Saturday, 26 September 2015

The Bell State a defence of naivety

In this post  I wish to defend a naive interpretation of the Bell State against some of the more wilder claims made for it in the literature

The Bell state (for those who have been asleep for the past 40 years) applies to a pair of particles emitted from a common source in opposite directions. The Bell State describes the joint spin of the two particles which because of the conservation of angular momentum must be zero. But until measurement occurs we do not know the spin of a given  particle so the probability amplitude takes the form

$$|B> = \frac{1}{\sqrt{2}}(|u1>|d2>-|u2> |d1>)$$

where u represents spin up and d represents spin down and the naive interpretation I want to defend is that this simply represents the conservation of angular momentum, so that once the spin of 1 particle is known by measurement, then the other spin is automatically known and must be equal and opposite. All the Bell state represents is the 50 % probability on measurement that the particle a person observes at one detector will either be measured with spin up or spin down. Repeated measurements will confirm the fact that at a particular detector the spin will be measured up or down 50% of the time and at the other detector the spin will be equal and opposite. It is well known that the Bell state is rotationally invariant so if one the particles are emitted at an angle $$\theta$$ with respect to the polarisation axis between the particles, the other particle will be emitted at an angle $$-\theta$$ with respect to the polarisation axis . Applying a bit of Pauli spin algebra to the Bell state is well known to give the correct correlation function as was spectacularly confirmed by Aspect in the early 1980's

So why all the fuss, apart from the intrinsic skill in being able to measure the relative spin states of two particles separated a long distance away, from the naive perspective I am offering this is just another routine success of quantum mechanics.

As many people will know, the fuss has arisen because it is claimed that this experiment encapsulates the essence of the debate that Bohr and Einstein had in the early 1930's about the interpretation of quantum mechanics and which culminated in the so called EPR paradox.

To cut a long story short the claim is made that because we don't know the spin state of a given particle before measurement, The Bell state represents a physical state of superposition so that the particles spin state is undetermined until measurement which then collapses the Bell state into one of the two possibilities. So that it is claimed the other particle instantly knows what spin take to take as soon as the first particles spin state has been determined. As the particles can in principle be millions of miles away, it is claimed that this means that some signal faster than the speed of light has been sent to the other particle. Thus there would appear to be a prima facia contradiction between special relativity and quantum mechanics.

Furthermore in his paper

 It is claimed that Bell showed that the 2 assumptions

a) that there were hidden variables that predetermined the spins of the particles

b) the assumption that the two detectors were independent of each other

Led to a contradiction with the predictions of quantum mechanics.

The failure of assumption a) would imply that quantum measurements cause reality to happen giving rise to the 'California Interpretation' and the failure of assumption b) gives rise to the idea  that a measurement of the spin of a particle at one detector affects the other. This is the so called 'Spooky action at a distance' On this reading (which is the one that seems to pervade the popular literature) The Aspect experiment would imply that quantum mechanics does either involve spooky action at a distance (non-locality)  or a form of non realism in which measurement determines properties of particles. What is worse in the popular literature, both claims are made about quantum mechanics when it is clear that only one of the two need be made. So that ever since the Aspect experiment quantum mechanics is seen in many circles  as being both non-realist and non-local.

I want to challenge not the assumptions that Bell made but the interpretation of their significance as they have become known in the popular literature.

First it should be stressed that Bell himself did not think quantum mechanics in itself showed non-locality or spooky action at a distance. If one reads the conclusion ( section VI) Bell states explicitly in the first sentence.

In a theory in which parameters are added to quantum mechanics to determine the results of individual measurements without changing the statistical predictions, there must be a mechanism whereby the setting of one measuring device can influence the reading of another instrument, however remote. Moreover the signal must propagate instantaneously, so that such a theory could not be Lorentz invariant. 

So what Bell thought he had proven was that hidden variable theories had to have some form of non-locality or spooky action at a distance not quantum mechanics. Indeed if one goes through the actual calculation of the correlation function measured in the Aspect experiment using quantum mechanics no mention is made of any form of signalling at all. All one does is apply a bit of Pauli Algebra to the Bell state and out pops the result almost by magic. It seems odd to claim non-local signalling to explain an experimental result when the correct quantum mechanical prediction makes no mention of it.

Lets consider the situation in a bit more detail

Quantum mechanics is essentially a statistical theory, the solution of Schrodinger's equation for a given problem gives rise to a probability amplitude the modulus squared of which gives rise to a probability density function via the Born rule. The solution to Schrodinger's equation is not a real wave existing in (3N+1)*S configuration space where N is the number of particles and S is the product of all the possible spin states of the particles involved. But effectively the square root of a probability density function and the (3N+1)*S configuration space is simply the probability sample space not a real physical space.

That being so the Bell state is nothing physical it is (as is any quantum superposition) a superposition of possible outcomes with coefficients representing the probability of one of the possible outcomes. When a measurement is made the so called collapse of the probability state vector is not a physical process, but a realization of one of the possibilities, just as say I can assign a probability state vector to a coin or a dice or any situation which results in a number of possible outcomes N

This is simply

$$ |Final> =\frac{1}{\sqrt{N}}{a_1|1a_>+.........a_n|N>} $$

where the a's represent the square roots of the probability of one of the outcomes being realized.

But the system, is not before measurement physically in some hybrid state of all the possibilities. All one can say is that one of the possibilities will be realized on measurement, not that measurement causes the system to jump from it's limbo state into one of the possible outcomes. The probability amplitude can be seen as akin to a Bayesian prior representing our best guess as to what the possible outcomes will be. Given that one can by definition only ever see a quantum system in one state speculation as to what happens before the act of measurement occurs is totally meaningless.

Furthermore if as the standard story has it the spin states of the particles are only determined at measurement then that would make a nonsense of the conservation of angular momentum. It would imply that the conservation of angular momentum only applies at the point of measurement. On the naive view I'm defending the relative orientations of the particles spin states is fixed at the point of emission but we wont know what spin state we will measure at a given detector. So whilst a given individual particles spin state is not determined, the relative orientation with respect to the other is, If that is not so then the conservation of angular momentum is meaningless and we would not be able to predict the outcome of experiments such as those at CERN. I hope then to have dispelled the myth that the Aspect experiment or any other such experiment involves spooky action at a distance.

Ok so what does entanglement or non-locality actually mean?  Again there is a fairly straightforward interpretation. If the joint probability state vector of a system of many objects is entangled that means that it cannot be written as the product of the individual objects probability state vectors. This is well known in statistics, If the probability density function of two or more variables cannot be written as the product of the probability density functions of the individual variables, then that is is an indication that the two variables are not independent of each other. For the Bell state we know that the individual particle spin states are not independent of each other, as they are equal and opposite because of the law of conservation of angular momentum. Again there is no need to invoke spooky action at a distance and I have to say Bell has misled a whole generation of physicists by invoking this in his conclusion. 

Is my interpretation consistent with the quantum formalism undoubtedly yes as I am using the Bell state to make the correct predictions as measured in the Aspect experiment. The question has to be why invoke spooky action at a distance or collapse of the wave packet when a simple statistical interpretation in accordance with the Born interpretation is all one needs to successfully predict the outcome of quantum mechanical experiments.

As a final point if Bell has really disproved the existence of hidden variables then we cannot go beyond the statistics to find out "what is really going on". The hope of going beyond the statistical predictions of quantum mechanics to predict the results of individual measurements is an illusion and "Nature really does play dice." 


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