quantum contextuality in a young-type interference experiment
TRANSCRIPT
PHYSICAL REVIEW A 89, 052106 (2014)
Quantum contextuality in a Young-type interference experiment
Gilberto Borges,1 Marcos Carvalho,1 Pierre-Louis de Assis,2 Jose Ferraz,1 Mateus Araujo,3 Adan Cabello,1,4
Marcelo Terra Cunha,5 and Sebastiao Padua1,*
1Departamento de Fısica, Universidade Federal de Minas Gerais, Caixa Postal 702, Belo Horizonte, Minas Gerais 30123-920, Brazil2CEA-CNRS-UJF Group Nanophysique et Semiconducteurs, Institut Neel, CNRS, Universite Joseph Fourier, 38042 Grenoble, France
3Faculty of Physics, University of Vienna, Boltzmanngasse 5, 1090 Vienna, Austria4Departamento de Fısica Aplicada II, Universidad de Sevilla, E-41012 Sevilla, Spain
5Departamento de Matematica, Universidade Federal de Minas Gerais, Caixa Postal 702, Belo Horizonte, Minas Gerais 30123-920, Brazil(Received 22 April 2013; published 8 May 2014)
We show that correlations between the detector positions at the two-photon Young interference plane exhibitcontextual behavior. Contextuality is demonstrated by showing the violation of the n-cycle noncontextualityinequalities [M. Araujo et al., Phys. Rev. A 88, 022118 (2013)] for any even number n of observables rangingfrom 4 to 14. These violations exclude noncontextual hidden-variable theories as an explanation of the conditionaltwo-photon Young pattern. Unlike recent contextuality experiments, ours is free of the compatibility loophole.
DOI: 10.1103/PhysRevA.89.052106 PACS number(s): 03.65.Ud, 42.50.Dv, 42.65.Lm
I. INTRODUCTION
Young’s double-slit interference experiment is 200 yearsold and is still the simplest experimental evidence of the wavebehavior of light and matter [1]. At the quantum level it hasbeen used to demonstrate the wave properties of single parti-cles, i.e., electrons [2,3], neutrons [4], and atoms [5], and largemolecules [6]. Feynman introduces the double-slit experimentwith single particles as “a phenomenon which is impossible,absolutely impossible, to explain in any classical way, andwhich has in it the heart of quantum mechanics. In reality,it contains the only mystery” [7]. Double-slit experimentswith two particles also show intriguing quantum features.Greenberger, Horne, and Zeilinger showed that in some specialgeometric conditions a photon pair produces a conditionalinterference pattern [8]. Conditional means that the probabilityof detecting the photon pair (i,s) at some detector positions xi
and xs depends on the sum or difference of these variables,i.e., xi ± xs . In other words, the maximum probability ofdetecting a photon, for example, s, at position xs depends onthe position xi where photon i is detected. This prediction wasexperimentally verified for photon pairs generated by sponta-neous parametric down-conversion crossing a double slit in anentangled transversal path state [9]. The word “conditional”here is important since, depending of the geometric conditionsinvolving the photon-pair source and the double slit, othertype of fringes, i.e., independent fringes, can be observed ina two-particle Young interference pattern. In this case, themaximum probability of detecting a photon, for example, s, atposition xs does not depend on the position xi where photoni is detected in coincidence [8,9]. Recent works of Gneitingand Hornberger have derived an entanglement criterion forthe two-particle Young interference experiment using modularvariables and detected from the nonlocal (conditional) interfer-ence pattern [10,11]. This criterion was verified experimentallyin [12].
A fundamental question can be made about the two-particleYoung interference pattern: Is it possible to explain the statis-tics of pair detections (xs,xi) by modeling such phenomenaby noncontextual theories, i.e., by considering models wherethe appearance of photon s (i) at xs (xi) does not depend onthe experimentalist choice of where to put (or even not toput) the other detector Di (Ds)? This question is answeredin the present work. The answer to this question is importantsince contextuality distinguishes classical correlations usedfor studying condensed matter and biological systems [13–16]from genuine quantum correlations [17] used in quantuminformation for secure communication and faster informationprocessing in quantum computers [18,19]. Contextuality,demonstrated through the violation of noncontextuality in-equalities, i.e., joint measurement correlation inequalitiessatisfied by noncontextual hidden-variable theories, has beenobserved in various physical systems, e.g., ions [20,21],photons [22–24], and neutrons [25]. By making the rightmeasurements, one can observe that two entangled photons in adouble-slit setup produce the maximum violation predicted byquantum theory of several members of a family of inequalitiesthat exactly separate noncontextual from contextual models.A method developed to verify it is discussed below. Arelation between the detector position at the Fourier plane andthe required measurement operator for the noncontextualityinequalities test is established.
