Quantum violation of variants of LGIs upto algebraic maximum for qubit system
QQuantum violation of variants of LGIs upto algebraic maximum for qubit system
A. K. Pan, Md. Qutubuddin, and Swati Kumari
National Institute Technology Patna, Ashok Rajpath, Patna, Bihar 800005, India
In 1985, Leggett and Garg formulated a class of inequalities for testing the compatibility betweenmacrorealism and quantum mechanics. In this paper, we point out that based on the same assump-tions of macrorealism that are used in the derivation of Leggett-Garg inequalities (LGIs) , there is ascope of formulating another class of inequalities different from standard LGIs. By considering thethree-time measurement scenario in a dichotomic system, we first propose an interesting variant ofstandard LGIs and show that its quantum violation is larger than the standard LGI. By extendingthis formulation to n -time measurement scenario, we found that the quantum violations of variantsof LGIs for a qubit system increase with n , and for a sufficiently large n algebraic maximum canbe reached. Further, we compare the quantum violations of our formulated LGIs with the standardLGIs and no-signaling in time formulation of macrorealism. I. INTRODUCTION
Since the inception of quantum mechanics (QM), it re-mains a debatable question how our everyday world viewof macrorealism can be reconciled with the quantum for-malism. Historically, this question was first pointed outby Schr ¨ o dinger [1] through his famous cat experiment.Since then, quite a number of attempts have been madeto pose the appropriate questions relevant to this issueand to answer that questions. One effective approachto encounter this issue is to experimentally realize thequantum coherence of Schr ¨ o dinger cat-like states of largeobjects [2]. Another approach within the formalism ofQM is the decoherence program [3]. It explains howinteraction between quantum systems and environmentleads to classical behavior, but does not by itself providethe desired ‘cut’ ( à la Heisenberg [8]). It is also arguedthat even if the decoherence effect is made negligible, thequantum behavior can be disappeared by the effect ofcoarse-graining of measurements[4]. Proposal has alsobeen put forwarded [5] to modify the dynamics of stan-dard formalism of QM allowing an unified description ofmicroscopic and macroscopic systems.However, the above mentioned attempts do not exactlyaddress the fundamental question whether macrorealismis, in principle, compatible with the formalism of QM.Macrorealism is a classical world view that asserts thatthe properties of macro-objects exist independently andirrespective of ones observation. Motivated by the Bell’stheorem [6], in 1985, Leggett and Garg [9] formulateda class of inequalities based on the notions of macroreal-ism, which provides an elegant scheme for experimentallytesting the compatibility between the macrorealism andQM.To be more specific, the notion of macrorealism con-sists of two main assumptions [9–11] are the following;
Macrorealism per se (MRps):
If a macroscopic sys-tem has two or more macroscopically distinguishable on-tic states available to it, then the system remains in oneof those states at all instant of time.
Non-invasive measurement (NIM):
