Tests of hidden variable models by the relaxation of the measurement independence condition
aa r X i v : . [ qu a n t - ph ] A p r Tests of hidden variable models by the relaxation of themeasurement independence condition
R. Rossi Jr. ∗ and Leonardo A. M. Souza † Universidade Federal de Vi¸cosa - Campus Florestal,LMG818 Km6, Minas Gerais, Florestal 35690-000, Brazil (Dated: April 19, 2018)
Abstract
Bell inequalities or Bell-like experiments are supposed to test hidden variable theories basedon three intuitive assumptions: determinism, locality and measurement independence. If one ofthe assumptions of Bell inequality is properly relaxed, the probability distribution of the singletstate, for example, can be reproduced by a hidden variable model. Models that deal with therelaxation of some condition above, with more than one hidden variable, have been studied inthe literature nowadays. In this work the relation between the number of hidden variables andthe degree of relaxation necessary to reproduce the singlet correlations is investigated. For theexamples studied, it is shown that the increase of the number of hidden variables does not allowfor more efficiency in the reproduction of quantum correlations.
Keywords: Causality; Bell Inequalities; Nonlocality ∗ Electronic address: [email protected] † Electronic address: [email protected] . INTRODUCTION One of the most intriguing features of quantum mechanics is that a multipartite statemay present genuine (and intrinsic) quantum correlations. Different from classical mechan-ics, quantum correlations can not be described by a model simultaneously consistent with:determinism, locality and measurement independence [1–4]. However, it is possible to re-produce quantum correlations in a model in which at least one of the previous assumptionsis partially relaxed [5–9]. For instance, the reproduction of the singlet state correlations,by the relaxation of the measurement independence condition (MIC), was studied in Refs.[7–9]. Different models have been considered and it was shown in Reference [9] a model withthe lowest degree of relaxation necessary to reproduce the statistics of the singlet.Acoording to the MIC, the measurement set variables ( x and y ), in the context of Bellinequality scheme, must be independent of the hidden variable. In the language of the theoryof causal models [10, 11], this statement means that there must be no causal connectionsbetween the hidden variable and x or y . Due to our lack of knowledge about the hiddenvariables, one can conceive models with a variety of them, interacting and affecting thevalues of the observable variables [12, 13].A question may arise: is it possible to use this freedom and consider a larger numberof hidden variables to reproduce the singlet statistics, through the violation of the MIC,in a more efficient way? The degree of violation of the MIC, that measures the efficiencyof a model to reproduce the singlet statistics, can be defined as the mutual information I ( λ : x, y ) [14]. Compared to all models shown in the literature, the model of reference [9]is the most efficient, due to the relation among the measurement set variables considered inthis reference.For models with more than one hidden variable the conditions for the violation of themeasurement independence are different from the conditions of the traditional model (withone hidden variable). Here we consider models with three hidden variables ( that werepresented in [12] and [13]). The violation of the MIC for these models is given when causalconnections among the hidden variables are assumed.In this work, we investigate the possibility of increase the efficiency of a model (withhidden variables) in reproducing the singlet probability, through the violation of the MIC,by growing the number of hidden variables. We show that the efficiency of the model2resented in [9] can not be increased just by adding hidden variables within our approach. II. REPRODUCING THE SINGLET CORRELATION
