Confined Contextuality in Neutron Interferometry: Observing the Quantum Pigeonhole Effect
Mordecai Waegell, Tobias Denkmayr, Hermann Geppert, David Ebner, Tobias Jenke, Yuji Hasegawa, Stephan Sponar, Justin Dressel, Jeff Tollaksen
CConfined Contextuality in Neutron Interferometry:Observing the Quantum Pigeonhole Effect
Mordecai Waegell,
1, 2
Tobias Denkmayr, Hermann Geppert, David Ebner, TobiasJenke, Yuji Hasegawa, Stephan Sponar, Justin Dressel,
1, 2 and Jeff Tollaksen
1, 2 Institute for Quantum Studies, Chapman University, Orange, CA 92866, USA Schmid College of Science and Technology, Chapman University, Orange, CA 92866, USA Atominstitut, TU-Wien, Stadionallee 2, 1020 Vienna, Austria Institut Laue-Langevin 6, Rue Jules Horowitz, 38042 Grenoble Cedex 9, France
Previous experimental tests of quantum contextuality based on the Bell-Kochen-Specker (BKS) theorem havedemonstrated that not all observables among a given set can be assigned noncontextual eigenvalue predictions,but have never identified which specific observables must fail such assignment. We now remedy this short-coming by showing that BKS contextuality can be confined to particular observables by pre- and postselection,resulting in anomalous weak values that we measure using modern neutron interferometry. We construct aconfined contextuality witness from weak values, which we measure experimentally to obtain a σ averageviolation of the noncontextual bound, with one contributing term violating an independent bound by more than σ . This weakly measured confined BKS contextuality also confirms the quantum pigeonhole effect, whereineigenvalue assignments to contextual observables apparently violate the classical pigeonhole principle. I. INTRODUCTION
Quantum contextuality, as introduced by Bell, Kochen andSpecker (BKS) [1, 2], forbids all observable properties of asystem from being predefined independently from how theyare observed. This phenomenon is one of the most counter-intuitive aspects of quantum mechanics, and finds itself at theheart of recent quantum information processing applications[3–8]. The BKS theorem is proved by exhibiting a
BKS-set of observables [9, 10] that contains geometrically related andmutually commuting subsets (or measurement contexts ) thatresult in a logical incompatibility: Any noncontextual hiddenvariable theory (NCHVT) that pre-assigns eigenvalues glob-ally to the entire BKS-set (i.e., non contextually) results ina contradiction with the predictions of quantum mechanics.That is, at least one eigenvalue in a global assignment to aBKS-set cannot be predefined without violating a constrainton the product of eigenvalues within some context, which wecall the contradictory context . See Appendix A for a reviewof the BKS theorem and how the BKS-sets used in this articleprove it.Previous contextuality experiments [11–13] have confirmedsuch a global contradiction. However, neither the BKS theo-rem, nor these experiments, specify which contexts are con-tradictory. In this article, using recently developed weakmeasurement techniques in neutron interferometry [14–20],we experimentally demonstrate which specific measurementcontext within a BKS-set (Fig. 1a) must contain contradic-tory value assignments, essentially confining the contextual-ity [21]. Like squeezing a balloon, we condition the BKS-setthrough pre- and postselection (Fig. 1b) to force the contra-diction to appear in a particular context (Fig. 1c) [22–24]. Re-markably, measuring the weak values [25] within that con-text explicitly reveals the contradiction that is left implicitin the original BKS proof. The measured weak values vi-olate the classical pigeonhole principle [26], and contradictNCHVT value assignments to the projectors in that context,which we call forbidden projectors . We show that the confine- ment of contextuality in the quantum pigeonhole effect forcessome of the forbidden projectors to have negative weak val-ues. The appearance of these negative weak values thus wit-nesses the confined contextuality, making the forbidden pro-jectors witness observables for contextuality. These witnessescorroborate recent results [27–29] that link negative projectorweak values to contextuality using Spekkens’ generalizationof contextuality [30], which encompasses the original notionof BKS.In our experiment, we witnessed the BKS-contextuality ofneutron spin. We measured the spin using neutron interferom-etry by performing path-dependent spin rotations, making thepath a weakly-coupled meter for the spin (Fig. 2); condition-ing the path measurements on spin postselections then revealsthe desired weak values [19]. We collected seventeen inde-
