PHYSICAL REVIEW A 88, 022105 (2013)

Two-photon interferometry and quantum state collapse

Art Hobson*

Department of Physics, University of Arkansas, Fayetteville, Arkansas 72701, USA(Received 8 January 2013; revised manuscript received 13 March 2013; published 8 August 2013)

The quantum measurement problem still finds no consensus. Nonlocal interferometry provides an unprece-dented experimental probe by entangling two photons in the “measurement state” (MS). The experiments showthat each photon measures the other; the resulting entanglement decoheres both photons; decoherence collapsesboth photons to unpredictable but definite outcomes; and the two-photon MS continues evolving coherently.Thus, when a two-part system is in the MS, the outcomes actually observed at both subsystems are definite.Although standard quantum physics postulates definite outcomes, two-photon interferometry verifies them tobe not only consistent with, but also actually a prediction of, the other principles. Nonlocality is the key tounderstanding this. As a consequence of nonlocality, the states we actually observe are the local states. Theseactually observed local states collapse, whereas the global MS, which can be “observed” only after the fact bycollecting coincidence data from both subsystems, continues its unitary evolution. This conclusion implies arefined understanding of the eigenstate principle: Following a measurement, the actually observed local stateinstantly jumps into the observed eigenstate. We also discuss and rebut objections to this.

DOI: 10.1103/PhysRevA.88.022105 PACS number(s): 03.65.−w, 42.50.−p

I. INTRODUCTION

As Weinberg recently reminded us [1], the measurementproblem remains a fundamental conundrum. To review theessentials, it is sufficient to consider two-state (“qubit”)systems. Suppose a qubit S, whose Hilbert space is spannedby orthonormal states |si〉 (i = 1,2), is in the superpositionstate,

|�〉S = (|s1〉 + |s2〉)/√

2. (1)

A measurement apparatus A, which may be micro- ormacroscopic, is designed to distinguish between states |si〉 bytransitioning into state |ai〉 if it finds S is in |si〉 (i = 1,2)[2]. Assume the detector is reliable, implying the |ai〉’sare orthonormal and that the measurement interaction doesnot disturb states |si〉—i.e., the measurement is “ideal” [3].When A measures S, the Schrodinger equation’s unitary timeevolution then leads to the “measurement state” (MS),

|�〉SA = (|s1〉|a1〉 + |s2〉|a2〉)/√

2, (2)

of the composite system SA following the measurement.Standard quantum physics postulates a definite but un-

predictable measurement outcome, either |a1〉 or |a2〉 andthat S suddenly “collapses” into the corresponding state |si〉.Equation (2) does not appear to resemble such a collapsedstate. The measurement problem is as follows: (1) How do wereconcile the measurement postulate’s definite outcomes withthe MS; and (2) how does the outcome become irreversiblyrecorded in light of the Schrodinger equation’s unitary (and,hence, reversible) evolution? This paper deals with only thefirst part, known as the “problem of definite outcomes,” aproblem that Leggett [4] expresses as follows: “The quantummeasurement paradox is that most interpretations of quantummechanics at the microscopic level do not allow definiteoutcomes to be realized, whereas, at the level of our human

*ahobson@uark.edu

consciousness it seems a matter of direct experience thatsuch outcomes occur.” The second part, which must takeinto account the macroscopic nature of either the apparatusor the wider environment, has been discussed in depth bySchlosshauer [2] and Zurek [5].

The MS is subtle: a bipartite (two-subsystem) entangled(nonfactorizable) coherent superposition. A crucial feature isthat, like all entangled states, it implies a nonlocal connectionbetween S and A [6]. This paper presents an experimentaland theoretical analysis of the MS that resolves the problemof definite outcomes. It shows that the MS actually is thecollapsed state of both S and A, even though it evolvedpurely unitarily. The formation of subsystems and the resulting“local” versus “global” distinction (a consequence of nonlocalentanglement) are crucial to understanding this: The localstates of S and of A, i.e., the states that we actually observedirectly at the locations of S and A (which could be lightyears apart), collapse into definite outcomes. The globalstate Eq. (2), which is observable only by the subsequentnonlocal collection of data (which could take years) showingthe experimental coincidences between the two subsystems,maintains its unitary and reversible time evolution. Thus, localsubsystem collapse is consistent with global unitary evolutionof the composite system.

