In the May 15, 1935 issue ofPhysical Review Albert Einsteinco-authored a paper with his two postdoctoral research associates atthe Institute for Advanced Study, Boris Podolsky and Nathan Rosen. Thearticle was entitled “Can Quantum Mechanical Description ofPhysical Reality Be Considered Complete?” (Einsteinetal. 1935). Generally referred to as “EPR”, this paperquickly became a centerpiece in debates over the interpretation ofquantum theory, debates that continue today. Ranked by impact, EPR isamong the top ten of all papers ever published inPhysicalReview journals. Due to its role in the development of quantuminformation theory, it is also near the top in their list of currently“hot“ papers. The paper features a striking case where twoquantum systems interact in such a way as to link both their spatialcoordinates in a certain direction and also their linear momenta (inthe same direction), even when the systems are widely separated inspace. As a result of this “entanglement”, determiningeither position or momentum for one system would fix (respectively)the position or the momentum of the other. EPR prove a general lemmaconnecting such strict correlations between spatially separatedsystems to the possession of definite values. On that basis they arguethat one cannot maintain both an intuitive condition of local actionand the completeness of the quantum description by means of the wavefunction. This entry describes the lemma and argument of that 1935paper, considers several different versions and reactions, andexplores the ongoing significance of the issues raised.

• 1. Can Quantum Mechanical Description of Physical Reality Be Considered Complete?
  • 1.1 Setting and prehistory
  • 1.2 The argument in the text
  • 1.3 Einstein’s versions of the argument
• 2. A popular form of the argument: Bohr’s response
• 3. Development of EPR
  • 3.1 Spin and The Bohm version
  • 3.2 Bell and beyond
• Bibliography
• Academic Tools
• Other Internet Resources
• Related Entries

1. Can Quantum Mechanical Description of Physical Reality Be Considered Complete?

1.1 Setting and prehistory

By 1935 conceptual understanding of the quantum theory was dominatedby Niels Bohr’s ideas concerning complementarity. Those ideas centeredon observation and measurement in the quantum domain. According toBohr’s views at that time, observing a quantum object involves anuncontrollable physical interaction with a measuring device thataffects both systems. The picture here is of a tiny object banginginto a big apparatus. The effect this produces on the measuringinstrument is what issues in the measurement “result”which, because it is uncontrollable, can only be predictedstatistically. The effect experienced by the quantum object limitswhat other quantities can be co-measured with precision. According tocomplementarity when we observe the position of an object, we affectits momentum uncontrollably. Thus we cannot determine both positionand momentum precisely. A similar situation arises for thesimultaneous determination of energy and time. Thus complementarityinvolves a doctrine of uncontrollable physical interaction that,according to Bohr, underwrites the Heisenberg uncertainty relationsand is also the source of the statistical character of the quantumtheory. (See the entries on theCopenhagen Interpretation and theUncertainty Principle.)

Initially Einstein was enthusiastic about the quantum theory. By 1935,however, while recognizing the theory’s significant achievements, hisenthusiasm had given way to disappointment. His reservations weretwofold. Firstly, he felt the theory had abdicated the historical taskof natural science to provide knowledge of significant aspects ofnature that are independent of observers or their observations.Instead the fundamental understanding of the quantum wave function(alternatively, the “state function”, “statevector”, or “psi-function”) was that it only treatedthe outcomes of measurements (via probabilities given by the BornRule). The theory was simply silent about what, if anything, waslikely to be true in the absence of observation. That there could belaws, even probabilistic laws, for finding things if one looks, but nolaws of any sort for how things are independently of whether onelooks, marked quantum theory as irrealist. Secondly, the quantumtheory was essentially statistical. The probabilities built into thestate function were fundamental and, unlike the situation in classicalstatistical mechanics, they were not understood as arising fromignorance of fine details. In this sense the theory wasindeterministic. Thus Einstein began to probe how strongly the quantumtheory was tied to irrealism and indeterminism.

He wondered whether it was possible, at least in principle, to ascribecertain properties to a quantum system in the absence of measurement.Can we suppose, for instance, that the decay of an atom occurs at adefinite moment in time even though such a definite decay time is notimplied by the quantum state function? That is, Einstein began to askwhether the formalism provides a description of quantum systems thatis complete. Can all physically relevant truths about systems bederived from quantum states? One can raise a similar question about alogical formalism: are all logical truths (or semantically validformulas) derivable from the axioms. Completeness, in this sense, wasa central focus for the Göttingen school of mathematical logicassociated with David Hilbert. (See entry on Hilbert’s Program.) Werner Heisenberg, who had attended Hilbert’s lectures, picked upthose concerns with questions about the completeness of his own,matrix approach to quantum mechanics. In response, Bohr (and otherssympathetic to complementarity) made bold claims not just for thedescriptive adequacy of the quantum theory but also for its“finality”, claims that enshrined the features ofirrealism and indeterminism that worried Einstein. (See Beller 1999,Chapters 4 and 9, on the rhetoric of finality and Ryckman 2017,Chapter 4, for the connection to Hilbert.) Thus complementarity becameEinstein’s target for investigation. In particular, Einstein hadreservations about the uncontrollable physical effects invoked by Bohrin the context of measurement interactions, and about their role infixing the interpretation of the wave function. EPR’s focus oncompleteness was intended to support those reservations in aparticularly dramatic way.

Max Jammer (1974, pp. 166–181) locates the development of theEPR paper in Einstein’s reflections on a thought experiment heproposed during discussions at the 1930 Solvay conference. (For moreon EPR and Solvay 1930 see Howard, 1990 and Ryckman, 2017, pp.118–135.) The experiment imagines a box that contains a clockset to time precisely the release (in the box) of a photon withdeterminate energy. If this were feasible, it would appear tochallenge the unrestricted validity of the Heisenberg uncertaintyrelation that sets a lower bound on the simultaneous uncertainty ofenergy and time. (See the entry on theUncertainty Principle and also Bohr 1949, who describes the discussions at the 1930conference.) The uncertainty relations, understood not just as aprohibition on what is co-measurable, but on what is simultaneouslyreal, were a central component in the irrealist interpretation of thewave function. Jammer (1974, p. 173) describes how Einstein’s thinkingabout this experiment, and Bohr’s objections to it, evolved into adifferent photon-in-a-box experiment, one that allows an observer todetermine either the momentum or the position of the photonindirectly, while remaining outside, sitting on the box. Jammerassociates this with the distant determination of either momentum orposition that, we shall see, is at the heart of the EPR paper. CarstenHeld (1998) cites a relatedcorrespondence with Paul Ehrenfest from 1932 in which Einstein described an arrangement for the indirectmeasurement of a particle of massm using correlations with aphoton established through Compton scattering. Einstein’s reflectionshere foreshadow the argument of EPR, along with noting some of itsdifficulties.

Thus without an experiment onm it is possible to predictfreely, at will,either the momentumor the positionofm with, in principle, arbitrary precision. This is thereason why I feel compelled to ascribe objective reality toboth. I grant, however, that it is not logically necessary.(Held 1998, p. 90)

Whatever their precursors, the ideas that found their way into EPRwere discussed in a series of meetings between Einstein and his twoassistants, Podolsky and Rosen. Podolsky was commissioned to composethe paper and he submitted it toPhysical Review in March of1935, where it was sent for publication the day after it arrived.Apparently Einstein never checked Podolsky’s draft before submission.He was not pleased with the result. Upon seeing the published version,Einstein complained that it obscured his central concerns.

For reasons of language this [paper] was written by Podolsky afterseveral discussions. Still, it did not come out as well as I hadoriginally wanted; rather, the essential thing was, so to speak,smothered by formalism [Gelehrsamkeit]. (Letter from Einstein to ErwinSchrödinger, June 19, 1935. In Fine 1996, p. 35.)

Unfortunately, without attending to Einstein’s reservations, EPR isoften cited to evoke the authority of Einstein. Here we willdistinguish the argument Podolsky laid out in the text from lines ofargument that Einstein himself published in articles from 1935 on. Wewill also consider the argument presented in Bohr’s reply to EPR,which is possibly the best known version, although it differs from theothers in important ways.

