Remarks on the Quantum de Finetti Theorem for Bosonic Systems

M. Lewin, P. T. Nam, N. Rougerie
2014 Applied Mathematics Research eXpress  
The quantum de Finetti theorem asserts that the k-body density matrices of a N -body bosonic state approach a convex combination of Hartree states (pure tensor powers) when N is large and k fixed. In this note we review a construction due to Christandl, Mitchison, König and Renner [8] valid for finite dimensional Hilbert spaces, which gives a quantitative version of the theorem. We first propose a variant of their proof that leads to a slightly improved estimate. Next we provide an alternative
more » ... roof of an explicit formula due to Chiribella [7] , which gives the density matrices of the constructed state as a function of those of the original state. It is a well-known fact that Hartree states, i.e. projections onto tensor powers Ψ = u ⊗N ∈ H N (for u a normalized vector in H), play a very special role in the physics of bosonic systems. Indeed, since bosons, contrarily to fermions, do not satisfy the Pauli exclusion principle, there is a possibility for many particles to occupy the same quantum state, which is the meaning of the ansatz u ⊗N . It is in fact the case for non interacting particles in the ground state of a one-body Hamiltonian, or in the thermal state at low enough temperature. This is the famous Bose-Einstein condensation phenomenon.
doi:10.1093/amrx/abu006 fatcat:tuedebblerey7cpbncdslc3ljy