On the violation of L u ¨ ders bound of macrorealist and noncontextual inequalities
OOn the violation of L ¨ u ders bound of macrorealist and noncontextual inequalities Asmita Kumari, Md. Qutubuddin, and A. K. Pan
National Institute of Technology Patna, Ashok Rajpath, Patna, Bihar 800005, India
In a recent Letter [PRL, 113, 050401 (2014)], it is shown that the quantum violation of a three-time Leggett-Garg inequality (LGI) for a dichotomic qutrit system can exceed the L ¨ u ders bound.This is obtained by using a degeneracy breaking projective measurement rule which the authorstermed as von Neumann rule. Such violation can even approach the algebraic maximum in theasymptotic limit of system size. In this paper, we question the implication of such violation ofL ¨ u ders bound and its conceptual relevance in LG scenario. We note an important fact that thebasis for implementing the proposed von Neumann rule for a degenerate observable is non-uniqueand show that the violation of L ¨ u ders bound is crucially dependent on the choice of basis. Further,we demonstrate the violation of L ¨ u ders bound of the simplest non-contextual inequality (NCI) whichis in contrast to the reasoning provided in the aforementioned Letter. This result further raises thedoubts regarding the validity of the proposed rule as a viable projective measurement. We discussthe relevance of such results with respect to the usual quantum violation of LGI and NCI. I. INTRODUCTION
In their epoch-making paper, Einstein, Podolsky andRosen [1] had raised the fundamental question about theincompleteness of quantum mechanical description of na-ture by the ψ -function. The realist hidden variable mod-els are those which assume to provide the ‘complete state’of the system with the aid of so-called hidden variablesalong with the quantum state. In this regard, the cen-tral question is what constraint a realist model of quan-tum phenomena has to satisfy to be compatible with theempirically verifiable predictions of quantum mechan-ics (QM). In his pioneering work in 1964, Bell [2] firstpointed out that any realist model reproducing the QMhas to be nonlocal. Since then, extensive studies (see,for reviews, [3, 4]) have been performed for revealing thenonlocal character of QM both theoretically and experi-mentally.Another constraint, known as contextuality, was dis-covered by Kochen and Specker [5]. According to theassumption of non-contextuality in a realist hidden vari-able model, the definite value assignment to a measure-ment is independent of the compatible measurementsthat are being performed jointly or sequentially. Kochenand Specker theorem demonstrates that such a value as-signment is in contradiction with the statistics of QM fora certain set of observables. The original proof of theKochen and Specker theorem involves a complex struc-ture using 117 vectors. Later, simplified versions havebeen proposed by Peres [6] and Mermin [7]. These proofsare converted into testable inequalities, valid for any non-contextual realist model, but violated by QM [8, 9]. Thenotion of contextuality has also been extensively studiedboth theoretically [10, 11] and experimentally [12].Of late, the macrorealist models and its compatibilitywith the QM has also been attracting increasing atten-tion. Such a line of study was first put forwarded byLeggett and Garg [13] by introducing a set of inequali-ties (henceforth, LGIs) for testing the compatibility be-tween the classical world view of macrorealism and QM. These inequalities play similar role to that of the Bellinequalities in testing local hidden-variable models butinvolving a single system subjected to the measurementsat different times. Leggett and Garg inequalities are de-rived by considering the assumptions of macrorealism perse and non-invasive measurability . According to the as-sumption of macrorealism per se , at any instant of timea system remains in one of it’s macroscopically distinctontic state, whereas the non-invasive measurability en-sures that the ontic state of the system remains unin-fluenced by the measurement and dynamics [13, 14]. Inrecent times, a flurry of theoretical studies have been re-ported [15–25] and a number of experiments have beenconducted by using different systems [26–33].The quantum violations of local, non-contextual andmacrorealist inequalities are witnesses of non-classicality.Improved quantum violation of a given inequality is thusthe signature of