Independence and L ́evy Processes in Quantum Probability 111.2.1 Distinguishing features of classical and quantum
probability
There are several fundamental differences between a classical (or
‘commutative’) probability space (Ω,F,P) and a quantum
mechanical (or ‘noncommutative’) probability space such as
(B(H),〈ψ,·ψ〉).
An ‘event’ is the most elementary question that we can ask in an
experiment, that is a random variable or observable that can take
only two values (denoted by 1 and 0, or by ‘true’ and ‘false’). In a
classical probability space events are described by (measurable)
subsetsE ⊆ Ω, which can be identified with their indicator
functions (^1) E, whereas in quantum probability spaces they are
given by closed subspaces, which are identified with their
orthogonal projections. In classical probability events form a
Boolean lattice, but in quantum probability in general they form a
non-Boolean (or non-distributive) lattice. This has consequences
for the logical structure of quantum mechanics that can be
surprising.
Recall thatE∨Fdenotes thejoinor the smallest upper bound of
EandF, that is, ifEandFare events in a classical probability space,
that is, measurable sets, thenE∨Fdenotes the union ofEandF,
and ifEandFare events in a quantum probability space, that is,
orthogonal projections, thenE∨Fis the projection on the closed
subspace generated by the ranges ofEandF.
The meet E∧F is the greatest lower bound of E and F. In
classical probability this is the intersection ofEandF; in quantum
probability this is the projection on the intersection of the ranges of
EandF.
Theorem 1.2.8 Let E,F be projections on H such that EF 6 =FE. Then
E∨F6≤E+F.
From this theorem we can derive the existence of states such that
the join of two non-commuting projections has a probability that
exceeds the sum of the probabilities of the two projections.
Corollary 1.2.9 Let E,F be projections on H s.t. EF 6 =FE. Then, for
some stateρ,
trρ(E∨F)6≤trρE+trρF.
Let us give a simple example in the setting of Example 1.2.7.