Tests against noncontextual models with measurement disturbances
TTests against noncontextual models with measurement disturbances
Jochen Szangolies,
1, 2, ∗ Matthias Kleinmann, † and Otfried G¨uhne ‡ Institut f¨ur Theoretische Physik III, Heinrich-Heine-Universit¨at D¨usseldorf, D-40225 D¨usseldorf, Germany Naturwissenschaftlich-Technische Fakult¨at, Universit¨at Siegen,Walter-Flex-Straße 3, D-57068 Siegen, Germany
The testability of the Kochen-Specker theorem is a subject of ongoing controversy. A central issueis that experimental implementations relying on sequential measurements cannot achieve perfectcompatibility between the measurements and that therefore the notion of noncontextuality does notapply. We demonstrate by an explicit model that such compatibility violations may yield a violationof noncontextuality inequalities, even if we assume that the incompatibilities merely originate fromcontext-independent noise. We show, however, that this problem can be circumvented by combiningthe ideas behind Leggett-Garg inequalities with those of the Kochen-Specker theorem.
PACS numbers: 03.65.Ta, 03.65.Ud
I. INTRODUCTION
Bell’s theorem [1] is a famous no-go result that pro-vides constraints on the program of interpreting quan-tum mechanics as an incomplete theory in the sense ofEinstein, Podolsky, and Rosen [2]. It is expressed via in-equalities that are fulfilled by any local realistic theory,but which are predicted to be violated by quantum me-chanics. Experimentally, it is indeed found that quantummechanics violates these inequalities for certain entan-gled states [3, 4]. Similar to Bell, Leggett and Garg [5]have proposed inequalities that are fulfilled by theoriesthat satisfy a criterion of macroscopic realism , meaningthat a system always occupies one of the states accessi-ble to it. Under the further assumption of measurementnon-invasiveness, the correlations between measurementsperformed on the system at different points in time obeya bound that is violated by quantum mechanics.A third no-go result is provided by the Kochen-Speckertheorem [6, 7]. Essentially, it replaces Bell’s assumptionof locality with the condition of noncontextuality: theoutcome of a measurement on a system should not de-pend on other compatible measurements performed onthe same system. Here, two measurements are calledcompatible, if they can be measured simultaneously orin a temporal sequence without any disturbance. TheKochen Specker result contains Bell’s theorem as a spe-cial case in which the measurements are performed atspatial separation [8]. It is, however, also applicable tosingle quantum systems; consequently, entanglement isnot necessary for violations of noncontextuality. In fact,violations of Kochen-Specker inequalities occur for allquantum systems of dimension d ≥
3, independent ofthe initial quantum state [6].However, in contrast to Bell’s theorem, the Kochen-Specker theorem does not readily lend itself to experi- ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] mental tests of quantum mechanics. The direct testabil-ity is stymied by the fact that, during each experiment,only one measurement context is accessible at any giventime. This limitation can be overcome, however, becausesimilarly to Bell’s theorem, the Kochen-Specker theoremcan be expressed using inequalities, though it was origi-nally not cast into this form [9, 10]. This permits testingquantum contextuality by using measurements that arecarried out sequentially.But even in this formulation, the question of experi-mental testability poses further difficulties. The reasonfor this lies in the notion of contextuality, which only ap-plies in the case of compatible observables. But in anexperiment, this condition cannot be fulfilled perfectly;indeed, even measuring the same observable twice mayyield different results. This is due to the unavoidablepresence of noise during the measurement process, whichleads to disturbances of the state. It has been arguedthat this