The definite onticstate of the macrosystem is determined without affecting the state itself or its possible subsequent dynamics.It is reasonable to assume that the systems in oureveryday world, in principle, obeys the aforementionedassumptions of a macrorealistic theory. Based onthese assumptions, the standard Leggett-Garg inequal-ities (LGIs) are derived. Such inequalities can be shownto be violated in certain circumstances, thereby implyingthat either or both the assumptions of MRps and NIMis not compatible with all the quantum statistics. In re-cent times, a flurry of theoretical studies on macrorealismand LGIs have been reported [12–23] and a number of ex-periments have been performed by using various systems[24–29].Let us encapsulate the simplest LG scenario. Considerthat the measurement of a dichotomic observable ˆ M isperformed at three different times t , t and t ( t ≥ t ≥ t ) . In Heisenberg picture, this in turn implies thesequential measurement of the observables ˆ M , ˆ M and ˆ M corresponding to t , t and t respectively. Fromthe assumption of MRps and NIM, one can derive the astandard LGI is given by 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) ≤ (1)Here (cid:104) M M (cid:105) = (cid:80) m ,m = ± m , m P ( M m , M m ) andsimilarly for other temporal correlation terms. By rela-beling the measurement outcomes of each M i as M i = − M i with i = 1 , , and , three more standard LGIs canbe obtained.Instead of three times, if the measurement of M isperformed n times, then the standard LGI for the n -measurement LG strings can be written as K n = (cid:104) ˆ M ˆ M (cid:105) + ... + (cid:104) ˆ M n − ˆ M n (cid:105) − (cid:104) ˆ M ˆ M n (cid:105) (2)The inequality (2) is bounded as [23] follows. If n isodd, − n ≤ K n ≤ n − for n ≥ and if n is even, − ( n − ≤ K n ≤ n − for n ≥ . For n = 3 , one simplyrecovers inequality (1).For a two-level system, the maximum quantum valueof K n is ( K n ) maxQ = n cos πn . For n = 3 , ( K ) maxQ = 3 / .Thus for a three-time standard LG scenario involving adichotomic observable, the temporal Tsirelson bound of a r X i v : . [ qu a n t - ph ] J un K is / . It is proved [12] that this bound is irrespectiveof the system size.Within the standard framework of QM, the maximumviolation of CHSH inequality [6] is restricted by theTsirelson bound[7], which is significantly less than thealgebraic maximum of the inequality. The algebraic max-imum may be achieved in post-quantum theory but notin QM. LGIs are often considered to be the temporal ana-log of Bell’s inequality. However, it has been shown [19]that for a degenerate dichotomic observables in a qutritsystem, the quantum value of K goes up to . andcan even reach to algebraic maximum in the asymp-totic limit of the system size. Such amount of violationis achieved by invoking a degeneracy breaking projectivemeasurement which they termed as von Neumann rule.Recently, two of us have argued [31] that such a viola-tion of temporal Tsirelson bound has no relevance to theusual violation of LGIs.The purpose of the present paper is to provide im-proved quantum violation of macrorealism for qubit sys-tem. We argue that by keeping the assumptions ofmacrorealism intact, there is scope for formulating in-equalities different from the standard LGIs. We notehere an important observation that due to the sequentialnature of the measurement, the LG scenario is flexiblethan CHSH one. Such flexibility allows us to formulatenew variants of standard LGIs. For the simplest case ofthree-time measurement scenario, we first formulate aninteresting variant of LGI and show that our proposedinequality provides considerably larger quantum viola-tion compared to the standard LGIs. We then formulatemore variants of standard LGIs by increasing number ofmeasurements n and show that the quantum violationincreases with n . For sufficiently large n , the quantumvalues of variants of LGIs reach its algebraic maximum,even for qubit system. Such variants of LGIs thus provideimproved test of macrorealism than standard LGIs. Fur-ther, in terms of no-disturbance (coined as no-signalingin time in LG scenario), we discuss how the variants ofLGIs are conceptually elegant and can be