Independence and L ́evy Processes in Quantum Probability 15randomness does not disappear, except for some special
observables (those for which the state vector is an eigenvector).
Entanglement of states of a composed system is a new
phenomenon in quantum physics that is behind many of the
(apparent) paradoxes. It is also used as a kind of resource in
quantum computation and quantum communications. For more
information, see [Par03] and the references therein.
Not necessarily reversible dynamics can more generally be
described by completely positive maps. See Adam Skalski’s
lecture ‘Quantum dynamical systems from the point of view of
noncommutative mathematics’ in this volume.
1.3 Why do we Need Quantum Probability?
Owing to the spectral theorem the descriptions of an individual
observable as an element in a quantum probability space or by
random variable defined on a classical probability space are
equivalent, see also Example 1.2.4. This extends to commuting
families of observables, but not to non-commuting observables. In
this section we shall discuss several experiments and theorems
that suggest that it is indeed impossible to describe quantum
mechanics ‘in a reasonable way’ via classical probability spaces.
1.3.1 Mermin’s version of the EPR experiment
In this section we will discuss the EPR experiment, named after
Einstein, Podolsky and Rosen, in a version introduced by Mermin
[Mer85], see also [Pen89].
We consider a quantum system that consists of two particles
that can be described by the two-dimensional Hilbert spaceC^2. We
keep the notation from Example 1.2.7. The joint system is therefore
described by the four-dimensional Hilbert space
C^4 ∼=C^2 ⊗C^2 ∼=span{
|jk〉=|j〉⊗|k〉;j,k=0, 1}
.We assume that the particles are prepared in the state with state
vector
ψ=1
√
2(
| 01 〉−| 10 〉)
.If the particles are electrons, then this state is a superposition of first
particle ‘up’ and second particle ‘down’, and first particle ‘down’
and second particle ‘up’, the total spin is therefore zero.