inherent difficulty, together with a related is-sue concerning the finite precision of real measurements,nullifies the physical significance of the Kochen-Speckertheorem [11–13].Different strategies have been proposed to overcomethis problem. In Ref. [13], the modification of Kochen-Specker inequalities through the introduction of errorterms was considered (see also Ref. [14]). Given thatthe measurement-induced disturbances are cumulative,these terms compensate for the violations of compatibil-ity. A related approach, addressing a similar loophole inexperimental tests of Leggett-Garg inequalities, was pro-posed in Ref. [15]. On the other hand it was suggestedin Ref. [16] to perform experiments on separate qutritsin order to forestall the possibility of violations of com-patibility.In this paper, we take a different approach. First, weconsider the question: what does an experimentally ob-served violation of a noncontextuality inequality licenseus to conclude? By proposing an explicit model captur-ing the effects of noise-induced compatibility violations,we argue that to conclude contextuality from the vio-lation alone is difficult to justify: the model producesviolations of noncontextuality inequalities while being in- a r X i v : . [ qu a n t - ph ] M a y dependent of the measurement context, and thus, non-contextual in this sense. In particular, we show that eventhe introduction of error terms as proposed in Ref. [13]cannot settle the issue.We then propose a way to circumvent this problem bytaking into account the ideas of Leggett and Garg: im-posing a suitable time-ordering onto the measurements,it turns out to be possible to formulate inequalities thatcannot be violated within our framework, and thus, allowto rule out more general hidden-variable models underrealistic experimental conditions.This paper is organized as follows. In Section II, webriefly recall the notions of compatibility and contextu-ality. Then, we propose an explicit classical noise modelcapable of inducing compatibility violation in such a wayas to violate contextuality inequalities. In Section III weshow that the model cannot be ruled out by previousapproaches. To remedy this problem, in Section IV wepropose new inequalities, utilizing ideas from Leggett andGarg. These inequalities allow to rule out more generalhidden variable models. II. NONCONTEXTUAL MODELS
As a basis for our investigations we take a variant ofthe well-known Clauser-Horne-Shimony-Holt (CHSH) in-equality [17] (cid:104) χ CHSH (cid:105) = (cid:104) AB (cid:105) + (cid:104) BC (cid:105) + (cid:104) CD (cid:105)−(cid:104) DA (cid:105) NCHV ≤ QM ≤ √ . (1)Each of the observables A , B , C , D has outcomes ±
1, and (cid:104) AB (cid:105) denotes the average over many repetitions obtainedby measuring first A , then B , and then multiplying theresults. If we assume that the observables in each con-text (cid:104) AB (cid:105) , etc., are compatible, then NCHV denotes theclassical (noncontextual hidden-variable) bound, i.e., thevalue obtained if each of the observables is assumed tohave a fixed value independently of which context it ismeasured in. The bound QM denotes the maximal valueachievable in quantum mechanics [18]. The question nowis: suppose one experimentally observes (cid:104) χ CHSH (cid:105) >
2. Isthis sufficient to conclude contextual behavior?First, we need to make the notions of compatibilityand noncontextuality precise. Consider some observ-ables O = { A, B, C, . . . } . Compatibility then means thatwithin any sequence of measurements composed of theseobservables, the observed value does not depend on thepoint at which it is measured within the sequence. Thatis, for any sequence of compatible measurements C , theobserved value of O at the i th point in the sequence, v i ( O |C ), does not depend on i , i.e., v i ( O |C ) = v j ( O |C ) forall i and j . This formalizes the notion that measurementof one observable does not influence the measurement ofany other observable.Then, any