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Is the EPR paradox really a paradox?

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1998 Eur. J. Phys. 19 307

(http://iopscience.iop.org/0143-0807/19/3/015)

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Eur. J. Phys. 19 (1998) 307–311. Printed in the UK PII: S0143-0807(98)90320-8

Is the EPR paradox really a paradox?

A Tartaglia †Dipartimento di Fisica, Politecnico, Corso Duca degli Abruzzi 24, I-10129, Torino, Italy

Received 6 January 1998

Abstract. The EPR paradox and the meaning of the Bell inequality are discussed. It is shownthat if the quantum objects are considered as carrying ‘instruction leaflets’ with them, tellingthem what to do when meeting a measurement apparatus, then any paradox disappears. In thisview the quantum state is characterized by the prescribed behaviour rather than by the specificvalue a parameter assumes as a result of an interaction.

1. Introduction

A famous paper by Einstein, Podolsky and Rosen (hereafter abbreviated to EPR) [1] thatappeared in 1935 gave rise to a lively debate on the meaning and foundation of quantummechanics which has continued to the present day. EPR pointed out that something must bemissing in the quantum theory, because applying it to a thought experiment they analysedled to a paradox, or at least to a contradiction of the Heisenberg uncertainty principle. Threedecades later, Bell proved his celebrated inequality [2] and, applying it to the EPR experiment,concluded that in some cases quantum mechanics violates it and that its violation would implya breakdown either of the principle of reality or of the principle of locality (provided one wouldnot be inclined to give up causality).

Different variants of the EPR experiment have been considered, such as that proposed byBohm [3] using pairs of spin-12 particles prepared in a singlet state. Finally, in the 1980s, Aspectand his group actually performed a series of EPR-type experiments measuring the polarizationof pairs of photons produced in an SPS atomic cascade [4]. They verified that the result of theexperiments was correctly predicted by quantum theory and that in some configurations Bell’sinequality was actually violated.

All this strengthened the idea that quantum mechanics is indeed paradoxical in characterand revived a debate whose ingredients are mainly the supposed non-locality of quantumphenomena, if not the existence of some sort of superluminal interaction between distantquantum objects.

Now what is rather surprising in this story is that the so-called EPR paradox mainly appearsto be due to a misuse of the Heisenberg principle which should have been seen from the verybeginning. A vague idea of this may be found in the answer by Bohr [5] to EPR’s paper.

The claim that there was no paradox and that its apparent existence was simply aconsequence of an incorrect application of the rules of probability was first made by Accardi [6],who developed an axiomatic approach to what he called quantum probability [7].

In the present paper I shall show why the EPR paradox is not a real paradox and why theresult of Aspect’s experiments does not imply any violation of the principle of locality. In

† E-mail: tartaglia@polito.it

0143-0807/98/030307+05$19.50 © 1998 IOP Publishing Ltd 307

308 A Tartaglia

section 2 the basic properties of a quantum object or system are reviewed. In section 3 theEPR-type experiments are analysed, while section 4 contains a discussion of the meaning ofthe Bell inequality. Finally, section 5 summarizes the whole paper.

2. Quantum objects

A quantum object is something that can exist in a set of different physical states.Mathematically these states can be eigenstates of a number of different operators. Practicallythe effect of any quantum operator corresponds to the interaction with a given apparatus whichinduces the object to enter one of the eigenstates of the corresponding operator.

We know that operators exist which are mutually incompatible in the sense that they donot admit common eigenstates. The commutator of two such operators differs from zero: ifA andB are the two operators, applying them in the orderAB leaves the system in a statedifferent to that in which it is left after applying them in the orderBA.

All this means that when the system is in an eigenstate ofA, the physical quantity associatedwith that operator assumes one of its eigenvalues, saya, but the quantity associated withBis undefined. Conversely, when the system is in an eigenstate ofB the quantity it expressesassumes the eigenvalueb, but now the quantity associated withA is undefined.

This is of course the Heisenberg principle. It is appropriate to stress that there is nothingmysterious, exotic or peculiar to quantum physics in this. It simply states, in a refined andmathematically precise way, the fact that anyone may be standing or seated, but no-one maysimultaneously be standing and seated.

3. The EPR paradox

In their thought experiment EPR consider a pair of particles moving in opposite directions andprepared in such a way that their total momentum is zero. If we measure the momentum ofone of them we know the momentum of the otherwithout measuring it. Now if we determinethe position of the second particle, we end up knowing both the positionand momentum ofone and the same object, thus contradicting the Heisenberg principle.

