3-9 EPR and Bell s theorem. EPR Bohm s version. S x S y S z V H 45

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1 1 3-9 EPR and Bell s theorem EPR Bohm s version S x S y S z n P(n) n n P(0) 0 0 V H 45 P(45) D S D P(0) H V

2 2 ( ) Neumann EPR n P(n) EPR PP(n) n EPR ( ) But even at this stage there is essentially the question of an influence on the very conditions which define the possible types of predictions regarding the future behavior of the system. Bohr, N., Can Quantum Mechanical Description of Physical Reality be Considered Complete?, Physical Review, 48,

3 Bell EPR 2 n n n 2n n n 4 3 The change in the wave function by a measurement is not just local but instantaneously pervades the whole space, which seems to contradict the assertion of relativity theory. This paradox led Einstein and Schrö dinger, among others, to reject quantum theory as a complete theory. As long as one is dealing with only one observable at a time, there is essentially no difference between classical and quantum probability. However, there appears differences when one consider questions involving more than two observables at the same time, such as joint distributions or conditional probabilities. Interference effects arise and they make the quantum probabilities quite different from the classical ones. Lemma 1 For any events Q, R, and S, P(Q/R) + P(R/S) P(Q/S). ( P(Q/R) means the probability of the event that Q occurs but R does not.) (proof) Easy. Corollary 2 For n 2 events Q 1, Q 2,...,Q n, we have the inequality P(Q j /Q j+1 ) P(Q 1 /Q n ). ( Here is from j = 1 to j = n -1.) (proof) Induction on n. Proposition 3 For any events Q, R, and S, D(R, S) = P(R S) - P(R S) satisfies: (1) D(R, R) = 0; (2) D(R, S) = D(S, R) (symmetry); (3) D(Q, S) D(Q, R) + D(R, S) (the triangle inequality). ( In topology a function D satisfying above three conditions is called a distance function or metric. Proposition 3 has been known to mathematicians since 1920s, but its relevance to quantum theory was first appreciated by John Bell in 1964.)

4 4 (proof) (1) and (2) are routine. (3) is proved by using Lemma 1. It is helpful to think the polarization of light when confused by the paradoxes of quantum measurements. The filter only transmits light polarized at the appropriate angle, the measurement has affected what is measured. When we treat two filters case and three filters case, we have to be careful applying the probability calculation in such cases. There is a dangerous probabilistic trap. two filters case Malus law: a proportion cos 2 of the light transmitted by the first filter will also pass through a second filter polarized at an angle to the first. When is /2, no light can pass through both filters. three filters case The insertion of a third filter must result in even more photons being stopped. But we can have the result: P(Q/R) + P(R/S) - P(Q/S) < 0. The event S does not have the same meaning in P(Q/S) as it does in P(R/S). In the case of four filters, P(Q/R) + P(R/S) + P(S/T) - P(Q/T) < 0. (*) Locality assumption of EPR: The transmission of photon B by a filter r B really does depend only on the photon and the filter and not on anything else which has happened to A. Event R B is well-defined in the sense of locality. Proposition 4 Let Q A and S A be the events that a photon A is transmitted by the polaroid filters q A or s A, respectively, and similarly R B and T B the events that photon B is transmitted by filters r B or t B. Then for some filter angles the quantum mechanical probabilities satisfy P(Q A /R B ) + P(R B /S A ) + P(S A /T B ) < P(Q A /T B ). (proof) By (*). From Corollary 2 and Proposition 4 we have a contradiction. Bell was the first to realize that quantum mechanics gives predictions which are inconsistent with the inequalities for classical probability derived in Lemma 1. The experiments about this were done by Freedman and Clauser, and Aspect. ( ) Quantum nonlocality Generally, physical systems have global properties which continue to evolve globally even as the system becomes spatially separated - when the parts of the system are measured then they will manifest correlations which embody the previous global state. Might these correlations be the result of local processes, perhaps involving other, unknown or hidden facts about nature? No. (This is precisely what John Bell proved, in 1964.) The history of nonlocality Einstein, Podolsky and Rosen first focused attention on correlated, spatially extended quantum systems in 1935, though in their argument they assumed locality in order to find fault with quantum theory. 25 years later Bell s result showed that EPR s assumption was mistaken. In 1989, Greenberger, Horne and Zeilinger sharpened Bell s results further by considering correlated states with 3 or more entangled particles. There is a relation between the kind of entangled states considered in these proofs and the phenomenon of quantum computation. The full extent of nonlocality as a physical fact is not well understood - for example, does a superfluid exhibit nonlocality? Generally, almost any collapse of a wavefunction appears to be nonlocal: is this an artifact of our description?