set of compatible observables C is calleda context . A theory is called noncontextual , if for allobservables O and for all contexts C , C (cid:48) the observed value FIG. 1: Schematic representation of a sequence of measure-ments. Measurements A and B are performed sequentiallyon a system whose (hidden-variable) state λ evolves stochas-tically as indicated. Time runs left to right. is independent of the context, i.e., v ( O |C ) = v ( O |C (cid:48) ).Note that through the definition of a context, the notionof noncontextuality explicitly depends on compatibility.To approach the question, we construct a counterex-ample given by a simple model for noise-induced distur-bances of the hidden-variable states. These hidden states λ i are assumed to completely specify all possible exper-imental outcomes. In the present case, they can thusbe indexed by the dichotomic outcomes of measurementsof the observables O = { A, B, C, D } : a given state isspecified uniquely by a set of values v ( O ) ∈ {± } for all O ∈ O . For ease of notation, this set of values may beinterpreted as a binary number, whose decimal value isused to index the state, i.e., λ = λ (++ − +) denotes thestate that produces the measurement results A = +1, B = +1, C = − D = +1. The model can begeneralized by considering states that are convex combi-nations of the value attributions λ i , such that the mostgeneral state can be written as a mixture (cid:80) (2 n ) − i =0 p i λ i ,where (cid:80) (2 n ) − i =0 p i = 1 and n denotes the number of ob-servables.The dynamics of this model now is such that after ev-ery measurement, the system may randomly execute atransition to a different state. Note that this transitiondoes not depend on which measurement was carried out.This models the effect of noise introduced during mea-surement, i.e., after a noisy interaction with the system,further measurements will in general yield different re-sults. We will now show that this is equivalent to theintroduction of compatibility violations in a realistic ex-periment, and, crucially, that these violations may leadto false positives in Kochen-Specker tests.Consider the evolution depicted in Fig. 1: a measure-ment of the observable A is made on a system in the state λ , consequently producing the result +1. Subsequently,the observable B is measured, yielding −
1. Then, thesystem undergoes a state transition to λ , and a subse-quent measurement of A yields −
1. Thus, compatibilityis violated.Of course, this model cannot suffice to capture allquantum mechanical effects; in particular, for a priori incompatible observables, it is easy to show that its be-havior differs from that of quantum mechanics: take ameasurement sequence such as
AAA . Without distur-bances, both quantum mechanics and the model predictthat the same result will be repeated three times; allow-ing for noise influences, there will be a small probabil-ity of disagreement. Measuring
ACA , however, since A and C are not compatible, quantum mechanics predictsthat the result for the second measurement of A must berandom, while in our model, it will agree with the firstresult up to possible probabilistic state changes (i.e., inour model, the probability distribution from which thevalue of A is drawn will not differ whether it is the thirdmeasurement in the sequence AAA or in the sequence
ACA ). However, Kochen-Specker tests are always car-ried out within compatible sets of observables, and, sincewe are (for the moment) only investigating what can beconcluded from such a test alone, this is not our concernhere. Our main point is that this simple model can in-validate some ideas to make Kochen-Specker tests robustagainst noise.Let us now consider what happens during a measure-ment of the left-hand side of Eq. (1) if violations of com-patibility are present. Then, if we denote by A i the ob-served value of A , given that the hidden variable state is λ i , (cid:104) χ CHSH (cid:105) can be calculated as follows: (cid:104) χ CHSH (cid:105) = (cid:88) i,j ( A i B j + B i C j + C i D j − D i A j ) p ij ≡ (cid:88) i,j K ij p ij , (2)where the p ij denote the probability that the evolutionof the system is λ i → λ j , that is, that the state duringthe first measurement was λ i , which transitioned to λ j before the second one, and we have introduced the quan-tity K ij = A i B j + B i C j + C i D j − D i A j . The maximum K max of the coefficients K ij provides the upper bound (cid:104) χ CHSH (cid:105) = (cid:88) i,j K ij p ij ≤ max ij { K ij } ≡ K max . (3)Each K ij represents the value of χ CHSH , given the hid-den variable evolution λ i → λ j . It is easy to check that K , = 4: λ = (+ + + +) and λ = ( − + + +), and thus, (cid:104) AB (cid:105) = (cid:104) BC (cid:105) = (cid:104) CD (cid:105) = +1, while (cid:104) DA (cid:105) = −
1. Hence,a simple model that after each measurement changes thesystem’s state from λ to λ will maximally violate theCHSH inequality; if the change happens only with a cer-tain probability p , obviously any value between 2 and 4can be achieved.It should be noted that despite the evolution of thehidden variable, this model is noncontextual in the sensethat whether or not a state transition is effected does notdepend on the measured context. It thus seems surprisingthat this model can violate the CHSH inequality, appar-ently indicating contextual behavior. However, strictly speaking, noncontextuality simply does not apply in thiscase, as it is defined only under the assumption of perfectcompatibility. III. CONNECTION WITH PREVIOUS WORKS
An approach to rein in the effects of compatibility vi-olations was proposed in Ref. [13]. There, several classesof error terms were proposed, such that additional mea-surements may be performed in order to quantify thedegree of failure of a priori compatible observables tobe compatible in the actual experiment, i.e., the degreeof influence a measurement of A has on the compati-ble measurement B , for example. We will concentrate,for the moment, on the first class of error terms fromRef. [13], which are those that have been experimentallyimplemented.Based on an assumption of noise cumulation, that is,an assumption that additional measurements always leadto additional noise and thus a worse violation of compat-ibility, the inequality (cid:104) χ CHSH (cid:105) − p err [ BAB ] − p err [ CBC ] − p err [ DCD ] − p err [ ADA ] ≤ , (4)holds [13]. Here for instance p err [ BAB ] is the probabilitythat the second measurement of B in the sequence BAB disagrees with the first one.However, it is clear that the model we discuss does notobey the assumption of cumulative noise: for an evolu-tion such as λ → λ → λ , clearly both measurementsof B in the sequence BAB agree, but if B were measuredin the second place of the sequence, then it would haveyielded a value opposite to the first. Thus, the model isnot necessarily constrained by Inequality (4); and in fact,since the error terms all vanish for such an evolution, itis clear that the model can violate it.Alternatively, it may be noted that while the originalCHSH-inequality is only concerned with measurementsequences of length 2, the error terms contain only se-quences of length 3, and thus, can only provide informa-tion about the system’s behavior during such sequences.This criticism holds for the other two classes of errorterms in Ref. [13] as well. IV. MODIFIED INEQUALITIES
However, another approach, which does not need anyadditional measurements or further assumptions, is pos-sible. This amounts to essentially applying the ideas ofLeggett and Garg to contextuality inequalities. Ratherthan employing the original inequalities proposed inRef. [5], it is convenient for our purposes to use aslightly different formulation. Consider two differentmeasurements C and C (cid:48) , performed at two points intime. Then, C ( C + C (cid:48) ) + C (cid:48) ( C − C (cid:48) ) = ±
