EPR paradox

The EPR paradox arises in a thought experiment which shows that quantum mechanics leads to very counter-intuitive and paradoxical consequences. It is named after Einstein, Podolsky, and Rosen, who published the idea in 1935. It is also referred to as the EPRB paradox after Bohm, who converted the idea into something that was nearer to being experimentally testable.

Contents

1 The paradox defined

2 How the EPR paradox affects our understanding of particles

3 A color metaphor

4 Mathematics of the EPR paradox

5 Modern perspectives on the EPR paradox

6 References

7 External links

8 See also

The paradox defined

The EPR paradox draws attention to a phenomenon predicted by quantum mechanics known as quantum entanglement, in which measurements on spatially separated quantum systems can instantaneously influence one another. As a result, quantum mechanics violates a principle formulated by Einstein, known as the principle of locality or local realism, which states that changes performed on one physical system should have no immediate effect on another spatially separated system.

The principle of locality is persuasive, both on intuitive grounds and because it seems at first sight to be a natural outgrowth of the theory of special relativity. According to relativity, information can never be transmitted faster than the speed of light, or causality would be violated. Any theory which violates causality would be deeply unsatisfying, and probably internally inconsistent. However, a detailed analysis of the EPR scenario shows that quantum mechanics violates locality without violating causality, because no information can be transmitted using quantum entanglement.

Nevertheless, the principle of locality appeals powerfully to physical intuition, and Einstein, Podolsky and Rosen were unwilling to abandon it. They suggested that quantum mechanics is not a complete theory, just an (admittedly successful) statistical approximation to some yet-undiscovered description of nature. Several such descriptions of quantum mechanics, known as "local hidden variable theories" were proposed. These deterministically assign definite values to all the physical quantities at all times, and explicitly preserve the principle of locality.

Of the several objections to the prevailing interpretation of the quantum mechanics spearheaded by Einstein, the EPR paradox was the subtlest. It is at present considered to have been unsuccessful, the existence of hidden variables having been refuted experimentally and the EPR "paradox" taken to be fully resolved within the current interpretation of the theory. The belief that entanglement is a real phenomenon has led to a radical shift in thinking about 'what is reality' and what is a 'state of a physical system'. First, a review of the history:

How the EPR paradox affects our understanding of particles

Before 1936, the generally accepted view was that a particle, such as an electron, has measurable properties such as a position and a momentum but 'we cannot know both' at the same time. This view is present in some explanations of the Heisenberg uncertainty principle. In such an explanation, the 'more exactly we measure the position', the 'more we disturb the particle' and its momentum becomes that much less certain. The numerical measure of uncertainty satisfies Heisenberg's principle, but this (local realistic) interpretation is rejected in professional circles, though it still lives in popular books.

The shift was caused by the EPR thought experiment, which has shown how to measure the property of a particle, such as a position, without disturbing it. In today's terminology, we would say that they did the determination by measuring the state of a distant but entangled particle. According to quantum mechanics, the state of our particle will instantly change even though we did not disturb it in any local way. It is called a paradox, since it conflicts with our classical intuition —specifically, with the principle of locality.

The very concept of 'entanglement' also conflicts with our intuition the same way. One possibility is that quantum mechanics is wrong. However, experiments have been interpreted as showing that entanglement does occur, and applications in the fields of quantum cryptography and quantum computation are currently under development. In quantum cryptography, an entangled signal is sent down a communications channel making it impossible to intercept and rebroadcast that signal without leaving a trace. In quantum computation, entangled states allow simultaneous computations to occur in one step.

We could argue that the EPR paper 'discovered' entanglement. The concept, also called 'nonlocal behaviour' and (jokingly) 'quantum weirdness' has no classical analogy. It is the fact that QM treats two particles, which interacted in the past (and so became entangled) and then separated spatially (i.e., 'flew apart'), as one object. When one such particle is changed, the other will change too (instantly). Einstein called this behavior 'spooky action at a distance', and considered it unacceptable. Before it was accepted as real and inevitable by most physicists, one escape route had to be closed, namely the possible existence of 'hidden parameters'.

John Bell is currently considered to have closed that escape route. The setup of the EPR experiment and Bell's theorem are described in separate pages. Here we proceed historically and first describe Bohm's contributions and then explain the conceptual meaning of the hidden parameter using a parable of color.

A color metaphor

Bohm substituted measurement of spin coordinates for measurement of momentum and position. The classical analogy of spin of a photon is polarization of light, which is quite familiar. However, the mathematical description of this property in quantum mechanics is complex. The experiment measuring spin is, however, easier than the original EPR setup.

We now describe an EPR-like experiment using the words 'red' and 'cyan' for 'spin up' and 'spin down'. This is a variant of an expository device used by Bell in his 1981 Bertlmann's socks paper. Note that in the experiment described below, hidden variables do account for the perfect correlation of observed values.

Alice sees billiard ball is cyan

Imagine that a single white particle splits into two, one cyan and one red. (Here the color (spin) is conserved and red+cyan=white).

