Ambiguous measurements, signalling and violations of Leggett-Garg inequalities
aa r X i v : . [ qu a n t - ph ] O c t Ambiguous measurements, signalling and violations of Leggett-Garg inequalities
Clive Emary
Joint Quantum Centre (JQC) Durham-Newcastle,School of Mathematics, Statistics and Physics, Newcastle University,Newcastle-upon-Tyne, NE1 7RU, United Kingdom
Ambiguous measurements do not reveal complete information about the system under test. Theirquantum-mechanical counterparts are semi-weak (or in the limit, weak-) measurements and herewe discuss their role in tests of the Leggett-Garg inequalities. We show that, whilst ambiguousmeasurements allow one to forgo the usual non-invasive measureability assumption, to derive anLGI that may be violated, we are forced to introduce another assumption that equates the invasiveinfluence of ambiguous and unambiguous detectors. Based on this assumption, we derive signallingconditions that should be fulfilled for the plausibility of the Leggett-Garg test. We then proposean experiment on a three-level system with a direct quantum-optics realisation that satisfies allsignalling constraints and violates a Leggett-Garg inequality.
I. INTRODUCTION
The Leggett-Garg inequalities (LGIs) [1] were con-structed as tests of macrorealism , as defined by the threeassumptions (the first two as stated in Ref. [1]; the thirdwas made explicit in e.g. Refs. [2–4]): (A1)
Macroscopic realism per se : A macroscopic systemwith two or more macroscopically distinct statesavailable to it will at all times be in one or theother of these states; (A2) Noninvasive measurability (NIM) at the macro-scopic level: It is possible, in principle, to deter-mine the state of the system with arbitrarily smallperturbation on its subsequent dynamics; (A3)
Arrow of time : The outcome of a measurement onthe system cannot be affected by what will or willnot be measured on it later.The inequalities that follow from these assumptions havebeen the subject of much work that was reviewed a fewyears ago in Ref. [5] with many theoretical [6–12] andexperimental [13–21] developments having taken placesince.Of the three assumptions, (A2) is particularly vexa-tious since, while it assumes NIM to hold in principle,non-invasivity must also be seen to hold true in practice,otherwise any violation of an LGI can be assigned to someunwitting invasivity of the measurement [22]. This is the“clumsiness loophole” of Wilde and Mizel [23]. And, sincequantum-mechanical measurements are in fact invasive,NIM is a counterfactual and can never be ruled out em-pirically.Leggett and Garg’s proposal [1] for dealing with thissituation was to use ideal negative measurements, andthese have been employed in various recent experiments[13, 19, 21, 24, 25]. This approach, however, just shiftsthe locus of any presumed non-invasivity away from thesystem itself and onto some degree of freedom in its envi-ronment. There have also been a number of attempts to formulate LGIs under different assumptions [26–30], butthese too must ultimately suffer from similar loopholes.This issue of invasivity relates to an important differ-ence between the LGIs and the formally-similar Bell’sinequalities, and this is the issue of signalling [4, 9]. Forspacelike-separated observers Alice and Bob in a Belltest, we have the no-signalling condition P ( A ) − X B P ( A, B ) = 0 , (1)where P ( A ) is the probability that Alice obtains result A and P ( A, B ) is the joint probability of result A for Aliceand B for Bob. Thus, the influence of Bob’s measurementis statistically undetectable to Alice (and vice versa). Inthe LGI setting, there is no external physical principlesuch as locality to which we can appeal that enforces alack of signalling between the two measurements. Let usdefine δ ( n ) = P ( n ) − X n P ( n , n ) , (2)where n and n are outcomes of measurements at times t and t > t , to quantify the signalling in a LGI context.Under the assumptions (A1–3), these signalling quanti-fiers should be zero, just as in the Bell’s test. Indeed the no-signalling-in-time (NSIT) equalities δ ( n ) = 0 [4], orones very similar to them [31], have been discussed astests of macrorealism themselves [4, 7, 12, 32]. Thesestudies show that, generically, quantum-mechanical vi-olations of an LGI are accompanied by a violation ofNSIT conditions (see Ref. [33] and also Sec. II). Fromthis, a macrorealist would conclude that, since there isexperimental evidence that the measurements can sig-nal forward in time, they are invasive and (A2) does nothold in practise. Thus, observation that the NSIT equal-ities hold may be taken as a necessary [but by no meanssufficient] condition that our measurements appear non-invasive. NSIT also restores the symmetry between LGIsand Bell inequalities.A number of ways of