By Forman Sinnickson Acton

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4, we can obtain an interesting correspondence between states and points of the unit ball in R3. We shall identify states with their corresponding density operators. 5. 24) where r\ + r\ + r\ < 1. The state D is pure if and only ifr\ + r§ + r | = 1. Proof. Let D = α\σ\ + α 2 σ 2 Η- α 3 σ 3 -f 04/ be an arbitrary self-adjoint matrix. 4. Now D is a density operator if and only if A1*1 > 0 and λ + + λ " = 1. The second condition holds if and only if a± = \. The first condition is then equivalent to a\ + a% + a 3 < \.

16) that observables which commute can be simultaneously measured with any degree of accuracy. For this reason, we call observables or events that commute com patible. Now it is easy to show that a collection of quantum events in L{H) mutually commute if and only if they are contained in a Boolean σ-algebra in L{H) [Gudder, 1979]. The imbedding h: Σ -+ L{H) of the previous para graph shows that the events and random variables of classical probability theory correspond to compatible observables and quantum events.

R>\ It is easy to check that F is a distribution function. Let μ be the corre sponding probability measure on (R,J5(R)). We shall show that μι =Φ- μ. 22, it is sufficient to show that Fi>(\) —► F(X) at continuity points of F. Let λ be a fixed continuity point of F. For any rational r > X we have lim supFi/(A) < lim Fi>(r) = F(r). i'—►oo %'—>oo For any rational r\ < X we obtain lim infF lim Fif(n) t'—*oo %'—>oo = F(r±). Since F is continuous at A, for any ε > 0 there exists rationale ri < X < r such that F(r) < F(ri) -f ε.