ZZIZIZ IZZ +1 +1 +1
ZZIXXIYYIZIZXIXYIY IZZIXXIYY (a) BKS 3-spin Square (b) Pre- / postselection (c) Paradoxical values +1+1+1 +1+1+1+1 +1 +1-1 -1 -1
Figure 1. Confining Bell-Kochen-Specker (BKS) contextuality in the3-spin Square. Each row or column (measurement context) of theSquare mutually commutes. (a) According to quantum mechanics,the product of the three 3-spin measurement outcomes in each rowis +1 (thin line), while their product in each column is − (thickline). (b) A particular preparation and postselection fixes the valuesof two rows. (c) In any noncontextual hidden variable theory, theremaining values must be − , which confines the BKS contradictionto the top row (blue dashed line). This also demonstrates the quan-tum pigeonhole paradox: all pairs in the row appear anticorrelated,which violates the classical pigeonhole principle. Weak measure-ments confirm the paradox, revealing the correlation of each pair tobe − . ± . , − . ± . , and − . ± . , from leftto right. a r X i v : . [ qu a n t - ph ] O c t pendent data sets of neutron spin measurements, indexed n =1 , . . . , . We use these single-spin data sets to show con-fined contextuality within the N -spin Wheel BKS-sets [10](see Appendix A) for odd numbers of spins N = 3 , , . . . , ,using data sets n = 1 , . . . , N and the following simplifica-tion. While most contexts in an N -spin Wheel are entangled,we use separable pre- and postselection to fix the eigenvaluesof certain observables (as in Fig. 1b), and by the definition ofnoncontextuality, any NCHVT must assign the same eigenval-ues to those observables in the entangled contexts as it does inthe separable contexts — just as in [22]. This would be trueeven if we were considering N N -spin Wheel BKS-set. Weare not claiming this is the same as performing genuine N -spin measurements; we are claiming that N single-spin mea-surements are sufficient to reveal the contradiction inherent inthe N -spin Wheel BKS set, between quantum mechanics andNCHVTs.Weak values do not appear shot-by-shot in our experiment,but only as conditioned averages from ensembles of identi-cally preselected and postselected data. The fact that we areconstructing N -spin weak values from single-spin weak valuemeasurements may seem odd, but this construction is gen-erally valid for any averages from probability distributionsdescribing independent (separable) systems — including ourdistinct sets of neutron measurements. That is, the completeset of collected single-neutron data was naturally divided into17 smaller and independent subsets, each collected sequen-tially in time to minimize experimental drift, and each chosento be sufficiently large to achieve acceptable statistical errorfor estimating a single-spin weak value. The number 17 waslimited only by the total collected statistics, which was lim-ited by the neutron flux from the reactor and the stability ofthe experimental setup. The measured N -spin witnesses vio-late their noncontextuality bounds by (cid:38) σ , showing that thecontextuality was indeed confined in our experiment (Fig. 3).One particular 5-spin weak value exceeded its bound by morethan σ . The N -spin confined contextuality observed herealso violates the classical pigeonhole principle for putting N pigeons in 2 boxes. Our experiment verifies the quantum pre-dictions, and rules out NCHVTs of quantum mechanics.This article is organized as follows. In Section II, we dis-cuss how to confine contextuality to particular contexts withpre- and postselection, and the relationship between such con-fined contextuality and the quantum pigeonhole effect. InSection III we show how the confined contextuality of theWheel family of BKS-Sets permits the construction of N -spincontextuality witnesses that may be factored into measurablesingle-spin weak values. In Section IV, we detail the experi-mental procedure used to measure the single-spin weak values using neutron interferometry. In Section V, we summarizethe main results for the N -spin contextuality witnesses. Weconclude in Section VI. For completeness, we also includetwo appendices. In Appendix A, we describe the constructionof the Wheel family of BKS-sets used in the main text. InAppendix B, we provide additional details about the experi-mental determination of the single-spin weak values used toconstruct the results reported in the main text. II. CONFINING BKS CONTEXTUALITY