achieving LGI violations withoutsignalling have been discussed. Much has been writtenabout weak (or semi-weak) measurements [34, 35] and theviolation of the LGIs [36–43]. As Halliwell makes clear[9], weakly-measured quasiprobalities naturally have theNSIT property and can violate LGI inequalities. The im-portance of weak measurements in unifying spatial andtemporal correlations, and hence LG and and Bell in-equalities, has been discussed in e.g. Ref. [44]. Outsidethe weak-measurement paradigm, the work of Georgeet al. [33] (see also Ref. [45]) stands out as having re-ported a measurement of an LGI violation whilst obeyingthe relevant no-signalling conditions (George et al. usedthe language of the measurements being “non-disturbing”rather than no-signalling). These results, however, camein a very specific setting, viz the quantum 3-box paradox,leaving open the question as to whether their results rep-resent a peculiarity of this model, or whether similar sit-uations can be found in other, perhaps even macroscopic,systems.In this paper we describe a general approach to LGIviolations using a set of ambiguous measurements [46]that are realised quantum-mechanically by a particularclass of POVM. We show that with the quasiprobabilitiesinferred from such measurements we are able to violatean LGI inequality without making the NIM assumption.However, in order to justify use of these quasiprobabitiesas proxies for the real thing, we find we must make an al-ternative assumption that equates the invasive influenceof ambiguous and unambiguous detectors on a macrorealstate. We will call this assumption Equivalently-invasivemeasureability (EIM). We believe that an assumptionalong these lines tacitly underlies previous work on weakmeasurements and the LGI. While it may seem hard tojustify a priori that two potentially very different detec-tors are invasive in the same way (although, see later),this assumption leads to a testable consequence, namelythat the signalling quantifiers for both detectors shouldbe equal. These conditions we call equal-signalling intime (ESIT) and they can be tested empirically. We notehere that the assumption of EIM in the presence of ful-filled NSIT conditions is similar to the non-collusion ofadroit measurements discussed by Wilde and Mizel [23].From our considerations we arrive at several conclu-sions: (i) Violations of any single LGI are always accom-panied by violations of NSIT (this is a slight generali-sation of a result given in [33]); (ii) There exists specificcombinations of quantum dynamics and ambiguous mea-surements that can both violate LGIs and satisfy ESIT.In the examples we discuss here, ESIT is realised throughthe NSIT conditions for both ambiguous and unambigu-ous each being exactly zero. We thus show that theresults of George can be generalised to arbitrary sys-tems, provided we choose the dynamics and measure-ments appropriately. (iii) In general, violations of LGIin the weak-measurement limit are accompanied by a vi-olation of ESIT (and would thus be unconvincing to amacrorealist). This must always be the case when ourmeasurements involve just two outcomes. However, wealso show that there exist specific weak-measurement sce- narios with multiple outcomes in which ESIT remainsintact.This paper proceeds as follows. In Sec. II we derive amodified version of the LGI without making the NIM as-sumption and show that directly-measured probabilities,even those from quantum mechanics, can never violateit. In Sec. III we then introduce our ambiguous mea-surements, discuss the necessity of the EIM assumptionand construct our ambiguously-measured LGI. We thenanalyse this quantum-mechanically in Sec. IV and showhow the ambiguous LGI can be violated whilst ESIT ispreserved in general. In Sec. V we discuss a concreteexample of our formalism, eminently-realisable in termsof the quantum optics set up pursued in Refs. [19, 20].In Sec. VI we make the connection with weak measure-ments, before discussing the significance of our results inSec. VII.
II. THE LGI WITHOUT THE NON-INVASIVEMEASURABILITY ASSUMPTION