more non-classicality. The present pa-per questions the issue of improved quantum violationof LGIs that is proposed in a recent work [21]. Thesimplest Bell’s inequality is the CHSH one [34] whosequantum violation is restricted by the Cirelson’s bound[35]. LGIs are often considered structurally analogous toBell’s inequalities. But, Budroni and Emary [21] haverecently shown that the former can be violated up to itsalgebraic maximum. Specifically, by considering a degen-eracy breaking projective measurement scenario termedas ‘von Neumann rule’, they showed that the quantumviolation of the LGI can exceed the L ¨ u ders bound andcan even approach the algebraic maximum at the asymp-totic limit of the system size. Such amount of violationof CHSH inequalities may be achieved in post-quantumtheories but not in QM. For dichotomic observables, themaximum quantum violation of a three-time LGI is . if L ¨ u ders projection rule [36] is used, irrespective of thesystem size [37]. Authors in [21] have shown that for di-chotomic observables in a qutrit system and for suitablechoice of initial quantum state, the use of such a degener-acy breaking rule provides the quantum violation of LGI . exceeding the L ¨ u ders bound. Using semi-definiteprogramming they have shown that such a violation can a r X i v : . [ qu a n t - ph ] J un reach 2.21. Such a result is verified in recent experi-ment [32, 33] using qutrit system. Additionally, theyremarked that the L ¨ u ders bound of a non-contextual in-equality (NCI) cannot be violated in this manner. Insupport of their claim, they mentioned that since in con-textuality test the joint measurements are assumed forcommuting observables, then which state update rule isbeing used becomes irrelevant even when the measure-ments are performed sequentially [21].Before proceeding further, let us first encapsulate theessence of L ¨ u ders rule and the von Neumann rule pro-posed in [21]. Consider an observable ˆ A having discreteeigenvalues a , a , a ... a m with degree of degeneracies x , x , x ... x m respectively. Let P αm = | φ αm (cid:105)(cid:104) φ αm | is theprojection operator associated with m th eigenvalue where α denotes the degeneracy. The von Neumann projectionrule breaks the degeneracy, so that, the reduced densitymatrix can be written as ρ v = (cid:80) m,α P αm ρP αm where ρ isthe initial density matrix of the system. As already indi-cated, ρ v is not unique for degenerate observable. On theother hand, the L ¨ u ders projection rule respects the de-generacy. The reduced density matrix in this case can bewritten as ρ l = (cid:80) m P m ρP m where P m = (cid:80) x m α =1 | φ αm (cid:105)(cid:104) φ αm | [38, 39]. Then, for an observable with degenerate eigen-values, the von Neumann rule provides the reduced den-sity matrix less coherent than that is obtained using theL ¨ u ders rule. For non-degenerate observable both therules are identical. Throughout our paper by von Neu-mann rule we refer the discussion in this paragraph toavoid any confusion.In this paper, we critically examine the reason of such aviolation of L ¨ u ders bound of LGIs and question the valid-ity of the so-called von Neumann rule in LG scenario. Wefirst point out that the choice of basis for applying thatrule is not unique. There can be infinitely many choicesof basis for implementing it, even for a dichotomic ob-servable in qutrit system. Thus, in a sequential measure-ments of two degenerate observables, the reduced stateafter the first measurement becomes non-unique whichmay then produce different results of sequential measure-ments of the same two observables. We indeed show thathow different choice of von Neumann basis provides dif-ferent amount of quantum violation of a given LGI. Weprovide some detail calculations to show how for a givenstate and observable the choice of von Neumann basisprovide different amount of violations of LGIs.Further, we study the quantum violation of non-contextual inequality (NCI). As opposed to the claim in[21], we show that the use of the von Neumann rule vio-lates the L ¨ u ders bound of NCI. Surprisingly, we considerthe simplest NCI involving three commuting dichotomicobservables pertaining to a qutrit system. In such choicesof observables the violation of L ¨ u ders bound of NCI isnot expected. Similar to the case of LGIs, we find differ-ent choices of von Neumann basis that provide differentamounts of quantum violations of a given NCI and a