BKS contextuality confinement follows from theAharonov-Bergmann-Lebowitz (ABL) formula [31], whichgives the probability of obtaining a particular strong mea-surement outcome j , between a preparation | ψ (cid:105) and apostselection (cid:104) φ | . The outcome j corresponds to a projec-tion operator Π j that is part of a complete measurementbasis B (i.e., context) such that (cid:80) j ∈B Π j = I . TheABL formula can be expressed in terms of weak values (Π j ) w = (cid:104) φ | Π j | ψ (cid:105) / (cid:104) φ | ψ (cid:105) [21], P ABL (Π j = 1 | ψ, φ, B ) = | (Π j ) w | (cid:80) k ∈B | (Π k ) w | . (1)It then follows from (cid:80) k ∈B (Π k ) w = 1 that P ABL (Π j =1 | ψ, φ, B ) = 1 implies (Π j ) w = 1 . Furthermore, if B contains only two outcomes, then the converse also follows: (Π j ) w = 1 implies P ABL (Π j = 1 | ψ, φ, B ) = 1 . As shownin Ref. [24], the ABL formula constrains any NCHVT since aprojection with an ABL probability of 1 must also be assigneda value of 1 in any NCHVT. Thus, in this case, measuring aprojector weak value (Π j ) w of implies that any NCHVTmust also assign Π j a value of — and a value of to allprojectors orthogonal to Π j .Specifying to N independent neutron spins, we use I, X, Y, Z to denote the independent spin components (Paulimatrices). We prepare the spins in the product state | ψ (cid:105) = | + X (cid:105) ⊗ N (all X eigenvalues +1), and postselect onto the prod-uct state | φ (cid:105) = | + Y (cid:105) ⊗ N . Since the predictions of productsof X and Y by an NCHVT must be consistent with theseboundary conditions, only products involving Z are left unde-termined (see Fig. 1b). The ABL rule then determines thesevalues, as we now explain.For our specific case of N > spins, consider a productof any two spin operators ZZ , with spectral decomposition ZZ = (+1)Π even + ( − odd in terms of the rank-2 parityprojectors, Π even = Π + ⊗ Π + + Π − ⊗ Π − , (2) Π odd = Π + ⊗ Π − + Π − ⊗ Π + , with Π ± ≡ |± Z (cid:105)(cid:104)± Z | = (1 ± Z ) / . Given | ψ (cid:105) and | φ (cid:105) defined above, (Π even ) w = 0 and (Π odd ) w = 1 , and thus ( ZZ ) w = − . The ABL rule in Eq. (1) then implies ZZ = − for all pairs of spins in any NCHVT, as illustrated in Fig. 1.This pairwise constraint is the quantum pigeonhole effect [26]. To see this, let the spin eigenstates |± Z (cid:105) correspond totwo boxes in which pigeons may be placed. The projectors inEq. (2) describe definite numbers of pigeons in each box, up toan exchange of boxes; i.e., Π even denotes two pigeons in onebox, while Π odd denotes one pigeon in each. The pigeonholeprinciple states that if N > pigeons are placed in two boxes,then at least one box must contain multiple pigeons. However,the constraint ZZ = − for all pairs implies that, regardlessof how many pigeons are placed in the two boxes, no twopigeons are ever in the same box! III. WITNESSING BKS CONTEXTUALITY
Following the pigeon analogy, all NCHVT assignments ofdefinite numbers of pigeons to each box are forbidden. Theprojectors corresponding to such forbidden assignments for N = 3 are Π (3)0 = Π + ⊗ Π + ⊗ Π + + Π − ⊗ Π − ⊗ Π − , (3) Π (3)1 = Π + ⊗ Π + ⊗ Π − + Π − ⊗ Π − ⊗ Π + , Π (3)2 = Π + ⊗ Π − ⊗ Π + + Π − ⊗ Π + ⊗ Π − , Π (3)3 = Π + ⊗ Π − ⊗ Π − + Π − ⊗ Π + ⊗ Π + , which are the invariant eigenspaces of the first row of Fig. 1( Π (3)0 indicates all three pigeons in one box, while Π (3)1 , , arethe permutations of two and one). Any NCHVT assigns 0 toall forbidden projectors. We call complete sets of forbiddenprojectors like this contextual bases .Crucially, such forbidden BKS value assignments manifestas anomalous projector weak values (with real part outside therange [0 , ) in the contextual basis of Eq. (3) [21]—classicalassignments of pigeons to boxes must respect the range [0 , .An anomaly indicates contradictory noncontextual value as-signments to the corresponding context. Thus, the forbiddenprojectors constitute witness observables such that negativeweak values imply confined BKS contextuality, and contradictthe assignment of 0 by an NCHVT. These witnesses promotethe logical contradiction of the quantum pigeonhole effect intoan experimentally robust inequality.For the explicit example in Fig. 1, the weak value of Π (3)0 is (Π (3)0 ) w = (cid:89) n =1 Z ( n ) w (cid:89) n =1 − Z ( n ) w − , (4)with | ψ (cid:105) and (cid:104) φ | above, where each n is a distinct spin, and Z w = (cid:104) + Y | Z | + X (cid:105) / (cid:104) + Y | + X (cid:105) = i is a purely imaginarysingle-spin weak value, implying ( ZZ ) w = ( Z ) w ( Z ) w = − . This example also illustrates a subtle point about theconnection between anomalous weak values and contextual-ity. As discussed above, a projector weak value for a sepa-rable composite system with a negative real part is a witnessof contextuality. However, the projector weak values for eachspin are (Π ± ) w = (1 ± Z w ) / e ± iπ/ / √ , which havepositive real parts. Nevertheless, the product of three suchweak values, (Π ⊗ ± ) w = (Π ± ) w = e ± i π/ / √ , has a nega-tive real part, enabling the contextuality witness. In this sense, the observation of a nonzero phase for a projector weak valueon a single system already implies a contextuality witness ona larger composite system.The logic for the above construction for N = 3 generalizesto odd N > (see the family of Wheel BKS-sets [10]). Thatis, all classical assignments of N pigeons to 2 boxes are for-bidden. Analogously to Fig. 1, the contextuality is confinedto a context in the N -spin Wheel-set consisting of N pairwiseobservables ZZ arranged in a ring. All pairs ( ZZ ) w = − asbefore, since each Z w = i . We label the invariant eigenspaceprojectors corresponding to this context by defining the N -digit binary sequences x ( N ) j , j ∈ . . . N − − , e.g., x (3)0 =(0 , , , x (3)1 = (0 , , , x (3)2 = (0 , , , x (3)3 = (0 , , .The weak values of the N − forbidden projectors (witnessobservables) in this contextual basis are then (Π ( N ) j ) w = N (cid:89) n =1 − x ( N ) j,n Z ( n ) w N (cid:89) n =1 − ( − x ( N ) j,n Z ( n ) w , (5)where x ( N ) j,n is the n th digit of x ( N ) j . As in Eq. (4), these pro-jector weak values may be computed from N single-spin Z w values. This great simplification enables us to construct allforbidden projectors for any number of spins by measuringsingle-spin Z w . All projector weak values then evaluate to (Π ( N ) j ) w = ± − ( N − / , with a sign depending on the index.Finally, we construct an unbiased contextuality witness C ( N ) , using all N − rank-2 projectors in an N -spin contex-tual basis, that aggregates the contextuality of the entire basis, C ( N ) = I ( N ) − N − − (cid:88) j =0 s j Π ( N ) j , (6)with s j = sign [ Re (Π ( N ) j ) w ] , using the predicted value of (Π ( N ) j ) w . Regardless of the signs s j , if all ≤ Re (Π ( N ) j ) w ≤ , then Re C ( N ) w ≥ . Observing Re C ( N ) w < is thus anexperimental witness of confined BKS-contextuality. Thischoice of the signs s j optimizes C ( N ) w by accumulatinganomalous parts of the weak values (below 0 or above 1), pro-ducing the ideal values Re C ( N ) w = 1 − ( N − / . IV. EXPERIMENTAL PROCEDURE
In our experiment, we measure the weak value Z w of theneutron spin in the z -direction using an interferometer. Theneutron’s path is used as a pointer to measure both the real andimaginary parts of Z w . This approach has already been suc-cessfully used to completely determine weak values of mas-sive systems [19]. The experiment was conducted at the in-strument S18 at the high flux research reactor of the InstituteLaue-Langevin (ILL) in Grenoble, France. The experimentalsetup is depicted in Fig. 2.A perfect silicon crystal selects neutrons with a wavelengthof λ = 1 . ˚A ( λ/λ ∼ . ) by Bragg reflection froma white neutron beam [19]. Between the monochromator Figure 2. Experimental setup. The unpolarized neutron beam passesa magnetically birefringent prism (P) that