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Posted by on Dec 29, 2012 in Quantum Mechanics Research | 0 comments

Consistency of Bell’s (CHSH) Inequalities and two dimensional spin

I have been saying in my blogs that if spin has two axes of quantization, then all the quantum weirdness dissolves and the EPR paradox is reconciled. This is not some new change or addition to quantum mechanics, and there is nothing classical about it. The only deviation from the usual application of quantum mechanics is that a single spin is isolated and there is no measuring probe. That is, space is isotropic. So the only conceptual change I am making is the following:

Quantum mechanics is a theory of measurement, but not of Nature, and can be extended to states that exist beyond our ability to measure.

Nonetheless this spin model has the following properties:

  • The two axes of quantization are indistinbuishable (in isotropy)
  • Heisenberg Uncertainty is satisfied
  • Cannot measure two axes simultaneously
  • Fair sampling is satisfied (each EPR pair gives two coincidences)
  • All the hidden states are pure and dispersion free (locally real)
  • Straightforward su(2) spin algebra
  • Entanglement does not persist to space like separation
  • Nature is local
  • Nature is real
  • Explains three photon experiments, teleportation, without entanglement
  • Separates entangled states into product states
  • Reconciles the EPR paradox
  • Each axis has a maximum √2 correlation between an EPR pair
  • Bell’s Inequalities violated
  • The sum of two experiments gives 2√2 exactly
  • The 2D spin becomes the usual 1D spin in the presence of a probe.

The only real change is the repudiation of Bell’s Theorem which states that

Any hidden variable theory that underpins quantum mechanics must be non-local.

This conclusion is wrong if spin has two axes of quantization instead of one.

A relevant point of the 2D spin is each EPR pair is capable of two coincidences: one in the z-rep. and the other in the x-rep. Therefore in one experiment, only half the coincidences can be detected and this changes the experimental value of the correlation to,

Understanding the new factor of 2 is at the core of reconciling the EPR paradox and the 2D spin. It also implies that it is experimentally impossible to determine if spin has one axis of quantization or two. This is because half the coincidences available are counterfactual.

A crucial test of the 2D spin is that it gives exactly the correlation that is found from quantum mechanics by calculating one EPR coincidence at a time. In a nutshell, the 2D spin has eigenstates of |± √2 > which are a superposition of the two axes.  These exist only within one isolated spin. In addition the states corresponding to a filter in the direction a are  |±,a >.

Now simply transform both of these states to the same laboratory frame and work out the matrix elements,

(1)

These probabilities are complementary meaning that the z rep. (even quadrants) and the x rep. (odd quadrants) cannot be simultaneously detected.  Two filter settings are required, one in the and the other in the d direction.

In EPR experiments products are taken and these determine the coincidences,

This equation gives the probability of a +/-click at Alice and a -/+ click at Bob in the even quadrants. Calculating all products over the range of local hidden variables gives exactly one half of the correlation, see Figure 1.

Figure 1: Results of one EPR calculation or experiment giving exactly one half the correlation predicted from quantum mechanics

Likewise a second experiment can be done in the odd quadrants (where the second axis lies) and the coefficients change from a+- to a-+.

This gives exactly the same as the first part, both of which fit to  give one half of the quantum correlation, E(a,b)=-½cosθab . The sum of the two experiments gives the full quantum correlation and this value exceeds the CHSH equation giving exactly 2√2. (A Java program for this calculation will soon be released)

CHSH consistent with the 2D spin.

In this blog I want to discuss the coefficients which are obtained from this 2D spin model (click to enlarge)

(2)

These coefficients can be related to the CHSH form of Bell’s Inequalities. In other words the 2D spin probabilities and the CHSH equation are in tandem so when the probabilities are maximum, or minimum, then so is the CHSH form of Bell’s Inequalities. That is the correlation displayed in the 2D model is identical to that displayed in the CHSH equation.

The extreme points occur when

 and

The coefficients, a+- and a-+ in the probabilities of Eq.(1) determine the correlation between EPR pairs.  With a little trigonometry (see the end of this blog for details) three vectors are found in each coefficient in Eq.(2)  giving:

(3)

The two vectors b and c are orthogonal (see end) and correspond to the orientation of the 2D spin by the LHV θ, φ;

Since the angles are arbitrary, (i.e. orientation of one 2D spin) place b and c anywhere, like in Figure 2. Note the directions b±c. These sum to a resultant, b±c = √2n± and since b and c are orthogonal, this vector bisects the quadrants formed by b and c, to give a vector of length √2 or normalized

At this point it is worth recalling that the 2D spin resonates between its quadrants and Figure 2 can be compared to Figure 3. Note the 2D resonance spin lies along the two directions, cf. b±c. If  b+c lies in an even (odd) quadrant then b-c lies in an odd (even) and these vectors have magnitude √2. The two probabilities in Eq.(3) are related to Bell’s expression for his inequalities,

(4)

and the sum gives the CHSH equation,

Figure 2 Filter settings that maximize the correlation between EPR pairs.

It is important to point out that Bell’s Inequalities have not been derived here.  Rather it is shown that the correlation in the 2D spin  takes the same form and is consistent with Bell’s expressions when the quantum correlation, E(a,b)=-a.b, is used.

There are, however, some restrictions here that are not in the most general form of the CHSH expression. First the two directions b±c are orthogonal and second the vectors a and d correspond to filter directions. No experiment can apply two filters in different direction at the same time. Measuring in only one direction is a consequence of the Heisenberg Uncertainty Principle. The CHSH equation reflects correlation in two perpendicular axes, and the maximum occurs when a+-=±1 or a-+=±1, but not simultaneously.

In this way the usual Bell expression for the inequalities is given for each axis by Eq.(4).  These give violations in two experiments.  In the z representation, the maximum violation is found when b and c are 45 degrees displaced from a.  In the x representation, and lying in an adjacent quadrant, the maximum violation is found when b is 45 degrees from d and c is 135 degrees from d, see Fig. 2.

The two vectors, b±c, are not normalized and sum to a vector with magnitude √2., see Figure 2

In this entry I have shown that the geometry of the 2D spin is reflected in the CHSH equation. It becomes geometrically clear the setting of filter angles at 45 degrees maximizes both CHSH equation and the probabilities of clicks for the 2D spin (Figure 2).

Figure 3: Depiction of the 2D spin and its states which are superpositions of the two axes of quantization. In the absence of a measuring probe the spin can resonate between its quadrants.

One the other hand, it should be possible to start the other way round. Take CHSH equation and find a vector of length √2 which maximizes the CHSH. I will do that in my next blog.

__________________________

Proof of Eq.(3): (click to enlarge)

Note that the two vectors, n=b and n’=c are at right angles. This follows from my 2D spin. The vectors n and n’ are renamed to b and c in order to relate to the CSHS equation.

Between EPR pairs, the two vectors,  b and c, must be the same, i.e. the Local Hidden Variables of the two spins in an EPR pair must be the same.

The equation for the other term a-+ is found in exactly the same way. However we know the second term cannot be measured simultaneously with the first, so we use a different filter direction, d

 

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