Noncommutative Mathematics for Quantum Systems

Independence and L ́evy Processes in Quantum Probability 19

Aa,Ab,Ac,Ba,Bb,Bc

such that

P(Ax=−Bx) = 1

for allx∈{a,b,c}and

P(AX 1 =BX 2 ) =

1

2

for X 1 ,X 2 two independent random variables with uniform
distribution over{a,b,c}, independent ofAa,Ab,Ac,Ba,Bb,Bc.
Let us verify that this is indeed impossible. The first condition
obviously impliesAx = −Bxforx∈ {a,b,c}. Therefore, we can
only choose half of the random variablesAa,Ab,Ac,Ba,Bb,Bc, say,
Aa,Ab,Ac.
If we choose allA’s equal to ‘+1’, then allB’s are equal to ‘−1’,
and, no matter how Alice and Bob choose the directions of their
measurements, they will always find opposite values.
If we choose

(Aa,Ab,Ac) = (+1,+1,− 1 ),

then

(Ba,Bb,Bc) = (−1,−1,+ 1 ),

and we see that Alice and Bob get the same result only if they
measure in the directions(a,c),(b,c),(c,a),(c,b), that is, only for
four of the nine possible combinations.
The remaining cases are similar, for any choice of values for
(Aa,Ab,Ac)the probability that Alice and Bob get the same result

is at most^49 , if they randomly choose their directions. Therefore, it
is impossible to reproduce the quantum probabilistic joint
probabilities with classical random variables. No matter how we
define(Aa,Ab,Ac), the probability that Alice and Bob observe the
same result when they choose their directions uniformly and
independently can never exceed^49.
Therefore, if quantum mechanics is correct, as all experiments
have confirmed until now, then it is impossible to assign values to
the observables, independently of the measurements we choose.
It is possible to define 18 random variables (A(x,y),

B(x,y))(x,y)∈{a,b,c} 2 that do produce the correct joint probabilities,