Independence and L ́evy Processes in Quantum Probability 7
on a quantum system in the stateρcan only yield values that belong
to the spectrum ofX. A valueλ∈σ(X)occurs with probability
pλ=tr(ρEλ)
where tr denotes the trace. If the observed value isλ, then the state
‘collapses’ to
ρ ̃λ=
EλρEλ
tr(ρEλ)
,
that is, the state of the quantum system after the measurement is
described by the density matrixρ ̃λ.
As eachλ∈σ(X)occurs with probability
ρλ=Pρ(X=λ) =tr(ρEλ),
we get
Eρ(X) = ∑
λ∈σ(X)
λpλ=tr(ρX)
for the expectation and
Varρ(X) =Eρ
(
(X−Eρ(X))^2 ) =tr(ρX^2 )−
(
tr(ρX)
) 2
for the variance of the observableXin the stateρ.
The simplest experiments in quantum mechanics can produce
only two possible outcomes, like the measurement of the spin of
a fermion (like, for example the electron) in a fixed direction in a
Stern-Gerlach-type experiment, or sending a single photon through
a polarization filter. Such experiments are described by the two-
dimensional Hilbert spaceC^2.
Example 1.2.7 (Spin of a spin-^12 particle or polarization of a
photon) ConsiderH=C^2. As vectors that differ only by a phase
define the same state, we can assume that the first component of a
state vector inC^2 is not negative. Therefore, the most general state
vector is of the form
u(θ,φ) =cos
θ
2
| 0 〉+eiφsin
θ
2
| 1 〉=
(
cosθ 2
eiφsinθ 2
)