suit-able choice of basis provide the quantum violation of NCI2. A LGI consists of statistical correlations of sequen-tial measurements of two observables. It seems thought-provoking why the violation of L ¨ u ders bound occurs whenone uses von Neumann reduction rule. To understandthis issue let us consider a degenerate observable A . Themeasurement of which naturally uses L ¨ u ders projectionrule. In order to forcefully implement the von Neumannprojection rule [38], one needs to perform a non-detectivemeasurement of another observable (say, A (cid:48) ) prior to A , where A and A (cid:48) are commuting. Hence, the mea-sured statistics of A remains same irrespective of the factwhether it is measured solely or after A (cid:48) . However, thecrucial fact is that the reduced density matrices for thosetwo aforementioned measurement scenarios are differentwhose effect can be detectable if a subsequent measure-ment of another observable, say A (may or may not becommuting with A ), is performed separately on the tworeduced density matrices. Then, the statistics of such asequential measurement can be different for two differ-ent density matrices because A (cid:48) may not be commutingwith A in general. Moreover, different von Neumannbasis produce different reduced density matrices. Similarargument holds good for other two sequential measure-ments involving the LGI.It is then intuitively clear that a kind of additional non-classicality is being introduced through the von Neumannprojection rule which provides the violation of L ¨ u dersbound of LGIs and also of NCIs. In other words, onemay say that it is a kind of quantum contextuality in-duced violation of L ¨ u ders bound of LGIs and NCIs whichcannot be considered as the violation of LGIs and NCIsin its usual sense. Note that, the prior observable A (cid:48) isnowhere included in deriving the classical bounds of LGIsor NCIs. The inclusion of A (cid:48) s in the realist model for-mulation may provide a different classical bound of therelevant inequalities. Then, it is certainly erroneous totreat the violation of L ¨ u ders bound by von Neumann pro-jection rule in the same footing as the quantum violationof usual LGIs and NCIs.This paper is organized as follows. In Sec.II, we showthe difference between L ¨ u ders and von Neumann rules insequential measurement of two degenerate qutrit observ-ables and in Sec.III, we first show that given a particularvon Neumann basis, the suitable intermediate evolutionscan improve the maximum quantum violations of LGIsfor a given state and observable and then we explicitlydemonstrate that the violation of L ¨ u ders bound is depen-dent on the choices of von Neumann basis. In Sec. IV,we demonstrate that quantum violations of NCIs exceed-ing the L ¨ u ders bound if von Neumann projection rule isused. We discuss the implications of our study in Sec. V. II. L ¨ u DERS RULE, VON NEUMANN RULEAND SEQUENTIAL MEASUREMENT
For a qutrit system, let us now assume two dichotomicdegenerate observables ˆ A and ˆ B such that P αm and P βn aretheir respective projectors. Here α , β are degeneracy and m, n = ± are the eigenvalues of ˆ A and ˆ B respectively. Forthe particular case of dichotomic observable in a qutritsystem α, β = 1 , . Then the sequential measurement ˆ A and ˆ B provides (cid:104) ˆ A ˆ B (cid:105) seq = (cid:88) m,n = ± mnP ( P αm P βn ) (1)For L ¨ u ders rule the joint probability P ( P αm P βn ) = T r [ P m ρP m P n ] where P m = (cid:80) α P αm , P n = (cid:80) β P βn . Onthe other hand, using von Neumann rule the joint prob-ability is given by P ( P αm P βn ) = T r [ (cid:80) α,β P αm ρP αm P βn ] .The sequential measurement by using L ¨ u ders rule pro-vides (cid:104) ˆ A ˆ B (cid:105) Lseq = 12 (
T r [ ρ ˆ A ˆ B ] + T r [ ρ ˆ B ˆ A ]) (2)which is irrespective of the dimension of the system. But,using von Neumann projection rule, for example, in thecase of dichotomic observable in qutrit system, we obtain (cid:104) ˆ A ˆ B (cid:105) Vseq = (cid:104) ˆ A ˆ B (cid:105) Lseq − T r [( P α + ρP α + + P α + ρP α + ) ˆ B ] (3)Here P α + and P α + are the projectors with same eigen-value m = +1 . It is then evident from Eq.(3) that thereis an extra term along with (cid:104) ˆ A ˆ B (cid:105) Lseq which is in gen-eral non-zero and responsible for the violation of L ¨ u dersbound. Importantly, while (cid:104) ˆ A ˆ B (cid:105) Lseq is basis independent,the quantity