In 1990, two-photon interferometry experiments demon-strated a new way to experimentally test nonlocality (i.e.,Bell’s inequality) for entangled photon momentum eigenstatesrather than, as in earlier nonlocality tests by Aspect andothers, for entangled photon polarization eigenstates. Thesetwo-photon states were precisely the MS. Importantly, thephase angles ϕS between |s1〉 and |s2〉 and ϕA between |a1〉 and|a2〉 could both be varied independently at either subsystem.These experiments in which two photons (call them S andA) were entangled in the MS offer a unique window on theMS. The experiments were logically equivalent to two coupleddouble-slit experiments in which the entanglement caused eachphoton to act as a which-path (or momentum) detector for theother photon. We discuss these experiments’ relevance to themeasurement problem.

022105-11050-2947/2013/88(2)/022105(5) ©2013 American Physical Society

ART HOBSON PHYSICAL REVIEW A 88, 022105 (2013)

The experimental results (Sec. II) show that A decoheresS, whereas, S decoheres A, removing the interferences thatwould be seen if the photons were not entangled and collapsingboth S and A as viewed locally into incoherent unpredictablebut definite outcomes. This shows that the MS is the collapsedstate exhibiting definite local outcomes. But at the global level,the experiment finds a two-body state with coherent nonlocalinterferences. Thus, the experiments demonstrate the coherentunitary nature of the global evolution and the local collapse ofS and A into definite outcomes.

Section III analyzes the MS theoretically. The MS’s nonlo-cal aspect is crucial because experiment (Sec. II) and theory(Sec. III) both show that the states actually observed at bothsubsystems are the “reduced” states, not the global MS. Theseactually observed outcomes are not coherent superpositionsbut are, instead, incoherent and definite. The global MSremains a coherent superposition. Similar conclusions havebeen stated [7] and have been suggested [8,9], but thesehave been overlooked. Hopefully, the experimental facts andanalysis presented here will reinforce these arguments. The“modal interpretations” [10–12] have long postulated theseconclusions, but there is no need for a separate postulatebecause these results follow from the other standard quantumpostulates.

Section IV rebuts previous analyses disputing these re-sults, including misconceptions about Schrodinger’s cat andthe charge that decoherence cannot solve the measurementproblem. Section V summarizes the conclusions.

II. THE MEASUREMENT STATE: EXPERIMENTS

In the ordinary two-slit experiment with a single pho-ton, we see nonlocality in the photon’s discontinuous andinstantaneous jumps between a self-interfering pure stateand a noninterfering mixed state depending, respectively, onwhether a which-path detector is absent or is present. Theinstantaneous and nonlocal nature of these jumps is clear inthe “delayed-choice” experiment [13]: A random decision tonot entangle or to entangle the photon with a which-pathdetector is made quickly after the photon has entered theinterferometer but before there has been time to send a signal tothe detector. The photon expands to a superposition over bothpaths [14] the instant the detector switches off and collapsesto a mixture the instant the detector switches on. This andsimilar recent experiments [15] leave little doubt, today, ofinstantaneous (or at any rate “faster than light”) state collapseupon measurement.

Thus, the many nonlocality experiments during recentdecades are likely to be relevant to understanding the MS.Nonlocal two-photon interferometry experiments by Rarityand Tapster [16,17] and, independently, Ou et al. [18] probe theMS. Such experiments, described below, can determine whichstates are incoherent with unpredictable but definite outcomesand which are coherent superpositions. Importantly, in theseRarity, Tapster, and Ou et al. (RTO) experiments, both S andA are microscopic, so both subsystems can be fully analyzedwith rigorous quantum theory, and the phases of S and A canbe varied. As we will see, the experiments demonstrate eachlocal subsystem to be in an incoherent mixture of unpredictablebut definite outcomes. But coincidences of paired photons

ϕ ϕ

⟩

a

a s

s

⟩

⟩⟩

FIG. 1. (Color online) The RTO experiments. A source emits twoentangled photons A and S into two paths with one path phase shifted.Mirrors M and BSs recombine the beams. Without entanglement, eachphoton would exhibit coherent interference as a function of ϕA or ϕS .Entanglement causes the photons to be in incoherent states havingdefinite outcomes. The coherence is, however, preserved in the globalmeasurement state.