The most-studied LG correlator involves dichotomicobservable Q = ± and reads K ≡ h Q Q i + h Q Q i − h Q Q i , (3)where Q i = Q ( t i ) is the measurement outcome at times t > t > t . Under assumptions A1-3 above, Leggettand Garg showed that K ≤ . We want to investigate thiscorrelator without assumption (A2). To simplify mat-ters, we first assume the coincidence of the measurementat t and our preparation step [11, 13, 19, 40]. Declaring Q = +1 , the LG correlator becomes K = h Q i + h Q Q i − h Q i . (4)We assume that our measurements unambiguously revealone of M different outcomes, each of which we associatewith a different “macroscopically-distinct” state. We al-lot a Q -value to each via q ( n ) = ± with ≤ n ≤ M [47]. In terms of the probabilities P ( n i ) of obtaining re-sult n i at time t i , the simple expectation values in K read h Q i i = P n i q ( n i ) P ( n i ) . Under assumptions (A1)and (A3) [but not (A2)], adding a measurement at time t does not affect the result at t and so we can write h Q i = P n n q ( n ) P ( n , n ) , where P ( n , n ) is thejoint probability of measuring n and n . We then usethe signalling quantifiers δ ( n ) in Eq. (2) to eliminate P ( n ) from K . The result is K = X n ,n [ q ( n ) + q ( n ) q ( n ) − q ( n )] P ( n , n ) − X n q ( n ) δ ( n ) . (5)The first term here is what we get under the standardderivation of the LGI with NIM. The second term isnew and describes the effects of the invasiveness of ourmeasurements. Taking a maximally-adverse position, in-dependent maximisation of these two terms yields ourmodified NIM-free LGI [48] K ≤ with ∆ ≡ X n | δ ( n ) | . (6)The idea is, therefore, to make measurements of both K and ∆ and compare them with this inequality. Itis immediately clear, however, that as long as P ( n , n ) and P ( n ) form two sets of genuine probabilities, thenEq. (6) can never be violated. This holds just as well forprobabilities obtained quantum-mechanically, and indeedirrespective of whether the measurements are projectiveor more general. III. AMBIGUOUS MEASUREMENTS
The foregoing makes clear that we are never going toviolate Eq. (6) with directly-measured probabilities. Theonly remaining possibility is therefore to replace the mea-sured probabilities with quasiprobabilities in a way that amacrorealist would feel was a fair substitution. We main-tain that this can only be the case when we perform twoexperiments and compare them. The first experimentproceeds as above with our detector at time t giving un-ambiguously one of ≤ n ≤ M outcomes. These resultsare repeatable. The second experiment analyses the samesystem, but with the detector at time t being ambigu-ous [46]. This detector gives one of ≤ α ≤ M A resultswith the key property that repeated measurements do notnecessarily lead to the same outcome. The macrorealistwill view this measurement as only revealing incompleteinformation about the “real state” of the system.We then look to relate the two experiments. By fol-lowing a measurement of unambiguous result n with anambiguous measurement, we obtain the conditional prob-abilities c αn that state n gives response α . Using theseresults and Bayes’ rule, the macrorealist would be happyto write the probability of obtaining result α as P ( α ) = X n c αn P ( n ) . (7)We use the notation P to denote a probability that isnot measured directly but rather inferred. Collecting co-efficients c αn into matrix c and assuming M A ≥ M suchthat the ambiguous measurements give us sufficient in-formation to reconstruct P ( n ) , we write P ( n ) = X α d nα P ( α ) , (8)where d nα are elements of the left-inverse of c , i.e. d · c = . Note that we can not measure the quantities d nα directly, as we do not know what it means to prepare in“state α ”.In analogy with Eq. (8), we next write down the in-ferred joint probability P ( n , n ) = X α d n α P ( n , α ) . (9) For a macrorealist to agree that this inferred probabil-ity is same as the directly-measured one, P ( n , n ) , heor she would have to assume that the evolution of thesystem from state n is the same with the ambiguous de-tector in place as it would be had we actually measuredresult n with the unambiguous detector. We codify thisassumption as (A2*) Equivalently-invasive measureability (EIM): Theinvasive influence of ambiguous measurements onany given macroreal state is the same as that ofunambiguous ones,and make it here as an alternative to NIM.The degree of signalling due to the ambiguous mea-surements is quantified by δ A ( n ) ≡ P ( n ) − X α P ( n , α ) . (10)Inserting P n d nα = 1 from the conservation of probabil-ity and using Eq. (9), we obtain δ A ( n ) = P ( n ) − X n P ( n , n ) . (11)Thus, as a consequence of (A2*), the macrorealist willexpect the signalling quantifiers δ ( n ) and δ A ( n ) to bethe same. Let us quantify this by defining the “signallingdifferences” D ( n ) ≡ δ ( n ) − δ A ( n ) , (12)which, under EIM, obey D ( n ) = 0; ∀ n . (13)In analogy with no-signalling in time, we dub these con-ditions the “equal-signalling-in-time” (ESIT) equalities.To obtain an LGI that may be violated, we take Eq. (5)and replace the measured probabilities with the inferredones, Eq. (9). This yields the correlator K A = X n ,n ,α [ q ( n ) + q ( n ) q ( n ) − q ( n )] d n α P ( n , α ) − X n q ( n ) δ A ( n ) , (14)which involves measured quantities only. Perceiving thiscorrelator to be equivalent to Eq. (5), the macrorealistwould expect K A and δ A ( n ) to be related in the sameway as the original K and δ ( n ) . Thus we obtain K A ≤ A ; ∆ A = X n | δ A ( n ) | . (15)This we will refer to as the ambiguously-measured LGI.The important point is that the macrorealist would onlywrite this inequality down if they were convinced firstlyof the existence of the states n (for which we need theunambiguous measurements), and secondly that assump-tion (A2*) is valid. IV. QUANTUM FORMULATION