1.2 The argument in the text

The EPR text is concerned, in the first instance, with the logicalconnections between two assertions. One asserts that quantum mechanicsis incomplete. The other asserts that incompatible quantities (thosewhose operators do not commute, like thex-coordinate ofposition and linear momentum in directionx) cannot havesimultaneous “reality” (i.e., simultaneously real values).The authors assert the disjunction of these as a first premise (laterto be justified): one or another of these must hold. It follows thatif quantum mechanics were complete (so that the first assertionfailed) then the second one would hold; i.e., incompatible quantitiescannot have real values simultaneously. They take as a second premise(also to be justified) that if quantum mechanics were complete, thenincompatible quantities (in particular coordinates of position andmomentum) could indeed have simultaneous, real values. They concludethat quantum mechanics is incomplete. The conclusion certainly followssince otherwise (if the theory were complete) one would have acontradiction over simultaneous values. Nevertheless the argument ishighly abstract and formulaic and even at this point in itsdevelopment one can readily appreciate Einstein’s disappointment.

EPR now proceed to establish the two premises, beginning with adiscussion of the idea of a complete theory. Here they offer only anecessary condition; namely, that for a complete theory “everyelement of the physical reality must have a counterpart in thephysical theory.” The term “element“ may remind oneof Mach, for whom this was a central, technical term connected tosensations. (See the entry onErnst Mach.) The use in EPR of elements of reality is also technical butdifferent. Although they do not define an “element of physicalreality” explicitly (and, one might note, the language ofelements is not part of Einstein’s usage elsewhere), that expressionis used when referring to the values of physical quantities(positions, momenta, and so on) that are determined by an underlying“real physical state”. The picture is that quantum systemshave real states that assign values to certain quantities. SometimesEPR describe this by saying the quantities in question have“definite values”, sometimes “there exists anelement of physical reality corresponding to the quantity”.Suppose we adapt the simpler terminology and call a quantity on asystemdefinite if that quantity has a definite value; i.e.,if the real state of the system assigns a value (an “element ofreality”) to the quantity. The relation that associates realstates with assignments of values to quantities is functional so thatwithout a change in the real state there is no change among valuesassigned to quantities. In order to get at the issue of completeness,a primary question for EPR is to determine when a quantity has adefinite value. For that purpose they offer a minimal sufficientcondition (p. 777):

If, without in any way disturbing a system, we can predict withcertainty (i.e., with probability equal to unity) the value of aphysical quantity, then there exists an element of realitycorresponding to that quantity.

This sufficient condition for an “element of reality” isoften referred to as the EPRCriterion of Reality. By way of illustration EPR point to those quantities for which thequantum state of the system is an eigenstate. It follows from theCriterion that at least these quantities have a definite value;namely, the associated eigenvalue, since in an eigenstate thecorresponding eigenvalue has probability one, which we can determine(predict with certainty) without disturbing the system. In fact,moving from eigenstate to eigenvalue to fix a definite value is theonly use of the Criterion in EPR.

With these terms in place it is easy to show that if, say, the valuesof position and momentum for a quantum system were definite (wereelements of reality) then the description provided by the wavefunction of the system would be incomplete, since no wave functioncontains counterparts for both elements. Technically, no statefunction—even an improper one, like a delta function—is asimultaneous eigenstate for both position and momentum; indeed, jointprobabilities for position and momentum are not well-defined in anyquantum state. Thus they establish the first premise: either quantumtheory is incomplete or there can be no simultaneously real(“definite”) values for incompatible quantities. They nowneed to show that if quantum mechanics were complete, thenincompatible quantities could have simultaneous real values, which isthe second premise. This, however, is not easily established. Indeedwhat EPR proceed to do is odd. Instead of assuming completeness and onthat basis deriving that incompatible quantities can have real valuessimultaneously, they simply set out to derive the latter assertionwithout any completeness assumption at all. This“derivation” turns out to be the heart of the paper andits most controversial part. It attempts to show that in certaincircumstances a quantum system can have simultaneous values forincompatible quantities (once again, for position and momentum), wherethese are definite values; that is, they are assigned by the realstate of the system, hence are “elements of reality”.

They proceed by sketching an iconic thought experiment whosevariations continue to be important and widely discussed. Theexperiment concerns two quantum systems that are spatially distantfrom one another, perhaps quite far apart, but such that the totalwave function for the pair links both the positions of the systems aswell as their linear momenta. In the EPR example the total linearmomentum is zero along thex-axis. Thus if the linearmomentum of one of the systems (we can call it Albert’s) along thex-axis were found to bep, thex-momentumof the other system (call it Niels’) would be found to be−p. At the same time their positions alongxare also strictly correlated so that determining the position of onesystem on thex-axis allows us to infer the position of theother system alongx. The paper constructs an explicit wavefunction for the combined (Albert+Niels) system that embodies theselinks even when the systems are widely separated in space. Althoughcommentators later raised questions about the legitimacy of this wavefunction, it does appear to guarantee the required correlations forspatially separated systems, at least for a moment (Jammer 1974, pp.225–38; see also Halvorson 2000). In any case, one can model thesame conceptual situation in other cases that are clearly well definedquantum mechanically (seeSection 3.1).

At this point of the argument (p. 779) EPR make two criticalassumptions, although they do not call special attention to them. (Forthe significance of these assumptions in Einstein’s thinking seeHoward 1985 and also section 5 of the entry onEinstein.) The first assumption (separability) is that at the time whenthe systems are separated, maybe quite far apart, each has its ownreality. In effect, they assume that each system maintains a separateidentity characterized by a real physical state, even though eachsystem is also strictly correlated with the other in respect both tomomentum and position. They need this assumption to make sense ofanother. The second assumption is that oflocality. Giventhat the systems are far apart, locality supposes that “no realchange can take place” in one system as a direct consequence ofa measurement made on the other system. They gloss this by saying“at the time of measurement the two systems no longerinteract.” Note that locality does not require that nothing atall about one system can be disturbed directly by a distantmeasurement on the other system. Locality only rules out that adistant measurement may directly disturb or change what is counted as“real“ with respect to a system, a reality thatseparability guarantees. On the basis of these two assumptions theyconclude that each system can have definite values (“elements ofreality”) for both position and momentum simultaneously. Thereis no straightforward argument for this in the text. Instead they usethese two assumptions to show how one could be led to assign positionand momentum eigenstates to one system by making measurements on theother system, from which the simultaneous attribution of elements ofreality is supposed to follow. Since this is the central and mostcontroversial part of the paper, it pays to go slowly here in tryingto reconstruct an argument on their behalf.

Here is one attempt. (Dickson 2004 analyzes some of the modalprinciples involved and suggests one line of argument, which hecriticizes. Hooker 1972 is a comprehensive discussion that identifiesseveral generically different ways to make the case.) Locality affirmsthat the real state of a system is not affected by distantmeasurements. Since the real state determines which quantities aredefinite (i.e., have assigned values), the set of definite quantitiesis also not affected by distant measurements. So if by measuring adistant partner we can determine that a certain quantity is definite,then that quantity must have been definite all along. As we have seen,theCriterion of Reality implies that a quantity is definite if the state of the system is aneigenstate for that quantity. In the case of the strict correlationsof EPR, measuring one system triggers a reduction of the joint statethat results in an eigenstate for the distant partner. Hence anyquantity with that eigenstate is definite. For example, sincemeasuring the momentum of Albert’s system results in a momentumeigenstate for Niels’, the momentum of Niels’ system is definite.Likewise for the position of Niels’ system. Given separability, thecombination of locality and the Criterion establish a quite generallemma; namely,when quantities on separated systems have strictlycorrelated values, those quantities are definite. Thus the strictcorrelations between Niels’ system and Albert’s in the EPR situationguarantee that both position and momentum are definite; i. e., thateach system has definite position and momentum simultaneously.

EPR point out that position and momentum cannot be measuredsimultaneously. So even if each can be shown to be definite indistinct contexts of measurement, can both be definite at the sametime? The lemma answers “yes”. What drives the argument islocality, which functions logically to decontextualize the reality ofNiels’ system from goings on at Albert’s. Accordingly, measurementsmade on Albert’s system are probative for features corresponding tothe real state of Niels’ system but not determinative of them. Thuseven without measuring Albert’s system, features corresponding to thereal state of Niels’ system remain in place. Among those features area definite position and a definite momentum for Niels’ system alongsome particular coordinate direction.