permits only spin-up neu-trons to fulfill the Bragg condition for entering the interferometer. Toprevent depolarization, a magnetic guide field (GF) is applied overthe whole setup. A DC coil (DC1) aligns the incoming neutron spinalong the positive x direction. Inside the interferometer, the neutronssplit into two paths (P1) and (P2), where two spin rotators (SRs) canindependently rotate the neutron spin in the xy plane. A cadmiumslab (CD) can optionally block one of the paths. To tune the relativephase χ between the path eigenstates, a phase shifter (PS) is insertedinto the interferometer. After the interferometer, a second DC coil(DC2) mounted on a translation stage, in combination with a polar-izing supermirror (A), postselects a specific spin component. Theneutrons are detected by He detectors (O & H). and the interferometer crystal, two magnetically birefringentprisms (P) split the unpolarized beam in two beams, one withthe neutron spin aligned parallel to the positive z -directionand one aligned antiparallel. Even though the angular separa-tion is just four seconds of arc (exaggerated in Fig. 2), onlythe beam with spin up component fulfills Bragg’s condition atthe interferometer’s first plate. The degree of polarization isabove 99% with the neutron spin state given by | + Z (cid:105) . Theother beam passes through unaffected and does not furthercontribute to the experiment.A DC coil (DC1) in front of the interferometer generatesa constant magnetic field B y in y -direction. After enteringthe coil, the neutron experiences a non-adiabatic field changeand its spin starts to precess around B y . If the magnetic fieldmagnitude is adjusted accordingly, the neutron spin will turnby exactly π/ in the coil. This changes the initial spin statefrom | + Z (cid:105) to | + X (cid:105) , completing the spin preselection.At the first interferometer plate, the beam is coherently splitby amplitude division. In each path, (P1) and (P2), a spin rota-tor (SR)—small coil in a Helmholtz configuration—producesa weak magnetic field in the ± z direction. To prevent thermalstress on the interferometer, the coils are water cooled. Theweak magnetic fields lead to path-dependent spin rotationsaround the field axis, causing (weak) entanglement betweenthe spin and path degrees of freedom of each neutron. Forall measurements, the angle of rotation was set to α = 15 ◦ .The infidelity I = sin α between the partial path states cor-responding to z = ± quantifies the measurement strength.Our weak measurement has infidelity I = 0 . [19], com-pared to a strong measurement with α = 90 ◦ and I = 1 .Between the second and final interferometer plate, a sap-phire phase shifter (PS) is inserted. A phase shifter in com-bination with a Cadmium beam block (CD) mounted on a ro-tational stage provides full control over the neutron’s path for the pointer readout. The phase shifter can change the pathstate in the equatorial plane of the Bloch sphere, while thebeam block permits access to the path eigenstates at the poles.At the final interferometer plate, the two paths are recom-bined. A second DC coil (DC2), in combination with a CoTisupermirror array [32, 33] (A), enables arbitrary spin-statepostselection. The neutrons are detected by He counter tubes(O) and (H). Of the two outgoing ports, only (O) is analyzedto postselect the spin state (cid:104) + Y | .All measurements were performed using an IN/OUTmethod. For each fixed phase shifter position, the intensity isrecorded with the spin-path coupling field turned on (IN), andthen the intensity is recorded with the coupling field turnedoff (OUT). Background intensities are also recorded in or-der to calibrate the counters. This method permits the spin-independent relative phase χ to be determined for the pathpostselection state ( | P (cid:105) + e iχ | P (cid:105) ) / √ . After curve-fittingan intensity scan over χ on the Bloch sphere equator (seeAppendix B), the intensities for the y -eigenstates at points χ = π/ , π/ are identified by inserting the phase valuesobtained from the OUT measurements into the IN measure-ment fit functions. These intensities determine the real part ofthe spin weak value Re Z w (Eq. (19) of [19]). This methodalso maximally reduces the