considered bet-ter candidates for experimentally testing the macroreal-ism compared to standard LGIs.This paper is organized as follows. In Sec.II, we pro-pose variant of LGI for three-time measurement scenarioand demonstrate that it provide larger quantum viola-tion compared to standard LGI. By increasing the num-ber of measurements ( n ), in Sec.III, we formulate twomore variants of LGIs. We show that for a qubit system,the quantum violation of our variants of LGIs increasewith n and can even reach algebraic maximum for large n limit. In Sec.IV, we compare variant of LGIs withstandard LGIs and no-signaling in time conditions. Wesummarize our results in Sec.V. II. VARIANTS OF LGIS IN THREE-TIMEMEASUREMENT SCENARIO
We start by noting that the standard LGIs is a par-ticular class of inequalities but is not unique one. Theflexibility of LG scenario allows us to formulate variantsof LGIs different from the standard LGI given by Eq. (1).We ensure that the assumptions of MRps and NIM usedin the derivation of standard LGI remains the same.Let us again consider the three-time LG scenario in-volving measurement of dichotomic observables ˆ M , ˆ M and ˆ M in sequence. Now, instead of three two-timecorrelation functions used in Eq.(1), we consider a three-time correlation function (cid:104) ˆ M ˆ M ˆ M (cid:105) , a two-time func-tion (cid:104) ˆ M ˆ M (cid:105) and finally (cid:104) ˆ M (cid:105) . Using them, we proposean inequality is given by K = (cid:104) ˆ M ˆ M ˆ M (cid:105) + (cid:104) ˆ M i ˆ M j (cid:105) − (cid:104) ˆ M k (cid:105) ≤ (3)where i, j = 1 , , with j > i . We call those inequalitiesas variant of LGIs. It is crucial to note again that, theassumptions of MRps and NIM remain same as in thederivation of standard LGIs.The inequalities (3) are violated by QM. In order toshowing this, we take one inequality by choosing i, j and k are , and respectively, and consider the qubit stateis given by | ψ ( t ) (cid:105) = cosθ | (cid:105) + exp( − iφ ) sinθ | (cid:105) (4)with θ ∈ [0 , π ] and φ ∈ [0 , π ] . The measurement observ-able at initial time t is taken to be Pauli observable ˆ σ z .The unitary evolution is given by U ij = exp − iω ( t j − t i ) σ x and ω is coupling constant. For simplicity, we consider τ = | t i +1 − t i | and g = ωτ .The quantum mechanical expression of K is given by ( K ) Q = cos 2 g (4 cos g cos 2 θ ) + sin 4 g sin 2 θ sin φ − g cos 2 θ (5)which is state-dependent in contrast to the quantumvalue of standard LGI.To compare with the standard LGIs, let us write thequantum expression of K is given by ( K ) Q = 2 cos 2 g − cos 4 g (6)which is independent of the state.If the values of the relevant parameters are taken as g = 1 . , θ = 2 . and φ = π/ , the quantum valueof K is . , thereby violating the inequality (3). Themaximum quantum value ( K ) can be shown to be fordifferent coupling constants in between the evolutions.For simplicity, here we take same coupling constant g .The quantum value of K is then larger than ( K ) maxQ =3 / . The expressions ( K ) Q and ( K ) Q are plotted inFig.(1).Thus, if the larger violation of an inequality is con-sidered to be an indicator of more non-classicality, thenthe variant of LGI captures the notion of macrorealismbetter than the standard LGIs. π π π π - g K K Figure 1. The quantities ( K ) Q and ( K ) Q given byEq.(5) and Eq.