We now describe the situation quantum-mechanically.Let ρ i be the system density matrix at time t i , andlet the time evolution from time t i to t j be given by ρ j = Ω ji [ ρ i ] = U ji ρU † ji with U ji the appropriate unitaryoperator. Our unambiguous measurement is describedwith orthogonal projectors Π n , ≤ n ≤ M that obey P n Π n = and Π n Π n ′ = δ nn ′ Π n , thus ensuring therepeatability of the measurement. In this case the sig-nalling quantifiers are given by δ QM ( n ) = X n,n ′ = n X ( n , n, n ′ ) . (16)with X ( n , n, n ′ ) ≡ Tr { Π n Ω [Π n ρ Π n ′ ] } .In the ambiguous case, the measurement is describedby a POVM with elements that we take to be sums ofprojection operators F α = X n c αn Π n , (17)and have a clear interpretation in terms of the states n .With P ( α ) = Tr { F α ρ } and P ( n ) = Tr { Π n ρ } , this im-mediately reproduces Eq. (7). The POVM elements areassociated with the Kraus operators as F α = M α with M α = M † α = P n √ c αn Π n , such that the joint proba-bilities are given by P ( n , α ) = Tr { Π n Ω [ M α ρ M α ] } .This gives the ambiguous signalling quantifiers as δ QMA ( n ) = X n,n ′ = n γ ( n, n ′ ) X ( n , n, n ′ ) (18)with γ ( n, n ′ ) = 1 − P α √ c αn c αn ′ . Similarly, fromEq. (14), the LG correlator is K QMA = X n n [ q ( n ) + q ( n ) q ( n ) − q ( n )] P ( n , n ) − X n q ( n ) δ QMA ( n )+ X n ,n [ q ( n ) + q ( n ) q ( n ) − q ( n )] κ ( n , n ) , with κ ( n , n ) ≡ X n,n ′ = n Γ( n , n, n ′ ) X ( n , n, n ′ ) , (19)and Γ( n , n, n ′ ) ≡ P α d n α √ c αn c αn ′ . The important re-sult here is that, whereas the first two terms in K QMA are exactly what we get in the unambiguous case [seeEq. (5)], a third term appears which is not directly re-lated to the signalling quantifiers. This new term opensup the possibility of violating Eq. (15).If we look at the signalling difference, however, we findthat D QM ( n ) = X n,n ′ = n [1 − γ ( n, n ′ )] X ( n , n, n ′ ) . (20) Unless we can set these quantities to zero, the macro-realist can conclude that (A2*) does not hold, and anyviolation of the ambiguously-measured LGI is due to thenon-comparability of the two experiments. V. INVERTED MEASUREMENTS AND AQUANTUM-OPTICS REALISATION
Is it possible to satisfy ESIT D ( n ) = 0; ∀ n and stillviolate Eq. (15)? We answer this question by consideringa simple measurement scheme which we call an “invertedmeasurement”. The idea is that, whereas the unambigu-ous detector identifies the system as being in state n ,the inverted-measurement detector identifies it as beingin any state other than n . So, with three unambiguousoutcomes n ∈ { A, B, C } , our inverted-measurement de-tects the three disjunctions: A ∪ B , B ∪ C , and A ∪ C .From these, a macrorealist would have no qualms infer-ring the (quasi-)probabilities P ( A ) = P ( A ∪ B ) + P ( A ∪ C ) − P ( B ∪ C ) , etc. Such a detector has M A = M andis described by the matrices c = 1 M − J − ) ; d = J − ( M − , (21)where is the unit matrix and J is a matrix of ones.With this detector, we obtain ( M − δ QMA ( n ) = δ QM ( n ) , with δ QM ( n ) as in Eq. (16), and thus D QM ( n ) = ( M − δ A ( n ) . (22)Thus, if we can find a quantum dynamics that obeysNSIT, then ESIT will be automatically satisfied. Fur-thermore, with this measurement set-up, the terms re-sponsible for LGI violations read κ ( n , n ) = − δ QMA ( n )+ X n [ X ( n , n , n ) + X ( n , n, n )] , (23)which remain finite even when δ QMA ( n ) = 0 . Thus thisscheme offers a route to satisfy NSIT for both measure-ments, ESIT along with it, and still violate Eq. (15).We now consider a three-level system as the lowest-dimensional system for which inverse measurementsmake sense. We label the states n ∈ { A, B, C } , choosemeasurement assignments q ( A ) = − q ( B ) = q ( C ) = 1 ,and initialise the system in state ρ = | C ih C | . Timeevolution is governed by U = U = U with U = φ sin φ − sin φ cos φ × cos χ χ − sin χ χ × cos θ sin θ − sin θ cos θ
00 0 1 , (24)with parameters φ , χ and θ . U t t t A (a)(b) BC (c) UUUUU