T r [( P α + ρP α + + P α + ρP α + ) ˆ B ] is dependenton the choice of von Neumann basis leading to the basisdependent violation of L ¨ u ders bound. We show how thechoice of basis provides different amount of violation ofLGIs and NCIs. III. VIOLATION OF L ¨ u DERS BOUND OF LGIS
Let us assume a suitable dichotomic observable ˆ M at t = t , and its measurement on the state of a macroscopicsystem produce a definite outcome or − at any instantof time, as per the assumption of macrorealism per se .In LG scenario, the measurement of ˆ M is performed onmacroscopic system at three different times t , t and t ( t > t > t ) leads the measurement observables ˆ M , ˆ M and ˆ M . The notion of non-invasive measurability en-sure the existence joint probability of different outcomes P ( M ± M ± M ± ) and uninfluenced marginal effect on priorand future measurements.Based on the aforementioned two assumptions, theLGIs are derived as K = (cid:104) ˆ M ˆ M (cid:105) + (cid:104) ˆ M ˆ M (cid:105) − (cid:104) ˆ M ˆ M (cid:105) ≤ (4) K = (cid:104) ˆ M ˆ M (cid:105) − (cid:104) ˆ M ˆ M (cid:105) + (cid:104) ˆ M ˆ M (cid:105) ≤ (5) K = −(cid:104) ˆ M ˆ M (cid:105) + (cid:104) ˆ M ˆ M (cid:105) + (cid:104) ˆ M ˆ M (cid:105) ≤ (6) where (cid:104) ˆ M r ˆ M s (cid:105) is a correlation function of dichoto-mous observable ˆ M r and ˆ M s , where r, s = 1 , and with r < s . In QM, the ˆ M r and ˆ M s are unitar-ily connected as ˆ M s = U ∆ t rs ˆ M r U † ∆ t rs , where U ∆ t rs = e iH ( t r − t s ) . For a quantum state ρ t the correlationfunctions (cid:104) ˆ M r ˆ M s (cid:105) = (cid:80) m,n = ± mnP ( M mr M ns ) where thejoint probability can be obtained by using L ¨ u ders ruleis P ( M mr M ns ) = T r [ U ∆ t rs P m U ∆ t r ρ t U † ∆ t r P m U † ∆ t rs P n ] .Here P m = (cid:80) µ Π µm , P n = (cid:80) ν Π νn and µ, ν are therelevant degeneracies. On the other hand, joint prob-ability for the von Neumann rule is P ( M mr M ns ) = T r [ (cid:80) µ,ν U ∆ t rs Π mµ U ∆ t r ρ t U † ∆ t r Π mµ U † ∆ t rs Π nν ] .We consider the same observable ˆ M = | (cid:105)(cid:104) | + | (cid:105)(cid:104) |−| (cid:105)(cid:104) | as in [21], where | (cid:105) = (1 , , T , | (cid:105) = (0 , , T and | (cid:105) = (0 , , T are the eigenvectors with eigenval-ues ( − , , ) respectively. However, ˆ M can also bedecomposed as ˆ M = | (cid:48) (cid:105)(cid:104) (cid:48) | + | (cid:48) (cid:105)(cid:104) (cid:48) | − | (cid:105)(cid:104) | where | (cid:105) = (1 , , , | (cid:48) (cid:105) = ξ | (cid:105) + (cid:112) − ξ | (cid:105) and | (cid:48) (cid:105) = (cid:112) − ξ | (cid:105) − ξ | (cid:105) ) are also eigenvectors having eigenval-ues ( − , , respectively. For ξ = 1 , former decompo-sition can be recovered which is the case considered byBudroni and Emary [21]. Importantly, L ¨ u ders measure-ment is independent of ξ but it plays a crucial role in vonNeumann rule as seen from Eq.(3). Different values of ξ thus implement different von Neumann measurements.The evolution is governed by the Hamiltonian H = γ ˆ J x , where γ the coupling constant and ˆ J x is the angularmomentum operator along ˆ x direction. Then the unitaryevolution can be written as U ∆ t rs = e iγ ˆ J x ∆ t rs . We denote g = γ ∆ t and g = γ ∆ t . Usually, in LG scenariothe coupling strengths are taken to be same, i.e., g = g . Here we take g (cid:54) = g which further improves theamount of maximum quantum violation. For qubit casedifferent values of g and g do not improve the maximumquantum violation.Before proceeding further, let us first briefly recapit-ulate the essence of results in [21]. By considering thesame state | ψ t (cid:105) = (0 , , T , the quantum mechanicalexpression of K (say, K v b ) using von Neumann basiswith ξ = 1 and g = g = g derived by them is given by[21] K v b = 116 (cid:2) g ) −
20 cos(2 g ) + 3 cos(4 g ) (cid:3) (7)The maximum quantum value of Eq. (7) is . at g = 1 . , clearly exceeding the L ¨ u ders bound . . They[21] have further showed that the violation increases withthe increment of system size and approaches algebraicmaximum in asymptotic limit of system size.Now, instead of same coupling constant, if we take g (cid:54) = g , we have K v = 12 (cid:104) sin ( g ) + cos( g ) + 2 cos ( g ) cos( g )+ cos ( g ) sin ( g ) + cos( g ) (cid:0) ( g ) (cid:1) − g + g ) − sin ( g + g ) (cid:105) (8) Quantum violations of various LGIs for ξ = 1 LGI Max.Value g g K l . π π K v b .
75 1 .
31 1 . K v .
91 0 .
98 1 . K l π πK v b π πK v . − π/ π/ K l . π − π K v b π πK v .
