matrix of the system is canonical, for the overwhelming majority of wave functions in the subspace corresponding to the energy interval encompassed by the microcanonical ensemble. This clarifies, expands and justifies remarks made by Schrödinger in 1952. Key words: canonical ensemble in quantum theory; density matrix; typical wave function. A quantum system in thermal equilibrium at inverse temperature β is described by the canonical density matrix ρβ = 1 Z exp(−βH(S) ) (1) where H (S) is the system Hamiltonian and Z = tr exp(−βH (S)). The usual justification for (1) is that it is the reduced density matrix of the system when it is weakly coupled to a heat bath, and the composite system is described by the microcanonical density matrix at a suitable total energy E