Noncommutative Mathematics for Quantum Systems

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

)