Bell’s inequality, entanglement and the demise of local reality – II

Einstein saw that quantum mechanics predicted what Alice and Bob found — when they had their machines’ switches set the same (both to zero, both to plus, etc.) the same color light flashed on the two machines (when the photons arrived at each machine at exactly the same time.  He called this ‘spooky action at a distance’. Einstein also hated the fact that quantum mechanics gave only statistical predictions about what would happen (rather than an exact prediction).  He felt that physics should do better and exactly describe an underlying reality.  This is why he called quantum mechanics ‘incomplete’.  There were elements of reality that it was missing.

In the famous Einstein Podolsky Rosen paper mentioned in the first post on Zeilinger’s book (see a criterion for just when physics was describing something real was given.  Reality as defined in the EPR paper is just this — if you can make a measurement which consistently produces the same result (within experimental error) you are measuring something real, which exists outside your measurement.  It doesn’t have to be that complicated.  If every time you look up in the sky and see the moon, the moon is there whether you are looking or not. Certainly when Alice and Bob had their switches set the same way, Alice could always predict what Bob would get.  To Einstein this meant that they were measuring something real.  This ‘reality’ also gets around the ‘spooky action at a distance’ and is consistent with locality — which is simply the idea that any measurement Alice made (locally) could not affect any measurement Bob could make (somewhere else).

So if each photon carried instructions about what to do for each of the 3 settings of the switches, the pair of identical photons which were simultaneously emitted by the black box between Alice and Bob under the Danube would produce the experimental results found.  Except they don’t and herein lies the genius of John Bell.   Call the instructions engraved on each photon about what to do at each switch setting, hidden variables.  You need 2 for each switch setting (e.g. make the machine flash green or make it flash red).

We can’t measure the hidden variables directly, but they correspond to something real by Einstein’s criterion.  Here is where Zeilinger’s explanation is just brilliant.  He abandons physics entirely and explains the Bell inequalities in terms of biology — identical twins rather than identical photons.  Identical twins can both be tall or short, both male or both female, both be redheads or blonds.  If you know that you’re dealing with identical twins (and we do) finding a tall redhead means that your partner will find a tall redhead.  Zeilinger doesn’t know too much molecular biology, so he calls the hidden biologic variables ‘genes’ but that’s OK.  We know of at least 80 regions in our genome which affect height rather than one.

So instead of red at a setting of + we’ll have tall, instead of green at a setting of + we’ll have short

Instead of red at a setting of 0 we’ll have male, instead of green at a setting of 0 we’ll have female

Instead of red at a setting of –  we’ll have redheads, instead of green will have blods

Clearly there are 8 kinds of twins.

Instead of Alice and Bob measuring photons with their machines, a door opens in each lab and in walks one of the of identical twins.

It’s very clear that the following equality holds

Number  of Tall/Male twins = Number of Tall/Male/Redheaded twins + Number of Tall/Male/Blond twins    (equation #1)

Clearly the number of Tall/Male/Redheaded twins is less than the number of Tall/Redheaded twins (since Tall/Female/Redheads also count).  All we’ve done is relax the sex requirement. (inequality #1)

Similarly the number of Tall/Male/Blond twins is less than the number of Male/Blond twins (since Short/Male/Blond twins also counts). (inequality #2)

So abbreviating number of Tall/Male/Redheaded twins as #TMR  etc. etc. Equation #1 becomes #TM = #TMR + #TMB

Inequality #1 becomes #TMR less than #TR.  Inequality #2 becomes #TMB less than #MB

Substituting the inequalities in equation #1 we get

#TM less than #TR + #MB

Since we’re dealing with identical twins things should be no different if Alice and Bob measure different aspects of the twins (se Alice measures Tall/Short — corresponding to a setting of the switch to +) and Bob measures the sex (corresponding to a switch setting of zero).  Essentially this is the locality assumption.  Both twins have identical hidden variables, and  measuring one type of hidden variable has absolutely no effect on the measurement of another hidden variable.

Back to the Zeilinger book.  When Alice has her switch on + (e.g. + 30 degrees) and Bob has his on 0 zero — They get both red (corresponding to TM) 75% of the time (see p. 140 of the book).  When Alice has her switch on + and Bob on – (e.g. -3o degrees)  They both get red 25% of the time (this corresponds to TR).  Finally Alice puts her switch on 0 (for Male/Female) Bob on minus and they look to see what percentage of the time Alice gets Male and Bob gets Blond.  This happens 25% of the time.

So we have .75 less than .25 + .25 for the hidden variable theory assuming locality (that Bob’s measurements don’t affect Alice’s and vice versa). The reality of the experimental results torpedoes Einstein’s local realism.  Not only that, but quantum mechanics predicts the values (.75, .25 .25) actually measured in this combination of switches. .75 is cos(30)^2.  see Feynman Vol. III p. 11 – 10 for how quantum mechanics gets this..

An astounding result that people are still coming to grips with.  Now you’ve seen it (even if you don’t believe it).  For where the .75 and .25 come from look at the book review mentioned earlier.

The last post in the series will concern how Bell did it (behind everyone’s back) and some more implications.  How sad that he was struck down by an intracranial aneurysm in his early 60s and never received the Nobel he so richly deserved.  But he died in ’90 before a lot of subsequent experimentation confirmed the early (and somewhat inconsistent) results.

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  • Jilan  On August 30, 2013 at 3:42 pm

    Mmm, seems to me the strange result only arises as the argument supposes the quantum properties are like classical properties and directly analogous to the probability (eg number of boys with brown hair etc). This is just the case in QM. The property is related to the wave function and we know that probabilities, correlations etc are the square of these. Try applying square roots to all the terms in the Bells inequalities and you will see no violations of the equations!

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