demonstrate that the global state remains coherent. Thus, bothsubsystems are in unpredictable but definite states, whereas,the composite system remains in a pure-state superposition!The experiments demonstrate that the global MS is actuallythe collapsed state. Some experts (Sec. IV) argue that thefully collapsed incoherent mixture represented by the densityoperator,

(|a1〉|s1〉〈s1|〈a1| + |a2〉|s2〉〈s2|〈a2|)/2 (3)

is the state we actually see upon measurement. They arecontradicted by the fact that Eq. (3) has no entanglement and,thus, no nonlocal channel, whereas, the delayed-choice ex-periment [13] shows that measurement establishes a nonlocalchannel. We will see that there is no contradiction between theMS Eq. (2) and the collapse postulate.

In the RTO experiment, a photon was “parametricallydown-converted” into two entangled photons S and A, whichmoved through separate two-slit interference experiments andimpacted separate detection screens. More accurately, theexperiments used interferometers equipped with beam splitters(BSs), variable phase shifters, and photon detectors, but thisis equivalent to two slits and a screen for each photon. Theexperiments were performed with photon pairs, but the resultswould be the same if they were performed with materialquanta.

S and A emerged from the down-converter in the entangled“Bell state” Eq. (2) (Fig. 1), implying that each photonacted as an ideal von Neumann detector for the otherphoton. Each photon emerged from the down-converter intwo superposed beams; call S’s two beams |si〉 and A’s twobeams |ai〉 (i = 1,2). One beam of S was then phase shiftedthrough an angle ϕS , and one beam of A was phase shiftedthrough ϕA, analogous to observing off-center at S’s and A’sviewing screens in the equivalent two-slit experiment. S’stwo beams were then combined at a BS whose outputs weremonitored by photon detectors and similarly for A’s beams.See Refs. [19–21] and my conceptual physics textbook [22]for more on these experiments.

In an ensemble of two-photon trials, neither A’s nor S’sdetector showed local (i.e., as actually observed at eitherdetector) interference. This verified that each photon actedas a which-path detector for the other photon, decohering bothphotons and creating definite local outcomes, just like theone-photon two-slit experiment with a which-path detectorat the slits. As observed locally, each photon impacted itsdetectors randomly with 50:50 probability regardless of ϕS

022105-2

TWO-PHOTON INTERFEROMETRY AND QUANTUM STATE . . . PHYSICAL REVIEW A 88, 022105 (2013)

and ϕA. There was no sign of the local interference that wouldoccur were S and A not entangled—a textbook example ofmeasurement-induced decoherence.

But how then was unitary evolution preserved, despitelocal collapse and loss of coherence? The answer: RTO’snonlocal “fourth-order” (involving all four photon paths)pattern showed the coherence predicted by Eq. (2). A coin-cidence detector revealed coherence as a function of the phasedifference ϕ = ϕS − ϕA between S’s phase angle and A’sphase angle! The photons “knew” each others’ phase angles,despite their arbitrarily large separation [22]. This certainlyappears to be nonlocal, and indeed, the results violate Bell’sinequality, verifying nonlocality [16]. “The entangled state ofEq. (2) shows a definite phase relation between the two-particlestates, namely, |s1〉|a1〉 and |s2〉|a2〉 but no definite phaserelations between single-particle states” [19] (the quotationuses the present paper’s notation).

Special relativity implies that local incoherence must bethe case because, otherwise, a change in ϕS would, byaltering ϕ = ϕS − ϕA, instantly alter A’s outcomes ina manner that A could detect locally even if A were inanother galaxy. This would violate “Einstein causality:” S

could send an instantaneous signal to A. Thus, ϕ mustbe “camouflaged” at both detectors, implying that neitherphoton shows interference [23,24]. So quantum entanglementdelicately protects causality while permitting an instantaneousnonlocal channel between subsystems.

Thus, experiments confirm that, when SA is in the MS,the two-body states |s1〉|a1〉 and |s2〉|a2〉 are coherentlysuperposed, whereas, the states of S and A are incoherentlymixed. As we will see next, standard quantum physics (butwith no collapse postulate) correctly predicts all of this.