Quantum contextuality may be the key to understandingquantum correlations and, by extension, quantum theory fromfundamental principles [26,27]. It has been shown to be afundamental resource of quantum systems generating advan-tages in quantum information applications. Several authorshave shown that the origin of the speedup in some quantumcomputation schemes is quantum contextuality. Howard et al.showed that quantum contextuality is a critical resource forquantum speedup within the model for fault-tolerant quantumcomputation: magic state distillation [28]. Raussendorf hasshown that all measurement-based quantum computations thatcompute a nonlinear Boolean function with a high probabilityare contextual [29]. Nonlocality of quantum theory is aspecial case of contextuality where the unexpected context
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GILBERTO BORGES et al. PHYSICAL REVIEW A 89, 052106 (2014)
dependence is on the choice of measurements performed on aremote physical system. When only local quantum operationsand shared randomness are available, nonlocality emerges asa quantifiable resource in communication complexity [30] andpractical developments such as device-independent quantumkey distribution [28,31] and randomness certification [32].Without requiring entanglement or spatially separated com-posite systems, quantum contextuality provides protectionagainst attacks to quantum cryptographic protocols such asBB84 in which complementarity is imitated with classicalresources [33] and can be used to certify randomness [34].Quantum contextuality can also be used to increase the numberof classical messages that can be sent without error through aclassical channel [35].
In this paper we present experimental data that demonstratethat the conditional Young two-particle interference patternis contextual in different measurement scenarios, that is,when different pairs of compatible observables are measured.The two-photon Young interference pattern is used to testexperimentally the recently found n-cycle noncontextualityinequalities, the violation of which indicates the presenceof contextual correlations [36]. These inequalities containlinear functions of n correlations between pairs of observableschosen in a cyclic order.
The problem of finding the conditions that separate contex-tual from noncontextual correlations for an arbitrary measure-ment scenario is computationally intractable and the solution isknown only for a few scenarios [37]. A measurement scenariois defined as a set of observables and the subsets that arejointly measurable. The solution to this problem is given as aset of noncontextuality inequalities that define the boundaryof the set of noncontextual correlations. The most famous setsof noncontextuality inequalities are the ones that characterizethe Clauser-Horne-Shimony-Holt (CHSH) [38] and Klyachko-Can-Binicioglu-Shumovsky (KCBS) [39] scenarios, whichwere, respectively, the first nonlocality and contextualityscenarios to be completely characterized. Inequalities withspecific n were measured with a nonmaximally entangledstate for n = 6 [40] and n = 42 [41] observables, with thegoal of testing Hardy’s nonlocality without inequalities [42].Recently, CHSH and KCBS scenarios have been understood asthe n = 4 and 5 cases of a more general scenario, the n-cyclescenario. This measurement scenario consists of n dichotomicobservables Oj , with j = 0, . . . ,n − 1 and such that Oj
and Oj+1 (with the sum modulo n) are jointly measurable.The corresponding sets of inequalities have been found forarbitrary n [36]. This provides a valuable tool to investigatecontextuality and how it evolves with the number of settings.For this specific scenario, the quantum violation occurs for alln and only technical reasons make the observation of violationsharder for large n.
In this paper we demonstrate experimentally the violationsof a whole family of tight noncontextuality inequalitiesdescribed in Ref. [36] for any even number of settings rangingfrom n = 4 to 14. Tight means that these inequalities preciselydelimitate the border between the sets of noncontextualand contextual correlations. This work studies a differentaspect of the two-particle conditional Young interferencepattern and shows experimentally the contextual characteristicof it.
