Measuring a certain property of one of the particles, allows one to predict with certainty a property of the other particle a hundred miles away.
Quantum Mechanics and Non-Locality
By Travis Norsen (February 2002, Part 2 of 5)

[OBJECTIVE SCIENCE.COM] In order to bring out some of these difficulties, Einstein, Podolsky, and Rosen (EPR) suggested a clever thought experiment. [2]  Their idea was to consider a pair of entangled particles created together, but moving off in opposite directions.  Such a pair could be created, for example, in the decay of an unstable spin-0 particle.  Due to the conservation of angular momentum, the spins of the two daughter particles would have to be perfectly anti-correlated, meaning simply that if one particle's spin is "up" along a given axis, the other must be "down" along that same axis in order for the total angular momentum to add to zero.   In the language of QM, we say that the wave function of the pair would have to be in the "singlet" state:

                                                                         (1)

where the "+1" and "-1" refer to the component of the angular momentum along the z-axis.  ("+1" means spin "up"; "-1" means spin "down".)  The subscripts A and B refer to the two particles, one traveling, say, to the left toward Alice, and the other traveling to the right toward Bob.  In words, this wave function is a superposition of two states: one in which particle A is spin up (+1) along the z-axis and particle B is spin down (-1) along the z-axis; and another in which particle A is down and particle B is up.  Because of this superposition, according to QM we are not able to attribute a definite value of the z-axis spin component to either particle.  This is exactly analogous to the example mentioned above, in which a "spread out" wave function precludes one from attributing a definite position to a particle. 

The point of considering this specific entangled state involving spins is that, according to QM, this superposition is maintained as the particles fly apart in opposite directions.  But after they have separated by some emotionally satisfying large distance (a hundred miles, say) Alice can measure the z-component of the spin of her particle.  If she finds "+1", then she can immediately conclude that Bob's particle has "-1" as its spin component along the z-direction, and vice versa.  Measuring a certain property of one of the particles, allows one to predict with certainty a property of the other particle a hundred miles away.   

According to standard QM, this predictive ability results because the measurement causes the system to "collapse" into one or the other of the definite states that were previously superposed.  Before the measurement, each particle separately had no well-defined spin properties; the measurement then forced the system to actualize one of the two potentialities represented in the entangled wave function, Eq. (1).  In particular, Alice's measurement of her particle's spin z-component causes Bob's particle to collapse into a definite spin state -- despite the fact that they are separated by a vast expanse of space.  This surprising implication is called "non-locality" because it seems to imply that Alice's measurement of her particle's spin properties has a causal effect on a particle (namely Bob's) which is not nearby.  [3] 

Of course, Alice could also choose to measure the spin along, say, the x-axis, and the same analysis will apply.  If she finds "+1" for the x-component of the spin of her particle, then one can immediately conclude with certainty that the x-component of the spin of Bob's particle will be "-1" (and vice versa).   

Contrary to the claim of QM that the system exists in an indefinite state of limbo until the measurement is made, the obvious inference from all of this is that the separate particles do in fact have pre-existing, well-defined properties. [4]  At least, that is the obvious inference given two assumptions:  (1) that the whole concept of "limbo", i.e., states which fail to possess identity, should be rejected and (2) that measurements and/or particles cannot affect each other "non-locally," i.e., across miles of distance in barely any time at all.  Given these assumptions, the fact that one can predict the outcome of a distant experiment with certainty must mean that the distant particle had its own properties all along. 

In the last several decades, it has become possible to test these issues with actual experiments. The theoretical interpretation of these tests is the subject of this series of articles. 

Let us begin by writing down the predictions of quantum mechanics.  We have already seen that, so long as the experimenters on either side measure the spin of their respective particles along the same spatial direction, their results will be completely anti-correlated.  That is, whenever Alice obtains "+1," Bob will obtain "-1," and vice versa.  (This kind of correlation is required by the conservation of angular momentum.) 

Yet nothing requires Alice and Bob to measure along the same axes.  In general, Alice may choose to measure the spin along some arbitrary direction, call it , while Bob may choose to measure the spin along a different direction, .  Using the quantum mechanical wave function written above, it is simple to calculate the probability that both experimenters measure "+1" along their chosen measurement directions.  The result is: 

      (2)

where  is the angle between the directions  and .  (In case the notation is at all unclear, this should be read as "The probability for Alice to measure "+1" along the direction  and for Bob to measure "+1" along the direction , is one half times the square of the sine of half the angle between the two directions.") 

Thus, in the case that , we get zero as expected.  The particle spins are perfectly anti-correlated, so a "+1" for Alice means a guaranteed "-1" for Bob, hence no probability of two "+1"s.  And in the case where  and are in exactly opposite directions, we get the correct probability of 1/2.  (The particles will with certainty be either both "+1" or both "-1," with equal probability for the two cases.)  For arbitrary angles in between there is a smooth  dependence.  So this result is plausible, but the point here isn't (yet) whether it's right or wrong, but simply that this is the prediction of the quantum mechanical theory.  We will use this result later to compare QM to a proposed model in terms of "local hidden variables" -- the kind of model that the EPR argument above suggests we should try to construct. 


Further Reading:
Hidden Variable Theories
By Travis Norsen (March 2002, Part 3 of 5)
As discussed in my previous article, in the QM picture the "collapse of the wave function" is needed because the entire system exists in an indeterminate limbo until a measurement is made, at which time the system suddenly takes on well-defined values.  This is, of course, crazy, so we are motivated (like EPR) to hypothesize the existence of some hidden variables which supplement the description given by QM.


References and Notes:

[2]  "Can Quantum-Mechanical Description of Physical Reality be Considered Complete?"  Einstein, Podolsky, and Rosen.  Originally published in Physical Review, 47, 777-80 (1935).  Reprinted in "Quantum Theory and Measurement", Wheeler and Zurek, ed. 

[3]  In their paper, EPR explicitly rejected the idea of non-locality, and therefore argued that this kind of situation proves the incompleteness of the quantum mechanical description.  Clearly, they said, Alice's measurement cannot actually affect Bob's distant particle; hence, Bob's particle must have had well-defined spin properties all along.  And so the quantum mechanical description (the entangled wave function which doesn't attribute a definite z-direction spin component to Bob's particle) must be incomplete.  

The purpose of the rest of this paper is to show that the assumption of locality is in fact unwarranted – it is contradicted by observed experimental facts.  Of course, this doesn't necessarily entail that EPR's conclusion – that QM is incomplete – is wrong.  There are several distinct arguments for this conclusion which are not undermined by the reality of non-locality.

[4]  This was precisely the conclusion of Einstein et al in the EPR paper. 

Copyright 2002 Objective Science. All rights reserved. Permission granted to link to this article only. 

Back to Objective Science Home