Actual experiments rather use pairs of objects prepared in a singlet state or photons emittedin an SPS atomic transition, so that their spins are antiparallel (in the first case) or polarizationvectors equal (in the second). Once the two objects are well separated, each of them istested, using an appropriate analyser, for the spin component or polarization state along agiven direction. Now using the EPR argument, if the measurements are performed alongdifferent directions (for instance orthogonal directions) we conclude that the spin componentsor polarization states of one object along different directions can be simultaneously determined,in one case by direct measurement and in the other by a counterfactual but (apparently) soundinference.

The weak point in this is the idea of the ‘element of reality’ (to use EPR’s words)whose value is being tested or inferred. Einstein (and many others after him) thought ofthe experimentally testable properties of a quantum system as being things carried around bythe system: they are objectively there, whether we measure them or not. Now this idea appearsto be in contrast to experience, because of the ‘paradoxes’ it generates.

Actually the situation is different. We may think that a quantum object (which is welldefined and existent, independently from the fact of being under measurement or not) carriesits properties as an ‘instruction leaflet’ telling it what to do when encountering a measuringapparatus or in general undergoing an interaction with something else.

In Aspect’s experiments the two photons, in the act of separating, each receive their‘instructions’ (whose content depends on the way they are prepared). When they meet ananalyser they behave accordingly: realism is safe and Einstein’s locality is also safe. If the

Is the EPR paradox really a paradox? 309

two objects meet two experimental sets of the same kind, so are the results (in the case of spincomponents, they may have the same or opposite sign, depending on the initial preparation).If, however, one of the members of the pair is led to interact with a different type of apparatus,its ‘instruction card’ tells it to behave differently.

Now the counterfactual inference, which in the EPR argument led to the ‘paradox’, tells uswhat the object in question would have done if it had been subjected to an experiment differentfrom the one actually performed; nothing more than this. Our quantum object possessesnothing other than its unique and not contradictory ‘instruction leaflet’: no two incompatiblebehaviours can be adopted at the same time. Thus the Heisenberg principle is safe.

This way of reading things resolves the EPR paradox and has no particular subtlety in it;furthermore it is neither particularly exotic nor confined to the quantum domain.

To express ideas of the same kind as those presented in this paper Accardi uses a nicemetaphor, namely that of chameleons [8]. A chameleon is a well defined non-quantum ‘object’which has the property of assuming the colour of the support on which it rests. Now supposeyou have a pair of twin chameleons, which you separate far from each other, and then putone of them on a leaf: it will become green. We can conclude, without experiment, that thesecond chameleon, if it had been put on a leaf,would have becomegreen too, but not that itis green. Actually, suppose someone puts the second chameleon on a piece of wood: it willbecome brown. No-one, knowing the result of both experiments, would conclude that thesecond chameleon is green and brown at the same time. The point is that the chameleon doesnot possess a colour, but rather the capacity to assume a colour according to the environmenton which it is laid. Everyone will agree: (i) that the chameleons are perfectly real and exist atany moment; (ii) that the phenomenon we verified is purely local; (iii) that there is no logicalcontradiction to be accounted for.

There is no reason that what is true for chameleons (or other classical and macroscopicsystems) cannot be true for electrons or photons. On the contrary, there are good reasons forit also to be true for them.

4. The Bell inequality

The Bell inequality has been considered as a cornerstone in the understanding of the natureand properties of quantum objects and phenomena. This concept has been rather emphaticallyexpressed in the statement by Stapp that Bell’s is ‘the most profound discovery of science’ [9].

Bell originally applied his analysis to the EPR experiment. Perhaps the clearest way toexpound it is to consider Aspect’s version of that experiment. The core of the argument movesfrom the results of three sets of measurements, two actually being performed and the thirdbeing inferred on the basis of the original preparation of the pair of objects being tested. Theprocedure is rather idealized and the physical quantities and features of the experiment are sodevised that the issues of each of the three actual or hypothetical measurements (let us callthema, b andc) may only be±1. Everything then rests on the identity [10]:

a(b − c) ≡ ±(1− bc). (1)

Repeating the experiment many times and then considering statistical averages of theresults, one arrives at

|〈ab〉 − 〈ac〉| 6 1− 〈bc〉 (2)

which is the Bell inequality.Using quantum mechanics to evaluate the averages one sees both by calculation (as Bell

himself did) and by experiment (as done by Aspect and co-workers) that in some situations(2) is violated.