II. METHODS
To certify contextuality we use the violation of the noncon-textuality inequalities [36]
� =n−2∑j=0
〈OjOj+1〉 − 〈On−1O0〉NCHV� n − 2 (1)
for even n � 4, whereNCHV� n − 2 indicates that n − 2 is the
highest value allowed for noncontextual correlations. Anyexperimental test of a noncontextuality inequality shouldsatisfy two conditions. One is that the observables whosecorrelations are considered in the inequality should be jointlymeasurable, i.e., compatible [43]. For inequality (1), thismeans that Oj and Oj+1 should be jointly measurable forany j = 0, . . . ,n − 1. The second requirement is that everyobservableOj has to be measured using the same experimentalconfiguration in every context [44]. For testing inequality (1)this means that, for any j = 0, . . . , n − 1, the experimentalconfiguration used for measuring Oj should be the same whenOj is jointly measured with Oj+1 and when Oj is jointlymeasured with Oj−1. To ensure that these two requirementsare achieved in our experiments, we use a two-photon systemin which each photon encodes a photon path qubit initiallyprepared in the state
|φ+〉 = 1√2
(|00〉 + |11〉) (2)
and the detection plane implement measurements of theobservables
Oj ={
O(θj ) ⊗ I2 for even j
I2 ⊗ O(−θj ) for odd j,(3)
where I2 is the identity in the Hilbert space of one qubit andθj is jπ/n.
The fact that the measurements represented by Oj andOj+1 are always performed on different particles ensuresjoint measurability, avoiding the so-called compatibility loop-hole [43,45] present in recent experiments [20–25]. MeasuringOj together with Oj+1 or with Oj−1 does not require anychange in the experimental configuration for Oj , so we canguarantee that every observable is measured the same way inevery context (Fig. 1). For the prepared state (2) the maximalquantum value of � is attained for the observables
O(θj ) = cos(jπ/n)σx + sin(jπ/n)σy, (4)
where σx and σy are Pauli matrices.
III. EXPERIMENTAL IMPLEMENTATION
State |φ+〉 can be produced using a double slit (DS)to encode two qubits in the transversal path of photonpairs generated by spontaneous parametric down-conversion(SPDC) [46,47]. In the near field of the double-slit aperture,the quantum state is described by [48,49]
|φ+〉 = 1√2{|0〉s |0〉i + |1〉s |1〉i}, (5)
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O0
O1
O2O3
O4
O5
O0
O4
O1O3Detector ADetector B
FIG. 1. (Color online) Shown in the top left corner is a graphof compatibility for the n = 6 scenario. The vertices Oi representdichotomic observables and adjacent vertices represent jointly mea-surable observables. On the right is an example of the experimentalimplementation of two correlations O0O1 (blue) and O3O4 (green).In our experiment, different detector positions correspond to theimplementation of different observables.
where s (i) refers to signal (idler) photons and |j 〉 is the stateof the photon that crossed the slit j (j = 0,1). In Fig. 2 weshow a schematic view of our experimental apparatus, whichis divided in two parts, the first being the state preparation andthe second the detection setup.
For the state preparation, a continuous 50-mW diode laser,operating at 405 nm, is focused in the center of a 2-mm BiB3O6
crystal (BiBO) by a spherical lens (not shown in Fig. 2),f1 = 30 cm, generating collinear type-I SPDC pairs of photons.The photon pairs (λ = 810 nm) and the laser beam propagatealong the z direction and, immediately after the crystal, adichroic mirror (DM) reflects the pump beam and transmitsthe down-converted photons. The DS is placed 40 cm from theBiBO crystal, perpendicular to the z direction; each slit of theDS has width 2a = 80 μm, along the x direction; the center tocenter separation is d = 160 μm. Between the crystal and the
FIG. 2. (Color online) Experimental setup. (a) State preparation.A cw laser generates collinear photon pairs in a BiBO crystal. Anonconfocal telescope, consisting of a spherical lens (SL) and of acylindrical lens (CL), projects the magnified image of the crystal inthe DS plane x direction. A dichroic mirror (DM) reflects the pumpbeam. (b) Detection system. The lens FL is used to project the DS farfield in the detection plane. Here BS is a 50:50 beam splitter and DA
and DB are APD detectors.
DS, a linear optical setup controls the quantum correlationsof the two photons in the aperture plane [48]. This setup iscomposed of a cylindrical lens (CL) of focal length fCL =5 cm and a spherical lens (SL) with focus fSL = 20 cm.This scheme projects a magnified image of the crystal centeronto the DS plane in the x direction. With this experimentalconfiguration, we are able to control the correlations of thedown-converted photons such that the photon pair alwayspasses through the same slit of the DS, thus forming theentangled state represented in Eq. (2). The length of the DS’slarger dimension along the y direction is 8 mm, which ismuch larger than the down-converted beam width and can beconsidered infinite.