44 2 . − . Table I. The quantities K l , K l and K l denote the max-imum values when L ¨ u ders measurement is performed, and K v b , K v b and K v b are the values when von Neumann mea-surement with ξ = 1 is performed and g = g as used in [21]. K v , K v and K v are calculated for ξ = 1 but g (cid:54) = g . which naturally reduces to Eq.(7) at g = g . Inter-estingly the maximum value of K v in Eq.(8) is . for g = 0 . and g = 1 . . Thus, instead of using g = g = g , the consideration of two different interme-diate couplings improved the quantum violation of LGI K . However, if the maximum quantum value of K and K is calculated by using ξ = 1 and g = g = g ,we have ( K v b ) max = 1 and ( K v b ) max = 1 . Then, no vi-olation of LGIs given by Eq. (5) and (6) can be obtainedin this case. We demonstrate that for a suitable choiceof g and g even for ξ = 1 , the maximum values of K v and K v can be largely improved as given in Table I.Let us now examine if suitable choice of ξ can fur-ther increases the quantum violation of LGIs. The QMexpressions K v , K v and K v for arbitrary ξ, g and g are given in Eqs. (A1), (A2) and (A3) respectively. Itcan be seen from Table. II that, K v = 2 ≥ K v b when ξ = √ . Thus, the choice of von Neumann basis plays animportant role. However, Budroni and Emary [21] foundthe quantum violation of LGI 2.21 by using SDP for allpossible choices of basis. Here we wanted to explicitlyshow that the violation of LGI is basis dependent if vonNeumann measurement rule is taken for state reduction.It may also be possible that for a particular value of ξ no violation of any of the LGIs occurs but for the samestate and same observable the violation can be consid-erably large for different value of ξ . Note that, for theabove choices of state and observables, the value of K v is lower than L ¨ u ders bound for any choices of ξ .Next, in order to showing the effect of basis choicefor the violation of L ¨ u ders bound of K we consider astate | ψ t (cid:105) = (1 , , T and calculate K v for that state.The quantum expressions of K v is given in Eq.( A ) ofthe Appendix. The quantum expressions of K v , K v and K v given by Eqs. (A1), (A2) and (A4) respectivelyare plotted in Fig.1. We found that ( K v ) max = 2 for Quantum violations of various LGIsLGI Max.Value g g ξK v .
91 0 .
98 1 .
85 1 K v . π π √ K v .
44 2 . − .
73 1
Table II. The quantities K v , K v and K v are the quantumvalues of LGIs expression when von Neumann measurementis performed for different values of ξ , g and g . ξ K v K v K v Figure 1. (color online): The quantum mechanical expressionsof K v and K v (calculated for | ψ t (cid:105) = (0 , , T ) and K v (cal-culated for | ψ t (cid:105) = (1 , , T ) given by (A1), (A2) and (A4)respectively are plotted as a function of ξ for fixed values of g and g , as given in Table II. g = g = π , when ξ = √ . More suitable choice of state,observables and basis can provide the result obtained in[21] using SDP. The important point we wanted to ex-plicitly pointed out here that the choice of basis playscrucial role in von Neumann measurement scenario andthe suitable choice of which can violate the L ¨ u ders boundof LGIs by a considerably large amount.In the next section we demonstrate the violation ofL ¨ u ders bound of non-contextual inequalities (NCIs) anddiscuss the subtleties involved regarding von Neumannprojection rule and discuss its relevance in the violationof realist inequalities. IV. VIOLATION OF L ¨ u DERS BOUND OF NCIS
Let ˆ A , ˆ A and ˆ A be the three mutually commut-ing dichotomic observables. Given a quantum state, themeasurement statistics of ˆ A is independent of whethermeasuring with ˆ A or ˆ A . Similar arguments holds goodfor ˆ A and ˆ A . This feature is the non-contextuality inQM. In an non-contextual realist model, it is assumedthat the individual measured values say, v ( A ) (fixed bya hidden variable) follows the same context independenceas in QM. So that, v ( A ) is independent of v ( A ) or v ( A ) . We can then write that in a non-contextual real-ist hidden variable model the following three inequalitiesare satisfied. β = (cid:104) ˆ A ˆ A (cid:105) + (cid:104) ˆ A ˆ A (cid:105) − (cid:104) ˆ A ˆ A (cid:105) ≤ , (9) β = (cid:104) ˆ A ˆ A (cid:105) − (cid:104) ˆ A ˆ A (cid:105) + (cid:104) ˆ A ˆ A (cid:105) ≤ , (10) β = −(cid:104) ˆ A ˆ A (cid:105) + (cid:104) ˆ A ˆ A (cid:105) + (cid:104) ˆ A ˆ A (cid:105) ≤ , (11)It is evident that none of the above inequalities willbe violated by QM, if the L ¨ u ders rule is used. Thisis due to the fact that the triple-wise joint probability P ( A , A , A ) exists whose suitable marginal correctlyprovides all pair-wise joint probabilities satisfied by QM.Thus, the L ¨ u ders bound of those NCIs is also 1. In theirpaper [21], Budroni and Emary remarked that the L ¨ u dersbound of NCIs can not be violated by using von Neumannrule. We first demonstrate that their inference is not cor-rect and then discuss the suitabilities involved in the vonNeumann rule.For this, let us consider the observables ˆ A i = I − | α i (cid:105)(cid:104) α i | with (cid:104) α i | α j (cid:105) = δ ij where i, j = 1 , , . Wetake specific examples where | α (cid:105) = ( − , , T / √ , | α (cid:105) = (1 , , T / √ and | α (cid:105) = (0 , , T . Eigen-states of ˆ A may be written as | a (cid:105) = ( − , , T / √ , | a (cid:105) = (1 , , T / √ and | a (cid:105) = (0 , , T with eigen-values ( − , , respectively. Similarly, the eigenstatesof ˆ A can be written as | b (cid:105) = (1 , , T / √ , | b (cid:105) =( − , , T / √ and | b (cid:105) = (0 , , T with eigenvalues ( − , , respectively, and the eigenstates of ˆ A are | c (cid:105) = (0 , , T , | c (cid:105) = (0 , , T and | c (cid:105) = (1 , , T with eigenvalues ( − , , respectively.As we have already pointed out that the choices ofbasis for invoking the von Neumann projection rule isnot unique. For example, for observable ˆ A , the eigen-states can be chosen as | a (cid:105) = ( − , , T / √ , | a (cid:48) (cid:105) = (cid:15) | a (cid:105) + √ − (cid:15) | a (cid:105) and | a (cid:48) (cid:105) = √ − (cid:15) | a (cid:105) − (cid:15) | a (cid:105) witheigenvalues ( − , , respectively. Similarly, one canchoose the eigenstates of ˆ A are | b (cid:105) = (1 , , T / √ , | b (cid:48) (cid:105) = λ | b (cid:105) + √ − λ | b (cid:105) and | b (cid:48) (cid:105) = √ − λ | b (cid:105) − λ | b (cid:105) with eigenvalues ( − , , respectively, and the eigen-state of ˆ A are | c (cid:105) = (0 , , T , | c (cid:48) (cid:105) = δ | c (cid:105) + √ − δ | c (cid:105) and | c (cid:48) (cid:105) = √ − δ | c (cid:105) − δ | c (cid:105) with eigenvalues ( − , , respectively. Then there are infinite number of basischoice for implementing von Neumann measurement fordifferent choices of (cid:15), λ, δ ∈ [0 , .We examine how different values of (cid:15) , λ and δ play im-portant role for improving the violaton of different NCIs.By considering the state | ψ (cid:105) = (sin( θ ) sin( φ ) , cos( θ ) sin( φ ) , cos( φ )) T the quantum expression β l , β l and β l are calculatedby using L ¨ u ders projection rule is given by β l = 1 − (cid:104) cos( φ ) + sin( θ ) sin( φ ) (cid:105) (12) Quantum violations of NCIsNCI Max.value φ θ (cid:15) λ δβ v π/ . . β v π/ π/ . . β v π/ π/ Table III. The quantities β v , β v and β v denote the quantumvalues of different NCIs using von Neumann projection rule. β l = 1 − (cid:104) cos( φ ) − sin( θ ) sin( φ ) (cid:105) (13)and β l = 1 − ( θ ) sin ( φ ) (14)It is then straightforward to see that the maximum val-ues of β l , β l and β l cannot exceed for any possiblechoices of θ and φ , as expected.The quantum expressions β v , β v and β v for von Neu-mann rule are given by Eq.(A5),(A6) and (A7) respec-tively in the Appendix A. We maximize β v , β v and β v by suitably choosing the relevant parameter θ, φ, λ and δ as given in Table. IV. In Fig.2, we plotted β v , β v and β v as a function of θ with fixed values of φ, (cid:15), λ and δ asgiven in the Table. IV. We found that quantum values ofall β v , β v and β v reach , exceeding L ¨ u ders bound . - π - π π π - θ β v β v β v Figure 2. (color online). The quantum mechanical expressions β v , β v and β v given by Eq. (A5), (A6) and (A7) respectivelyare plotted as a function of θ for fixed values of φ, (cid:15), λ and δ given by Table IV. We thus exhibited