310 A Tartaglia

Now, according to Bell and to many people after him, the only hypotheses underlying theidentity (1) are

(a) realism

(b) determinism

(c) locality.

(3)

Since (1) proves to be false and (2) violated, at least one of the hypotheses, it is said, mustalso be false. The one usually doomed to be abandoned is the third, i.e. locality (often calledEinstein locality). This introduces action at a distance and consequently violates the principleof relativity, forcing people to formulate various kinds of arguments intended to demonstratethat the action at a distance is actually there but cannot really be used to transmit informationat a speed higher than that of light.

Actually, however, there is a fourth hidden (or, more accurately, implicit) hypothesisunderlying Bell’s reasoning: this is the assumption that the property being measured issomething possessed as such by the quantum object and being there at any moment. Thismeans that, even if, say,a andc on the one hand, andb andc on the other cannot be measuredsimultaneously, provided a rule exists enabling us to guess what the result would have been ifc had been measuredinsteadof a (or c insteadof b), we are entitled to use botha andc (or bandc) at the same time in (1).

The situation is different if we think of quantum objects as carrying, so-to-speak, recipebooks rather than cakes, as put in other words in the previous section. A recipe produces acake only when the right kitchen and ingredients are met and it has no meaning to correlatea cake (= the result of a measurement) with a recipe different from that being used. Usingthis metaphor, the identity (1) is now deprived of meaning and so is the ensuing inequality (2).The logic of recipe books is different from that of cakes.

There is nothing strange in this, nor peculiar to the quantum world, as Accardi’s chameleonsshow, and all three conditions (3) are fully satisfied.

5. Conclusion

From the very beginning quantum mechanics subverted deeply rooted ways of thinking andmental habits. This enveloped its phenomena and theory in a halo of mystery, giving rise to adecades-long debate and a host of different interpretations of quantum phenomena calling intoplay free will, the consciousness of the experimenter and things like that. Now that the shockof the subversion of the simple beliefs of nineteenth-century physics is far behind us, a calmre-assessement of some of the ‘strange’ behaviours of quantum objects shows that they areneither really that strange nor confined to the quantum world. An example is the famous EPRparadox, which is indeed no paradox provided we accept that the state of a quantum object isdefined by what I called an ‘instruction leaflet’ or a ‘recipe book’. In other words, the spin orthe polarization are a behaviour exhibited by a quantum object when interacting, rather than a‘colour’ or a label attached to a particle.

Looking at things from this viewpoint, we see that the conditions of realism, determinismand localism are generally satisfied (as we reasonably expect) and that some features of thequantum phenomena may again find their counterpart in the macroscopic domain.

This does not mean that everything is clear and solved with quantum mechanics: themeaning of the wavefunction, its collapse, and the quantization of the objects it describes arestill beyond our imagination and, at least in my case, our understanding. Nonetheless, we mayclaim that part of the mystery is only apparent and comes from a kind of long-lasting prejudice.

In conclusion, quantum measurements and the story of the violation of the Bell or similarinequalities tell us that the objects of the quantum world are not like boxes containing spin,polarization vector, etc, as buttons, pins, pearls and the like, but rather like programmedmachines capable of different behaviours according to the physical conditions locally triggering

Is the EPR paradox really a paradox? 311

them. To recall Accardi’s beautiful metaphor I would say that the quantum particles are‘camleons’ rather than ‘boxons’.

References

[1] Einstein A, Podolsky B and Rosen N 1935Phys. Rev.47777[2] Bell J S 1964Physics1 195[3] Bohm D 1951Quantum Theory(New York: Prentice-Hall)[4] Aspect A, Grangier P and Roger G 1982Phys. Rev. Lett.4991

Aspect A, Dalibard G and Roger G 1982Phys. Rev. Lett.491804[5] Bohr N 1935Phys. Rev.48696[6] Accardi L 1984 The probabilistic roots of the quantum mechanical paradoxesThe Wave–Particle Dualismed

S Dineret al (Dordrecht: Reidel) pp 297–330[7] Accardi L 1982 Foundations of quantum probabilityRend. Sem. Mat. Univ. Polit. Torino(Torino: Levrotto and

Bella) pp 249–70[8] Accardi L 1997Urne e camaleonti(Milano: Il Saggiatore)[9] Stapp H P 1975Nuovo CimentoB 29270

[10] Peres A 1993Quantum Theory: Concepts and Methods(Dordrecht: Kluwer) pp 160–7