Noncommutative Mathematics for Quantum Systems

12 Noncommutative Mathematics for Quantum Systems

Example 1.2.10 LetH = C^2 ,ψ = | 0 〉,u = cosθ 2 | 0 〉+sinθ 2 | 1 〉,

v=cosφ 2 | 0 〉+sinφ 2 | 1 〉, withθ,φ∈[0,π), and denote byEandF
the projections ontoCuandCv, resp. Ifφ 6 =θ, then the join ofE
andFis the identity and therefore trρ(E∨F) =1. ForEandFwe
have

tr(ρE) =cos

θ

2

and tr(ρF) =cos

φ

2

.

Choosing bothθandφclose toπ, we can make these probabilities
arbitrarily small.

Another consequence of the non-commutativity in quantum
probability is the existence of lower bounds on the product of the
variances of non-commuting observables.

Theorem 1.2.11 (Heisenberg Uncertainty Relation) X,Y observables,
ρa state. Then

Varρ(X)Varρ(Y) ≥

(

trρ

1

2

(XY+YX)

) 2

+

(

trρ

i

2

(XY−YX)

) 2

≥

1

2

(

trρi[X,Y]

) 2

.

This theorem can be demonstrated by applying the Kadison–
Schwarz inequality (stated in Section 2.6.1 in Skalski’s lecture in
this volume) to trρ(X+zY)∗(X+zY)and carefully choosing the
value forz, see also [Par03, Proposition 1.2.3].
The bracket in Heisenberg’s Uncertainty Relation denotes the
commutator,[X,Y] = XY−YX. If we apply this relation to the
position operatorQand the moment operatorP, then we have
[P,Q] =i I, whereIdenotes the identity operator, and

Varρ(Q)Varρ(P)≥

1

2

that is, there exists no state in which the variances of position and
momentum are both small.
Another difference concerns the structure of the set of extreme
points of the set of states of a classical and a quantum probability
space, see also the dictionary ‘classical↔quantum” below.

Theorem 1.2.12 (Extreme points) The set of extreme states (that is,
pure states) on an n-dimensional complex Hilbert space is a real manifold
of dimension 2 n− 2.