Here we apply some tools developed in the theory of causal models [10, 11] that aresuitable for analysis of systems with hidden variables. This was also done in references [15–17]. We consider three different causal models and study the reproduction of singlet statecorrelations through violation of the measurement independence condition. From now on,any variable λ or λ i represents a different hidden variable. A. The First Causal Model (Bell Scheme)
In a traditional Bell experiment two subsystems, which may have interacted previously,are spatially separated and measured by two observers: Alice and Bob. The variable x and y are the setting variable, they describe the possible measurements that can be chosen by Aliceand Bob, respectively. The variables a ( b ) represents the possible outcome of measurementsof x ( y ). FIG. 1: One hidden-variable model. The causal structure shows: (a) no causal connections between λ and x or y , therefore no violation of MIC (b) The hidden variable λ is connected to x and y , thisis a violation of MIC Bell’s theorem present the possibility to test experimentally a theory satisfying three3ssumptions (here written in the language of causal model):I) The value of the variable a ( b ) is the join effect of a hidden variable λ and the settingvariable x ( y ).II) The experiments performed by Alice and Bob are space-like separated events, therefore a and b are statistically independent given λ , x and y . Or in mathematical terms P ( a, b | x, y, λ ) = P ( a | x, λ ) P ( b | y, λ ).III) The measurement setting variables are independent of λ , that is the measurementindependence condition. It can be written as P ( x, y | λ ) = P ( x, y ) which is equivalentto P ( λ | x, y ) = P ( λ ).Using the assumptions I, II and III, one can write the conditional probability P ( a, b | x, y ) as: P ( a, b | x, y ) = Z dλP ( a, b | x, y, λ ) P ( λ | x, y ) , = Z dλP ( a | x, λ ) P ( b | y, λ ) P ( λ ) . (1)There is a conflict between the separable form of P ( a, b | x, y ) in Equation (1) and the pre-dictions of quantum theory. This conflict is experimentally verified by violations of Bellinequality [18].A singlet state (in the computational basis | ψ singlet i = √ [ | i − | i ]) is a maximallyentangled state and can be used in experiments that show violation of a Bell inequality (forinstance [18]). Therefore, a model that satisfy assumptions I, II and III can not be used topredict the conditional probability P S ( a, b | x, y ) measured in a system prepared in a singletstate, which is [9]: P s ( a, b | x, y ) = 1 − ab ( x · y )4 . (2)If one of the assumptions (I, II or III) is relaxed one can reproduce, within a hiddenvariable model, the probability P S ( a, b | x, y ) of the singlet state. Here we consider relaxationon the measurement independence condition (MIC). Some models have been proposed [6–9]in which the authors investigate the degree of relaxation of the MIC necessary to reproduce P S ( a, b | x, y ) for the singlet state. In the context of causal models, the relaxation of the MICis represented by the causal connections between λ and x , y as shown in the directed acyclicgraph (DAG) of Figure 1(b), i.e. there may be an explicit dependence of the measurementapparatus concerning the hidden variable. 