2, and thus A = σ z ⊗ B = ⊗ σ z C = σ z ⊗ σ z a = ⊗ σ x b = σ x ⊗ c = σ x ⊗ σ x α = σ z ⊗ σ x β = σ x ⊗ σ z γ = σ y ⊗ σ y TABLE I: The Peres-Mermin square, with the Pauli matrices σ i , and the 2 × . The observables in allrows and columns commute, and the product of all rows andthe first two columns is equal to , while for the last column, Ccγ = − . (cid:104) C (cid:48) C (cid:105) + (cid:104) CC (cid:105) + (cid:104) CC (cid:48) (cid:105) − (cid:104) C (cid:48) C (cid:48) (cid:105) ≤ , where (cid:104) CC (cid:48) (cid:105) de-notes the correlation between C , measured at t , and C (cid:48) ,measured at t .We can now impose a similar time-ordering of observ-ables on Eq. (1), to get (cid:104) χ CHSH (cid:105) = (cid:104) AB (cid:105) + (cid:104) CB (cid:105) + (cid:104) CD (cid:105) − (cid:104) AD (cid:105) ≤ . (5)It is not hard to see that for Eq. (5), K ij ≤ i, j ): if the first three terms are equal to +1, the fourth isnecessarily equal to +1, as well. Hence, our model cannotviolate Inequality (5), despite the violation of compatibil-ity. Since in the case of a trivial evolution of the hiddenvariables, i.e. an evolution that leaves the state invariant,we recover the usual notion of (sequential) noncontextu-ality, an experimental test of Eq. (5) constitutes a test ofquantum contextuality robust against the compatibilityloophole.It should be noted that the CHSH-inequality is notthe only one that can be modified to hold in the caseof compatibility violations: another important inequalityproposed in Ref. [9] is based on the Peres-Mermin square([19]; see Table I). Using the same reasoning as in theCHSH-case, the inequality (cid:104) χ PM (cid:105) = (cid:104) ABC (cid:105) + (cid:104) cab (cid:105) + (cid:104) βγα (cid:105) + (cid:104) Aaα (cid:105) + (cid:104) βBb (cid:105) − (cid:104) cγC (cid:105) ≤ (cid:104) χ PM (cid:105) = 6can be reached. Again, it is here the ordering of the mea-surement sequences that matters: the original inequalityproposed in Ref. [9] followed the ordering indicated inTable I; but in this form, it is not hard to see that theinequality can be violated easily by our model. Inter-estingly, the ordering proposed here is also useful if theMermin-Peres inequality should be used for estimatingthe dimension of a quantum system [20].The importance of this scenario is that this inequal-ity is state-independent, that is, one does not require aspecial quantum state for the violation (as is the casefor the CHSH-inequality). Furthermore, an experimentusing sequential measurements on trapped ions alreadyimplemented this scenario by measuring the observablesin Table I in all possible permutations [14]. This ex-periment focused on the violation of the inequality asoriginally proposed in Ref. [9], and using this data, theobserved value for Eq. (6) is (cid:104) χ PM (cid:105) = 5 . (cid:104) χ KCBS (cid:105) = (cid:104) AB (cid:105) + (cid:104) BC (cid:105) + (cid:104) CD (cid:105) + (cid:104) DE (cid:105) + (cid:104) EA (cid:105) NCHV ≥ − QM ≥ − √ , (7)which exhibits a quantum violation even for a sin-gle qutrit system, as demonstrated experimentally inRef. [21], cannot be rearranged appropriately. Never-theless, our approach can be generalized: the modifiedinequality (cid:104) AB (cid:105) + (cid:104) CB (cid:105) + (cid:104) CD (cid:105) + (cid:104) ED (cid:105) + (cid:104) EA (cid:105) − (cid:104) AA (cid:105) NCHV ≥ − QM ≥ − √ A i must equal E j , as must A j ; how-ever, then A i = A j , and thus, (cid:104) AA (cid:105) = 1. This shows thateven in the case of a single qutrit a Kochen-Specker testruling out our model can be undertaken. However, oneshould note that due to this modification, the relativequantum violation shrinks, since the absolute violationstays the same, while the absolute value of the classicalexpectation increases. Finally, it should be noted that asimilar inequality like Eq. (8) has already been used inRef. [21] in order to compensate for the fact that in thissetup the observable A was implemented in two differentways.In fact, a recently proposed state-independent inequal-ity