There are a host of discussions on this topic some suggested reading:

The New Physics: Essay 13èConceptual Foundations of Quantum Mechanics (Abner Shimony)

 

Stanford Encyclopedia of Philosophy

 http://plato.stanford.edu/entries/bell-theorem/

 

 

 

Entanglement

 

A decaying atom of Cesium emits two correlated photons.

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At laboratory A an experiment is set up to measure the photon’s polarization. The laboratory chooses to describe the arriving photon wrt an x-y system such that

 

 

while in laboratory B the photon is characterized by

 

 

To characterize the complete sytem we will need to use a Hilbert space that is a product of the space which describes and.

 

Basis:

 

The total product state for the two photons and is

 

 

Let us examine this state as the two experimenters measure. One can set up a beam splitting experiment similar to the filters discussed earlier and again detect which path the incoming photon chooses.

 

Experiment-A measures the polarization of a photon to be  while experiment B measures a result . 

 

After the A measurement we know the system should be described by

 

 

The measurement of part of the system requires that has be in the state andhas not been measured.  After the B-experiment the system is

 

 

Now the initial state of the system is determined by the decay of cesium. The process emits the photons back to back with momentum and energy conservation. To conserve angular momentum the photons spin is oppositely directed. If we describe the system using cirularlly polarized states the there are only two possibilities so the wavefunction becomes

where

 

The system must respect the fact that the two particles are Bosons.  The formal method of enforcing this condition is to require that there be no difference in the wavefunction if the two photons are exchanged.

 

Boson and Fermion states:

 

 

general two particle system

Identical Fermions

Identical Bosons

Notice that the antisymmetry of the fermion wavefunction makes the state go to zero if the particles are assumed to be in the same state but that does not happen for Bosons.

 

Antisymmetric and symmetric refer to the sign of the wavefunction under particle exchange. This is similar to the way parity behaves. There are symmetric (even) and antisymmetric states (odd) when one makes the change that but there are states that are neither.  All possible states can be written in terms of linear combinations of even and odd states. Similarly any general two particle state (electron-proton) can be broken into a symmetric and antisymmetric part under particle exchange.  When the two particle are the same some of these states are not allowed. For Bosons only the symmetric states (+) are allowed for Fermions only the antisymmetric states are allowed (-).

 

Cesium will decay such that each state of polarization is the same for each direction requires so the .

 

The anti symmetry of the wavefunction determines the relative phase. Thus the decay of  Cesium results in a two photon state:

 

 

 

Whenever a measurement of the photon in A finds  the measurement in B must find the photon in state .   In order to see the extreme requirements of quantum mechanics the two experiments will be set at large distances from the cesium and each other.  This means that when the photon arrives at either A or B it will be impossible to exchange information between the two fast enough to correlate the measurements causally.  Thus if one experiment finds a result and a standard view of causality is taken the photon arriving at B must arrive with the state defined.

 

Relativity is sometimes said to require that nothing may travel faster than the speed of light.  One must be a bit careful, the proper statement is that information cannot travel faster than the speed of light.  Consider the intersection of two lines that are moving towards each other at a small angle.

The point of intersection moves at an arbitrary velocity and indeed when the lines are parallel the point can have an infinite velocity.

For the case of the two experiments relativity requires that no information can be sent between the two experiments based on the result of an experiment.

 

Considering the above intersection point, one can synchronize two events by requiring that they occur at the time the lines cross.  Is this information transportation? Imagine that people at event1 realize that they are not ready and want to tell event 2 to start a few nanoseconds after the lines crosses.  Now there is no way to change the start of event 2 if the problem arose on a time scale that is less than d/c where d is the distance between the events.  Agreeing before hand to synchronize is not transmitting information.

 

Thus the results of experiment A and B can be used to synchronize certain events but this doesn’t transfer any information about A to B or vice versa.

 

So the fact that the two experiments require a correlated result is not problematic if one imagines that the intial state is sometimes   and sometimes .  However the state that we originally described was a linear combination of both states at all times.  We need to test if the arriving state is truly a linear combination or not.  To do this we can rotate the one of the measurements.

 

 

With respect to this basis the state becomes:

 

 

 

This is an example of how one can compute probabilities for outcomes. The key to seeing the impact of quantum mechanics is to realize that a possible outcome at either site requires the full state. The interference of x-y polarizations that we explored a few weeks ago required that the “analyzer” have a complete set of  basis states in order to get the correct result. Similarly as one explores outcomes at either A or B one finds that the complete quantum state must be present in order to get the correct results when measuring states x-y, x’-y’ or R-L.

 

In discussing these ideas the philosophical content of QM is perhaps clarified by considering what the quantum state tells us.

 

Shimony uses some descriptors that help discuss QM.

 

Students at the end of this course should be able to read and discuss these ideas.