III. THE MEASUREMENT STATE: THEORY

Einstein was perhaps the first to recognize nonlocality,although he used it to argue for an alternative view of quantumphysics. During the 1927 Solvay Conference, Einstein notedthat “a peculiar action-at-a-distance must be assumed to takeplace” when the Schrodinger field passes through a single slit,diffracts as a spherical wave, and strikes a detection screen.Theoretically, when the interaction localizes as a small flashat an isolated point on the screen, the field instantly vanishesover the rest of the screen. Supporting de Broglie’s theory thatsupplemented the Schrodinger field with particles, Einsteincommented “if one works only with Schrodinger waves, theinterpretation of psi, I think, contradicts the postulate ofrelativity” [14], [25]. Since that time, however, the peacefulcoexistence of quantum nonlocality and special relativity hasbeen demonstrated [23,24].

It’s well known today that Bell’s theorem shows quantumphysics contains a nonlocal element. Many experiments,culminating in those of Aspect [26,27], show (subject toonly a few remaining highly speculative loopholes) that thequantum predictions agree with nature; moreover, nature isnonlocal even if it should turn out that quantum physicsis wrong. The nonlocality of the entangled MS has beenverified directly not only in RTO’s two-photon interferometryexperiment where both subsystems are photons, but also in thesingle-photon two-slit delayed-choice experiment where the

second subsystem is a which-slit detector that can be quicklyand randomly inserted [13].

S and A could be separated by light years (the currentdistance record for nonlocality demonstrations is 147 km [28]),but random changes in S’s phase shifter are, nevertheless,predicted to show up in A’s data with no time delay. We musttake this nonlocality into account when analyzing the MS;even though S and A might be next to each other in an actualexperiment, they could be widely separated without changingany essentials of the experiment.

Consider a series of trials of the RTO experiment in whicheach trial consists of a quick random change in S’s phaseshifter and the recording of data in both S’s and A’s detectors.Neither an observer at S nor an observer at A necessarily hasany direct experimental knowledge of coincidences becausethe two could be light years apart. The only directly observeddata are local. Global data are available only later.

Haroche and Raimond [29] have weighed in on this questionof when a two-part system should be considered a compositeof two subsystems. They advise us that a system should beconsidered single whenever the binding between its parts ismuch stronger than the interactions involved in its dynamics.By this criterion, a system and its measuring apparatus (whichare not necessarily bound together at all) must be consideredseparate subsystems.

Theorem. When a bipartite system is in the MS, neithersubsystem is in a local superposition.

Proof. Define the pure-state density operator ρSA =|�〉SASA〈�|. As is well known, the expectation value of anyobservable Q of SA can be found from 〈Q〉 = Tr(ρSAQ). Forany observable QS operating on S alone, the MS implies

〈QS〉 = TrSA(ρSAQS) = TrS(QS)

= 〈s1|QS |s1〉 + 〈s2|QS |s2〉, (4)

because

TrAρSA = �k〈ak|ρSA|ak〉=|s1〉〈s1| + |s2〉〈s2| = IS (5)

(the identity operator acting in S’s Hilbert space). If S

was in a local superposition β|s1〉 + γ |s2〉 (β and γ arenonzero), the expectation of the observables QS = |s1〉〈s2| +|s2〉〈s1| and PS = i|s1〉〈s2| − i|s2〉〈s1| would be 2 Re(βγ ∗)and 2 Im(βγ ∗), respectively. Now, 2 Re(βγ ∗) is zero onlyif amplitudes β and γ are 90◦ out of phase, and in this case,2 Im(βγ ∗) is nonzero, so at least one of these two observableshas a nonzero expectation. But Eq. (4) predicts zero for theexpectations of both QS and PS—a contradiction. Thus, S

cannot be in a local superposition. The same argument goesfor A. �

Schrodinger’s cat provides an example. Schrodinger fa-mously considered a cat, a radioactive nucleus, and a devicethat automatically releases poison gas, killing the cat when thenucleus decays. If the nucleus S was isolated, then its state ata time of one half-life would be given by Eq. (1) where |s1〉and |s2〉 are the undecayed and decayed states, respectively.But the nucleus + cat composite system is described bythe MS, with A being the cat and |a1〉, |a2〉 representing itsalive and dead states, respectively. Thus, the theorem saysthat Schrodinger’s cat is not in a superposition β|a1〉 + γ |a2〉(β and γ are nonzero) of dead and alive states and similarly

022105-3

ART HOBSON PHYSICAL REVIEW A 88, 022105 (2013)

for the nucleus. Neither an observer at A nor an observer at S

sees a superposition. So an observer at A must see either analive or a dead cat, an observer at S must see an undecayedor a decayed nucleus, and a global observer sees the expectedcorrelations between these outcomes.