In the penultimate paragraph of EPR (p. 780) they address the problemof getting real values for incompatible quantities simultaneously.

Indeed one would not arrive at our conclusion if one insisted that twoor more physical quantities can be regarded as simultaneous elementsof reality only when they can be simultaneously measured or predicted.… This makes the reality [on the second system] depend upon theprocess of measurement carried out on the first system, which does notin any way disturb the second system. No reasonable definition ofreality could be expected to permit this.

The unreasonableness to which EPR allude in making “the reality[on the second system] depend upon the process of measurement carriedout on the first system, which does not in any way disturb the secondsystem” is just the unreasonableness that would be involved inrenouncing locality understood as above. For it is locality thatenables one to overcome the incompatibility of position and momentummeasurements of Albert’s system by requiring their joint consequencesfor Niels’ system to be incorporated in a single, stable realitythere. If we recallEinstein’s acknowledgment to Ehrenfest that getting simultaneous position and momentum was “notlogically necessary”, we can see how EPR respond by making itbecome necessary once locality is assumed.

Here, then, are the key features of EPR.

• EPR is about the interpretation of state vectors (“wavefunctions”) and employs the standard state vector reductionformalism (von Neumann’s “projection postulate”).
• The Criterion of Reality affirms that the eigenvalue corresponding to the eigenstate of asystem is a value determined by the real physical state of thatsystem. (This is the Criterion’s only use.)
• (Separability) Spatially separated systems have real physical states.
• (Locality) If systems are spatially separate, the measurement (or absence ofmeasurement) of one system does not directly affect the reality thatpertains to the others.
• (EPR Lemma) If quantities on separated systems have strictlycorrelated values, those quantities are definite (i.e., have definitevalues). This follows from separability, locality and the Criterion.No actual measurements are required.
• (Completeness) If the description of systems by state vectors were complete, thendefinite values of quantities (values determined by the real state ofa system) could be inferred from a state vector for the system itselfor from a state vector for a composite of which the system is apart.
• In summary, separated systems as described by EPR have definiteposition and momentum values simultaneously. Since this cannot beinferred from any state vector, the quantum mechanical description ofsystems by means state vectors is incomplete.

The EPR experiment with interacting systems accomplishes a form ofindirect measurement. The direct measurement of Albert’s systemyields information about Niels’ system; it tells us what wewould find if we were to measure there directly. But it does thisat-a-distance, without any physical interaction taking place betweenthe two systems. Thus the thought experiment at the heart of EPRundercuts the picture of measurement as necessarily involving a tinyobject banging into a large measuring instrument. If we look back atEinstein’s reservations about complementarity, we can appreciatethat by focusing on an indirect, non-disturbing kind of measurementthe EPR argument targets Bohr’s program for explaining centralconceptual features of the quantum theory. For that program relied onuncontrollable interaction with a measuring device as a necessaryfeature of any measurement in the quantum domain. Nevertheless thecumbersome machinery employed in the EPR paper makes it difficult tosee what is central. It distracts from rather than focuses on theissues. That was Einstein’s complaint about Podolsky’stext in his June 19, 1935 letter to Schrödinger.Schrödinger responded on July 13 reporting reactions to EPR thatvindicate Einstein’s concerns. With reference to EPR hewrote:

I am now having fun and taking your note to its source to provoke themost diverse, clever people: London, Teller, Born, Pauli, Szilard,Weyl. The best response so far is from Pauli who at least admits thatthe use of the word “state” [“Zustand”] forthe psi-function is quite disreputable. What I have so far seen by wayof published reactions is less witty. … It is as if one personsaid, “It is bitter cold in Chicago”; and anotheranswered, “That is a fallacy, it is very hot in Florida.”(Fine 1996, p. 74)

1.3 Einstein’s versions of the argument

If the argument developed in EPR has its roots in the 1930 Solvayconference, Einstein’s own approach to issues at the heart of EPR hasa history that goes back to the 1927 Solvay conference. (Bacciagaluppiand Valentini 2009, pp. 198–202, would even trace it back to1909 and the localization of light quanta.) At that 1927 conferenceEinstein made a short presentation during the general discussionsession where he focused on problems of interpretation associated withthe collapse of the wave function. He imagines a situation whereelectrons pass through a small hole and are dispersed uniformly in thedirection of a screen of photographic film shaped into a largehemisphere that surrounds the hole. On the supposition that quantumtheory offers a complete account of individual processes then, in thecase of localization, why does the whole wave front collapse to justone single flash point? It is as though at the moment of collapse aninstantaneous signal were sent out from the point of collapse to allother possible collapse positions telling them not to flash. ThusEinstein maintains (Bacciagaluppi and Valentini 2009, p. 488),

the interpretation, according to which |ψ|² expresses theprobability thatthisparticle is found at a given point,assumes an entirely peculiar mechanism of action at a distance, whichprevents the wave continuously distributed in space from producing anaction in two places on the screen.

One could see this as a tension between local action and thedescription afforded by the wave function, since the wave functionalone does not specify a unique position on the screen for detectingthe particle. Einstein continues,

In my opinion, one can remove this objection only in the followingway, that one does not describe the process solely by theSchrödinger wave, but that at the same time one localizes theparticle during propagation.

In fact Einstein himself had tried this very route in May of 1927where he proposed a way of “localizing the particle” byassociating spatial trajectories and velocities with particlesolutions to the Schrödinger equation. (See Belousek 1996 andHolland 2005; also Ryckman 2017.) Einstein abandoned the project andwithdrew the draft from publication, however, after finding thatcertain intuitive independence conditions were in conflict with theproduct wave function used by quantum mechanics to treat thecomposition of independent systems. The problem here anticipates themore general issues raised by EPR over separability and compositesystems. This proposal was Einstein’s one and only flirtation with theintroduction of hidden variables into the quantum theory. In thefollowing years he never embraced any proposal of that sort, althoughhe hoped for progress in physics to yield a more complete theory, andone where the observer did not play a fundamental role. “Webelieve however that such a theory [“a complete description ofthe physical reality”] is possible” (p. 780). Commentatorshave often mistaken that remark as indicating Einstein’s predilectionfor hidden variables. To the contrary, after 1927 Einstein regardedthe hidden variables project — the project of developing a morecomplete theory by starting with the existing quantum theory andadding things, like trajectories or real states — an improbableroute to that goal. (See, for example, Einstein 1953a.) To improve onthe quantum theory, he thought, would require starting afresh withquite different fundamental concepts. At Solvay he acknowledges Louisde Broglie’s pilot wave investigations as a possible direction topursue for a more complete account of individual processes. But thenhe quickly turns to an alternative way of thinking, one that hecontinued to recommend as a better framework for progress, which isnot to regard the quantum theory as describing individuals and theirprocesses at all and, instead, to regard the theory as describing onlyensembles of individuals. Einstein goes on to suggest difficulties forany scheme, like de Broglie’s and like quantum theory itself, thatrequires representations in multi-dimensional configuration space.These are difficulties that might move one further toward regardingquantum theory as not aspiring to a description of individual systemsbut as more amenable to an ensemble (or collective) point of view, andhence not a good starting point for building a better, more completetheory. His subsequent elaborations of EPR-like arguments are perhapsbest regarded asno-go arguments, showing that the existingquantum theory does not lend itself to a sensible realistinterpretation via hidden variables. If real states, taken as hiddenvariables, are added into the existing theory, which is then tailoredto explain individual events, the result is either an incompletetheory or else a theory that does not respect locality. Hence, newconcepts are needed. With respect to EPR, perhaps the most importantfeature of Einstein’s reflections at Solvay 1927 is his insight that aclash between completeness and locality already arises in consideringa single variable (there, position) and does not require anincompatible pair, as in EPR.

