tr(ρ) = 1, at least one of the eigenvalues must be positive. The sign of the other eigenvalue is
then determined by the sign of det(ρ), since this is just the product of the two eigenvalues. The
determinant is readily computed, and we find,
det(ρ) =1
4
(1−a^2 ) (16.18)Thus, positivity of the eigenvalues will be guaranteed by
ρ=1
2
(
I+a·σ)
|a|≤ 1 (16.19)This space is referred to as theBloch ball, parametrized by|a| ≤1. Pure states, for whichρis a
rank 1 projection operator, precisely correspond to the boundary|a|= 1 sphere, called theBloch
sphere. On the other hand, any density operator with|a|<1 corresponds to a mixed state. The
state is unpolarized|a|= 0, and partially polarized for 0<|a|<1. This gives a geometrical
presentation of the space of density matrices for the two-state system.
16.4 Non-uniqueness of state preparation
The set of all density operators for a given system forms aconvex set. Thus, ifρ 1 andρ 2 are two
arbitrary density operators, then the operator
ρ(λ) =λρ 1 + (1−λ)ρ 2 (16.20)is again a density operator for anyλsuch that 0≤λ≤1. This is shown by verifying thatρ(λ)
indeed satisfies the three defining properties of a density operator.
The physical interpretation of this property leads to one ofthe key distinctions between classical
and quantum information theory. Consider an experimental set up where the system may be
prepared either in the state associated withρ 1 or in the state corresponding toρ 2. We assign a
probabilityλthat the system be prepared in stateρ 1 and a probability 1−λthat it be prepared
in stateρ 2. The ensemble average of the expectation value of any observableAis then obtained
by taking the quantum mechanical expectation value, weighted by the probabilities for preparation
either in stateρ 1 or in stateρ 2 ,
〈A〉=λtr(ρ 1 A) + (1−λ)tr(ρ 2 A) = tr(
ρ(λ)A)
(16.21)Thus, the expectation value of any observable are indistinguishable from what we would obtain if
they had been computed directly in the stateρ(λ). In fact, our two-state example of the preceding
subsection illustrated already this property. The totallyunpolarized density operator could have
been gotten either by equal populations ofSz=± ̄h/2, or equal populations ofSx=± ̄h/2, and in
the final state, you could never tell the difference. Clearly, for any mixed state, there would be an
infinite number of different ways of preparing the state, unless the state was actually pure and then
there is only a single way. For the sake of completeness, we note that if the eigenvalues ofρare
distinct, then there isa unique way of preparing the system in terms mutually orthogonal density
operators, but allowing for non-orthogonal density operators, the preparation is again not unique.
This situation is to be contrasted with the preparation of a classical ensemble, which is unique,
given the probabilities.