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.

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