4n the models presented in references [6–9] the authors considered the relation P ( a, b | x, y ) = R dλP ( a, b | x, y, λ ) P ( λ | x, y ) and chose suitable expressions for the conditionalprobabilities P ( a, b | x, y, λ ) and P ( λ | x, y ) to reproduce P S ( a, b | x, y ). The relation among thevariables λ, x, y , determine the degrees of relaxation on the MIC. Therefore, each modelshave different degrees of relaxation on the MIC. The model with smaller degree of relaxationis the one of reference [9] (to our knowledge).In the traditional Bell’s scheme (Figure 1(a)), there is only one hidden variable λ and aviolation of the MIC appear when this variable has a causal link with x , y as shown in Figure1(b). If we consider causal models with more than one hidden variable, different causal linksamong x , y and the hidden variables may be considered. Do these new structures allow usto reproduce the singlet probability P S ( a, b | x, y ) with a degree of relaxation smaller thanthe one in reference [9]? To partially answer this question we consider two models given inreference [12, 13]. The choice of this two models allow us to concentrate on the role playedby the number of hidden variables. B. The Second Causal Model
In this section we consider the model presented in [12], whose causal structure is shownin Figure 2(a), and here we show how one can reproduce the singlet probability P S ( a, b | x, y )whithin this model. Notice that the exogenous variables are the hidden variables, i.e. λ , λ and λ . In the traditional Bell scheme – Figure 1(a) – the violation of the MIC is representedby the causal connections among the variables x , y and λ , as it is shown in Figure 1(b). Withthe violation of the MIC, x and y cease to be exogenous variables, they become descendent[10] of λ . In the model of Reference [12], the violation of the MIC is represented by thecausal connections among λ , λ and λ , as shown in Figure 2(b), where λ and λ cease tobe exogenous variables and become descendent of λ .The conditional probability P ( a, b | x, y ) is given by: P ( a, b | x, y ) = Z Z Z dλdλ dλ P ( a, b | x, y, λ, λ , λ ) P ( λ, λ , λ | x, y ) , (3)from Bayes’ theorem we can write P ( λ, λ , λ | xy ) = P ( x, y | λ, λ , λ ) P ( λ, λ , λ ) P ( x, y ) = P ( x, y | λ, λ , λ ) P ( λ | λ , λ ) P ( λ , λ ) P ( x, y ) . (4)5 IG. 2: Three hidden-variable model. The causal structure shows: (a) no causal connectionsbetween λ and λ or λ , therefore no MIC violation; (b) now the hidden variable λ is connectedto λ and λ , this is a clear MIC violation. The causal Markov condition applied to the DAG of Figure 2(a) gives the relations: P ( a, b | x, y, λ, λ , λ ) = P ( a, b | x, y, λ ) (5) P ( x, y | λ, λ , λ ) = P ( x, y | λ , λ ) . (6)Working out equations (3) to (6), and using the definition P ( λ , λ | x, y ) = P ( x, y | λ , λ ) P ( λ , λ ) /P ( x, y ) we obtain: P ( ab | xy ) = Z Z Z dλdλ dλ P ( a, b | x, yλ ) P ( λ , λ | x, y ) P ( λ | λ , λ ) . (7)In the causal structure shown in Figure 2(a), variables λ , λ and λ are exogenous,therefore the causal Markov condition also return us the relation P ( λ | λ , λ ) = P ( λ ). Toinvestigate the relaxation of the MIC, let us consider the DAG shown in Figure 2(b). In thiscausal structure λ and λ are not exogenous and P ( λ | λ , λ ) = P ( λ ).In Reference [9] the author calculates, for the model in Figure 1(b), the degree of re-laxation of the MIC necessary to reproduce the probability P S ( a, b | x, y ) of the singlet. Inthis work the author considers a particular relation among the variables λ , x and y , andthe degree of MIC obtained depends on this relation. To calculate the degree of relaxationfor the second model and compare with the result obtained in Reference [9], we considerthe same relation among the exogenous variables, and substitute the measurement settingvariables x and y by the hidden variables λ and λ :6 ( λ | λ , λ ) = 14 π · λ .λ ) sign[( λ.λ )( λ.λ )]1 + (1 − φ λ λ /π ) sign[( λ.λ )( λ.λ )] , (8)where φ λ λ represents the angle between the measurement directions λ and λ . To repro-duce the singlet statistics we also consider the relations: P ( a, b | λ, x, y ) = δ a,A ( λ,x ) δ b,B ( λ,y ) (9) P ( λ , λ | x, y ) = δ λ ,x δ λ ,y (10)where A ( λ, x ) = sign( λ · x ) and B ( λ, y ) = − sign( λ · y ). Therefore, substituting Equationsfrom (8) to (10) in Equation (7) we obtain the singlet probability: P ( a, b | x, y ) = P S ( a, b | x, y ) . (11) C. The Third Causal Model