influences of phase drift in the in-terferogram (due to unavoidable instability of the apparatus).To determine the imaginary part Im Z w , it is also necessary topostselect neutron path eigenstates (Eq. (20) of [19]), whichis accomplished by blocking one path at a time. If an inten-sity is recorded while path P is blocked, a postselection ontothe state | P (cid:105) is performed, and vice versa. For our choice ofpre- and postselection, the expected weak value is Z w = i .The negligible phase shift observed between the IN and OUTinterferograms confirms that Re Z w ≈ . In contrast to that,the imaginary part shifts the pointer state towards the Blochsphere poles, changing the relative path intensities.In the experiment the weak values Z w of 17 individual spinswere determined with high precision. To extract Z w from oneneutron spin data set, two χ -scans were recorded, as well astwo single intensities. Together with the required backgroundmeasurements, a total collection time of ∼ seconds wasneeded to determine the real and imaginary part of each Z w . V. RESULTS
The measured Z w are used to construct the pairwise anti-correlations ( ZZ ) w ≈ − (see Appendix B), and the N -spinwitnesses in Eq. (5) and (6). Fig. 3a,b shows the final re-sults that violate the noncontextuality bound Re C ( N ) w ≥ .Fig. 3c,d shows final results that violate independent noncon-textuality bounds Re (Π ( N ) j ) w ≥ . The contextuality wit-nesses C ( N ) w and (Π ( N ) j ) w were calculated using Eqs. (6) and(5), respectively, for all odd numbers of spins from N = 3 to17. Note that the pair of forbidden projectors Π (5)0 and Π (13)0 have the remarkable geometric property that first order errorsvanish when Z w = i , explaining the small statistical stan-dard deviation σ observed in the experimental data. The cho- − − − − − − − − − (a) (c)(b) (d) − − − − − − − − − − − − − − − − Figure 3. Experimental results witnessing confined contextuality for N -spin Wheel BKS-sets, for N = 3 , , , . . . , . While noncontextualhidden variable predictions for the witness observables lie above the classical bound of 0 (red, dashed), quantum predictions (blue, solid) andexperimental data (orange, error bars showing one standard deviation) violate this bound. (a) Unbiased witnesses C ( N ) w given in Eq. (6) thattest an entire context, vs. N . (c) Exemplary set of projector witnesses (Π ( N ) j ) w from within a context, as given in Eq. (5), with specific indices j for each N given in the text. (b,d) Violation of classical boundary in units of statistical standard deviations σ , corresponding to (a) and (c),respectively. sen witnesses for other N in Fig. 3c,d are the projectors Π (3)0 , Π (7)1 , Π (9)3 , Π (11)7 , Π (15)1 , and Π (17)3 . The data for N = 5 ismost statistically significant, with Re C (5) w = − . ± . vi-olating the bound of 0 by ∼ σ , and Re (Π (5)0 ) w = − . ± . by ∼ σ . VI. CONCLUDING REMARKS
We have experimentally shown the confinement of contex-tuality within a BKS-set of observables to a particular mea-surement context, using modern techniques in neutron inter-ferometry to measure weak-valued contextuality witnesses.Using N -spin Wheel BKS-sets [10], we have reduced theproblem of witnessing contextuality to weakly measuring aparticular context, consisting of neighboring pairs of observ-ables ZZ arranged in a ring, with the remaining observablesin the BKS-set fixed by a particular pre- and postselection. Itfollows that ( ZZ ) w = − for all such pairs, implying anticor-relation that violates the classical pigeonhole principle [26].Moreover, the weak values of the invariant subspace projec-tors (Π ( N ) j ) w of this context contain anomalies, witnessingthe failure of classical value assignments. Our unbiased con-textuality witness C ( N ) w uses all such projector weak valueswithin the context to witness the same failure.Unlike the implicit global contradictions inherent to exist- ing BKS experiments [11–13], our method confines the ap-parent contradiction to a particular context, where its physicalconsequences may be explicitly revealed through weak mea-surements. Notably, unlike existing approaches to demon-strating BKS-contextuality [27], our witness does not requireentangled preparations or measurements, or indeed any inter-action between the different spins at all. The entangled mea-surement contexts that would normally be required have val-ues that are forced by the pre- and postselection according tothe geometry of the BKS-set itself, so they need not be mea-sured. In this way, confining the contextuality serves to sim-plify its experimental observation. Such a simplification notonly raises interesting foundational questions [26], but mayalso suggest future quantum information processing applica-tions [3, 6]. ACKNOWLEDGMENTS