This solves the problem of outcomes, at least for idealmeasurements: The MS implies that observers at S and A finddefinite states |si〉 and |ai〉, respectively, not superpositions.This prediction, based on standard quantum physics (however,with a collapse that is not postulated but is, instead, derived),is experimentally confirmed by the RTO experiments.

But the MS is a pure-state superposition. It shows a nonlocalrelationship between the phases of the two separated photons.As experiment (Sec. II) and theory [19] show, photon pairsexhibit a sinusoidal dependence on the phase difference ϕ =ϕS − ϕA between the individual photon phases ϕS and ϕA.This nonlocal phase relationship implies the coherence of theglobal MS.

The decoherence concept [2] is useful here. In the standarddefinition, decoherence refers to a loss of coherence amongthe terms of a superposition caused by measurementlike (i.e.,entangling) interactions with the wider environment. But thedecoherence can equally refer to the loss of coherence causedby laboratory measurements. In both the environmental and thelaboratory cases, entanglement turns coherent superpositionsinto incoherent mixtures. In the case of an ideal measurement,decoherence occurs instantly when the apparatus and theobserved system form the MS. This is carefully examinedin the delayed-choice experiment [13] and is found to be trulynonlocal and instantaneous (or at any rate “faster than light”).

To my knowledge, Rinner and Werner [7] were the firstto publish a clear statement that the MS directly implies theobserved collapse, although others [8,9] have suggested thisresult.

These results imply a refined understanding of the standardeigenstate principle according to which the state immediatelyfollowing a measurement is an eigenstate of the measuredobservable. Because of the bipartite entangled nature of theMS, one must ask “which state: the global MS, or thelocal state?” Clearly, the global state, Eq. (2), following ameasurement is not an eigenstate of the measured observable.In the eigenstate principle, the word state needs to beunderstood as the local state, or the “state that is actuallyobserved.” Thus, the standard quantum principles (minus thecollapse postulate) imply that the observed states of quantumsystems evolve by two mechanisms: the unitary Schrodingerevolution, punctuated by instantaneous collapse into a newstate upon entanglement.

IV. ASSESSMENT OF PREVIOUS ANALYSES

There are widespread claims that Schrodinger’s cat is not ina definite alive or dead state but is, instead, in a superposition ofthe two. van Kampen, for example, writes “The whole systemis in a superposition of two states: one in which no decay hasoccurred and. . .one in which it has occurred. Hence, the stateof the cat also consists of a superposition. . . |cat〉 = a|life〉 +b|death〉. . . .The state remains a superposition until an observerlooks at the cat” [30].

Bell [31], referring to van Kampen’s description, agrees,stating that “After the measurement the measuring instrument,according to the Schrodinger equation, will admittedly bein a superposition of different readings. . .|cat〉 = a|life〉 +b|death〉.” Many textbooks make the same assessment [32].In fact, “Cat states have now come to refer to any quantumsuperposition of macroscopically distinct states” [33]. ButSec. III rigorously proved the MS to be inconsistent withsuperpositions of the individual subsystems. This invalidatesthe above claims.

The formulation |cat〉 = a|life〉 + b|death〉 obscures thenonlocal entanglement. The cat state must be written as|cat〉 = a|live cat〉|undecayed nucleus〉 + b|dead cat〉|decayednucleus〉. This entangled state actually is the collapsed stateof both the cat and the nucleus, showing definite outcomes.Contrary to van Kampen’s and some others’ opinions,“looking” at the outcome changes nothing, beyond informingthe observer of what has already happened. Entanglementis the formal representation of looking. Once the compositesystem is in the MS, the looking has already occurred.

