Independence and L ́evy Processes in Quantum Probability 17for any 0≤θ≤π, 0≤φ< 2 π. This means that Alice and Bob
will observe that the two results ‘+1’ and ‘−1’, or ‘up’ and ‘down’,
are equally likely, no matter what direction they choose for their
experiment. However, after Alice has observed the spin in some
direction, saySA(θ,φ), then the state has to be updated according
to the result of her experiment. If Alice observes ‘+1’, that is, ‘up’ in
the direction of her choice, say, then, according to von Neumann’s
‘Collapse’ postulate, the state after the experiment is
(
|u(θ,φ)〉〈u(θ,φ)|⊗id
)
ψ
∣
∣∣
∣(|u(θ,φ)〉〈u(θ,φ)|⊗id)ψ∣
∣∣
∣=u⊥(θ,φ)⊗u(θ,φ),and if she observes ‘−1’, that is, ‘down,’ then the new state is
(
|u⊥(θ,φ)〉〈u⊥(θ,φ)|⊗id
)
ψ
∣
∣∣
∣(|u⊥(θ,φ)〉〈u⊥(θ,φ)|⊗id)ψ∣
∣∣
∣=u(θ,φ)⊗u⊥(θ,φ)(where we neglected the phase, since it does not change the state).
This calculation becomes particularly easy, if we use Equation
(1.3.1) to writeψin terms ofu(θ,φ)andu⊥(θ,φ).
Therefore, if Bob now measures the spin of his particle in some
direction, saySB(θ′,φ′), then he has to use the stateu⊥(θ,φ)or
u(θ,φ), and his probability to observe ‘+1’ or ‘−1’ depends on the
angle between the vectors
−cosφsinθ
−sinφsinθ
−cosθ
or
cosφsinθ
sinφsinθ
cosθ
, resp.and
cosφ′sinθ′
sinφ′sinθ′
cosθ′
,according to Equation (1.2.1).
We pretended here that Alice’s measurement takes place before
Bob’s; however, in fact the joint probabilities of Alice and Bob’s
measurements do not depend on the order of their measurements.
Their observables commute and are therefore independent;
therefore, the results are the same if they measure simultaneously
or one after the other.