Following the publication of EPR Einstein set about almost immediatelyto provide clear and focused versions of the argument. He began thatprocess within few weeks of EPR, in the June 19 letter toSchrödinger, and continued it in an article published thefollowing year (Einstein 1936). He returned to this particular form ofan incompleteness argument in two later publications (Einstein 1948and Schilpp 1949). Although these expositions differ in details theyall employ composite systems as a way of implementing indirectmeasurements-at-a-distance. None of Einstein’s accounts contains theCriterion of Reality nor the tortured EPR argument over when values of a quantity can beregarded as “elements of reality”. The Criterion and these“elements” simply drop out. Nor does Einstein engage incalculations, like those of Podolsky, to fix the total wave functionfor the composite system explicitly. Unlike EPR, none of Einstein’sarguments makes use of simultaneous values for complementaryquantities like position and momentum. He does not challenge theuncertainty relations. Indeed with respect to assigning eigenstatesfor a complementary pair he tells Schrödinger “ist mirwurst”—literally, it’s sausage to me; i.e., he couldn’tcare less. (Fine 1996, p. 38). These writings probe an incompatibilitybetween affirming locality and separability, on the one hand, andcompleteness in the description of individual systems by means ofstate functions, on the other. His argument is that we can have atmost one of these but never both. He frequently refers to this dilemmaas a “paradox”.

In the letter to Schrödinger of June 19, Einstein points to asimple argument for the dilemma which, like the argument from the 1927Solvay Conference, involves only the measurement of a single variable.Consider an interaction between the Albert and Niels systems thatestablishes a strict correlation between their positions. (We need notworry about momentum, or any other quantity.) Consider the evolvedwave function for the total (Albert+Niels) system when the two systemsare far apart. Now assume a principle of locality-separability(Einstein calls it aTrennungsprinzip—separationprinciple): Whether a determinate physical situation holds for Niels’system (e.g., that a quantity has a particular value) does not dependon what measurements (if any) are made locally on Albert’s system. Ifwe measure the position of Albert’s system, the strict correlation ofpositions implies that Niels’ system has a certain position. Bylocality-separability it follows that Niels’ system must already havehad that position just before the measurement on Albert’s system. Atthat time, however, Niels’ system alone does not have a statefunction. There is only a state function for the combined system andthat total state function does not single out an existing position forNiels’ system (i.e., it is not a product one of whose factors is aneigenstate for the position of Niels’ system). Thus the description ofNiels’ system afforded by the quantum state function is incomplete. Acomplete description would say (definitely yes) if a quantity ofNiels’ system had a certain value. (Notice that this argument does noteven depend on the reduction of the total state function for thecombined system.) In this formulation of the argument it is clear thatlocality-separability conflicts withthe eigenvalue-eigenstate link, which holds that a quantity of a system has a value if and only ifthe state of the system is an eigenstate (or a proper mixture ofeigenstates) of that quantity with that value as eigenvalue. The“only if” part of the link would need to be weakened inorder to interpret quantum state functions as complete descriptions.(See the entry onModal Interpretations and see Gilton 2016 for a history of the eigenvalue-eigenstatelink.)

This argument rests on the ordinary and intuitive notion ofcompleteness as not omitting relevant truths. Thus, in the argument,the description given by the state function of a system is judgedincomplete when it fails to attribute a position to the system incircumstances where the system indeed has a position. Although thissimple argument concentrates on what Einstein saw as the essentials,stripping away most technical details and distractions, he frequentlyused another argument involving more than one quantity. (It isactually buried in the EPR paper, p. 779, and a version also occurs inthe June 19, 1935 letter to Schrödinger. Harrigan and Spekkens,2010 suggest reasons for preferring a many-variables argument.) Thissecond argument focuses clearly on the interpretation of quantum statefunctions in terms of “real states” of a system, and noton any issues about simultaneous values (real or not) forcomplementary quantities. It goes like this.

Suppose, as in EPR, that the interaction between the two systems linksposition and also linear momentum, and that the systems are far apart.As before, we can measure either the position or the momentum ofAlbert’s system and, in either case, we can infer (respectively) aposition or a momentum for Niels’ system. It follows from thereduction of the total state function that, depending on whether wemeasure the position or the momentum of Albert’s system, Niels’ systemwill be left (respectively) either in a position eigenstate or in amomentum eigenstate. Suppose too that separability holds, so thatNiels’ system has some real physical state of affairs. If localityholds as well, then the measurement of Albert’s system does notdisturb the assumed “reality” for Niels’ system. However,that reality appears to be represented by quite different statefunctions, depending on which measurement of Albert’s system onechooses to carry out. If we understand a “completedescription” to rule out that one and the same physical statecan be described by state functions with distinct physicalimplications, then we can conclude that the quantum mechanicaldescription is incomplete. Here again we confront a dilemma betweenseparability-locality and completeness. Many years later Einstein putit this way (Schilpp 1949, p. 682);

[T]he paradox forces us to relinquish one of the following twoassertions:
(1) the description by means of the psi-function is complete
(2) the real states of spatially separate objects are independent ofeach other.

It appears that the central point of EPR was to argue that anyinterpretation of quantum state functions that attributes realphysical states to systems faces these alternatives. It also appearsthat Einstein’s different arguments make use of different notions ofcompleteness. In the first argument completeness is an ordinary notionthat amounts to not leaving out any relevant details. In the second,completeness is a technical notion which has been dubbed“bijective completeness“ (Fine 1996 ): no more than onequantum state should correspond to a real state. These notions areconnected. If completeness fails in the bijective sense, and more thanone quantum state corresponds to some real state, we can argue thatthe ordinary notion of completeness also fails. For distinct quantumstates will differ in the values they assign to certain quantities.(For example, the observable corresponding to the projector on a statetakes value 1 in one case but not in the other.) Hence each will omitsomething that the other affirms, so completeness in the ordinarysense will fail. Put differently, ordinary completeness impliesbijective completeness. (The converse is not true. Even if thecorrespondence of quantum states to real states were one-to-one, thedescription afforded by a quantum state might still leave out somephysically relevant fact about its corresponding real state.) Thus adilemma between locality and “completeness“ in Einstein’sversions of the argument still implicates ordinary completeness. Forif locality holds, then his two-variable argument shows that bijectivecompleteness fails, and then completeness in the ordinary sense failsas well.

As we have seen, in framing his own EPR-like arguments for theincompleteness of quantum theory, Einstein makes use ofseparability andlocality, which are also tacitly assumed in the EPR paper. Using the languageof “independent existence“ he presents these ideas clearlyin an article that he sent to Max Born (Einstein 1948).

It is … characteristic of … physical objects that theyare thought of as arranged in a space-time continuum. An essentialaspect of this arrangement … is that they lay claim, at acertain time, to an existence independent of one another, providedthese objects “are situated in different parts of space”.… The following idea characterizes the relative independence ofobjects (A and B) far apart in space: external influence on A has nodirect influence on B. (Born, 1971, pp. 170–71)

In the course of his correspondence with Schrödinger, however,Einstein realized that assumptions about separability and localitywere not necessary in order to get the incompleteness conclusion thathe was after; i.e., to show that state functions may not provide acomplete description of the real state of affairs with respect to asystem. Separability supposes that there is a real state of affairsand locality supposes that one cannot influence it immediately byacting at a distance. What Einstein realized was that separability wasalready part of the ordinary conception of a macroscopic object. Thissuggested to him that if one looks at the local interaction of amacro-system with a micro-system one could avoid having to assumeeither separability or locality in order to conclude that the quantumdescription of the whole was incomplete with respect to itsmacroscopic part.

This line of thought evolves and dominates over problems withcomposite systems and locality in his last published reflections onincompleteness. Instead he focuses on problems with the stability ofmacro-descriptions in the transition to a classical level from thequantum.

the objective describability of individual macro-systems (descriptionof the “real-state”) can not be renounced without thephysical picture of the world, so to speak, decomposing into a fog.(Einstein 1953b, p. 40. See also Einstein 1953a.)

In the August 8, 1935 letter to Schrödinger Einstein says that hewill illustrate the problem by means of a “crude macroscopicexample”.