In this section the causal model studied in reference [13] and shown in Figure 3(a) isconsidered. In this model, the condition equivalent to the measurement independence canbe written as: P ( λ, λ , λ ) = P ( λ ) P ( λ ) P ( λ ). FIG. 3: Three hidden-variable model. The causal structure shows: (a) no causal connectionsbetween λ and λ or λ , therefore no MIC violation; (b) the hidden variable λ is connected to λ and λ , a MIC violation. To investigate the degree of relaxation necessary to reproduce the singlet statistics, weconsider the causal structure given in Figure 3(b). The causal Markov condition permit us7o write: P ( a, b | x, y, λ, λ , λ ) = P ( a, b | λ, λ , λ ) (12) P ( x, y | λ, λ , λ ) = P ( x, y | λ , λ ) . (13)The conditional probability P ( a, b | x, y ) can be written as: P ( a, b | x, y ) = Z Z Z dλdλ dλ P ( a, b | x, y, λ, λ , λ ) P ( λ, λ , λ | x, y ) , = Z Z Z dλdλ dλ P ( a, b | λ, λ , λ ) P ( λ, λ , λ | x, y ) . (14)Again from Bayes’ theorem we can write: P ( λ, λ , λ | x, y ) = P ( x, y | λ, λ , λ ) P ( λ, λ , λ ) P ( x, y ) . (15)Using the definition of joint probability P ( λ, λ , λ ) = P ( λ | λ , λ ) P ( λ , λ ) and Equation(13), we can write: P ( λ, λ , λ | x, y ) = P ( x, y | λ , λ ) P ( λ , λ ) P ( x, y ) P ( λ | λ , λ ) = P ( λ , λ | x, y ) P ( λ | λ , λ ) (16)In order to reproduce the probability of the singlet state P S ( a, b | x, y ), we use the strategyof the previous section. The variables involved in the MIC for this model are λ , λ and λ ,and we consider the same relation among them, substituting x and y by λ and λ (since x and y are not ascendant of any variable in this model, see Figure 3(b)), as it was done inthe previous section. Then we obtain: P ( λ | λ λ ) = 14 1 + ( λ .λ ) sign[( λ.λ )( λ.λ )]1 + (1 − φ λ λ /π ) sign[( λ.λ )( λ.λ )] . (17)To reproduce the singlet statistics we also consider the relations: P ( a, b | λ, λ , λ ) = δ a,A ( λ,λ ) δ b,B ( λ,λ ) (18) P ( λ , λ | x, y ) = δ λ ,x δ λ ,y , (19)where A ( λ, λ ) = sign( λ · λ ) and B ( λ, λ ) = − sign( λ · λ ). Again, working out Equations(8) to (10), and substituting in Equation (7), we obtained the singlet probability: P ( a, b | x, y ) = P S ( a, b | x, y ) . (20)8 II. DEGREE OF RELAXATION OF MEASUREMENT INDEPENDENCE CON-DITION
In this section we compare the degree of relaxation of the MIC for the causal modelsrepresented in Fig. 1(b), Fig. 2(b) and Fig. 3(b). Some measures of the relaxation degreeof the MIC have been considered in the literature [7, 9, 14, 19–23], we follow [14] and usemutual information as our figure of merit. For the models shown in Figure 2(b) and inFigure 3(b) the violation of the MIC is due to the relations among variables λ, λ and λ ,therefore, the degree of violation of the MIC is given by the mutual information: I ( λ , λ : λ ) = I ( λ : λ , λ ) = H ( λ ) − H ( λ | λ , λ ) , (21)where, H ( φ ) is the usual Shannon entropy related to some variable φ . From Equation (21)we can see that the degree of relaxation of MIC depends only on the conditional probability p ( λ | λ , λ ) which are the same in the models shown in Fig 2(b) and Fig. 3(b). Therefore,the degree of relaxation will be the same for both models.To compare the models with more than one hidden variable (Fig. 2(b) and Fig. 3(b))with the traditional one (Fig. 1(b)), let us consider the difference I ( λ : λ , λ ) − I ( λ : x, y ),where I ( λ : x, y ) represents the mutual information among the variables of interest in themodel represented by Fig. 1(b). In this way, we obtained: I ( λ : λ , λ ) − I ( λ : x, y ) = − H ( λ | λ , λ ) + H ( λ | x, y ) (22)= − X λ,λ ,λ p ( λ | λ , λ ) log [ p ( λ | λ , λ )] + X λ,x,y p ( λ | x, y ) log [ p ( λ | x, y )] . Now we do need to make a digression and re-direct our attention back to the hiddenvariables domain, which we call Ω. From Equations (10) and (19) we