QuantumPhysics.dvi

Since the superpositions involved here are incoherent, theweightswimay be thought of as classical
probability weights assigned to each population. Note thatwe do not require that the various
different pure states|ψi〉be orthogonal to one another, since it should certainly be possible to
superimpose pure states which are not orthogonal to one another.

The following properties of the density operator immediately result,

1. Self-adjointness :ρ†=ρ;


2. Unit trace : Tr(ρ) = 1;


3. Non-negative :〈ψ|ρ|ψ〉≥0 for all|ψ〉∈H.

Conversely, any operator inHwith the above three properties is of the form of (16.6) with nor-
malization (16.5), but some subtleties must be noted. Self-adjointness guarantees thatρcan be
diagonalized in an orthonormal basis|φi〉, with real eigenvaluespi,

ρ=

∑

i

|φi〉pi〈φi| (16.7)

Non-negativity then implies thatpi≥0 for alli, and unit trace forces

∑

i

pi= 1 (16.8)

Note that|φi〉need not coincide with|ψi〉, not even up to a phase, andpineed not coincide with
wi. Thus, a given density operator will have several equivalent representations in terms of pure
states. We shall now illustrate this phenomenon by some examples.

Pure states are also described by the density operator formalism. A pure state|ψj〉corresponds
to the case where the mixture actually consists of only a single pure state,wj= 1, andwi= 0 for
alli 6 =j. In that case,ρ^2 =ρ, so that the density operator is a projection operator, namely onto
the pure state|ψj〉. Conversely, a density operatorρ(i.e. an operator which satisfies the above
three properties) which is also a projection operatorρ^2 =ρcorrespond to a pure state. It is easiest
to show this in the orthonormal representation ofρderived in (16.7). The conditionρ^2 =ρthen
implies

∑

i

|φi〉pi(pi−1)〈φi|= 0 (16.9)

As a result, we must havepi(pi−1) = 0 for alli, so that eitherpi= 0 orpi= 1. The normalization
condition (16.77) then implies thatpican be equal to 1 only for exactly a single pure state. Note
that the density operator is an economical way to describe pure states, since the normalization of
the states, and the omission of the overall phase have automatically been incorporated.