violated by a single qutrit system [22] can be treatedin the same way. This inequality features 13 observables { A , . . . , A } , and the form that yields the maximumquantum violation is [23, 24] (cid:88) i Γ i (cid:104) A i (cid:105) + (cid:88) ij Γ ij (cid:104) A i A j (cid:105) ≤ , (9)where the coefficients are as follows: Γ i = 1 for i ∈{ , , , . . . , } , Γ i = 2 for i ∈ { , , , , } , and Γ i = 3for i ∈ { , } ; Γ ij = − i, j ) ∈ { (1 , , , , , , , , , , , , , , , , } , Γ ij = − i, j ) ∈ { (2 , , (2 , , (2 , , (3 , , (3 , , (5 , , (6 , } ,and Γ ij = 0 else. By checking all possible hidden variableevolutions one verifies that the modified inequality [25] (cid:88) i Γ i (cid:104) A i (cid:105) + (cid:88) ij Γ ij (cid:104) A i A j (cid:105) + 4 (cid:88) i (cid:104) A i A i (cid:105) ≤
68 (10)cannot be violated by noncontextually evolving models.However, since the maximum quantum value in this caseis only 69 + , the relative violation is reduced to ≈ . ≈ . V. CONCLUSION
We have provided a novel approach to the compatibil-ity problem in Kochen-Specker experiments. Using theidea of time-ordering, as first proposed by Leggett andGarg, we have derived new inequalities violated by quan-tum mechanics even in the case of imperfectly compatiblemeasurements. This shows that with a careful orderingof the measurements classical models can be ruled out,which cannot be excluded with existing approaches [13].Nevertheless, we are not claiming that our modified in- equalities allow a test of the Kochen-Specker theorem freefrom the compatibility loophole. Our results, however,show that with a simple reordering of the measurementsa significantly larger class of hidden-variable models canbe ruled out.We thank C. Roos for providing us with the datafrom their experiment, and C. Budroni and D. Bruß forvaluable discussions. This work has been supported bythe EU (Marie Curie CIG 293993/ENFOQI) and BMBF(projects QuOReP and QUASAR). [1] J.S. Bell, Physics , 195 (1964).[2] A. Einstein, B. Podolsky, N. Rosen, Physical Review ,777 (1935).[3] A. Aspect, P. Grangier, G. Roger, Phys. Rev. Lett. ,460 (1981); A. Aspect, P. Grangier, G. Roger, Phys. Rev.Lett. , 91 (1982); A. Aspect, J. Dalibard, G. Roger,Phys. Rev. Lett. , 1804 (1982).[4] G. Weihs, T. Jennewein, C. Simon, H. Weinfurter, A.Zeilinger, Phys. Rev. Lett. , 5039 (1998).[5] A.J. Leggett, A. Garg, Phys. Rev. Lett. , 857 (1985).[6] S. Kochen, E. P. Specker, J. Math. Mech. , 59 (1967).[7] J. S. Bell, Rev. Mod. Phys. , 447 (1966).[8] N. D. Mermin, Rev. Mod. Phys. , 803 (1993).[9] A. Cabello, Phys. Rev. Lett. , 210401 (2008).[10] A. A. Klyachko, M. A. Can, S. Binicio˘glu, A. S. Shu-movsky, Phys. Rev. Lett. , 020403 (2008).[11] D. A. Meyer, Phys. Rev. Lett. , 3751 (1999).[12] A. Kent, Phys. Rev. Lett. , 3755 (1999).[13] O. G¨uhne, M. Kleinmann, A. Cabello, J-˚A. Larsson, GKirchmair, F. Z¨ahringer, R. Gerritsma, C. F. Roos, Phys.Rev. A , 022121 (2010).[14] G. Kirchmair, F. Z¨ahringer, R. Gerritsma, M. Klein-mann, O. G¨uhne, A. Cabello, R. Blatt, C. F. Roos, Na-ture , 494 (2009).[15] M. Wilde, A. Mizel, Found. Phys. , 256 (2012). [16] A. Cabello, M. Terra Cunha, Phys. Rev. Lett. ,190401 (2011).[17] J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt,Phys. Rev. Lett. , 880 (1969).[18] B. S. Tsirelson, Lett. Math. Phys. , 93 (1980).[19] A. Peres, Phys. Lett. A , 107 (1990); N. D. Mermin,Phys. Rev. Lett. , 3373 (1990).[20] O. G¨uhne, C. Budroni, A. Cabello, M. Kleinmann, andJ.-˚A. Larsson, arXiv:1302.2266.[21] R. Lapkiewicz, P. Li, C. Schaeff, N.K. Langford, S.Ramelow, M. Wie´sniak, and A. Zeilinger, Nature (Lon-don) , 490 (2011).[22] S. Yu, C. H. Oh, Phys. Rev. Lett , 030402 (2012).[23] M. Kleinmann, C. Budroni, J.-˚A. Larsson, O. G¨uhne, A.Cabello, Phys. Rev. Lett. , 250402 (2012).[24] X. Zhang, M. Um, J. Zhang, S. An, Y. Wang, D.-l. Deng,C. Shen, L. Duan and K. Kim, Phys. Rev. Lett. ,070401 (2013).[25] There is no ordering ambiguity for the single measure-ments (cid:104) A i (cid:105)(cid:105)