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A Review: "Can Quantum-Mechanical Description of Physical Reality be Considered Complete?"

A. Einstein, B. Podolsky, and N. Rosen vs. N. Bohr

J. Hilts Department of Physics and Computer Science, Wilfrid Laurier University, Waterloo, ON, Canada N2L 3C5

(July 26, 2007; published date) The EPR paradox is a thought experiment which was first introduced by A. Einstein, B.

Podolsky, and N. Rosen through a paper with the above title.[1] Einstein, Podolsky, and Rosen, collectively known as EPR, used philosophical arguments based on locality, realism, and counterfactual definiteness to prove by contradiction that the above question is in the negative. One of the father’s of quantum mechanics, Neils Bohr, advocated the Copenhagen interpretation of quantum mechanics. It is well known that N. Bohr and A. Einstein did not share the same views regarding the interpretation of quantum mechanics and had many discussions regarding the “nature of nature”.[3] It is then no surprise that Bohr refuted the results of EPR, claiming an ambiguity in the assumptions made by EPR regarding the “criterion of physical reality”.

PACS numbers: 01.70.+w and 03.65.Ud

S the theory of quantum mechanics was being developed, the philosophy surrounding it was the focus of many debates and discussions. The implications of a statistical universe did not sit well with many physicists, made famous by Albert Einstein’s quote, “God does not play dice.” [Figure 1] He believed for every measured value of an observable, the observable had that value before the measurement took place.1 How do you know if the particle really did have that value before the measurement? The Copenhagen, or orthodox, interpretation of quantum mechanics regards this question as meaningless.[4] Physicists who hold this view claim that the particle did not have any value before the measurement; it was the act of measurement which made the particle “choose” a value. That is, the wave function describing the particle collapsed into a particular state.

A

Even stranger than how and why the particle did not have a value (which is still in debate [4]) is the concept of an entangled state. Using the

1 An observable is a property of a system that can be determined by a sequence of operations performed on the system. Every observable in Quantum Mechanics has an associated operator. See footnote 4.

following example I shall explain the concept of entanglement [Figure 2]: Suppose Alice, Bob, and Charlie would like to conduct an experiment using two spin-1/2 particles prepared in the singlet state:2

( )100100

21 −= (1)

These particles can be in either the spin up state, 0 , or the spin down state, 1 , represented by

the observables and , respectively. In this case spin can only be measured along the z-axis. Now suppose Alice and Bob are spatially separated and Charlie prepares these two particles in some manner (it doesn’t matter how just as long as he can reproduce the experimental procedure [5]) and sends one towards Alice, call it u

+zv −

zv

1, and one towards Bob, call it u2. Alice and Bob then make one measurement on their particles at the exact same time; it turns out that no matter what value Alice gets when she measures u1, Bob will always get

2 Refers to two or more particles prepared in a correlated state, such that their total angular momentum is zero.[7]

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the opposite value for his measurement. Thus, Alice can predict with certainty what value Bob will get, and vice versa. How did Bob’s particle know what value Alice’s particle had at the time of measurement? This result was dubbed “spooky action at a distance” by Einstein, but is formally known as non-locality.3 This result demonstrates the quantum mechanical phenomenon known as Quantum Entanglement; the quantum states of two or more objects have to be described with reference to each other.[8] Entanglement is proving to be very useful for Quantum Computation and Quantum Information as it realizes information processing tasks which are impossible or much more difficult with classical resources.[5]

FIG. 1. A cartoon depicting the conflicting views of Einstein and Bohr.

The fundamental arguments of EPR are based

on (1) the elements of physical reality and (2) the completeness of quantum mechanical theory. Even though EPRs argument is philosophical in nature they fail to address what the elements of physical reality are in this context. They instead make the assumption that if without in anyway disturbing the system the value of a particle can be known before measurement then that element corresponds to an element of physical reality.

3 Refers to the possibility of instant interaction between two distant particles. The Principal of Locality states that this is not possible and only a particle’s immediate surroundings can influence it.

EPR used a similar thought experiment to the one above by noting that (classically), at any given time, we know both the relative position of the particles to one another, x1-x2, and the total angular momentum of the system, p1+p2. Note that x1-x2 and p1+p2 are commutable.4[8]

Suppose now that Alice and Bob can measure either the position of their particle, x1 and x2, or the momentum, p1 and p2, respectively, along the z-axis. Since x and p are non-commuting operators, a state describing both of these observables cannot possess a definite value for each operator.5 If Alice decides to measure x, then from the above example we know with certainty the value of x for Bob’s particle, and hence it is an element of physical reality. Similarly, if Alice decides to measure p, then we know with certainty what the value of p will be for Bob’s particle. Since we cannot know both the x and p with certainty, Alice’s decision to measure the position or momentum has an instantaneous effect on the elements of physical reality at Bob’s location.[6]