One flies left, one right, and we do not know which is which.

When Alice, on the left, will notice (measure) that her particle is cyan, she will instantly know that Bob's measurement on the right, far far away, will be red.

"So!", you may say: "there is no paradox here!". The one which went left was always cyan, the one which went right was always red. Alice just did not know which was the case, until she did her measurement. There is no need for any 'instant synchronisation at a distance', no need for spooky action."

That is indeed an explanation of the result in this experiment, since we can see the observed colors are there all along. Now we can attempt to give an analogous explanation for the quantum mechanical properties of spin up or spin down. In that case we call it a 'hidden parameter' hypothesis. Why hidden? Because when you look at the mathematical object, the wave-function, which according to standard QM describes the 'state' of that particle, it does not have that definite color (e.g. spin) there. It is described by a superposition of colors; this superposition implies a possibility of red, and possibility of cyan. These possibilities for one component of spin are complementary to such possibilities for other components. Because they are complementary, just like position and momentum, they cannot both be determined at the same time. QM says they do not both exist. The superposition state is converted to a non-superposition state of red or cyan, when we measure it. Instantly, the other, entangled particle, has its possible color jump to cyan or red. To avoid that apparent weirdness, hidden parameter theory says it was there, it was red for x-component and cyan for y-component, (violating Heisenberg's principle) and we just were not able to see it.

Our intuition, molded by experience of the macroscopic world, leads us to believe that these hidden particle states must exist, because otherwise we would have to admit the 'spooky action at distance' which Einstein disliked. Bohm disliked it too and so he constructed a hidden parameter theory which did agree with the experiment and gave the same results as QM. However, an early mathematical proof by Von Neumann said that Bohm's supposed 'local realistic' theory was impossible.

Bell disliked 'action at distance' (also known as 'nonlocality') as well. He investigated and discovered two things:

1. von Neumann's proof of the impossibility of hidden variables was wrong.

2. Bohm's theory was actually non-local.

Eventually, he corrected von Neumann's error and generalised von Neumann's proof to a whole class of theories.

And so, in 1964, John Stewart Bell showed that the whole class of theories known as hidden variable theories have to be non-local if they are to agree with the quantum-mechanical prediction for ideal experiments. If such a theory is local it must satisfy Bell's inequality, but Quantum mechanics predicts that the inequality is not satisfied. Thus far experiments have been conducted, starting with an experiment in Orsay (France) by Alain Aspect in 1982, appear to confirm that Bell's inequality is violated, favoring Quantum mechanics over local hidden variable theories.

Mathematics of the EPR paradox

In the following discussion, we will consider a system consisting of a pair of two-level quantum particles. Two level means that each particle is described by a (two-dimensional) Hilbert space H; the system of two such particles is described by the tensor product Hilbert space $H \otimes H$ . Now consider an observable A for a single particle (for instance, spin along some axis). A has a pair of eigenvectors e₁, e₂ which for purposes of exposition we assume correspond to eigenvalues +1 and -1. This is the case for any of the spin operators

$A(\vec{s}) = s_1 \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} + s_2 \begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix} + s_3 \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}, \quad s_1^2 + s_2^2 +s_3^2 = 1$

For the system of two such particles, (which we will label with Alice and Bob who are the intended recipients) we have two observables A_Alice and A_Bob, corresponding to measurements by Alice and Bob.

Consider next a measurement of A_Alice. By general principles of measurement (discussed for instance in quantum mechanics, mathematical formulation of quantum mechanics and more formally in quantum logic), when A_Alice is measured the system state ψ collapses into an eigenvector of A_Alice. If +1 is measured, this means that immediately after measurement the system state will be the orthogonal projection of ψ onto the space of states of the form

$| e_1 \rangle \otimes | \phi\rangle \quad \phi \in H$

and similarly if -1 is measured, it will be the orthogonal projection onto

$| e_2 \rangle \otimes | \phi\rangle \quad \phi \in H$

So far, nothing untoward is apparent. Suppose however the system is prepared in what is called a spin singlet state, that is one of the form

$\frac{1}{\sqrt{2}}\left(| \psi \rangle \otimes | \phi \rangle - | \phi \rangle \otimes | \psi \rangle \right)$

Remark. For two-level quantum systems, there is up to an (irrelevant scale factor) only one spin singlet state.

The reason is that the space of such tensors is one-dimensional. To show one-dimensionality, observe that if T is a linear operator on the two-dimensional space H, by an easy computation

$| T \psi \rangle \otimes | T \phi \rangle - | T \phi \rangle \otimes | T \psi \rangle = \operatorname{det}(T) \left(| \psi \rangle \otimes | \phi \rangle - | \phi \rangle \otimes | \psi \rangle \right)$

Now if the system was produced in a spin singlet state, by the above remark that there is only one spin singlet state, we can express this state in the following form:

$\frac{1}{\sqrt{2}}\left(| e_1 \rangle \otimes | e_2 \rangle - | e_2 \rangle \otimes | e_1 \rangle \right).$

Alice measures spin +1

Thus if +1 is measured for A_Alice, then the new system state will be determined by orthogonal projection of the spin singlet state as stated above. This will be the state

$| e_1 \rangle \otimes | e_2 \rangle.$

Similarly, if -1 is measured, then the new system state will be

$|e_2 \rangle \otimes | e_1 \rangle$

Now this means that the measurement for A_Bob is now determined! It will be -1 in the first case or +1 in the second case.