The system is a substance in chemically unstable equilibrium, perhapsa charge of gunpowder that, by means of intrinsic forces, canspontaneously combust, and where the average life span of the wholesetup is a year. In principle this can quite easily be representedquantum-mechanically. In the beginning the psi-function characterizesa reasonably well-defined macroscopic state. But, according to yourequation [i.e., the Schrödinger equation], after the course of ayear this is no longer the case. Rather, the psi-function thendescribes a sort of blend of not-yet and already-exploded systems.Through no art of interpretation can this psi-function be turned intoan adequate description of a real state of affairs; in reality thereis no intermediary between exploded and not-exploded. (Fine 1996, p.78)

The point is that after a year either the gunpowder will haveexploded, or not. (This is the “real state” which in theEPR situation requires one to assume separability.) The statefunction, however, will have evolved into a complex superposition overthese two alternatives. Provided we maintain the eigenvalue-eigenstatelink, the quantum description by means of that state function willyield neither conclusion, and hence the quantum description isincomplete. For a contemporary response to this line of argument, onemight look to the program of decoherence. (SeeDecoherence.) That program points to interactions with the environment which mayquickly reduce the likelihood of any interference between the“exploded” and the “not-exploded” branches ofthe evolved psi-function. Then, breaking the eigenvalue-eigenstatelink, decoherence adopts a perspective according to which the (almost)non-interfering branches of the psi-function allow that the gunpowderis indeed either exploded or not. Even so, decoherence fails toidentify which alternative is actually realized, leaving the quantumdescription still incomplete. Such decoherence-based interpretationsof the psi-function are certainly “artful”, and theiradequacy is still under debate (see Schlosshauer 2007, especiallyChapter 8).

The reader may recognize the similarity between Einstein’sexploding gunpowder example and Schrödinger’s cat (Schrödinger 1935a, p. 812). In thecase of the cat an unstable atom is hooked up to a lethal device that,after an hour, is as likely to poison (and kill) the cat as not,depending on whether the atom decays. After an hour the cat is eitheralive or dead, but the quantum state of the whole atom-poison-catsystem at this time is a superposition involving the two possibilitiesand, just as in the case of the gunpowder, is not a completedescription of the situation (life or death) of the cat. Thesimilarity between the gunpowder and the cat is hardly accidentalsince Schrödinger first produced the cat example in his reply ofSeptember 19, 1935 to Einstein’s August 8 gunpowder letter. ThereSchrödinger says that he has himself constructed “anexample very similar to your exploding powder keg”, and proceedsto outline the cat (Fine 1996, pp. 82–83). Although the“cat paradox” is usually cited in connection with theproblem of quantum measurement (see the relevant section of the entryonPhilosophical Issues in Quantum Theory) and treated as a paradox separate from EPR, its origin is here as anargument for incompleteness that avoids the twin assumptions ofseparability and locality. Schrödinger’s development of“entanglement”, the term he introduced for thecorrelations that result when quantum systems interact, also began inthis correspondence over EPR — along with a treatment of what hecalled quantum “steering” (Schrödinger 1935a, 1935b;seeQuantum Entanglement and Information).

2. A popular form of the argument: Bohr’s response

The literature surrounding EPR contains yet another version of theargument, a popular version that—unlike any ofEinstein’s—features theCriterion of Reality. Assume again an interaction between our two systems linking theirpositions and their linear momenta and suppose that the systems arefar apart. If we measure the position of Albert’s system, we can inferthat Niels’ system has a corresponding position. We can also predictit with certainty, given the result of the position measurement ofAlbert’s system. Hence, in this version, the Criterion of Reality istaken to imply that the position of Niels’ system constitutes anelement of reality. Similarly, if we measure the momentum of Albert’ssystem, we can conclude that the momentum of Niels’ system is anelement of reality. The argument now concludes that since we canchoose freely to measure either position or momentum, it“follows” that both must be elements of realitysimultaneously.

Of course no such conclusion follows from our freedom of choice. It isnot sufficient to be able to choose at will which quantity to measure;for the conclusion to follow from the Criterion alone one would needto be able to measure both quantities at once. This is precisely thepoint that Einstein recognized in his1932 letter to Ehrenfest and that EPR addresses by assuming locality and separability. What isstriking about this version is that these principles, central to theoriginal EPR argument and to the dilemma at the heart of Einstein’sversions, are obscured here. Instead this version features theCriterion and those “elements of reality”. Perhaps thedifficulties presented by Podolsky’s text contribute to this reading.In any case, in the physics literature this version is commonly takento represent EPR and usually attributed to Einstein. This readingcertainly has a prominent source in terms of which one can understandits popularity among physicists; it is Niels Bohr himself.

By the time of the EPR paper many of the early interpretive battlesover the quantum theory had been settled, at least to the satisfactionof working physicists. Bohr had emerged as the“philosopher” of the new theory and the community ofquantum theorists, busy with the development and extension of thetheory, were content to follow Bohr’s leadership when it came toexplaining and defending its conceptual underpinnings (Beller 1999,Chapter 13). Thus in 1935 the burden fell to Bohr to explain what waswrong with the EPR “paradox”. The major article that hewrote in discharging this burden (Bohr 1935a) became the canon for howto respond to EPR. Unfortunately, Bohr’s summary of EPR in thatarticle, which is the version just above, also became the canon forwhat EPR contained by way of argument.

Bohr’s response to EPR begins, as do many of his treatments of theconceptual issues raised by the quantum theory, with a discussion oflimitations on the simultaneous determination of position andmomentum. As usual, these are drawn from an analysis of thepossibilities of measurement if one uses an apparatus consisting of adiaphragm connected to a rigid frame. Bohr emphasizes that thequestion is to what extent can we trace the interaction between theparticle being measured and the measuring instrument. (See Beller1999, Chapter 7 for a detailed analysis and discussion of the“two voices” contained in Bohr’s account. See tooBacciagaluppi 2015.) Following the summary of EPR, Bohr (1935a, p.700) then focuses on the Criterion of Reality which, he says,“contains an ambiguity as regards the meaning of the expression‘without in any way disturbing a system’.” Bohragrees that in the indirect measurement of Niels’ system achieved whenone makes a measurement of Albert’s system “there is no questionof a mechanical disturbance” of Niels’ system. Still, Bohrclaims that a measurement on Albert’s system does involve “aninfluence on the very conditions which define the possible types ofpredictions regarding the future behavior of [Niels’] system.”The meaning of this claim is not at all clear. Indeed, in revisitingEPR fifteen years later, Bohr would comment,

Rereading these passages, I am deeply aware of the inefficiency ofexpression which must have made it very difficult to appreciate thetrend of the argumentation (Bohr 1949, p. 234).

Unfortunately, Bohr takes no notice there of Einstein’s later versionsof the argument and merely repeats his earlier response to EPR. Inthat response, however inefficiently, Bohr appears to be directingattention to the fact that when we measure, for example, the positionof Albert’s system conditions are in place for predicting the positionof Niels’ system but not its momentum. The opposite would be true inmeasuring the momentum of Albert’s system. Thus his “possibletypes of predictions” concerning Niels’ system appear tocorrespond to which variable we measure on Albert’s system. Bohrproposes then to block the EPR Criterion by counting, say, theposition measurement of Albert’s system as an “influence”on the distant system of Niels. If we assume it is an influence thatdisturbs Niels’ system, then the Criterion could not be used, as inBohr’s version of the argument, in producing an element of reality forNiels’ system that challenges completeness.

There are two important things to notice about this response. Thefirst is this. In conceding that Einstein’s indirect method fordetermining, say, the position of Niels’ system does not mechanicallydisturb that system, Bohr departs from his original program ofcomplementarity, which was to base the uncertainty relations and thestatistical character of quantum theory on uncontrollable physicalinteractions, interactions that were supposed to arise inevitablybetween a measuring instrument and the system being measured. InsteadBohr now distinguishes between a genuine physical interaction (his“mechanical disturbance”) and some other sort of“influence” on the conditions for specifying (or“defining”) sorts of predictions for the future behaviorof a system. In emphasizing that there is no question of a robustinteraction in the EPR situation, Bohr retreats from his earlier,physically grounded conception of complementarity.