The detection system is set to project the optical Fouriertransform of the DS at the detection plane, as shown inFig. 2(b). In this scheme, we use a spherical lens (FL) of focallength fFL = 30 cm in the f -f configuration and two avalanchephotodiode (APD) detectors at the exit port of a balanced beamsplitter (BS). The detectors are placed 60 cm from the DS planeand equipped with interference filters, centered at 810 nm(10-nm full width at half maximum bandwidth). In front ofeach detector we have a pinhole with diameter 2b = 200 μm,for spatial filtering. The APD detectors are mounted intranslation stages and can be scanned in the x direction (DA)or in the z direction (DB). Coincidences between the detectorsare obtained with a homemade electronic circuit with 5.4 nsof temporal window. A two-dimensional 25 × 25 array ofcoincidence counts was obtained by scanning both detectorsin the x direction of the far-field plane, with a step length of100 μm and an acquisition time of 30 s for each point.
IV. EXPERIMENTAL RESULTS AND DISCUSSION
In order to test experimentally the violation of the inequal-ity (1), we first need to find the correspondence between thecoincidence counts and our observables, defined in Eq. (3),i.e., we need to know how the measurement operator isimplemented when the coincidence counts are acquired in theDS far field. The mathematical description of the measurementoperator in the transversal direction at the focal plane is
�(xs,xi) = �s(xs) ⊗ �i(xi), (6)
where each single system operation �ν(xν) (ν = i,s) is givenby
�ν(xν) =Aν(xν){I2 + sinc(κb)
[cos(κxν)σ ν
x + sin(κxν)σ νy
]},
(7)
where I2 is identity operator, κ = kpd/2fFL, kp is the pumpbeam wave number, 2b is the transversal dimension of thedetectors, and xν (ν = s,i) is the detector transversal position.The Pauli matrices σ ν
x and σ νy in Eq. (7) are written in terms
of the slit (or photon path) states |j 〉 (j = 0,1). The functionsAν(xν) are given by
Aν(xν) = kpa
2πfFLsinc2
(κxν
d
), (8)
which is the diffraction envelope of the interference pattern.The measurement operator is physically implemented bythe FL spherical lens in the f -f configuration and the
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APD detectors at the FL focal plane [see Fig. 2(b)]. Themeasurement operator �ν(xν) is equal to E−
ν (xν)E+ν (xν),
with E+ν (xν) [E−
ν (xν)] being the positive- (negative-) fre-quency electric-field operator at the detection plane, and iswritten in terms of the slit states |j 〉 (j = 0,1) [46,47,50].
The expected value of operator �(xs,xi) [see Eq. (6)]is proportional to the coincidence counts at the detectorpositions xs and xi . The connection between the mean valueof the jointly measurable dichotomic observables and theexperimental coincidence counts is given by
〈Os(θs) ⊗ Oi(θi)〉 = C (θs,θi) + C (θs − π,θi − π ) − C (θs − π,θi) − C (θs,θi − π )
C (θs,θi) + C (θs − π,θi − π ) + C (θs − π,θi) + C (θs,θi − π ), (9)
where C (θs,θi) is the coincidence count associated with the θs
and θi interference pattern angular position and Oν (ν = s,i)are given by
Oν = cos(θν)σx + sin(θν)σy. (10)
Equation (9) is general, for any pair of angles θi and θs . Theangular settings of Eqs. (3) and (9) are related to the detectorpositions by
θν = κxν, (11)
which explains how each pair of detector position relates toeach term in the inequality (1) (see again Fig. 1).
The violation of the noncontextuality inequalities for then cycle is achieved when we use n/2 different positions foreach detector, registering the coincidence counts to calculatethe correlations coefficients of Eq. (9). To better explain theprocedure utilized, let us use the n = 4 (CHSH) scenario as anexample. In this case, an optimal setting for the signal subsys-tem measurement apparatus is {θs} = {0,π/2} and for the idler{θi} = {π/4,3π/4}. To obtain the correlation coefficients it isalso necessary to use the coincidence counts obtained fromthe implementation of the orthogonal measurement operators.The subsystem angles are then changed by a factor of π ,i.e., {θs − π} = {−π,−π/2} and {θi − π} = {−3π/4,−π/4}.Since the scanning process does not generate all the optimalpoints for maximal quantum violation, we use the nearestavailable experimental data. This problem reduces the amountof violation, but not dramatically. As we increase the valueof n, the above procedure is repeated, until we reach then = 14 value, above which the angular separation betweentwo consecutive observables in the same subsystem is smallerthan the experimental resolution of the detectors position.