In an influential paper, Wigner derives the pure state Eq. (2)from the standard measurement analysis and contrasts it withEq. (3), which the collapse postulate seems to imply [34]. Thepredictions of Eqs. (2) and (3) are nearly identical. Their localpredictions for S and A are identical, and both predict globalcorrelations such that |si〉 occurs if and only if |ai〉 occurs. Theonly physical difference is that Eq. (2) is entangled and, thus,predicts a nonlocal channel that Eq. (3) does not predict. Andon this count, Eq. (3) cannot represent the state we actuallyobserve following a measurement because the delayed-choicetwo-slit experiment [13] shows that, following a measurement,there is, indeed, a nonlocal channel between S and A. Wignersuggests that the measurement problem would be solved ifEq. (3) could be derived using only the unitary Schrodingerevolution. He then proves that this is impossible, concludingthat the equations of quantum physics do not predict definiteoutcomes. But we have seen that Eq. (2) does, in fact, implydefinite observed outcomes.

Adler [35] states that “the quantum measurement problemconsists in the observation that Eq. (2) is not what is observedas the outcome of a measurement! What is seen is not thesuperposition of Eq. (2), but rather either the unit normal-ized state [|s1〉|a1〉] or the unit normalized state [|s2〉|a2〉].”Adler’s “either/or” phrase is equivalent to Eq. (3). Similarly,Bassi and Ghirardi [36] derive Eq. (2) as a consequence ofquantum physics and then claim that Eq. (2) contradicts themeasurement postulate. We find this to be untrue.

V. CONCLUSIONS

When an apparatus A performs an ideal measurement thatdistinguishes between the terms of a superposed system S, thecomposite system SA evolves into the entangled MS, Eq. (2).Experimental and theoretical analyses of this state show it tobe the locally collapsed state of both S and A. Despite theseincoherent local states, the global composite state maintainsits unitary evolution with the coherence of the originalsuperposition now transferred to this global superposition.This MS exhibits, both experimentally and theoretically, acoherent nonlocal relationship between the phases of the two

022105-4

TWO-PHOTON INTERFEROMETRY AND QUANTUM STATE . . . PHYSICAL REVIEW A 88, 022105 (2013)

separated (perhaps widely separated) photons: The pairedphotons show a sinusoidal dependence on the differenceϕ = ϕS − ϕA between their individual phases ϕS and ϕA.This transfer of coherence from the local to the global stateis a consequence of unitarity and is surely fundamental tounderstanding the second law of thermodynamics in terms ofquantum physics.

In short, for ideal measurements both experiment andstandard quantum theory imply an instantaneous collapse tounpredictable but definite outcomes. The MS is the global formof the collapsed state with no need for a separate “process1” or measurement postulate [3]. The other postulates ofquantum physics imply that, when systems become entangled,their observed states instantly collapse into unpredictable

but definite outcomes. In particular, Schrodinger’s cat isin a nonparadoxical definite state, alive when the nucleusdoes not decay and dead when it does. This solution ofthe problem of outcomes requires no assistance from otherworlds, human minds, hidden variables, collapse mechanisms,collapse postulates, or “for all practical purposes” arguments.

ACKNOWLEDGMENTS

S. L. Adler, M. Sassoli de Bianchi, R. Brooks, J.Gea-Banacloche, A. J. Leggett, M. Lieber, D. Mermin, P.Milonni, P. Pearle, S. Rinner, M. Schlosshauer, M. O. Scully,S. Singh, and R. Vyas provided useful exchanges. A reviewerprovided important clarification.

[1] S. Weinberg, Phys. Rev. A 85, 062116 (2012).[2] M. Schlosshauer, Decoherence and the Quantum-to-Classical

Transition (Springer, Berlin, 2007).[3] J. von Neumann, The Mathematical Foundations of Quantum

Mechanics, (Princeton University Press, Princeton, 1955), trans-lated by R. T. Beyer from the 1932 German edition, Chaps. 5and 6.

[4] A. J. Leggett, Science 307, 871 (2005).[5] W. H. Zurek, Phys. Today 44, 36 (1991); Phys. Rev. D 24, 1516

(1981); 26, 1862 (1982); Prog. Theor. Phys. 89, 281 (1993);Phys. Rev. A 76, 052110 (2007).