The second important thing to notice is how Bohr’s response needs tobe implemented in order to block the argument of EPR and Einstein’slater arguments that pose a dilemma between principles of locality andcompleteness. In these arguments the locality principle makes explicitreference to the reality of the unmeasured system: the realitypertaining to Niels’ system does not depend on what measurements (ifany) are made locally on Albert’s system. Hence Bohr’s suggestion thatthose measurements influence conditions for specifying types ofpredictions would not affect the argument unless one includes thoseconditions as part of the reality of Niels’ system. This is exactlywhat Bohr goes on to say, “these conditions constitute aninherent element of the description of any phenomena to which the term‘physical reality’ can be properly attached” (Bohr1935a, p. 700). So Bohr’s picture is that these“influences”, operating directly across any spatialdistances, result in different physically real states of Niels’ systemdepending on the type of measurement made on Albert’s. (Recall EPRwarning against just this move.)

The quantum formalism for interacting systems describes how ameasurement on Albert’s system reduces the composite state anddistributes quantum states and associated probabilities to thecomponent systems. Here Bohr redescribes that formal reduction usingEPR’s language of influences and reality. He turns ordinary localmeasurements into “influences” that automatically changephysical reality elsewhere, and at any distance whatsoever. Thisgrounds the quantum formalism in a rather magical ontologicalframework, a move quite out of character for the usually pragmaticBohr. In his correspondence over EPR, Schrödinger compared ideaslike that to ritual magic.

This assumption arises from the standpoint of the savage, who believesthat he can harm his enemy by piercing the enemy’s image with aneedle. (Letter to Edward Teller, June 14, 1935, quoted inBacciagaluppi 2015)

It is as though EPR’s talk of “reality” and its elementsprovoked Bohr to adopt the position of Moliere’s doctor who, pressedto explain why opium is a sedative, invents an inherent dormativevirtue, “which causes the senses to become drowsy.”Usually Bohr sharply deflates any attempt like this to get behind theformalism, insisting that “the appropriate physicalinterpretation of the symbolic quantum-mechanical formalism amountsonly to predictions, of determinate or statistical character”(Bohr 1949, p. 238).

Could this portrait of nonlocal influences automatically shaping adistant reality be a by-product of Bohr’s “inefficiency ofexpression”? Despite Bohr’s seeming tolerance for a breakdown oflocality in his response here to EPR, in other places Bohr rejectsnonlocality in the strongest terms. For example in discussing anelectron double slit experiment, which is Bohr’s favorite model forillustrating the novel conceptual features of quantum theory, andwriting only weeks before the publication of EPR, Bohr argues asfollows.

If we only imagine the possibility that without disturbing thephenomena we determine through which hole the electron passes, wewould truly find ourselves in irrational territory, for this would putus in a situation in which an electron, which might be said to passthrough this hole, would be affected by the circumstance of whetherthis [other] hole was open or closed; but … it is completelyincomprehensible that in its later course [the electron] should letitself be influenced by this hole down there being open or shut. (Bohr1935b)

It is uncanny how closely Bohr’s language mirrors that of EPR. Buthere Bohr defends locality and regards the very contemplation ofnonlocality as “irrational” and “completelyincomprehensible”. Since “the circumstance of whether this[other] hole was open or closed” does affect the possible typesof predictions regarding the electron’s future behavior, if we expandthe concept of the electron’s “reality”, as he appears tosuggest for EPR, by including such information, we do“disturb” the electron around one hole by opening orclosing the other hole. That is, if we give to “disturb”and to “reality” the very same sense that Bohr appears togive them when responding to EPR, then we are led to an“incomprehensible” nonlocality, and into the territory ofthe irrational (like Schrödinger’s savage).

There is another way of trying to understand Bohr’s position.According to one common reading (seeCopenhagen Interpretation), after EPR Bohr embraced a relational (or contextual) account ofproperty attribution. On this account to speak of the position, say,of a system presupposes that one already has put in place anappropriate interaction involving an apparatus for measuring position(or at least an appropriate frame of reference for the measurement;Dickson 2004). Thus “the position” of the system refers toa relation between the system and the measuring device (or measurementframe). (SeeRelational Quantum Mechanics, where a similar idea is developed independently of measurements.) Inthe EPR context this would seem to imply that before one is set up tomeasure the position of Albert’s system, talk of the position ofNiels’ system is out of place; whereas after one measures the positionof Albert’s system, talk of the position of Niels’ system isappropriate and, indeed, we can then say truly that Niels’ system“has” a position. Similar considerations govern momentummeasurements. It follows, then, that local manipulations carried outon Albert’s system, in a place we may assume to be far removed fromNiels’ system, can directly affect what is meaningful to say about, aswell as factually true of, Niels’ system. Similarly, in the doubleslit arrangement, it would follow that what can be said meaningfullyand said truly about the position of the electron around the top holewould depend on the context of whether the bottom hole is open orshut. One might suggest that such relational actions-at-a-distance areharmless ones, perhaps merely “semantic”; like becomingthe “best” at a task when your only competitor—whomight be miles away—fails. Note, however, that in the case ofordinary relational predicates it is not inappropriate (or“meaningless”) to talk about the situation in the absenceof complete information about the relata. So you might be the best ata task even if your competitor has not yet tried it, and you aredefinitely not an aunt (or uncle) until one of your siblings givesbirth. But should we say that an electron is nowhere at all until weare set up to measure its position, or would it be inappropriate(meaningless?) even to ask?

If quantum predicates are relational, they are different from manyordinary relations in that the conditions for the relata are taken ascriterial for the application of the term. In this regard one mightcontrast the relativity of simultaneity with the proposed relativityof position. In relativistic physics specifying a world-line fixes aframe of reference for attributions of simultaneity to eventsregardless of whether any temporal measurements are being made orcontemplated. But in the quantum case, on this proposal, specifying aframe of reference for position (say, the laboratory frame) does notentitle one to attribute position to a system, unless that frame isassociated with actually preparing or completing a measurement ofposition for that system. To be sure, analyzing predicates in terms ofoccurrent measurement or observation is familiar from neopositivistapproaches to the language of science; for example, in PercyBridgman’s operational analysis of physical terms, where the actualapplications of test-response pairs constitute criteria for anymeaningful use of a term (seeTheory and Observation in Science). Rudolph Carnap’s later introduction of reduction sentences (see theentry on theVienna Circle) has a similar character. Still, this positivist reading entails justthe sort of nonlocality that Bohr seemed to abhor.

In the light of all this it is difficult to know whether a coherentresponse can be attributed to Bohr reliably that would derail EPR. (Indifferent ways, Dickson 2004 and Halvorson and Clifton 2004 make anattempt on Bohr’s behalf. These are examined in Whitaker 2004 and Fine2007. See also the essays in Faye and Folse 2017.) Bohr may well havebeen aware of the difficulty in framing the appropriate conceptsclearly when, a few years after EPR, he wrote,

The unaccustomed features of the situation with which we areconfronted in quantum theory necessitate the greatest caution asregard all questions of terminology. Speaking, as it is often done ofdisturbing a phenomenon by observation, or even of creating physicalattributes to objects by measuring processes is liable to beconfusing, since all such sentences imply a departure from conventionsof basic language which even though it can be practical for the sakeof brevity, can never be unambiguous. (Bohr 1939, p. 320. Quoted inSection 3.2 of the entry on theUncertainty Principle.)

3. Development of EPR

3.1 Spin and The Bohm version

For about fifteen years following its publication, the EPR paradox wasdiscussed at the level of a thought experiment whenever the conceptualdifficulties of quantum theory became an issue. In 1951 David Bohm, aprotégé of Robert Oppenheimer and then an untenuredAssistant Professor at Princeton University, published a textbook onthe quantum theory in which he took a close look at EPR in order todevelop a response in the spirit of Bohr. Bohm showed how one couldmirror the conceptual situation in the EPR thought experiment bylooking at the dissociation of a diatomic molecule whose total spinangular momentum is (and remains) zero; for instance, the dissociationof an excited hydrogen molecule into a pair of hydrogen atoms by meansof a process that does not change an initially zero total angularmomentum (Bohm 1951, Sections 22.15–22.18). In the Bohmexperiment the atomic fragments separate after interaction, flying offin different directions freely to separate experimental wings.Subsequently, in each wing, measurements are made of spin components(which here take the place of position and momentum), whose measuredvalues would be anti-correlated after dissociation. In the so-calledsinglet state of the atomic pair, the state after dissociation, if oneatom’s spin is found to be positive with respect to the orientation ofan axis perpendicular to its flight path, the other atom would befound to have a negative spin with respect to a perpendicular axiswith the same orientation. Like the operators for position andmomentum, spin operators for different non-orthogonal orientations donot commute. Moreover, in the experiment outlined by Bohm, the atomicfragments can move to wings far apart from one another and so becomeappropriate objects for assumptions that restrict the effects ofpurely local actions. Thus Bohm’s experiment mirrors the entangledcorrelations in EPR for spatially separated systems, allowing forsimilar arguments and conclusions involving locality, separability,and completeness. Indeed, a late note of Einstein’s, that may havebeen prompted by Bohm’s treatment, contains a very sketchy spinversion of the EPR argument – once again pitting completenessagainst locality (“A coupling of distant things isexcluded.” Sauer 2007, p. 882). Following Bohm (1951) a paper byBohm and Aharonov (1957) went on to outline the machinery for aplausible experiment in which entangled spin correlations could betested. It has become customary to refer to experimental arrangementsinvolving determinations of spin components for spatially separatedsystems, and to a variety of similar set-ups (especially ones formeasuring photon polarization), as “EPRB”experiments—“B” for Bohm. Because of technicaldifficulties in creating and monitoring the atomic fragments, however,there seem to have been no immediate attempts to perform a Bohmversion of EPR.