can conclude that,in order to reproduce the singlet statistics, Ω must contain U (the set of unitary vectorsand the domain of x and y ). Due to our lack of knowledge about the hidden variables, thecardinality of Ω is not known, but since one is interested in reproduce P S ( a, b | x, y ) withinthe causal models framework we are working, the cardinality of Ω must be greater than orequal to the cardinality of U . If the cardinalities are equal I ( λ : λ , λ ) − I ( λ : x, y ) = 0,but if they are different, Equation (22) is non-zero: I ( λ : λ , λ ) − I ( λ : x, y ) = − X λ,λ = x,λ = y p ( λ | λ , λ ) log [ p ( λ | λ , λ )] . (23)9s p ( λ | λ , λ ) is a probability, we can write 0 ≤ p ( λ | λ , λ ) ≤ p ( λ | λ , λ )] ≤
0. In conclusion, we obtain the inequality: I ( λ : λ , λ ) − I ( λ : x, y ) ≥ I ( λ : λ , λ ) ≥ I ( λ : x, y ) . (25)Inequality (25) shows that the degree of MIC violation, necessary to reproduce the singletprobability P S ( a, b | x, y ) within models of Fig. 2(b) and Fig. 3(b), is greater than or equalto the one calculated for the model of Reference [9]. IV. CONCLUSIONS
In this work we studied causal models where the measurement independence condition(as soon as we are dealing with hidden variables theories) may not be satisfied. Three causalmodels were studied, the first one with one hidden variable, and the other two with 3 hiddenvariables. The model with one hidden variable was used to exemplify our approach and toobtain the probability distribution for the singlet state. In the following models we calculatethe probability distribution, and we were able to obtain the statistics for the singlet state.Finally, we quantified the degree of relaxation for the studied cases. We show that theincrease in the number of hidden variables, at least for the models studied in this work, doesnot allow the reduction of the mutual information needed to reproduce P S ( a, b | x, y ). Acknowledgments
The authors thanks Brazilian agencies CNPq and FAPEMIG for finantial support. [1] J.S. Bell, Physics 1, 195 (1964).[2] J.F. Clauser, M.A. Horne, A. Shimony and R.A. Holt, Phys. Rev. Lett. 23, 880 (1969).[3] M. Zukowski and C. Brukner, Phys. Rev. Lett. 88 210401 (2002)[4] E.G. Cavalcanti et al., Phys. Rev. Lett. 99, 210405 (2007).[5] C. Branciard et al., Nature Physics 4, 681 (2008).[6] C. Brans, Int. J. Theoret. Phys. 27, 219 (1988).
7] M. J. W. Hall, Phys. Rev. A 84, 022102 (2011).[8] J. Degorre, S. Laplante, J. Roland, Phys. Rev. A 72, 062314 (2005)[9] M. J. W. Hall, Phys. Rev. Lett. 105, 250404 (2010)[10] J. Pearl,
Causality: Models, Reasoning, and Inference
Causation, Prediction, and Search2nd ed. Cam-bridge, MA: MIT Press (2001)[12] C. Branciard, D. Rosset, N. Gisin, and S. Pironio Phys. Rev. A 85, 032119 (2012).[13] T. Fritz, New J. Phys. 14 103001 (2012).[14] J. Barrett, N. Gisin, Phys. Rev. Lett. 106, 100406 (2011)[15] M. S. Leifer and R. W. Spekkens, Towards a formulation of quantum theory as a causallyneutral theory of bayesian inference, Phys. Rev. A 88, 052130 (2013).[16] R. Rossi Jr, Phys. Rev. A 96, 012106 (2017).[17] Rafael Chaves, Gabriela Barreto Lemos, and Jacques Pienaar, eprint arXiv:1710.07323v2[quantph].[18] B. Hensen, H. Bernien, A.E. Drau, A. Reiserer, N. Kalb, M.S. Blok, J. Ruitenberg, R.F.L.Vermeulen, R.N. Schouten, C. Abelln, W. Amaya, V. Pruneri, M. W. Mitchell, M. Markham,D.J. Twitchen, D. Elkouss, S. Wehner, T.H. Taminiau, R. Hanson, Nature 526, 686 (2015).[19] M. Banik, M.D.R. Gazi, S. Das, A. Rai, S. J. Kunkri, Phys. A 45, 205301 (2012).[20] Koh, D. E., Hall, M. J. W., Setiawan, Pope, J. E., Marletto, C., Kay, A., Scarani, V., Ekert,A. E. Phys. Rev. Lett. 109, 160404 (2012).[21] Thinh, L. P., Sheridan, L., Scarani, V. Phys. Rev. A 87, 062121 (2013).[22] Pope, J. E., Kay, A. Phys.Rev. A 88, 032110 (2013).[23] Putz, G., Rosset, D., Barnea, T. J., Liang, Y.-C., Gisin, N. Phys. Rev. Lett. 113, 190404(2014)