(6) respectively are plotted against g . Thevalues of relevant parameters are θ = 2 . and φ = π/ . III. VARIANTS OF LGIS FOR n -TIMEMEASUREMENTS The above idea can be extended to n -time measure-ment scenario where n > . For example, if n = 4 , wecan formulate the a variant of LGI is given by K = (cid:104) ˆ M ˆ M ˆ M ˆ M (cid:105) + (cid:104) ˆ M ˆ M ˆ M (cid:105) − (cid:104) ˆ M (cid:105) ≤ (7)This inequality belongs to the same class of (3). Threemore inequalities of this class can be obtained by chang-ing the positions of ˆ M , ˆ M , ˆ M and ˆ M in the last twoterms of the inequality (7).Interestingly, for n = 4 , there can be another variantof LGI can be proposed as ˆ L = (cid:104) ˆ M ˆ M ˆ M (cid:105) + (cid:104) ˆ M ˆ M ˆ M (cid:105) − (cid:104) ˆ M ˆ M (cid:105) ≤ (8)Similar to the earlier case three more inequalities canbe obtained. If number of measurements is further in-creased, one finds more variants of LGIs.Now, by generalizing the above formulation for n -timemeasurement scenario we propose the following two in-equalities K n = (cid:104) ˆ M ˆ M ... ˆ M n (cid:105) + (cid:104) ˆ M ˆ M ... ˆ M n − (cid:105) − (cid:104) ˆ M n (cid:105) ≤ (9)and ˆ L n = (cid:104) ˆ M ˆ M ˆ M ˆ M ... ˆ M n − (cid:105) + (cid:104) ˆ M ˆ M ... ˆ M n (cid:105)− (cid:104) ˆ M ˆ M n (cid:105) ≤ (10)where (cid:104) ˆ M ... ˆ M n (cid:105) = (cid:80) m ,...,m n m ...m n P ( M m , ..., M m n n ) and similarly for other correlation. While inequality (9)belongs to the class of (3) and (7), the inequality (10)belongs to the other class of inequalities given by (8).But, both the n -time inequalities are derived from thesame assumptions of macrorealism. Next, we examine the quantum violation of inequal-ities (7) and (8) for the state given by Eq. (4). Thequantum mechanical expressions of K and L are re-spectively given by ( K ) Q = 12 (cid:0) g + 8 cos 2 g sin g cos 2 θ − g sin θ sin φ (cid:1) (11) ( L ) Q = 2 cos g cos 2 g cos 2 θ − cos 6 g + 12 sin 4 g sin 2 θ sin φ (12)The value of ( K ) Q is . at g = 1 . , θ = 1 . and φ = π/ and of ( L ) Q is . at g = 0 . , θ = 0 . ,and φ = π/ . However, the above values of ( K ) Q and ( L ) Q are not temporal Tsrilson bound of (7) and (8) ,which is not very important to our present purpose. Notethat, for a qubit system, the maximum quantum valueof standard four-time LGI is √ and its macrorealistbound is 2. Then, in four-time measurement scenario,the difference between quantum and macrorealist valuesis . . But, in the case of our variant of LGIs, we have ( K ) Q − K = 1 . and ( L ) Q − L = 1 . . It canalso be seen that ( K ) Q > ( K ) Q > ( K ) Q and ( L ) Q > ( K ) Q . Thus, by increasing the number of measurementsthe quantum violation of the variants of LGIs can beimproved compared to the quantum violation of standardthree or four-time LGIs.Further, we demonstrate that when n is sufficientlylarge, the quantum values of ( K n ) Q and ( L n ) Q reachalgebraic maximum of K n and L n respectively. For n -time sequential measurement, the calculation of corre-lation function in QM seems difficult task. In order totackle this problem, we derive a compact formula for n -time sequential correlation given in Eq.(A7) of AppendixA.For the qubit state given by (4), the quantum expres-sions of K n for even n is given by ( K n even ) Q = (cos 2 g ) n + (cos 2 g ) n − − (cid:0) cos 2( n − g cos 2 θ + sin 2( n − g sin 2 θ sin φ (cid:1) (13)and for odd n ( K n odd ) Q = (cos 2 g ) n − cos 2 θ + (cos 2 g ) n − − (cid:0) cos 2( n − g cos 2 θ + sin 2( n − g sin 2 θ sin φ (cid:1) (14)By considering g = π n , Eqs.(13) and (14) take the form ( K n even ) Q = (cid:0) cos πn (cid:1) n + (cid:0) cos πn (cid:1) ( n − cos 2 θ + cos πn cos 2 θ − sin πn sin 2 θ sin φ (15)and ( K n odd ) Q = (cid:0) cos πn (cid:1) n − cos 2 θ + (cid:0) cos πn (cid:1) ( n − ) + cos πn cos 2 θ − sin πn sin 2 θ sin φ (16)respectively. In the large n limit, both of them reducesto ( K n even ) Q = ( K n odd ) Q ≈ θ (17)Thus, when θ ≈ , the quantities ( K n even ) Q =( K n odd ) Q ≈ , i.e., the algebraic maximum of the in-equalities (9-10).Next, we calculate the quantum violation of the othervariant of LGI given by (10) for the state in Eq.