3.2 Bell and beyond

That was to remain the situation for almost another fifteen years,until John Bell utilized the EPRB set-up to construct a stunningargument, at least as challenging as EPR, but to a differentconclusion (Bell 1964). Bell considers correlations betweenmeasurement outcomes for systems in separate wings where themeasurement axes of the systems differ by angles set locally. In hisoriginal paper, essentially using the lemma from EPR governing strictcorrelations, Bell shows that correlations measured in different runsof an EPRB experiment satisfy a system of constraints, known as theBell inequalities. Later demonstrations by Bell and others, usingrelated assumptions, extend this class of inequalities. In certain ofthese EPRB experiments, however, quantum theory predicts correlationsthat violate particular Bell inequalities by an experimentallysignificant amount. Thus Bell shows (see the entry onBell’s Theorem) that the quantum statistics are inconsistent with the givenassumptions. Prominent among these is an assumption of locality,similar to the locality assumptions tacitly assumed in EPR and(explicitly) in the one-variable and many-variable arguments ofEinstein. One important difference is that for Einstein localityrestricts factors that might influence the (assumed) real physicalstates of spatially separated systems (separability). For Bell,locality is focused instead on factors that might influence outcomesof measurements in experiments where both systems are measured. (SeeFine 1996, Chapter 4.) These differences are not usually attended toand Bell’s theorem is often characterized simply as showing thatquantum theory is nonlocal. Even so, since assumptions other thenlocality are needed in any derivation of the Bell inequalities(roughly, assumptions guaranteeing a classical representation of thequantum probabilities; see Fine 1982a, and Malley 2004), one should becautious about singling out locality (in Bell’s sense, or Einstein’s)as necessarily in conflict with the quantum theory, or refuted byexperiment.

Bell’s results have been explored and deepened by various theoreticalinvestigations and they have stimulated a number of increasinglysophisticated and delicate EPRB-type experiments designed to testwhether the Bell inequalities hold where quantum theory predicts theyshould fail. With a few anomalous exceptions, the experiments appearto confirm the quantum violations of the inequalities. (Brunneretal 2014 is a comprehensive technical review.) The confirmation isquantitatively impressive, although not fully conclusive. There are anumber of significant requirements on the experiments whose failures(generally downplayed as “loopholes”) allow for models ofthe experimental data that embody locality (in Bell’s sense),so-called local realist models. One family of “loopholes”(sampling) arises from possible losses (inefficiency) between emissionand detection and from the delicate coincidence timing required tocompute correlations. All the early experiments to test the Bellinequalities were subject to this loophole, so all could be modeledlocally and realistically. (The prism and synchronization models inFine 1982b are early models of this sort. Larsson 2014 is a generalreview.) Another “loophole” (locality) concerns whetherNiels’ system, in one wing, could learn about what measurements areset to be performed in Albert’s wing in time to adjust its behavior.Experiments insuring locality need to separate the wings and this canallow losses or timing glitches that open them to models exploitingsampling error. Perversely, experiments to address sampling mayrequire the wings to be fairly close together, close enough generally,it turns out, to allow information sharing and hence local realistmodels. There are now a few experiments that claim to close bothloopholes together. They too have problems. (See Bednorz 2017 for acritical discussion.)

There is also a third major complication or “loophole”. Itarises from the need to ensure that causal factors affectingmeasurement outcomes are not correlated with the choices ofmeasurement settings. Known as “measurement independence”or sometimes “free choice”, it turns out that evenstatistically small violations of this independence requirement allowfor local realism (Putz and Gisin 2016). Since connections betweenoutcomes and settings might occur anywhere in the causal past of theexperiment, there is really no way to insure measurement independencecompletely. Suitably random choices of settings might avoid thisloophole within the time frame of the experiment, or even extend thattime some years into the past. An impressive, recent experiment pushesthe time frame back about six hundred years by using the color ofMilky Way starlight (blue or red photons) to choose the measurementsettings. (Handsteineret al 2017). Of course travelingbetween the Milky Way and the detectors in Vienna a lot of starlightis lost (over seventy per cent), which leaves the experiment wide opento the sampling loophole. Moreover, there is an obvious common causefor settings and outcomes (and all); namely, the big bang. With thatin mind one might be inclined to dismiss free choice as not seriouseven for a “loophole”. It may seem like anad hochypothesis that postulates a cosmic conspiracy on the part of Naturejust to the save the Bell inequalities. Note, however, that ordinaryinefficiency can also be modeled locally as a violation of freechoice, because an individual measurement that produces no usableresult can just as well be regarded as not currently available. Sinceinefficiency is not generally counted as a violation of localcausality or a restriction on free will, nor as a conspiracy (well,not a cosmic one), measurement dependence should not be dismissed soquickly. Instead, one might see measurement dependent correlations asnormal limitations in a system subject to dynamical constraints orboundary conditions, and thus use them as clues, along with otherguideposts, in searching for a covering local theory. (See Weinstein2009.)

Experimental tests of the Bell inequalities continue to be refined.Their analysis is delicate, employing sophisticated statistical modelsand simulations. (See Elkouss and Wehner 2016 and Graft 2016.) Thesignificance of the tests remains a lively area for criticaldiscussion. Meanwhile the techniques developed in the experiments, andrelated ideas for utilizing the entanglement associated with EPRB-typeinteractions, have become important in their own right. Thesetechniques and ideas, stemming from EPRB and the Bell theorem, haveapplications now being advanced in the field of quantum informationtheory — which includes quantum cryptography, teleportation andcomputing (seeQuantum Entanglement and Information).

To go back to the EPR dilemma betweenlocality and completeness, it would appear from the Bell theorem thatEinstein’s preference for locality at the expense of completeness mayhave fixed on the wrong horn. Even though the Bell theorem does notrule out locality conditions conclusively, it should certainly makeone wary of assuming them. On the other hand, since Einstein’sexploding gunpowder argument (or Schrödinger’s cat), along with his later arguments overmacro-systems, support incompleteness without assuming locality, oneshould be wary of adopting the other horn of the dilemma, affirmingthat the quantum state descriptions are complete and“therefore” that the theory is nonlocal. It may well turnout that both horns need to be rejected: that the state functions donot provide a complete description and that the theory is alsononlocal (although possibly still separable; see Winsberg and Fine2003). There is at least one well-known approach to the quantum theorythat makes a choice of this sort, the de Broglie-Bohm approach (Bohmian Mechanics). Of course it may also be possible to break the EPR argument for thedilemma plausibly by questioning some of its other assumptions (e.g.,separability, the reduction postulate, theeigenvalue-eigenstate link, or measurement independence). That might free up the remainingoption, to regard the theory as both local and complete. Perhaps someversion of theEverett Interpretation would come to occupy this branch of the interpretive tree, or perhapsRelational Quantum Mechanics.