EPR stated that if two systems were prepared to be identical and in one of which Alice measured x1 and in the other p1, one could simultaneously know both the position and the momentum of both particles. This obviously violates the Heisenberg Uncertainty Principle6, thus EPR “[were] forced to conclude that the quantum mechanical description of physical reality given by wave functions is not complete.”[1]

Neils Bohr did not agree with these results, stating that one cannot use the results from two

4 Suppose we have two operators A and B. If the commutator of A and B is non-zero, namely [A,B]=AB-BA≠0, then A and B cannot have simultaneous reality. The more precise we measure the value of A, the less precise we can know the value of B. 5 The operator x represents position and the operator

( )( )xihp ∂

∂= represents momentum. Note, [ ] hipx =, , which is known as the canonical commutator. 6 2

h≥ΔΔ px where xΔ is the uncertainty in the position and pΔ is the uncertainty in the momentum.

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different systems as if the measurements were made on one system; the wave function collapses into a specific state which describes the system at the time of, say, measuring the position of particle, and hence changes the value of the momentum of particle 1. He believed that there was a possibility of non-local interaction occurring between two particles, something Einstein could never accept.

Bohr saw an ambiguity in the assumptions of the criterion describing the elements of reality given by EPR; how can a particle interact with a system but not disturb it in anyway?7 In response to this Bohr devised his own thought experiment involving an apparatus comprised of two free particles and a rigid diaphragm with two parallel slits; through each of these slits one particle with given initial momentum passes independently of the other. We then have a free choice to measure either the position or the momentum after the particles have passed through the slit.

If the momentum of the diaphragm is measured before and after the passing of the particles through the slits then one can determine the total momentum of the two particles perpendicular to the slits, as well as the positions of the particles relative to one another.[2] Thus, the measurement of the momentum or position of one particle will determine the momentum or position of the other particle, respectively. We could define the position of a particle to be nothing more then a correlation between its behavior and some apparatus which defines the space frame of reference.

The conclusions Bohr drew from his experiment he said best himself and so I quote:

7 It is from this that Bohr describes a system as complementary, which arises in a system when one consider the circumstances under which one measures properties of that system; Bohr noted that this implies impossibility of any sharp separation between the behavior of atomic objects and the interaction with the measuring instruments which serve to define the conditions under which the phenomena appear.[6]

“By allowing an essentially uncontrollable momentum to pass from the first particle into the mentioned support, … we have by this procedure cut ourselves off from any future possibility of applying the law of conservation of momentum to the system consisting of the diaphragm and the two particles and therefore have lost our only basis for an unambiguous application of the idea of momentum in predictions regarding the behavior of the second particle. Conversely, if we choose to measure the momentum of one of the particles, we lose through the uncontrollable displacement inevitable in such a measurement any possibility of deducing from the behavior of this particle the position of the diaphragm relative to the rest of the apparatus, and have thus no basis whatever for predictions regarding the location of the other particle.”[2]

With these results Bohr claimed that the description of physical reality given by EPR was wrong. Their conclusion regarding the quantum mechanical incompleteness of the description of reality is thus also false.

FIG. 2. A visual representation of the Alice, Bob, and Charlie experiment. The conclusions of the EPR paper try to

resolve this paradox by stating that quantum mechanics is merely a statistical approximation of a more complete description of nature which has yet to be discovered. In this more complete description there exist variables pertaining to every element of physical reality. There must be, however, some unknown mechanism acting

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on these variables to give rise to the observed effects of "non-commuting quantum observables." Such a theory is called hidden variable theory. [6]

John S. Bell derived a set of inequalities, known as Bell Inequalities, which showed that the predications of quantum mechanics through the EPR thought experiment actually differed from the predictions of various hidden variable theories.[9] These predictions have much stronger statistical correlations between measurement results performed on different axes than the hidden variable theories.[6] These theories are generally non-local; recall the EPR paper used locality as one of their arguments.

Today most physicists believe that the EPR “paradox” is only a paradox because our classical intuitions do not correspond to physical reality in the realm of quantum mechanics.

Acknowledgements – I would like to thank Neil Sinclair and Dr. Shohini Ghose for the insightful discussions which helped me better understand this topic and Quantum Mechanics in general.

[1] A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777 (1935). [2] N. Bohr, Phys. Rev. 48, 696 (1935) [3] Malone, Michael, Michigan State University www.msu.edu/user/malonemi/lbs333/quantum03.html, 1998. [4] Griffiths, David J., Introduction to Quantum Mechanics, Second Edition. New Jersey, 2005. [5] Neilsen, Michael A. and Chuang, Isaac L., Quantum Computation and Quantum Information. Cambridge University Press, 2003. [6] Wikipedia.org – EPR Paradox [7] Wikipedia.org – Singlet State [8] Peres, A., Quantum Theory – Concepts and Methods. Kluwer Academic Publishers, New York, 2002. [9] Bell, J. S. "On the Einstein-Podolsky-Rosen Paradox." Physics 1, 195-200, 1964