(4). Thequantum expression of L n for even n is given by ( L n even ) Q = (cos 2 g ) n − cos 2 θ + (cos 2 g ) n − (cid:0) cos 2 g cos 2 θ + sin 2 g (cid:1) − cos 2( n − g (18)If n is odd, we have ( L n odd ) Q = (cos 2 g ) n − + (cos 2 g ) n − − cos 2( n − g (19)which is independent of the state.Similar to the earlier case, again by taking g = π n ,from Eqs. (18) and (19), we have ( L n even ) Q = (cid:0) cos πn (cid:1) n − cos 2 θ + cos πn + (cid:0) cos πn (cid:1) ( n − (cid:16) cos πn cos 2 θ + sin πn sin 2 θ sin φ (cid:17) (20)and ( L n odd ) Q = 2 (cid:0) cos πn (cid:1) n − + cos πn (21)For large n , the quantum value of ( L n odd ) Q is 3 which isindependent of the state and the qantity ( L n even ) Q ap-proaches the algebraic maximum when θ ≈ . TheEqs.(20) and (21) are plotted in Figure 2 to demonstratehow the quantum values of ( L n odd ) Q and ( L n even ) Q ap-proach to algebraic maximum with increasing the numberof measurements n . IV. COMPARING VARIANTS OF LGIS WITHOTHER FORMULATIONS OF MACROREALISM
Fine [33] theorem states that the CHSH inequalitiesare necessary and sufficient condition for local realism.Since standard LGIs are often considered to be the tem-poral analogue of CHSH inequalities one may expect thatthey also provide the necessary and sufficient conditionfor macrorealism. In recent works, Clemente and Kofler[30] showed that no set of standard LGIs can providethe necessary and sufficient condition for macrorealism.However, a suitable conjunction of no-signaling in time(NSIT) conditions provides the same. In this connec-tion, two of us [20] have shown that the Wigner formu-lation of LGIs are stronger than standard LGIs but theyalso do not provide necessary and sufficient condition formacrorealism. Against this backdrop, in this section, we n L n odd3 L n even3 Figure 2. The quantities ( L n odd ) Q and ( L n even ) Q givenby (20) and Eq.(21) respectively are plotted againstnumber of measurements n by taking θ = 0 . Boththe quantities approach algebraic maximum of theinequalities (9-10)for large n . shall analyze the status of our variant of LGIs for three-time measurement scenario. For this, let us first find theconnection between standard LGIs, NSIT condition andmacrorealism.NSIT condition is the statistical version of NIM con-dition. It is analogous to the no-signaling condition inBell’s theorem, however violation of NSIT condition doesnot provide any inconsistency with physical theories. Itsimply assumes that the probability of a outcome ofmeasurement remains unaffected due to prior measure-ment. Clearly, the satisfaction of all NSIT conditions inany operational theory ensures the existence of globaljoint probability condition P ( M m , M m , M m ) where m , m , m = ± and in such a case no violation of anyLGI can occur.A two-time NSIT condition can be written as N SIT (1)2 : P ( M m ) = (cid:88) m P ( M m , M m ) (22)which means that the probability P ( M m ) is unaffectedby the prior measurement of M . Similarly, a three-timeNSIT condition is given by N SIT (1)23 : P ( M m , M k ) = (cid:88) m P ( M m , M m , M m ) (23)Here P ( M m , M m , M m ) denotes the joint probabil-ities when all the three measurements are performed.Clemente and Kofler [30] have shown that a suitableconjunction of two-time and three-time NSIT conditionsprovides the necessary and sufficient condition for macro-realism, i.e., N SIT (2)3 ∧ N SIT (1)23 ∧ N SIT ⇔ M R (24)where MR denotes macrorealism. We first show howstandard LGIs do not provide necessary and sufficientcondition for macrorealism. Such an argument was firstinitiated in [14] and discussed in detail in [20]. But formaking the present work