Density Matrices

Abstract

In this chapter we want to introduce density matrices, also called density operators, which conceptually take the role of the state vectors discussed so far, as they encode all the (accessible) information about a quantum mechanical system. It turns out that the "pure" states, described by state vectors 1 | ψ on Hilbert space, are idealized descriptions that cannot characterize statistical (incoherent) mixtures, which often occur in the experiment , i.e. in Nature. These objects are very important for the theory of quantum information and quantum communication. More detailed information about the density matrix formalism can be found in [17]. 9.1 General Properties of Density Matrices Consider an observable A in the "pure" state | ψ with the expectation value given by A ψ = ψ | A | ψ , (9.1) then the following definition is obvious: Definition 9.1 The density matrix ρ for the pure state | ψ is given by ρ := | ψ ψ | This density matrix has the following properties: I) ρ 2 = ρ projector (9.2) II) ρ † = ρ hermiticity (9.3) III) Tr ρ = 1 normalization (9.4) IV) ρ ≥ 0 positivity (9.5) 1 Remark for experts: It is possible to find a vector representation for every given quantum mechanical state, even those represented by a density matrix. This can be done via the so-called GNS (Gelfand-Neumark-Segal) construction. This vector representation need not, however, be of any practical form and the concept of the density matrix is therefore inevitable. 159