Bibliography

• Bacciagaluppi, G., 2015, “Did Bohr understand EPR?” inF. Aaserud and H. Kragh (eds.),One Hundred Years of the BohrAtom (Scientia Danica, Series M, Mathematica et physica, Volume1), Copenhagen: Royal Danish Academy of Sciences and Letters, pp.377–396.
• Bacciagaluppi, G. and A. Valentini, 2009,Quantum Theory atthe Crossroads: Reconsidering the 1927 Solvay Conference,Cambridge: Cambridge University Press.
• Bednorz, A., 2017, “Analysis of assumptions of recent testsof local realism”,Physical Review A, 95: 042118.
• Bell, J. S., 1964, “On the Einstein-Podolsky-Rosenparadox”,Physics, 1: 195–200, reprinted in Bell1987.
• –––, 1987,Speakable and Unspeakable inQuantum Mechanics, New York: Cambridge University Press.
• Beller, M., 1999,Quantum Dialogue: The Making of aRevolution, Chicago: University of Chicago Press.
• Belousek, D. W., 1996, “Einstein’s 1927 unpublishedhidden-variable theory: its background, context andsignificance”,Studies in History and Philosophy of ModernPhysics, 27: 437–461.
• Bohm, D., 1951,Quantum Theory, New York: PrenticeHall.
• Bohm, D., and Y. Aharonov, 1957, “Discussion of experimentalproof for the paradox of Einstein, Rosen and Podolski”,Physical Review, 108: 1070–1076.
• Bohr, N., 1935a, “Can quantum-mechanical description ofphysical reality be considered complete?”,PhysicalReview, 48: 696–702.
• –––, 1935b, “Space and time in nuclearphysics”, Ms. 14, March 21, Manuscript Collection, Archive forthe History of Quantum Physics, American Philosophical Society,Philadelphia.
• –––, 1939, “The causality problem inatomic physics” in Bohr, 1996, pp. 303–322.
• –––, 1949, “Discussions with Einstein onepistemological problems in atomic physics” in Schilpp, 1949,pp. 199–241. Reprinted in Bohr, 1996, pp. 339–381.
• –––, 1996,Collected Works, Vol. 7,Amsterdam: North Holland.
• Born, M., (ed.), 1971,The Born-Einstein Letters, NewYork: Walker.
• Brunner, N.et al., 2014, “Bell nonlocality”,Reviews of Modern Physics, 86: 419–478.
• Dickson, M., 2004, “Quantum reference frames in the contextof EPR”,Philosophy of Science, 71: 655–668.
• Einstein, A. 1936, “Physik und Realität”,Journal of the Franklin Institute, 221: 313–347,reprinted in translation in Einstein 1954.
• –––, 1948, “ Quanten-Mechanik undWirklichkeit ”,Dialectica, 2: 320–324.Translated in Born 1971, pp. 168–173.
• –––, 1953a, “Einleitende Bemerkungenüber Grundbegriffe ”, in A. George, ed.,Louis deBroglie: Physicien et penseur, Paris: Editions Albin Michel, pp.5–15.
• –––, 1953b, “Elementare Überlegungenzur Interpretation der Grundlagen der Quanten-Mechanik ”, inScientific Papers Presented to Max Born, New York: Hafner,pp. 33–40.
• –––, 1954,Ideas and Opinions, NewYork: Crown.
• Einstein, A., B. Podolsky, and N. Rosen, 1935, “Canquantum-mechanical description of physical reality be consideredcomplete?”,Physical Review, 47: 777–780 [Einstein, Podolsky, and Rosen 1935 available online].
• Elkouss, D and S. Wehner, 2016, “ (Nearly) optimal P valuesfor all Bell inequalities ”,NPJ Quantum Information,2: 16026.
• Faye, J. and H. Folse, 2017,Niels Bohr and the Philosophy ofPhysics, London: Bloomsbury Academic.
• Fine, A., 1996,The Shaky Game: Einstein, Realism and theQuantum Theory, 2nd Edition, Chicago: University of ChicagoPress.
• –––, 1982a, “Hidden variables, jointprobability and the Bell inequalities”,Physical ReviewLetters, 48: 291–295.
• –––, 1982b, “Some local models forcorrelation experiments”,Synthese 50:279–94.
• –––, 2007, “Bohr’s response to EPR:Criticism and defense”,Iyyun, The Jerusalem PhilosophicalQuarterly, 56: 31–56.
• Gilton, M. J. R., 2016, “Whence the eigenstate-eigenvaluelink?”,Studies in History and Philosophy of ModernPhysics, 55: 92–100.
• Graft, D. A., 2016, “ Clauser-Horne/Eberhard inequalityviolation by a local model”,Advanced Science, Engineeringand Medicine, 8: 496–502.
• Halvorson, H., 2000, “The Einstein-Podolsky-Rosen statemaximally violates Bell’s inequality”, Letters inMathematical Physics, 53: 321–329.
• Halvorson, H. and R. Clifton, 2004, “Reconsidering Bohr’sreply to EPR.” In J. Butterfield and H. Halvorson, eds.,Quantum Entanglements: Selected Papers of Rob Clifton,Oxford: Oxford University Press, pp. 369–393.
• Handsteiner, J.et al , 2017, “ Cosmic Bell test:Measurement settings from Milky Way stars”,Physical ReviewLetters, 118: 060401.
• Harrigan, N. and R. W., Spekkens, 2010, “Einstein,incompleteness, and the epistemic view of quantum states”,Foundations of Physics, 40: 125–157.
• Held, C., 1998,Die Bohr-Einstein-Debatte: Quantenmechanik undPhysikalische Wirklichkeit, Paderborn: Schöningh.
• Holland, P., 2005, “What’s wrong with Einstein’s 1927hidden-variable interpretation of quantum mechanics?”,Foundations of Physics, 35: 177–196.
• Hooker, C. A., 1972, “The nature of quantum mechanicalreality: Einstein versus Bohr”, in R. G. Colodny, ed.,Paradigms and Paradoxes, Pittsburgh: University of PittsburghPress, pp. 67–302.
• Howard, D., 1985, “Einstein on locality andseparability.”Studies in History and Philosophy ofScience 16: 171–201.
• Howard, D., 1990, “‘Nicht Sein Kann Was Nicht SeinDarf’, or the Prehistory of EPR, 1909–1935”, in A. I.Miller (ed.),Sixty-Two Years of Uncertainty, New York:Plenum Press, pp. 61–111.
• Jammer, M., 1974,The Philosophy of Quantum Mechanics,New York: Wiley.
• Larsson, J.-A., 2014, “Loopholes in Bell inequality tests oflocal realism”,Journal of Physics A, 47: 424003.
• Malley, J., 2004, “All Quantum observables in ahidden-variable model must commute simultaneously”,PhysicalReview A, 69 (022118): 1–3.
• Putz, G. and N. Gisin, 2016, “Measurement dependentlocality”,New Journal of Physics, 18: 05506.
• Ryckman, T., 2017,Einstein, New York and London:Routledge.
• Sauer, T., 2007, “An Einstein manuscript on the EPR paradoxfor spin observables”,Studies in History and Philosophy ofModern Physics, 38: 879–887.
• Schilpp, P.A., (ed.), 1949,Albert Einstein:Philosopher-Scientist, La Salle, IL: Open Court.
• Schlosshauer, M., 2007,Decoherence and theQuantum-to-Classical Transition, Heidelberg/Berlin:Springer.
• Schrödinger, E., 1935a, “Die gegenwärtigeSituation in der Quantenmechanik”,Naturwissenschaften,23: 807–812, 823–828, 844–849; English translationin Trimmer, 1980.
• –––, 1935b, “Discussion of probabilityrelations between separated systems”,Proceedings of theCambridge Philosophical Society, 31: 555–562.
• Trimmer, J. D., 1980, “The present situation in quantummechanics: A translation of Schrödinger’s ‘catparadox’ paper”,Proceedings of the AmericanPhilosophical Society, 124: 323–338
• Weinstein, S. 2009, “Nonlocality without nonlocality”,Foundations of Physics, 39: 921–936.
• Whitaker, M. A. B., 2004, “The EPR Paper and Bohr’sresponse: A re-assessment”,Foundations of Physics, 34:1305–1340.
• Winsberg, E., and A. Fine, 2003, “Quantum life: Interaction,entanglement and separation”,Journal of Philosophy, C:80–97.

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