TABLE I. Values obtained for �: �expt is associated with the rawexperimental coincidence counts; �bd
expt is the bound value obtained bythe theory when we use the angles implemented by the experiment,i.e., γexpt = κ(xi − xs); and �bd
max is the maximal value allowed forquantum mechanics, which corresponds to using the angles γ = π/n.
n �expt �bdexpt �bd
max
4 2.73 ± 0.02 2.73 2.836 4.90 ± 0.02 5.11 5.208 7.02 ± 0.02 7.25 7.3910 8.80 ± 0.02 9.25 9.5112 10.82 ± 0.02 11.25 11.5914 12.67 ± 0.02 13.25 13.65
In Table I we report our experimental results for each n. Thetable shows three different values of �. The first one is obtainedusing the experimental coincidence counts, denoted by �expt.The second one, denoted by �bd
expt, is the value predictedby quantum mechanics assuming ideal equipment when weconsider that the angular separation between two operatorsthat define a context is γexpt = κ(xi − xs) instead of γ = π/n.In other words, this value of � is the one obtained when we usequantum mechanics, the angles experimentally implemented,and the maximally entangled state defined by Eq. (2), but noother source of imperfection (such as the preparation of anonmaximally entangled state). The third value �bd
max is thequantum bound using the ideal angular separation γ = π/n.It is important to have in mind that the limit for noncontextualcorrelations is given by n − 2, as is shown in inequality (1).Error bars are statistically calculated. It might seem surprisingthat the error does not increase with the number of settings. Aquick calculation shows us, though, that for an ideal setup thevariance of � is
(��)2 = 1
Nn sin2(π/n) � π2
nN, (12)
where n is the number of settings and N is the total number ofcounts. That is, the error actually decreases with the numberof settings.
4 6 8 10 12 14
0.75
0.8
0.85
0.9
0.95
1
n
(Ω+
n)/
2n
Ωbdmax
Ωexpt
Ωbdnc
FIG. 3. (Color online) Experimental results �expt, noncontextualbounds �bd
nc , and Tsirelson bounds �bdmax for the n-cycle scenario. In
all three cases we apply the transformation � �→ (� + n)/2n, so thevalues can be interpreted as probabilities of success in the n-cyclegame (see [51]).
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In Fig. 3 we plot the data from Table I transformed via� �→ (� + n)/2n. Incidentally, this renormalization allowsus to interpret these values as the probability of success inthe n-cycle game, which is a direct generalization of theprediction game proposed in [51]. As it is characteristic forthe families of n-cycle inequalities, the violations get smallerwith increasing n. However, it is interesting to stress that ourdata show significant violations for all n tested and if we werenot limited by the positions of the detectors this setup couldshow significant violations above n = 14.
V. CONCLUSION
In this work we provided an experimental verificationthat the conditional Young two-particle interference pattern iscontextual. The system of two photonic qubits, encoded in thetransversal modes of entangled photons after being transmittedby a double slit, creates a probability distribution that isincompatible with any noncontextual hidden-variable theory.Our experiment tests the violation of the noncontextualityinequalities for the n-cycle scenario for even n, recentlyreported in [36]. It is important to stress that this is anexperimental violation of noncontextuality inequalities for acompletely characterized infinite family of scenarios and that
we have observed experimental violations for even n up ton = 14. Unlike recent contextuality experiments, our is freeof the compatibility loophole, since the joint measurabilityof each pair of observables is enforced by encoding them asobservables on different photons.
With a single two-particle conditional double-slit interfer-ence experiment, we were able to demonstrate the violationof six different noncontextuality inequalities with a number ofsettings ranging from 4 to 14. In this way, we have observedcontextuality (in the sense of violation of the noncontextualityinequalities) in very good agreement with the predictions ofquantum mechanics for six different scenarios. Our theoreticaland experimental approaches can be used in different degreesof freedom of photonic systems or in massive particle systemsthat produce spatial conditional interference patterns [10,52].
ACKNOWLEDGMENTS
This work was supported by CNPq, CAPES, FAPEMIG,National Institute of Science and Technology in Quantum In-formation, the Science without Borders Program (CAPES andCNPq, Brazil), and Project No. FIS2011-29400 (MINECO,Spain) with FEDER funds.
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