FIG. 1. Sketch of a three-level system realised as opticalchannels A , B and C with non-trivial time evolution gener-ated by the blocks labelled U . We initialise by injecting aphoton into channel C . Three configurations are shown. (a)No measurement at t ; (b) Unambiguous measurement. Withblocking elements in channels B and C , detection of the pho-ton at t means that we can infer that the photon was inchannel A at time t . (c) Ambiguous measurement. Withonly channel C blocked, detection of the photon at t meansthat, from a macrorealistic point of view, the photon was ineither channel A or B at time t . A set-up like this was realised optically in Refs. [19, 20],and our inverted measurement scheme would have a par-ticularly straightforward implementation in this context,see Fig. 1. The system state is encoded in one of three op-tical channels. Measurements are made through a com-bination of photon detectors on the far right ( t ) andby placing a sequence of blocking elements in the opticalpaths at t . Projective measurement of the probabilities P ( n , A ) are unambiguously obtained by blocking twopaths at once (Fig. 1b) since if we block e.g. paths B and C , the photon must have passed through channel A at t to survive through to the detector. In contrast, ourinverted measurements are obtained by blocking just oneof the three channels (Fig. 1c). With a block in chan-nel C , say, a detection of a photon at t would lead amacrorealist to infer that the photon state must havebeen either A or B at time t .We then calculate δ QMA for this system and choose χ as a function of φ and θ such that δ QMA ( A ) = 0 . Since P n δ QMA ( n ) = 0 , we have δ QMA ( B ) = − δ QMA ( C ) suchthat when one of these two remaining NSIT indicators isset to zero, then all three are zero. Fig. 2 shows δ QMA ( B ) as a function of the two angles θ and φ . Marked in blackare the parameters for which δ QMA ( B ) = 0 . Fig. 3 showsthe corresponding LGI correlator K QMA , which takes val-ues up to a maximum of K QMA = 1 . . Overlaid on thisfigure are the no-signalling lines from Fig. 2. In many θ/π φ/π FIG. 2. The signalling quantifier δ QMA ( B ) as a function ofparameters θ and φ for ambiguous measurements of a three-level system. Parameter χ was chosen to set δ QMA ( A ) = 0 .The black lines indicate parameters for which δ QMA ( B ) = − δ QMA ( C ) = 0 and both NSIT and ESIT are obeyed. places, these lines coincide with the LG correlator takingthe value K QMA = 1 . However, there are also several re-gions where this is not the case, and in particular, in thetop left corner of this figure we see a no-signalling lineintersect a region with K QMA > . For these parametervalues, then, we have NSIT, ESIT and a violation of theLGI.Fig. 4 shows two cuts through Fig. 3. Fig. 4a revealsthe maximum value of K QMA when signalling is zero tobe K QMA = 1 . . Fig. 4b shows a straight cut through-Fig. 3 for fixed θ . On this plot we also show the quantity QMA , which represents the modified upper bound for K QMA . Only around the points where ∆ QMA is close tozero do we obtain LGI violations.
VI. WEAK MEASUREMENTS
As example of LGI violations with weak measurements,let us consider a detector ( M A = M ) described by c = 1 − ǫM J + ǫ ; d = 1 ǫ + ǫ − ǫM J . (25)Each detector response is biased towards a certain (un-ambiguous) outcome, but in the limit ǫ → this bias dis-appears and the measurement becomes weak. To leadingorder in ǫ , we obtain γ ( n, n ′ ) ≈ M ǫ and Γ( n , n, n ′ ) ≈ ( δ n,n + δ n ′ ,n ) . Thus, with this detector in the ǫ → limit, the ambiguous NSIT quantities become lim ǫ → δ QMA ( n ) = 0 , (26) θ/π φ/π FIG. 3. The ambiguously-measured LG correlator K QMA for our three-level system as a function of angles θ and φ .Red/orange colours correspond to K QMA > ; blue to K QMA < . The black lines are the no-signalling lines from Fig. 2 alongwhich δ QMA ( n ) = 0; ∀ n . In the top-left quadrant we see theno-signalling line intersect a K > region, such that we haveNSIT, ESIT and a violation of the LGI. and there is no signalling for the weak measurement.Meanwhile, for terms responsible for violation of the am-biguous LGI, we obtain lim ǫ → κ ( n , n ) = 12 X n = n [ X ( n , n , n ) + X ( n , n, n )] , which will be non-zero provided there are coherences be-tween basis states at time t . Indeed, with these resultswe can rewrite the LG correlator as lim ǫ → K QM A = Tr nh ˆ Q + n ˆ Q , ˆ Q o − ˆ Q i ρ o (27)where ˆ Q n = ˆ Q ( t n ) = U † n ( P m q ( m )Π m ) U n is the mea-sured operator in the Heisenberg picture at time t n and {· , ·} denotes the anticommutator. We thus arrive at theweakly-measured form of the LGI as discussed in e.g.Ref. [9]. From Fritz [49], we know that the maximumquantum-mechanical value of this quantity is identical tothat obtained in the projective case in the Lüders limit[47, 50], i.e. when the number of projectors is exactlytwo. Thus, we conclude that, in the weak-measurementcase, lim ǫ → K QM A ≤ QM Lüders ≤ , (28)where ∆ QM Lüders is the no-signalling quantity that would beobtained under a projective Lüders measurements. Sincethis will generally be non-zero, no-signalling violations ofthe weakly-measured LGI are possible.