the violation of L ¨ u ders bound ofNCIs even when the observables ˆ A , ˆ A and ˆ A are mu-tually commuting. This is in contrast to the claim in [21].Again, the different choice of von Neumann basis providesdifferent violation of L ¨ u ders bound of NCIs. Conceptu-ally, this result is very surprising. Since the NCIs givenby Eqs. (9 − are derived by assuming that the exis-tence of the joint probability distribution P ( A , A , A ) in a non-contextual theory. The quantum violation ofNCIs can only be obtained when P ( A , A , A ) does notexist in QM. Since ˆ A , ˆ A and ˆ A are mutually commut-ing then triple-wise joint probability exists in QM too.Hence, as we have already mentioned in introductory sec-tion, a kind of non-classicality is introduced through thevon Neumann rule which enables the quantum violationof NCIs given in Eqs.(9-11). In the next section, we pro-vide a detail discussions regarding the meaning of suchviolation and its relevance with respect to the usual vio-lation of realist inequalities. V. SUMMARY AND DISCUSSION
In this paper we questioned the conceptual relevanceof the violation of L ¨ u ders bound of LGIs and NCIs ob-tained through the degeneracy breaking projective rulefor state reduction. The improved violations of realist in-equalities are the signature of more non-classicality andany such improvement is useful for testing the concernedinequalities experimentally. For a dichotomic system theL ¨ u ders bound of three time LGIs is restricted to . .Budruni and Emary [21] have shown that for dichotomicobservables in qutrit system the use of von Neumannmeasurement projection rule for state reduction can pro-vide quantum violation of LGI that exceeds the L ¨ u dersbound . . However, they have considered a particularchoice of the von Neumann basis. We have pointed outthat the choice of basis for implementing the so calledvon Neumann rule for state reduction is not unique andthere can be infinitely many choices possible. We showedthat the violation of L ¨ u ders bound of LGIs is dependenton the choices of von neumann basis.LGIs are pertaining to the sequential measurement oftwo non-commuting observables. In contrast, in contex-tuality test the pair-wise joint measurements are assumedfor commuting observables. The NCIs we have consid-ered here involve three commuting observables. Sincetriple-wise joint probability distribution of those com-muting observables exist in QM, no violation of NCIsis expected if L ¨ u ders rule is used. Budroni and Emary[21] claimed that this feature is true even if one uses thevon Neumann projection rule for state reduction. Weshowed here that their inference is not correct and suit-able choice of von Neumann basis can provide a consid-erably large violation of L ¨ u ders bound of NCIs. One may find this result is particularly interesting for exper-imental testing as compared to LGI test because in thenon-contextuality test one does not require to guaran-tee the non-invasiveness condition of the measurement.However, the meaning of such violation remains ques-tionable.Let us again return to the issue regarding the impli-cations of such violations of L ¨ u ders bound of LGIs andNCIs. For this, we consider the NCI given by Eq.(9). Asalready mentioned, for implementing von Neumann rulein a sequential measurement of (cid:104) A A (cid:105) one has to mea-sure another observable A (cid:48) , prior to A . Note that, themeasurement of A (cid:48) is non-detective, i.e., only the statereduction due to the measurement of A (cid:48) is taken intoaccount. Similarly, an observable (cid:104) A A (cid:105) requires A (cid:48) tobe measured before A . Moreover, the different choiceof von Neumann basis requires different A (cid:48) and similarlydifferent A (cid:48) . There is whatsoever no reason to consider A (cid:48) and A (cid:48) are same in general. Then the initial den-sity matrices for the sequential measurements of (cid:104) A A (cid:105) and (cid:104) A A (cid:105) can be different. One can suitably choose A (cid:48) and A (cid:48) to obtain an improved violation of NCI exceedingthe L ¨ u ders bound. But, neither the observables A (cid:48) and A (cid:48) nor the state change information due to the measure-ment of them was included in deriving the realist boundof NCI given by Eq.