Noncommutative Mathematics for Quantum Systems

18 Noncommutative Mathematics for Quantum Systems

If they measure in directions that are at an angleθ, then joint
probabilities of Alice and Bob’s results are

‘+ 1 ’ for Bob ‘− 1 ’ for Bob

‘+ 1 ’ for Alice^1 −cos 4 θ^1 +cos 4 θ

‘− 1 ’ for Alice^1 +cos 4 θ^1 −cos 4 θ

In particular, if they measure the spin in the same direction, they
always find opposite results.
In Mermin’s version of the EPR experiment, Alice and Bob
receive a large number of particles prepared in the stateψ. They
agree on three directions, saya,b,c, that are at an angle of 120◦to
one another, randomly measure the spin in one of these directions,
record their results, and then compare them.
When they measure in the same directions, one of them will find
‘+1’ and the other will find ‘−1’, by the nature of the stateψ.
Suppose now that they measure in different directions, saya
andb. If Alice measures in directionaand finds ‘+1’, then the
particle on which Bob carries out his measurement arrives in a
state pointing in the opposite direction ofa, which has an angle of
60 ◦with the direction he has chosen. Therefore, he will find ‘+1’
with probability^34 =^1 +cos 60

◦

2 , and ‘−1’ with probability

1

4 =

1 −cos 60◦
2. Had Alice measured ‘−1’, then Bob’s particle will
point in the direction ofa, which has an angle of 120◦ with the
direction he has chosen. Therefore he will find ‘+1’ with
probability^14 =^1 +cos 120

◦

2 , and ‘−1’ with probability

3

4 =

1 −cos 120◦

2.

In any case, if they choose the directions independently and

with a uniform probability, their results agree with probability^12

and disagree with probability^12 , since

P(same result) =P(same result|same direction)P(same direction)

+P(same result|diff. direction)P(diff. direction)

= 0 ·

1

3

+

3

4

·

2

3

=

1

2

.

It is a remarkable fact that this behavior cannot be reproduced with
classical random variables, that is, there do not exist any classical
random variables