PSfrag replacements K Q M A K Q M A K Q M A θ/π φ/πφ/π FIG. 4. Two cuts through Fig. 3 for the inverted-measurement case plus corresponding results for the weakmeasurements of Sec. VI. (a)
The correlator K QMA as a func-tion of θ along the no-signalling line that goes through theorange region in Fig. 3. Since ∆ A = 0 along this line, theLGI reverts to K A ≤ and violations of Eq. (15) occur withinthe indicated red region. Black line: inverted measurement;blue line: weak measurement. (b) The correlator K QMA alongthe straight-line cut in Fig. 3 from φ = 0 to φ = π with θ = 0 . π . The red region shows the righthand side ofEq. (15) and, since generally we have signalling here, thisquantity is greater than one. Indeed, only near the maximumof K QMA does
QMA drop significantly such that we ob-tain a violation of Eq. (15). NB: the maximum of the K QMA curve here is slightly displaced from the no-signalling pointand thus has a value slightly higher than the no-signallingmaximum ( . vs. . ). (c) The same as panel (b) but forthe weakly-measured case with θ = 0 . π . Again the maxi-mum is slightly offset from no-signalling maximum ( . vs. . ). This, however, indicates a problem when the numberof outcomes for our unambiguous measurements is ac-tually M = 2 , because then the quantity ∆ QM Lüders inEq. (28) is exactly the same ∆ QM for the unambigu-ous measurements. Thus violations of the ambiguously-measured LGI imply violations of the unambiguous NSITequalities. In the M = 2 case, therefore, we have lim ǫ → D QM ( n ) = δ QM ( n ) = 0 for at least some n ,from which the realist would conclude that (A2*) is in-valid.Away from this M = 2 case, however, this argumentdoes not apply, and we may obtain ∆ QM = 0 whilst ∆ QM Lüders > , since they are different quantities. To showthat is the case, we return to the three-level system of thelast section for which M = 3 . For this model, we alreadyknow that ∆ QM = ∆ QMA = 0 along the lines shown inFig. 2. A plot of the weakly-measured LG correlator ofEq. (27) [not shown] then looks very similar to Fig. 3(but with less-pronounced maxima) and again shows aregion of LGI violation intersected by the no-signallingline. Fig. 4a shows the value of K QMA along this line, fromwhich we obtain a maximum violation of K QMA = 1 . (and thus ∆ QM Lüders = 0 . ). Fig. 4c also shows a cut forfixed θ through the maximum. VII. DISCUSSION
In Sec. II we saw that in attempting to derive an LGIwithout the NIM assumption (A2), we ensure that it cannever be violated. This is not surprising because a re-alistic description of nature by itself is not inconsistentwith the probabilities of quantum mechanics [2]. To ob-tain an inequality that we could violate, we consideredtwo measurements, one unambiguous and one ambiguous[51]. By comparing the two, and replacing a probabil-ity in an inequality derived for one experiment with a(quasi-) probability inferred from the other, we obtainedour “ambiguously-measured LGI” which, as we have seenin our examples, can be violated by quantum theory. Tojustify this switch, however, we had to introduce the EIMassumption (A2*), in which the invasivity of the measure-ments in each of the two experiments was taken to beequivalent. Making the role of this assumption explicitin LGI tests with ambiguous or weak measurements isone of the main results of this work.Whilst the EIM assumption may not seem particularlyplausible in the abstract [52], for certain detectors, itmight be. In particular, consider measurements that arerealised with a set of individual detectors, each of whichonly interacts with the system when it is in just one of themacro-real states (as in ideal negative measurements). Inthis case, then, the unambiguous measurement would beimplemented by using one of these detectors at a time,whereas the ambiguous measurement would involve usingmore than one, deployed in such a way that any knowl-edge of which particular detector had fired was lost. Sincein this case, the components of the ambiguous measure-ment are “just the sum” of those from the unambiguousone, one