(9). Similar argument holds good forother NCIs and for LGIs. Thus, the violations of L ¨ u dersbound of NCIs and LGIs has no bearing on the issue ofquantum violations of the realist inequalities.Note however that, the results provided here can becalculated within the framework of standard QM by suit-ably choosing the prior measurements and can then beexperimentally tested. It would then be appealing tostudy how such empirically verifiable statistics can bereproduced in an ontological model and would be inter-esting to see what additional constraint is needed to beimposed in such a model in order to be consistent withthe QM. This calls for further study. ACKNOWLEDGMENTS
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The quantum mechanical expressions of K , K and K given by Eq. (4), (5) and (6) respectively are calculatedfor initial state | ψ (cid:105) = (0 , , T with different intermediate evolutions between measurements are given by K v = 1 + (cid:16) − − ξ + 11 ξ + 9 ξ ( − ξ ) cos( g ) + cos( g ) (cid:0) − ξ + 21 ξ + (2 − ξ + 15 ξ ) cos( g ) (cid:1)(cid:17) sin ( g ( g − ξ + 2 ξ ) sin( g ) sin( g ) −
14 (1 − ξ + 6 ξ ) sin(2 g ) sin(2 g ) (A1) K v = 14 (cid:104) (cid:16) − − ξ + 3 ξ − g ) + ξ ( − ξ )(4 cos( g ) + 9 cos(2 g )) + cos( g ) (cid:0) − ξ + 21 ξ + 4(1 − ξ + 3 ξ ) cos( g ) + (2 − ξ + 15 ξ ) cos(2 g ) (cid:1)(cid:17) sin ( g − − ξ + 2 ξ ) sin( g ) sin( g ) + (1 − ξ + 6 ξ )sin(2 g ) sin(2 g ) (cid:105) (A2) K v = 14 (cid:104) (cid:16) − − ξ + ξ + ( − − ξ + 3 ξ ) cos( g ) + cos(2 g ) (cid:0) − ξ + 27 ξ + (6 − ξ + 33 ξ ) cos( g ) (cid:1)(cid:17) sin ( g − − ξ + 3 ξ ) cos( g ) sin ( g − g ) sin( g ) + 8 ξ ( − ξ )( − g ) cos( g )) sin( g ) sin( g )+ sin(2 g ) sin(2 g ) (cid:105) (A3)The quantum mechanical expressions of K v obtained for the state | ψ t (cid:105) = (1 , , T is given by K v = 116 (cid:20) ξ − ξ + 2 −
16 cos( g ) sin (cid:16) g (cid:17) (cid:0)(cid:0) ξ − ξ + 1 (cid:1) cos( g ) + ξ − ξ + 3 (cid:1) + 2 cos(2 g )+ 4 cos(2 g ) sin (cid:16) g (cid:17) (cid:0)(cid:0) ξ − ξ − (cid:1) cos( g ) + 3 ξ − ξ − (cid:1) −
16 sin( g ) sin( g ) − g ) sin(2 g ) − (cid:0) ξ − (cid:1) ξ cos(2 g ) + 4 (cid:0) ξ − ξ + 3 (cid:1) cos( g ) (cid:21) (A4)The quantum mechanical expressions of β , β and β given by Eq.(9), (10) and (11) respectively are calculatedfor the initial state | ψ (cid:105) = (sin( θ ) sin( φ ) , cos( θ ) sin( φ ) , cos( φ )) T . The expressions are given by β v = (cid:16) − (cid:15) − (cid:15) + ( − λ )2 λ + 2 δ (1 − δ ) (cid:112) − δ (cid:17) cos ( φ ) + (cid:16) ( (cid:15) − λ )( (cid:15) + λ )( − (cid:15) + λ ) + (1 − (cid:15) + 3 (cid:15) + 3 λ − λ ) cos(2 θ ) + 2 δ ( − δ ) (cid:112) − δ sin ( θ ) + √ (cid:0) (cid:15) (1 − (cid:15) ) (cid:112) − (cid:15) + λ (1 − λ ) (cid:112) − λ (cid:1) sin(2 θ ) (cid:17) sin ( φ ) − (cid:16) √ (cid:0) (cid:15) ( − (cid:15) ) (cid:112) − (cid:15) + λ ( − λ ) (cid:112) − λ (cid:1) cos( θ ) + sin( θ ) + 2( − (cid:15) + (cid:15) − λ + λ + 2 δ − δ ) sin( θ ) (cid:17) sin(2 φ ) (A5) β v = (cid:0) − (cid:15) + 2 (cid:15) + 2 λ − λ + 2 δ ( − δ ) (cid:112) − δ (cid:1) cos ( φ ) + (cid:16) − (cid:15) + (cid:15) − λ + λ + (1 − (cid:15) + 3 (cid:15) − λ + 3 λ ) cos(2 θ ) + 2 δ (1 − δ ) (cid:112) − δ sin ( θ ) + √ (cid:0) (cid:15) (1 − (cid:15) ) (cid:112) − (cid:15) + λ ( − λ ) (cid:112) − λ (cid:1) sin(2 θ ) (cid:17) sin ( φ )+ (cid:16) √ (cid:0) (cid:15) (1 − (cid:15) ) (cid:112) − (cid:15) + λ (1 − λ ) (cid:112) − λ (cid:1) cos( θ ) + sin( θ ) + 2( (cid:15) − (cid:15) − λ + λ + 2 δ − δ ) sin( θ ) (cid:17) sin(2 φ ) (A6) β v = (cid:0) − (cid:15) + 2 (cid:15) + ( − λ )2 λ + 2 δ ( − δ ) (cid:112) − δ (cid:1) cos ( φ ) − (cid:16) − (cid:15) + (cid:15) − λ + λ + (2 − (cid:15) + 3 (cid:15) − λ + 3 λ ) cos(2 θ ) + 2 δ ( − δ ) (cid:112) − δ sin ( θ ) + √ (cid:0) (cid:15) (1 − (cid:15) ) (cid:112) − (cid:15) + λ ( − λ ) (cid:112) − λ (cid:1) sin(2 θ ) (cid:17) sin ( φ )+ (cid:16) √ (cid:0) (cid:15) ( − (cid:15) ) (cid:112) − (cid:15) + λ ( − λ ) (cid:112) − λ (cid:1) cos( θ ) + ( − − (cid:15) + 2 (cid:15) + 2 λ − λ + 4 δ − δ ) sin( θ ) (cid:17) sin(2 φ ))