Remark. Note that the a similar conclusion holds for any observable with possible measurement outcomes λ₁ and λ₂. As soon as Alice determines a value λ₁ for A_Alice, Bob's observed value for A_Bob, will be λ₂.

As pointed out by Bell in his 1964 paper, this fact alone on the perfect anti-correlation between the values A_Alice and A_Bob does not contradict hidden variables. The contradiction obtained by Bell uses three observables.

Modern perspectives on the EPR paradox

Today, most physicists believe that local hidden variable theories are untenable and that the principle of locality does not hold. Therefore, the EPR paradox would only be a paradox because our physical intuition does not correspond to physical reality. However, the book is not closed yet on this issue.

Some people continue to investigate local realist theories exploiting the defects in actual experiments. QM experiments are different from experiments on a macroscopic scale, which are directly accessible to our senses. In QM we can count the clicks of a Geiger counter or the spots on a photographic plate and those results have to be interpreted by some abstract reasoning. There are assumptions explicitly made or hidden assumptions (experimental loopholes) which may be just artifacts of today's measuring devices or fundamental limits not fully accounted for by today's theory.

Others investigate flaws in Bell's original assumptions. Local hidden variable theories as defined by Bell are just one kind of local realist theory. Arthur Fine showed that a theory satisfies Bell's inequality if and only if it can be cast as a local hidden variable theory. However examples of local realist theories can be constructed which do not conform to the requirements of a hidden variable theory needed for the proof of Bell's inequality and such theories thus escape Bell's theorem. The key to this approach is that the local hidden variables must have a suitable mathematical structure to allow probabilities to be correctly recovered by integration or summation, however one can construct realist theories which do not allow such a structure.

The two approaches may in fact be related. Certain experimental loopholes involve missing values in experiments. These missing values may be the result of fundamental physical constraints. Similarly certain local realist theories that escape Bell's theorem do so by filtering out certain values in order to produce probabilities matching QM predictions instead of those of standard local hidden variable theories. It may be the case that the ultimate foundations of QM lie in the mathematical behaviour of fundamental physical constraints that result in missing values.

The topic remains active. [1] (http://dmoz.org/Society/Philosophy/Philosophy_of_Science/Physics/Seeds_and_Escapes).

References

Bell, J.S.: On the Einstein-Poldolsky-Rosen paradox. Physics 1, pp. 195-200 (1964), reproduced as Ch. 2 of Bell, J.S., The Speakable and Unspeakable in Quantum Mechanics, Cambridge University Press 1987

Bell, J.S.: Bertlmann's Socks and the Nature of Reality,Journal de Physique, Tome 42, 1981, reproduced as Ch. 16 of Bell, J.S., The Speakable and Unspeakable in Quantum Mechanics, Cambridge University Press 1987

Einstein, A.; Podolsky, B. and Rosen, N.: Can quantum-mechanical description of physical reality be considered complete? (http://prola.aps.org/pdf/PR/v47/i10/p777_1) Physical Review 47, 777 (1935)

Fine, A.: ?Hidden Variables, Joint Probability, and the Bell Inequalities?, Physical Review Letters 48, 5, pp. 291-294 (1982)

Fine, A.: ?Do Correlations need to be explained?? in Cushing & McMullin, eds (1986), Philosophical Consequences of Quantum Theory: Reflections on Bell?s Theorem, University of Notre Dame Press (1986)

Hardy, L.: Nonlocality for 2 particles without inequalities for almost all entangled states. Physical Review Letters 71: (11) pp. 1665-1668 (1993)

Mizuki, M.: A classical interpretation of Bell's inequality. Annales de la Fondation Louis de Broglie, Volume 26, no 4, 2001 683

Sakurai, J.J.: Modern Quantum Mechanics. Addison-Wesley, USA 1994, pp. 174-187, 223-232

Thompson, C. H.: The Chaotic Ball: An Intuitive Analogy for EPR Experiments (http://arXiv.org/abs/quant-ph/9611037), Foundations of Physics Letters 9, 357 (1996).

External links

A. Aspect: Bell's inequality test: more ideal than ever (http://www-ece.rice.edu/~kono/ELEC565/Aspect_Nature.pdf). Nature, vol 398, 18 March 1999
Original papers in pdf format (http://www.drchinese.com/David/EPR_Bell_Aspect.htm)
Recent doubts about conclusions (http://www.lns.cornell.edu/spr/2002-08/msg0043292.html)
Lasting Contribution of the EPR paper (http://www.ensmp.fr/aflb/AFLB-253/aflb253p309.pdf)