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The family of N -spin Wheel KS sets [10] prove the BKStheorem [1, 2] for all odd N ≥ , with the 3-spin Wheel pre-sented as the 3-spin Square [9] in the main text for compact-ness, and the 5-spin Wheel shown in Fig. 4. Each Wheel setcontains three rings composed of the N pairwise Pauli observ-ables ZZ , XX , and Y Y respectively, of neighboring pairsin a ring of N spins. Each Wheel also contains N ‘spokes,’which contain the three observables ZZ , XX , and Y Y for aparticular neighboring pair in the ring. Each ring and spokecontains a set of mutually commuting observables that definea joint measurement basis. The product of the observables ineach ring (spoke) is + I ( − I ), and thus quantum mechanicspredicts that the product of the measurement outcomes for theobservables in each ring (spoke) is +1 ( − ).A noncontextual hidden variable theory that assigns aneigenvalue prediction ± to each of the N observables mustviolate at least one of these product predictions, which provesthe BKS theorem. To see this, consider the overall productof the predicted eigenvalues along each ring and along eachspoke. According to the quantum predictions, this productmust be − , since there are odd number of spokes. However,for any noncontextual value assignment this product is +1 ,since each observable appears in one ring and one spoke, andthus all of the N eigenvalue predictions are squared in theoverall product.Preparation of | + X (cid:105) ⊗ N and postselection of | + Y (cid:105) ⊗ N fixes all of the pairwise XX and Y Y observables in the N -spin Wheel to have eigenvalue assignment +1 in a noncon-textual hidden variable theory, and the Aharonov-Bergmann-Lebowitz rule shows that all of the ZZ observables have as-signment − . This results in a violation of the classical pi-geonhole principle as well an apparent violation of the quan-tum prediction for the product in the N -spin pairwise ZZ -ringcontext. The joint eigenspaces of this context are the projec-tors Π ( N ) j of the main text, which together form the compositeobservable C ( N ) for the ring. These observables witness con-textuality when weakly measuring them reveals negative weakvalues. Appendix B: Experiment
To determine the weak value of the Pauli spin operator Z the spin degree of freedom is weakly coupled to the path de-gree of freedom [19]. As described in the main body of thepaper the weak value’s real part is then inferred from an inter-ference fringe, while two single intensity measurements arenecessary to determine the weak value’s imaginary part. Todetermine Z w three interference fringes are recorded:1. The OUT curve with no interaction, to evaluate thephase of the empty interferogram.2. The IN curve with a path-dependent spin rotation of α = 15 ◦ and a (weak) interaction strength of sin ( α ) =0 . in each of the interferometer’s arms, which yieldsI y ± . ZZIIIXXIIIYYIII IZZIIIXXIIIYYII IIZZIIIXXIIIYYIIIIYYIIIXXIIIZZ YIIIYXIIIXZIIIZ (a) 5-spin BKS Wheel
ZZIII IZZIIIIZZIIIIZZZIIIZ +1+1 +1+1+1+1+1 +1+1+1 (b) Pre- / postselection +1+1 +1+1+1+1+1 +1+1+1 -1 -1-1 -1 -1 (c) Paradoxical Values
Figure 4. The 5-spin Wheel: (a) The product of the five observablesin each ring is + I (thin line), and the product of the observables ineach spoke is − I (thick line). (b) A particular preparation and posts-election fixes the values of two rings. (c) In any noncontextual hiddenvariable theory, the remaining values must be − , which confines theBKS contradiction to the outer ring (blue dashed circle). This alsodemonstrates the quantum pigeonhole paradox: all pairs in the rowappear anticorrelated, which violates the classical pigeonhole princi-ple.