self-contained we encapsulatethe essence of the argument.Let us consider the pairwise marginal statistics of theexperimental arrangement when all three measurements( M , M and M ) are performed and introduce the fol-lowing quantity D ( M m , M m ) = P ( M m , M m ) − (cid:88) m P ( M m , M m , M m ) (25)which quantifies the amount of disturbance created (inother words, degree of violation of NSIT condition) bythe measurement M at t to the measurements of M and M at t and t respectively. Similarly, D ( M m , M m ) = P ( M m , M m ) − (cid:88) m P ( M m , M m , M m ) (26) D ( M m , M m ) = P ( M m , M m ) − (cid:88) m P ( M m , M m , M m ) (27)Note that, since no information can travel backward intime, D ( M m , M m ) = 0 in any physical theory. Fortwo-time measurements, we can define similar quantity,for example, D ( M m ) .Standard LGIs are derived by assuming the satisfac-tion of all NSIT conditions. But, in QM, the NSIT con-ditions are, in general, not satisfied. This, in fact, is thereason of the violation of LGIs in QM. It is then straight-forward to understand that the difference between K and ( K ) plays an important role for the violation ofLGI. Clearly, if K = ( K ) , , is satisfied, the LGI will not be violated. When all the three measurements areperformed for measuring each correlation, the expressionof K in inequality(1) can be written ( K ) = (cid:104) M M (cid:105) + (cid:104) M M (cid:105) − (cid:104) M M (cid:105) = 1 − α (28)where α = P ( M +1 , M − , M +3 ) + P ( M − , M +2 , M − ) .Using Eqs.(25) and (26) we can write K − ( K ) = (29) (cid:88) m = m D ( M m , M m ) − (cid:88) m = m D ( M m , M m ) − (cid:88) m (cid:54) = m D ( M m , M m ) + (cid:88) m (cid:54) = m D ( M m , M m ) Since (cid:80) D ( M m , M m ) = 0 , (cid:80) D ( M m , M m ) = 0 and K ≤ , from Eq.(29)we obtain (cid:88) m = m D ( M m , M m ) − (cid:88) m = m D ( M m , M m )+ ( K ) ≤ (30) By putting the value of ( K ) from Eq.(28) we have (cid:88) m = m D ( M m , M m ) − (cid:88) m = m D ( M m , M m ) ≤ α (31)We have thus written down the standard LGIs in termsof NSIT conditions. For the violation of standard LGI in(1) the relation (cid:88) m = m D ( M m , M m ) − (cid:88) m = m D ( M m , M m ) > α (32)needs to be satisfied in QM. This implies that for viola-tion of standard LGI at least one of the two three-timeNSIT conditions ( N SIT (1)23 and
N SIT ) required tobe violated. However, mere violations of NSIT conditionsdo not guarantee the violation of LGIs which depends onthe interplay between the violations of two NSIT condi-tions and on a threshold value α . Thus, NSIT conditionsare necessary for LGI but not sufficient [14, 20].Next, we compare our variant of LGIs with standardLGIs and NIST conditions. We found that violation ofvariant of LGIs can be shown to be larger than the stan-dard LGIs ( ( K ) Q > ( K ) maxQ ). Before writing variantof LGI in terms of NIST conditions, we note the follow-ing interesting points. Let us write one of the variant ofLGIs for three-time measurement scenario is given by K = (cid:104) ˆ M ˆ M ˆ M (cid:105) + (cid:104) ˆ M ˆ M (cid:105) − (cid:104) ˆ M (cid:105) ≤ (33)Since (cid:104) ˆ M ˆ M ˆ M (cid:105) = ( (cid:104) ˆ M ˆ M ˆ M (cid:105) ) and (cid:104) ˆ M ˆ M (cid:105) =( (cid:104) ˆ M ˆ M (cid:105) ) , then disturbance can ionly come due tothe term (cid:104) ˆ M (cid:105) . Intuitively one may then expect thatwhenever the quantity D ( M m ) defined as D ( M m ) = P ( M m ) − (cid:88) m ,m P ( M m , M m , M m ) (34)is positive, one may expect the violation of the variant ofLGI given by Eq.(33). Thus, one may infer that the NSITcondition N SIT (12)3 provides the necessary and sufficientcondition for variant of LGI. But, we shall shortly seethat similar to the case of standard LGI,
