Einstein Podolsky Rosen Correlations

I learned about this from a short story by StephenBaxter and Arthur C Clarke "The Wire Continuum." Apparently, an unknown quantum state can be disassembled into, then later reconstructed from, purely classical information using measurements called EinsteinPodolskyRosenCorrelations. They were projecting that this "classical information" could be sent over a telephone wire.

In 1935 Albert Einstein and two colleagues, Boris Podolsky and Nathan Rosen (EPR) developed a thought experiment to demonstrate what they felt was a lack of completeness in quantum mechanics. This so-called "EPR paradox" has led to much subsequent, and still on-going, research.

One of the principal features of QuantumMechanics is that not all the classical physical observables of a system can be simultaneously known with unlimited precision, even in principle. Instead, there may be several sets of observables which give qualitatively different, but nonetheless complete (maximal possible) descriptions of a quantum mechanical system. These sets are sets of "good quantum numbers," and are also known as "maximal sets of commuting observables." Observables from different sets are "noncommuting observables".

A well known example is position and momentum. You can put a subatomic particle into a state of well-defined momentum, but then you cannot know where it is. It's not just a matter of your inability to measure, but rather, an intrinsic property of the particle. Conversely, you can put a particle in a definite position, but then its momentum is completely ill-defined. You can also create states of intermediate knowledge of both observables: If you confine the particle to some arbitrarily large region of space, you can define the momentum more and more precisely. But you can never know both, exactly, at the same time.

[In fact, some of the above statements are not perfectly correct. For one thing, observables which don't commute can still have mutual eigenstates. Such subtleties are very important to those who examine the derivation of Bell's inequality in great detail in order to find hidden assumptions.]

Position and momentum are continuous observables. But the same situation can arise for discrete observables such as spin. The quantum mechanical spin of a particle along each of the three space axes is a set of mutually noncommuting observables. You can only know the spin along one axis at a time. A proton with spin "up" along the x-axis has undefined spin along the y and z axes. You cannot simultaneously measure the x and y spin projections of a proton. EPR sought to demonstrate that this phenomenon could be exploited to construct an experiment which would demonstrate a paradox which they believed was inherent in the quantum-mechanical description of the world.

They imagined two physical systems that are allowed to interact initially so that they subsequently will be defined by a single Schrodinger wave equation (SWE). [For simplicity, imagine a simple physical realization of this idea - a neutral pion at rest in your lab, which decays into a pair of back-to-back photons. The pair of photons is described by a single two-particle wave function.] Once separated, the two systems [read: photons] are still described by the same SWE, and a measurement of one observable of the first system will determine the measurement of the corresponding observable of the second system. [Example: The neutral pion is a scalar particle - it has zero angular momentum. So the two photons must speed off in opposite directions with opposite spin. If photon 1 is found to have spin up along the x-axis, then photon 2 must have spin down along the x-axis, since the total angular momentum of the final-state, two-photon, system must be the same as the angular momentum of the initial state, a single neutral pion. You know the spin of photon 2 even without measuring it.] Likewise, the measurement of another observable of the first system will determine the measurement of the corresponding observable of the second system, even though the systems are no longer physically linked in the traditional sense of local coupling.

However, QM prohibits the simultaneous knowledge of more than one mutually noncommuting observable of either system. The paradox of EPR is the following contradiction: For our coupled systems, we can measure observable A of system I [for example, photon 1 has spin up along the x-axis; photon 2 must therefore have x-spin down.] and observable B of system II [for example, photon 2 has spin down along the y-axis; therefore the y-spin of photon 1 must be up.] thereby revealing both observables for both systems, contrary to QM.

QM dictates that this should be impossible, creating the paradoxical implication that measuring one system should "poison" any measurement of the other system, no matter what the distance between them. [In one commonly studied interpretation, the mechanism by which this proceeds is 'instantaneous collapse of the wavefunction'. But the rules of QM do not require this interpretation, and several other perfectly valid interpretations exist.] The second system would instantaneously be put into a state of well-defined observable A, and, consequently, ill-defined observable B, spoiling the measurement. Yet, one could imagine the two measurements were so far apart in space that special relativity would prohibit any influence of one measurement over the other. [After the neutral-pion decay, we can wait until the two photons are a light-year apart, and then "simultaneously" measure the x-spin of photon 1 and the y-spin of photon 2. QM suggests that if, for example, the measurement of the photon 1 x-spin happens first, this measurement must instantaneously force photon 2 into a state of ill-defined y-spin, even though it is light-years away from photon 1.

How do we reconcile the fact that photon 2 "knows" that the x-spin of photon 1 has been measured, even though they are separated by light-years of space and far too little time has passed for information to have travelled to it according to the rules of Special Relativity? There are basically two choices. You can accept the postulates of QM as a fact of life, in spite of its seemingly uncomfortable coexistence with special relativity, or you can postulate that QM is not complete, that there was more information available for the description of the two-particle system at the time it was created, carried away by both photons, and that you just didn't know it because QM does not properly account for it.

So, EPR postulated that the existence of hidden variables, some so-far unknown properties, of the systems should account for the discrepancy. Their claim was that QM theory is incomplete; it does not completely describe the physical reality. System II knows all about System I long before the scientist measures any of the observables, thereby supposedly consigning the other noncommuting observables to obscurity. Furthermore, they claimed that the hidden variables would be local. No instantaneous action-at-a-distance is necessary in this picture, which postulates that each System has more variables than are accounted by QM. Niels Bohr, one of the founders of QM, held the opposite view and defended a strict interpretation, the Copenhagen Interpretation, of QM.

In 1964 John S. Bell proposed a mechanism to test for the existence of these hidden variables, and he developed his inequality principle as the basis for such a test. He showed that if BellsInequality was satisfied, then there could be no local hidden variable theory which accounted for it.

http://faculty.physics.tamu.edu/thw/projects/eprmain.html

http://www.freespace.net/~udo/LRM_3B/

I'm not sure why there is a lot of excitement about the "two photons" being separated in space. Are there really two photons, or is there one SWE, extended to macro spatial dimentions, which is interpreted as two photons (but isn't)? --AndyPierce

CategoryPhysics