could reasonably expect the influence on thesystem of the two measurements to be the same. But,more than consideration of any specific realisation, justas NIM leads to the NSIT conditions, so the EIM as-sumption leads to the ESIT conditions, and these can beexperimentally tested. And while successful NSIT/ESITtests can not exclude NIM/EIM, they would at least pro-vide the macrorealist with some level of empirical confi-dence that the experiment was functioning in conformitywith these principles.In this paper we have discussed the concrete example ofa three-level system, under both inverted and weak mea-surements, and found parameter regimes where it cansatisfy both NSIT and ESIT equalities whilst violatingan LGI. In the weak measurement case, it is importantto note that whilst the NSIT equality is guaranteed tohold for the weak measurement itself, this is not necessar-ily the case for the unambiguous part of the experimentand thus, generally, ESIT would not hold in these experi- ments. Only under certain model-specific circumstances,and only when the system dimension is greater than two,can both NSIT and ESIT be fulfilled. As we show in ap-pendix A, the three-box problem from the experiment ofGeorge et al. [33] can be understood within the frame-work discussed here and shows the required NSIT/ESITproperties.Faced, then, with the violation of an ambiguously-measured LGI, together with the satisfaction of ESIT,what would a macrorealist conclude? Certainly, thiswould would give more cause for thought than havingmeasured an LGI violation in a single experiment, asthere a measurement of the NSIT equalities would beenough to dismiss the measurements as signalling. Withthe ambiguous prescription and ESIT, the macrorealistwould be faced with either giving up the combinationof A1+A3, or finding an explanation for how two dif-ferent measurements can somehow conspire to give ex-actly the same degree of signalling and yet somehow in-fluence the system in very different ways. This problem iscompounded in the case where ESIT is satisfied throughNSIT being satisfied for both experiments, as in this caseboth experiments are individually non-signalling. Withthis, then, we arrive at a situation similar to that pre-sented by Wilde and Mizel [23] (with a recent realisation[18]). Whilst the measurement procedures in their workwere very different to those considered here (involvingdifferent bases), the problem created for the macrorealistis similar — in order to maintain a macrorealistic de-scription of the system, the macrorealist is left with hav-ing to explain away a collusion between two sets of mea-surements. When properly executed, then, ambiguously-measured LGIs provide a further way with we might nar-row the “clumsiness loophole”.
ACKNOWLEDGMENTS
We are grateful to George Knee, Jonathan Halliwell,KunKun Wang and Peng Xue for helpful discussions.
Appendix A: Quantum three-box problem
The quantum three-box problem [33, 45] can be cast inthe language used here. With the three states labelled asin Sec. V Alice’s measurement at time t is characterisedby two projectors Π (3) − = | C ih C | and Π (3)+1 = − | C ih C | ,which we have labelled with the respective q = ± as-signments and a superscript to distinguish them fromthe previous projectors. Bob’s measurement at time t can be characterised by a four-element POVM with c = 12 , (A1)and the assignments q ( A ) = q ( B ) = − q ( C ) = +1 . Forthe time evolution operators we take U = 1 √ √ − √ √ − −√ √ ; (A2) U = 1 √ √ −√ √ √ −√ . (A3)Finally, we need to consider a different, but essentiallyequivalent [5], version of the LGI: K ′ = h Q i + h Q Q i − h Q i ≥ − . (A4) Calculating with the above formalism gives δ QMA ( n ) = δ QM ( n ) = 0 such that Bob’s measurements are non-signalling and ESIT is satisfied, along with K ′ QMA = − / , such that the ambiguous LGI is violated. This isin agreement with Ref. [33]. [1] A. J. Leggett and A. Garg,Phys. Rev. Lett. , 857 (1985).[2] A. J. Leggett, Rep. Prog. Phys. , 022001 (2008).[3] J. Kofler and Č. Brukner,Phys. Rev. Lett. , 090403 (2008).