N SIT (12)3 pro-vides the necessary but not the sufficient condition.Using similar approach adopted for standard LGIs, weexpress the variant of LGI given by (33) in terms ofNSIT condition. Then, the expression of K in inequal-ity (33) can be written when all three measurements areperformed is given by ( K ) = (cid:104) ˆ M ˆ M ˆ M (cid:105) + (cid:104) ˆ M ˆ M (cid:105) − (cid:104) ˆ M (cid:105) = 1 − β (35)where β = P ( M +1 , M − , M +3 ) + P ( M − , M +2 , M +3 ) . Us-ing Eq.(34), we can write K − ( K ) = (cid:88) m D ( M m ) (36)Since K ≤ , using Eq.(35) we obtain (cid:88) m D ( M m ) ≤ β (37)For the violation of variants in inequality (3) the follow-ing relation needs to be satisfied in QM is given by (cid:88) m D ( M m ) > β (38)which is the variant of LGI written in terms of the N SIT (12)3 condition. It can be seen from (38) that mereviolation of
N SIT (12)3 do not provide the violation ofinequality (33), the value of (cid:80) m D ( M m ) needs togreater than a non-zero threshold value β . Thus, NSITcondition is necessary for the violation of variant of LGIbut not sufficient.Using the similar argument we can derive the condi-tion of violation of inequality (9) for n -number of mea-surements in terms of NSIT condition as (cid:88) m n D , .. ( n − ( M m n n ) > γ (39)where γ = P ( M m , M m ....M m n n ) + .... n − terms. Thequantity D , .. ( n − ( M m n n ) denotes the amount of dis-turbance caused by n − number of prior measurements.Intutively, it increases with the number of measurementsand becomes maximum when quantum value of the in-equality (39) reaches its algebraic maximum. V. SUMMARY AND DISCUSSION
The quantum violation of standard LGIs for a di-chotomic system is restricted by temporal Tsrilson boundwhich is significantly lower than the algebraic maximum.In this paper, we note an important observation thatthe standard LGIs are a class of inequalities but not theunique one. There is a scope of formulating new variantof inequalities based on the assumptions of MRps andNIM. For the simplest case of three-time measurementscenario, we first proposed new variants of LGIs whichare different from the standard LGIs. For a qubit system,we demonstrated that such macrorealist inequalities pro-vide larger quantum violation than standard LGIs. Byincreasing the number of measurements n , we proposedmore variants of LGIs. We found that the quantum vi-olation of variants of LGIs increase with the incrementof n . Interestingly, for a sufficiently large value of n , thequantum violation of variant of LGIs reach their algebraicmaximum. Thus, we obtained the quantum violation ofLGIs up to its algebraic maximum, even for a state inqubit system. Further, we have compared the variants ofLGIs proposed in our paper with the standard LGIs andNSIT condition. ACKNOWLEDGEMENTS
AKP acknowledges the support from Ramanujan Fel-lowship research grant (SB/S2/RJN-083/2014). MQ ac-knowledge the Junior Research Fellowship from SERBproject (ECR/2015/00026). [1] E. Schroedinger, Naturwissenschaften, , 807 (1935).[2] M. Arndt et al ., Nature, , 680 (1999).[3] H. D. Zeh, Found. Phys. 1, (1970), W. H. Zurek, Phys.Rev. D , 1862 (1982).[4] J. Kofler and C. Brukner, Phys. Rev. Lett. 99, 180403(2007).[5] G. C. Ghirardi, A. Rimini, and T. Weber, Phys. Rev.D , 470 (1986); A. Bassi and G. Ghirardi, Phys. Rep. , 257 (2003).[6] J. S. Bell, Physics , 195 (1964).[7] J. F. Clauser, M. A. Horne, A. Shimony and R. A. Holt,Phys. Rev. Lett. , 880, (1969).[8] W. Heisenberg, Zeitschrift f ¨ u r Physik, , 879 (1925).[9] A. J. Leggett and A. Garg, Phys. Rev. Lett. , 857(1985).[10] A. J. Leggett, J. Phys. Condens , R415 (2002).[11] A. J. Leggett, Rep. Prog. Phys. , 022001 (2008).