Quantum operations are formulated in terms of the density operator description of a quantum mechanical system. Rigorously, a quantum operation is a linear, completely positive map from the set of density operators into itself. In the context of quantum information, one often imposes the further restriction that a quantum operation E must be physical, that is, satisfy 0 ≤ ≤ Tr ⁡ ⁡ [ E ( ρ ρ ) ] ≤ ≤ 1 for any state ρ ρ .