[6] N. Gisin, Phys. Lett. A 154, 201 (1991).[7] S. Rinner and E. K. Werner, Central Eur. J. Phys. 6, 178 (2008).[8] M. O. Scully, R. Shea, and J. D. McCullen, Phys. Rep. 43, 485

(1978).[9] M. O. Scully, B. G. Englert, and J. Schwinger, Phys. Rev. A 40,

1775 (1989).[10] D. Dieks, Phys. Lett. A 142, 439 (1989).[11] D. Dieks, Phys. Rev. A 49, 2290 (1994).[12] O. Lombardi and D. Dieks, Stanford Encyclopedia of Philoso-

phy, http://plato.stanford.edu/entries/qm-modal/.[13] V. Jacques, E. Wu, F. Grosshans, F. Treussart, P. Grangier,

A. Aspect, and J.-F. Roch, Science 315, 966 (2007).[14] A. Hobson, Am. J. Phys. 81, 211 (2013).[15] F. Kaiser, T. Coudreau, P. Milman, D. B. Ostrowsky, and

S. Tanzilli, Science 338, 637 (2012); A. Peruzzo, P. Shadbolt,N. Brunner, S. Popescu, and J. L. O’Brien, ibid. 338, 634 (2012).

[16] J. G. Rarity and P. R. Tapster, Phys. Rev. Lett. 64, 2495(1990).

[17] J. G. Rarity and P. R. Tapster, Phys. Rev. A 41, 5139 (1990).[18] Z. Y. Ou, X. Y. Zou, L. J. Wang, and L. Mandel, Phys. Rev. Lett.

65, 321 (1990).[19] M. A. Horne, A. Shimony, and A. Zeilinger, Sixty-Two Years of

Uncertainty (Plenum, New York, 1990), p. 113.

[20] H. J. Bernstein, D. M. Greenberger, M. A. Horne, andA. Zeilinger, Phys. Rev. A 47, 78 (1993).

[21] K. Gottfried, Am. J. Phys. 68, 143 (2000).[22] A. Hobson, Physics: Concepts & Connections, 5th ed. (Pearson

Addison-Wesley, San Francisco, 2010), Chap. 13.[23] L. E. Ballentine and J. P. Jarrett, Am. J. Phys. 55, 696 (1987).[24] P. H. Eberhard and P. R. Ross, Found. Phys. Lett. 2, 127 (1989).[25] M. Jammer, The Philosophy of Quantum Mechanics (Wiley-

Interscience, New York, 1974), p. 115.[26] A. Aspect, J. Dalibard, and G. Roger, Phys. Rev. Lett. 49, 1804

(1982).[27] A. Aspect, Nature (London) 398, 189 (1999).[28] M. Lucibella, APS News 22, 6 (2013).[29] S. Haroche and J. M. Raimond, Exploring the Quantum: Atoms,

Cavities, and Photons (Oxford University Press, Oxford, 2006),p. 52.

[30] N. G. van Kampen, Physica A 153, 97 (1988).[31] J. Bell, Phys. World 3, 33 (1990).[32] L. Ballentine, Quantum Mechanics: A Modern Development

(World Scientific, Singapore, 1998); A. Galindo and P. Pascual,Mecanica Quantica I (Springer, Berlin, 1990), English languageversion; A. Goswami, Quantum Mechanics (Brown, Dubuque,IA, 1992); W. Greiner, Quantenmechanik. 1, Einfuehrung(Springer, Berlin, 1994), English language version; D. Griffiths,Introduction to Quantum Mechanics (Pearson/Prentice Hall,Upper Saddle River, NJ, 2005); A. Rae, Quantum Mechanics(Institute of Physics, Bristol, UK, 2002); R. Scherrer, Quan-tum Mechanics: An Accessible Introduction (Pearson/Addison-Wesley, San Francisco, 2006).

[33] J. Dunningham, A. Rau, and K. Burnett, Science 307, 872(2005).

[34] E. Wigner, Am. J. Phys. 31, 6 (1963).[35] S. L. Adler, Stud. Hist. Philos. Mod. Phys. 34, 135 (2003).[36] A. Bassi and G. C. Ghirardi, Phys. Rep. 379, 257 (2003).

022105-5