[12] C. Budroni, T. Moroder, M. Kleinmann, and O. Guhne,Phys. Rev. Lett., , 020403 (2013).[13] C. Emary, N. Lambert, and F. Nori, Phys. Rev. B ,235447 (2012).[14] O. J. E. Maroney and C. G Timpson, arxiv: 1412.613v1.[15] J. Kofler and C. Brukner, Phys. Rev. A , 052115(2013).[16] C. Budroni et al ., Phys. Rev. Lett. , 200403 (2015). [17] D. Saha et al ., Phys. Rev. A , 032117 (2015).[18] J. J. Halliwell, Phys. Rev. A , 022123 (2016).[19] C. Budroni and C. Emary, Phys. Rev.Lett. , 050401(2014).[20] S. Kumari and A. K. Pan, Euro. Phys. Lett. , 50002(2017).[21] S. Kumari and A. K. Pan, Phys. Rev. A , 042107(2017).[22] J. Kofler and C. Brukner, Phys. Rev. Lett. , 090403(2008).[23] C. Emary, N. Lambert and F. Nori, Rep. Prog. Phys. ,016001 (2014).[24] N.Lambert et al ., Phys. Rev. A , 012105 (2016).[25] M. E. Goggin et al. Proc. Natl. Acad. Sci. U.S.A. ,1256 (2011).[26] G. C. Knee et al ., Nat. Commun. , 606 (2012).[27] A. Palacios-Laloy et al ., Nat. Phys. , 442 (2010).[28] R. E. George et al ., Proc. Natl. Acad. Sci. U.S.A. ,3777 (2013).[29] G. C. Knee et al ., Nat. Commun. , 13253 (2016).[30] L. Clemente and J. Kofler, Phys. Rev. A, , 062103(2015); Phys.Rev.Lett. , 150401 (2016).[31] A. Kumari, Md. Qutubuddin and A. K. Pan (Submitted).[32] B. C. Cirel’son, Lett. Math. Physics , 93 (1980).[33] A. Fine,Phys. Rev. Lett. ,291(1982). Appendix A: General formula for calculating n -time sequential measurement Here we provide a general formula for calculating sequential correlation of n -time measurements of a dichotomicobservable. In LG scenario, the measurement of a dichotomic observable ˆ M having outcomes ± is performed at time t , t .... t n ( t < t < ... < t n ) , which, in turn, can be considered as the sequential measurement of the observables ˆ M , ˆ M .... ˆ M n respectively.Given a density matrix ρ , the correlation function for the sequential measurement of two observables ˆ M and ˆ M can be calculated by using the formula [19] (cid:104) ˆ M ˆ M (cid:105) seq = 12 T r (cid:104) ρ (cid:110) ˆ M , ˆ M (cid:111)(cid:105) (A1)where {} denotes anti-commutation.Here we generalize the above formula for n -time measurement scenario. For this, let us first consider the three-measurement scenario. The correlation function for three-time measurement can be written as, (cid:104) ˆ M ˆ M ˆ M (cid:105) seq = (cid:88) m ,m ,m = ± m m m P ( M m M m M m ) (A2)Let Π m M , Π m M and Π m M are projectors of observables ˆ M , ˆ M and ˆ M corresponding to the to eigenvalues m , m and m respectively. In QM, Eq.(A2) can then be written as, (cid:104) ˆ M ˆ M ˆ M (cid:105) seq = (cid:88) m ,m ,m = ± m m m T r [Π m M Π m M ρ Π m M Π m M Π m M ]= (cid:88) m ,m = ± m m T r [Π m M Π m M ρ Π m M Π m M Π + M ] − (cid:88) m ,m = ± m m T r [Π m M Π m M ρ Π m M Π m M Π − M ] (A3)Using ˆ M = Π + M − Π − M and putting the value of m = ± , we have (cid:104) ˆ M ˆ M ˆ M (cid:105) seq = (cid:88) m = ± m T r [(Π + M Π m M ρ Π m M Π + M ) . ˆ M ] − (cid:88) m = ± m T r [(Π − M Π m M ρ Π m M Π − M ) . ˆ M ] (A4)Since Π ± M = ( I ± ˆ M ) / , Eq.(A4) can be simplified as (cid:104) ˆ M ˆ M ˆ M (cid:105) seq = 12 (cid:88) m = ± m T r (cid:104) (Π m M ρ Π m M ) . (cid:110) ˆ M , ˆ M (cid:111)(cid:105) (A5)Adopting the similar to the procedures adopted above, further simplification provides (cid:104) ˆ M ˆ M ˆ M (cid:105) seq = 14 T r (cid:104) ρ (cid:110) ˆ M , (cid:110) ˆ M , ˆ M (cid:111)(cid:111)(cid:105) (A6)For the case of n -time measurements, we derive (cid:104) ˆ M ˆ M ....... ˆ M n − ˆ M n (cid:105) seq = 12 n − T r (cid:104) ρ (cid:110) ˆ M , (cid:110) ˆ M , ........, (cid:110) ˆ M n − , (cid:110) ˆ M n − , ˆ M n (cid:111)(cid:111)(cid:111)(cid:111)(cid:105)(cid:111)(cid:111)(cid:111)(cid:111)(cid:105)