5 5 Since the 1964 analysis of John Bell (Bell s theorem) it is widely recognized that in some sense the nonlocality is real - quantum mechanics is a much different theory than one could assemble with local parts. In the words of Henry Stapp (1977), The present formulation asserts that a theory entails a nonlocal connection if there is no conceivable way for the results in each region to be independent of the choice made in the other region. Quantum theory has such a nonlocal connection: That is what Bell actually discovered. One could turn the question around by viewing the quantum description as completely global, and ask what is the root of the apparent, provisional locality. What is at the root of nonlocality? That the guiding wave, in the general case, propagates not in ordinary three-space but in multidimensionalconfiguration space is the origin of the notorious nonlocality of quantum mechanics... John Bell, Speakable and unspeakable in quantum mechanics Einstein, Podolsky and Rosen I. In a 1935 Physical Review article entitled Can Quantum Mechanical Description of Physical Reality be Considered Complete? Albert Einstein, Boris Podolsky and Nathan Rosen opened what is now a 60 year old discussion of correlated multi-particle quantum systems. EPR discussed such systems as a way of critiquing what they called the completeness of the quantum description. In 1964 John Bell used David Bohm s version of EPR s argument to ironically rule against them: EPR s assumption of locality is found to be false, releasing quantum theory from any claim of incompleteness or inconsistency. The status of nonlocality in quantum mechanics is still being debated. II. EPR was attempting to embarrass some non-realist suppositions of quantum mechanics. They suggested considering two-particle systems which are correlated but become arbitrarily separated... EPR s starting point and central idea is that we can measure some aspect of one of the pair and subsequently predict outcomes of measuring corresponding aspects of the second one. EPR said that since this property of the second particle was predictable after the first measurement, it was therefore real. The counterfactual idea: But for that matter, EPR argued, we might just as well instead measure some other, perhaps contradictory (complimentary) property of the first particle, and in this case we would know some corresponding other fact about the second particle in the pair: this 'other' property of the second particle would be real. The assumption of locality: We at first go nowhere near the second particle (which might be at the other end of the galaxy by this time): we are only measuring the first particle to start with - EPR said that surely we could not alter "the real situation" of the second, distant particle, by measuring the first... EPR said, since a. we could make either measurement of the first particle, and b. since (by locality) this cannot itself influence the second particle, c. therefore the second particle must have pre-prepared real values for both (all) possible properties corresponding to measurements on the first particle. Yet in the case of complementary variables quantum theory denies that both properties can be simultaneously present. EPR concluded that since there were "real" properties of the world not even definable in quantum theory, quantum theory is incomplete. III. This conclusion of EPR s is based on an assumption of locality, and their recommendation favored the use of local underlying variables to complete quantum theory. The line of argument initiated by EPR was formally updated in 1964 when John Bell showed that the assumption of locality made by Einstein, Podolsky and Rosen was itself actually in contradiction with facts predicted by quantum mechanics. Bell summed it up this way in 1964, The paradox of Einstein, Podolsky, and Rosen was advanced as an argument that quantum mechanics could not be a complete theory but should be supplemented by additional variables. These additional variables were to restore to the theory causality and locality... That idea will be formulated mathematically and shown to be incompatible with the statistical predictions of quantum mechanics. Bell s theorem I. In 1964 John showed that the world according to quantum mechanics really is nonlocal. The universe is nonlocal at the level of individual events (even though the locally observable aggregates of the universe

6 6 predicted by quantum mechanics do not bear any traceable nonlocal signature). Bell inverted the EPR argument since EPR had used locality as an assumption - Bell s theorem converted EPR s local-realist position into a paradigm of what is not true. Bell s ingenious proof focused on EPResque 2-particle systems: pairs which are perfectly correlated along any given measurement axis - recall that EPR had taken this correlation as implying underlying variables, but Bell problematized this conclusion by examining some other similar correlations of the pairs: for example when measured along respective axes with a relative rotation though some angle... With a relative angle between them the correlation between the two measurements is no longer perfect but varies as the cosine of the angle. Bell explored the type of underlying variable models (with built in locality) which EPR had suggested, and showed that the entire class of such models could in this case not give the same cosine varying correlation as quantum theory predicted. In 1991 GHZ sharpened Bell s result by considering systems of three or more particles and deriving an outright contradiction among EPR s assumptions. II. Because of issues associated with relativity, such as the relativity of simultaneity, nonlocality is problematic - some people prefer to avoid this conclusion and so try to interpret violations of the Bell inequality as implying something else... Quantum theory looks nonlocal (e.g. I measure here and collapse everywhere) and Bell s theorem seems to resolve the question in the affirmative. In the language of wavefunction collapse, Bell-GHZ showed that wavefunctions collapse at a distance as surely as they do locally. The Kochen-Specker Theorem The Kochen-Specker (KS) theorem demonstrates that it is, in general, impossible to ascribe to an individual quantum system a definite value for each of a set of observables not all of which necessarily commute. Of course elementary quantum metaphysics insists that we cannot assign definite values to noncommuting observables; the point of the KS theorem is to extract this directly from the quantum-mechanical formalism, rather than merely appealing to precepts enunciated by the founders... The GHZ nonlocality proof Beyond Bell s Theorem... (I.) In 1991, GHZ fundamentally updated Bell's result, essentially by investigating Bell-like relationships in correlated systems of more than two particles. What they showed surpassed Bell's result by eliminating the statistical nature of the proof. They show a situation involving three particles where after measuring two of the three, the third becomes an actual test contrasting between locality and the quantum picture: a local theory predicts one value is inevitable for the third particle, while quantum mechanics absolutely predicts a different value. (So, we only have to run the experiment once.) This is really equivalent to older proofs about the modeling of the quantum state by underlying variables, as was pointed out by David Mermin - Kochen and Specher required over a hundred particles in their original proof, while GHZ have it down to three. Mermin's Exponential Bell-inequality Beyond Bell s Theorem... (II.) Mermin, N. David, Extreme Quantum Entanglement in a Superposition of Macroscopically Distinct States, Physical Review Letters, V65, 15 (1990) A Bell inequality is derived for a state of n spin-1/2 particles which superposes two macroscopically distinct states. Quantum mechanics violates this inequality by an amount that grows exponentially with n. The Bell-inequality is a measure of the difference in correlation structure of the classical and quantum descriptions. The first examples (Bell, 1964) involved only two-particle systems - even in this case the richness of the correlation in the quantum de scription exceeded that of any scenarios in the classical, localized

7 description. Mermin has shown that this measure of the difference between the systems grown expentially with the number of particle in the entangled state. 7