The whole is greater than the sum of its parts

The tile of today's post is a quote from Aristotle, but I want to illustrate this in the quantum formalism. Here I will refer to a famous Hardy paper: Quantum Theory From Five Reasonable Axioms. In there one finds the following definitions:

• The number of degrees of freedom, K, is defined as the minimum number of probability measurements needed to determine the state, or, more roughly, as the number of real parameters required to specify the state. 
• The dimension, N, is defined as the maximum number of states that can be reliably distinguished from one another in a single shot measurement.  

Quantum mechanics obeys \(K=N^2\) while classical physics obeys \(K=N\).

Now suppose nature is realistic and the electron spin does exist independent of measurement. From Stern-Gerlach experiments we know what happens when we pass a beam of electrons through two such devices rotates by an angle \(\alpha\): suppose we pick only the spin up electrons, on the second device the electrons are still deflected up \(\cos^2 (\alpha /2)\) percent of time and are deflected down \(\sin^2 (\alpha /2)\) percent of time . This is an experimental fact.

Now suppose we have a source of electron pairs prepared in a singlet state. This means that the total spin of the system is zero. There is no reason to distinguish a particular direction in the universe and with the assumption of the existence of the spin independent of measurement we can very naturally assume that our singlet state electron source produces an isotropic distribution of particles with opposite spins. Now we ask: in an EPR-B experiment, what kind of correlation would Alice and Bob get under the above assumptions?

We can go about finding the answer to this in three ways. First we can cheat and look the answer up in a 1957 paper by Bohm and Aharonov who first made the computation, This paper (and the answer) is cited by Bell in his famous "On the Einstein-Podolsky-Rosen paradox". But we can do better than that. We can play with the simulation software from last time. Here is what you need to do:

-replace the generating functions with:

function GenerateAliceOutputFromSharedRandomness(direction, sharedRandomness3DVector) {

    var cosAngle= Dot(direction, sharedRandomness3DVector);

    var cosHalfAngleSquared = (1+cosAngle)/2;

    if (Math.random() < cosHalfAngleSquared )

        return +1;

    else

        return -1;

};

function GenerateBobOutputFromSharedRandomness(direction, sharedRandomness3DVector) {

    var cosAngle= Dot(direction, sharedRandomness3DVector);

    var cosHalfAngleSquared = (1+cosAngle)/2;

    if (Math.random() < cosHalfAngleSquared )

        return -1;

    else

        return +1;

};

-replace the -cosine curve drawing with  a -0.3333333 cosine curve: 

boardCorrelations.create('functiongraph', [function(t){ return -0.3333333*Math.cos(t); }, -Math.PI*10, Math.PI*10],{strokeColor:  "#66ff66", strokeWidth:2,highlightStrokeColor: "#66ff66", highlightStrokeWidth:2});

replace the fit test for the cosine curve with one for with 0.3333333 cosine curve: 

var diffCosine = epsilon + 0.3333333*Math.cos(angle); 

and the result of the program (for 1000 directions and 1000 experiments) is:

So how does the program work? The sharedRandomness3DVector is the direction on which the spins are randomly generated. The dot product compute the cosine of the angle between the measurement direction and the spin, and from it we can compute the cosine of the half angle. The square of the cosine of the half angle is used to determine the random outcome. The resulting curve is 1/3 of the experimental correlation curve. Notice that the output generation for Alice and Bob are completely independent (locality).

But the actual analytical computation is not that hard to do either. We proceed in two steps.

Step 1: Let \(\beta\) be the angle between one spin \(x\) and a measurement device direction \(a\). We have: \(\cos (\beta) = a\cdot x\) and:

\({(\cos \frac{\beta}{2})}^2 = \frac{1+\cos\beta}{2} = \frac{1+a\cdot x}{2}\)

Keeping the direction \(x\) constant, the measurement outcomes for Alice and Bob measuring on the directions \(a\) and \(b\) respectively are:

++ \(\frac{1+a\cdot x}{2} \frac{1+b\cdot (-x)}{2}\) percent of the time

-- \(\frac{1-a\cdot x}{2} \frac{1-b\cdot (-x)}{2}\) percent of the time

+-\(\frac{1+a\cdot x}{2} \frac{1-b\cdot (-x)}{2}\) percent of the time 

-+\(\frac{1-a\cdot x}{2} \frac{1+b\cdot (-x)}{2}\) percent of the time 

which yields the correlation: \(-(a\cdot x) (b \cdot x)\)

Step 2: integrate \(-(a\cdot x) (b \cdot x)\) for all directions \(x\). To this aim align \(a\) on the z axis and have \(b\) in the y-z plane:

\(a=(0,0,a)\)

\(b=(0, b_y , b_z)\)

then go to spherical coordinates integrating using:

\(\frac{1}{4\pi}\int_{0}^{2\pi} d\theta \int_{0}^{\pi} \sin\phi d\phi\)

\(a\cdot x = \cos\phi\)

\(b\cdot x = b(0, \sin\alpha, -\cos\alpha)\cdot(\sin\phi \cos\theta, \sin\phi\sin\theta, \cos\phi)\)

where \(\alpha\) is the angle between \(a\) and \(b\).

Plugging all back in and doing the trivial integration yields: \(-\frac{\cos\alpha}{3}\)

So now for the moral of the story. the quantum mechanics prediction and the experimentally observed  correlation is  \(-\cos\alpha\) and not \(-\frac{1}{3} \cos\alpha\)

The 1/3 incorrect correlation factor comes from demanding (1) the experimentally proven behavior from two consecutive S-G device measurements, (2) the hypothesis that the electron spins exist before measurement, and (3) and isotropic distribution of spins originating from a total spin zero state.

(1) and (3) cannot be discarded because (1) is an experimental behavior, and (3) is a very natural demand of isotropy. It is (2) which is the faulty assumption.

If (2) is true then circling back on Hardy's result, we are under the classical physics condition: \(K=N\) which means that the whole is the sum of the parts. 

Bell considered both the 1/3 result and the one from his inequality and decided to showcase his inequality for experimental purposes reasons: "It is probably less easy, experimentally, to distinguish (10) from (3), then (11) from (3).". Both hidden variable models:

    if (Dot(direction, sharedRandomness3DVector) < 0)

        return +1;

    else

        return -1;

and 

    var cosAngle= Dot(direction, sharedRandomness3DVector);

    var cosHalfAngleSquared = (1+cosAngle)/2;

    if (Math.random() < cosHalfAngleSquared )

        return -1;

    else

        return +1; 

are at odds with quantum mechanics and experimental results. The difference between them is on the correlation behavior for 0 and 180 degrees. If we allow information transfer between Alice generating function and Bob generating function (nonlocality) then it is easy to generate whatever correlation curve we want under both scenarios (play with the computer model to see how it can be done).

So from realism point of view, which hidden variable model is better? Should we insist on perfect anti-correlations at 0 degrees, or should we demand the two consecutive S-G results along with realism? It does not matter since both are wrong. In the end local realism is dead.