3. One interference fringe with orthogonal preparation and − π π π πχ [rad]0510152025 I n t e n s i t y [ c p s ] I y + I y − I z + = 6 . ± .
14 [cps] I z − = 3 . ± .
11 [cps]Re[ Z w ] = 0 . ± . Z w ] = 1 . ± .
095 INOUT
Figure 5. Measured interferogram for one data set: Since the weakvalue’s real part is zero, no phase shift is seen between the IN and theOUT curve. I z ± are obtained by two single intensity measurements.Background has already been subtracted. Set
Re [ Z w ] Im [ Z w ] − . ± .
044 0 . ± . − . ± .
044 1 . ± . . ± .
045 1 . ± . − . ± .
045 0 . ± . − . ± .
044 0 . ± . − . ± .
045 1 . ± . . ± .
049 0 . ± . − . ± .
049 0 . ± . . ± .
051 0 . ± . − . ± .
050 0 . ± . . ± .
050 1 . ± . − . ± .
049 0 . ± . . ± .
049 0 . ± . . ± .
050 0 . ± . − . ± .
051 1 . ± . − . ± .
052 0 . ± . − . ± .
050 1 . ± . Figure 6. Experimentally determined weak values for 17 differentdata sets. We used the first N spin weak values shown here for ouranalysis of N -qubit contextuality witnesses in the main text. postselection spin states, which is then subtracted fromthe IN/OUT curve as an effective background.Additionally two single intensities with one or the otherbeam blocked are recorded (I z ± ), and again backgroundmeasurements with orthogonal preparation and postselectionstates are performed and subtracted from the signal. Figure 5shows a typical IN and OUT curve of one experimental run.The data for a complete phase-shifter scan of χ is fit to a sinefunction, which allows us to determine the intensity at the cor- Sets
Re [( ZZ ) w ] Im [( ZZ ) w ] − . ± . − . ± . − . ± .
140 0 . ± . − . ± . − . ± . − . ± . − . ± . − . ± . − . ± . − . ± . − . ± . − . ± . − . ± . − . ± .
135 0 . ± . − . ± .
130 0 . ± . − . ± . − . ± . − . ± . − . ± . − . ± . − . ± . − . ± .
113 0 . ± . − . ± . − . ± . − . ± . − . ± . − . ± . − . ± . − . ± . − . ± . − . ± . − . ± . − . ± . − . ± . − . ± .
129 0 . ± . − . ± . − . ± . − . ± . − . ± . − . ± . − . ± . − . ± . − . ± . Figure 7. Experimentally determined weak values for pairwise prod-ucts, showing anticorrelations between each neighboring pair inclosed rings of N spins, for all odd ≤ N ≤ . The pairwiseanticorrelations in these rings violate the classical pigeonhole princi-ple. rect values of χ , with statistical uncertainty. The measurementprocedure was repeated until altogether 17 different data setswere recorded. For each data set the real and imaginary part ofthe the Pauli spin operator’s weak value is extracted. The re-sults are listed in Fig. 6, and the relevant pairwise correlationsare listed in Fig. 7. It is also noteworthy that the errors of sets1 to 6 are smaller than the others due to a change in reactorpower. While the first six interferograms were recorded at apower of ∼ MW, for the last eleven a power was ∼43