[4] J. Kofler and Č. Brukner,Phys. Rev. A , 052115 (2013).[5] C. Emary, N. Lambert, and F. Nori, Rep. Prog. Phys. , 016001 (2014).[6] A. Asadian, C. Brukner, and P. Rabl,Phys. Rev. Lett. , 190402 (2014).[7] G. Schild and C. Emary,Phys. Rev. A , 032101 (2015).[8] S. Brierley, A. Kosowski, M. Markiewicz, T. Paterek, andA. Przysiężna, Phys. Rev. Lett. , 120404 (2015).[9] J. J. Halliwell, Phys. Rev. A , 022123 (2016).[10] J. J. Halliwell, Phys. Rev. A , 052131 (2016).[11] N. Lambert, K. Debnath, A. F. Kockum,G. C. Knee, W. J. Munro, and F. Nori,Phys. Rev. A , 012105 (2016).[12] L. Clemente and J. Kofler,Phys. Rev. Lett. , 150401 (2016).[13] C. Robens, W. Alt, D. Meschede, C. Emary, and A. Al-berti, Phys. Rev. X , 011003 (2015).[14] Z.-Q. Zhou, S. F. Huelga, C.-F. Li, and G.-C. Guo,Phys. Rev. Lett. , 113002 (2015).[15] C. Robens, W. Alt, C. Emary, D. Meschede, and A. Al-berti, Applied Physics B , 12 (2016).[16] G. C. Knee, K. Kakuyanagi, M.-C. Yeh, Y. Matsuzaki,H. Toida, H. Yamaguchi, S. Saito, A. J. Leggett, andW. J. Munro, Nat. Commun. , 13253 (2016).[17] J. Formaggio, D. Kaiser, M. Murskyj, and T. Weiss,Phys. Rev. Lett. , 050402 (2016).[18] E. Huffman and A. Mizel, arxiv:1609.05957 (2016).[19] K. Wang, C. Emary, X. Zhan, Z. Bian, J. Li, and P. Xue,arXiv:1701.02454 (2017).[20] K. Wang, G. C. Knee, X. Zhan, Z. Bian, J. Li, andP. Xue, Phys. Rev. A , 032122 (2017).[21] H. Katiyar, A. Brodutch, D. Lu, and R. Laflamme,New Journal of Physics , 023033 (2017).[22] A. Montina, Phys. Rev. Lett. , 160501 (2012).[23] M. Wilde and A. Mizel, Found. Phys. , 256 (2012).[24] G. C. Knee, S. Simmons, E. M. Gauger, J. J. Morton,H. Riemann, N. V. Abrosimov, P. Becker, H.-J. Pohl, K. M. Itoh, M. L. Thewalt, G. A. D. Briggs, and S. C.Benjamin, Nat. Commun. , 606 (2012).[25] H. Katiyar, A. Shukla, K. R. K. Rao, and T. S. Mahesh,Phys. Rev. A , 052102 (2013).[26] S. F. Huelga, T. W. Marshall, and E. Santos,Phys. Rev. A , R2497 (1995).[27] S. Foster and A. Elby, Found. Phys. , 773 (1991).[28] A. Elby and S. Foster, Phys. Lett. A , 17 (1992).[29] R. Lapiedra, Europhys, Lett. , 202 (2006).[30] M. Zukowski, arxiv:1009.1749 (2010).[31] C.-M. Li, N. Lambert, Y.-N. Chen, G.-Y. Chen, andF. Nori, Sci. Rep. , 885 (2012).[32] C. Emary, J. P. Cotter, and M. Arndt,Phys. Rev. A , 042114 (2014).[33] R. E. George, L. M. Robledo, O. J. E. Maroney, M. S.Blok, H. Bernien, M. L. Markham, D. J. Twitchen,J. J. L. Morton, G. A. D. Briggs, and R. Hanson,Proc. Natl. Acad. Sci. USA , 3777 (2013).[34] Y. Aharonov, D. Z. Albert, and L. Vaidman,Phys. Rev. Lett. , 1351 (1988).[35] A. G. Kofman, S. Ashhab, and F. Nori,Phys. Rep. , 43 (2012).[36] A. N. Jordan, A. N. Korotkov, and M. Büttiker,Phys. Rev. Lett. , 026805 (2006).[37] R. Ruskov, A. N. Korotkov, and A. Mizel,Phys. Rev. Lett. , 200404 (2006).[38] N. S. Williams and A. N. Jordan,Phys. Rev. Lett. , 026804 (2008).[39] A. Palacios-Laloy, F. Mallet, F. Nguyen, P. Bertet,D. Vion, D. Esteve, and A. N. Korotkov, Nat. Phys. , 442 (2010).[40] M. E. Goggin, M. P. Almeida, M. Barbieri, B. P.Lanyon, J. L. O’Brien, A. G. White, and G. J. Pryde,Proc. Natl. Acad. Sci. USA , 1256 (2011).[41] J. Dressel, C. J. Broadbent, J. C. Howell, and A. N.Jordan, Phys. Rev. Lett. , 040402 (2011).[42] Y. Suzuki, M. Iinuma, and H. F. Hofmann, New J. Phys. , 103022 (2012).[43] A. Bednorz, W. Belzig, and A. Nitzan, New J. Phys. ,013009 (2012).[44] S. Marcovitch and B. Reznik, arXiv:1103.2557 (2011);arXiv:1107.2186 (2011).[45] O. J. E. Maroney, arXiv:1207.3114 (2012). [46] J. Dressel and A. N. Jordan,Phys. Rev. A , 022123 (2012).[47] C. Budroni and C. Emary,Phys. Rev. Lett. , 050401 (2014).[48] We note that an alternative approach to including sig-nalling effects in LGIs has been discussed by Kujalaand Dzhafarov. See, for example, J. V. Kujala andE. N. Dzhafarov in Contextuality from Quantum Physicsto Psychology edited by E. Dzhafarov, S. Jordan, R. Zhang, and V. Cervantes (World Scientific, New Jersey,2015), pp. 287-308.[49] T. Fritz, New J. Phys. , 083055 (2010).[50] C. Budroni, T. Moroder, M. Kleinmann, and O. Gühne,Phys. Rev. Lett.111