Are Bell-tests only about local incompatibility?
aa r X i v : . [ qu a n t - ph ] J un Are Bell-tests only about local incompatibility?
Diederik Aerts and Massimiliano Sassoli de Bianchi
Center Leo Apostel for Interdisciplinary Studies,
Brussels Free University, 1050 Brussels, Belgium
E-Mails: [email protected] , [email protected] Abstract
The view exists that Bell-tests would only be about local incompatibility of quantum observablesand that quantum non-locality would be an unnecessary concept in physics. In this note, we empha-size that it is not incompatibility at the local level that is important for the violation of Bell-CHSHinequality, but incompatibility at the non-local level of the joint measurements. Hence, non-localityremains a necessary concept to properly interpret the outcomes of certain joint quantum measure-ments.
Keywords:
Bell-CHSH inequality; Bell-test, contextuality; complementarity; compatibility; non-locality; non-spatiality; marginal laws, no-signaling conditionsSome authors have argued that since Bell-CHSH inequality is only violated under the condition oflocal incompatibility of Alice’s and Bob’s observables, Bell-tests should only be considered as specialtests of incompatibility of said local observables, hence Bell’s introduction of the very notion of non-locality would be misleading and the term “non-locality” should be dismissed altogether [1, 2, 3, 4, 5].More precisely, addressing here the more specific point raised in [2], and following Khalfin and Tsirelson’salgebraic method [6], we observe that by taking the square of the CHSH operator C = A ⊗ ( B + B ′ ) + A ′ ⊗ ( B − B ′ ) , (1)then using the fact that the observables A , A ′ , B and B ′ have ± C = 4 I + [ A, A ′ ] ⊗ [ B, B ′ ] . (2)Since the inequality C ≤ I (which implies |h ψ | C | ψ i| ≤
2, for all ψ , which is the usual statementof the Bell-CHSH inequality; see for instance [7] for the details) can only be violated if [ A, A ′ ] = 0and/or [ B, B ′ ] = 0, one finds that the Bell-CHSH inequality cannot be violated if Alice’s and/or Bob’sobservables are compatible, i.e., commute. Based on this observation, one might be tempted to concludethat Bell-tests cannot truly highlight the presence of non-locality, but only of local incompatibility (localnon-commutability) of Alice’s and Bob’s measurements.The above reasoning is however incomplete, as it does not take into account the reason why local non-commutativity is necessary in the first place. To show this, let us start considering the situation whereAlice’s and Bob’s measurements are compatible, so that we have the commutation relations [ A, A ′ ] = 0and [ B, B ′ ] = 0. If so, one can in principle define a single measurement scheme for Alice, consisting1n jointly measuring the two observables A and A ′ , as well as a single measurement scheme for Bob,consisting in jointly measuring the two observables B and B ′ . Let us denote A and B the observablesassociated with these two bigger local measurements, performed by Alice and Bob, respectively. If A and A ′ are 2-outcome observables, this means that A is associated with the 4 outcomes ( A , A ′ ),( A , A ′ ), ( A , A ′ ), ( A , A ′ ), so that the outcome-probabilities for the two sub-measurements A and A ′ can be deduced as marginals of the outcome-probabilities of such bigger local measurement; and thesame holds true for Bob’s observable B , associated with the 4 outcomes ( B , B ′ ), ( B , B ′ ), ( B , B ′ )and ( B , B ′ ).If we additionally assume that the measurement defined by jointly executing Alice’s and Bob’smeasurements is properly described in terms of a tensor product observable A ⊗ B , as is usually done instandard quantum mechanics, i.e., by the product of the two commuting observables
A ⊗ I and I ⊗ B ,it is clear that the overall experimental situation can be described in terms of a single measurement,defined by the action of Alice and Bob jointly performing A and B . Such single measurement wouldproduce the following 16 possible outcomes:(( A , A ′ ) , ( B , B ′ )) , (( A , A ′ ) , ( B , B ′ )) , (( A , A ′ ) , ( B , B ′ )) , (( A , A ′ ) , ( B , B ′ )) , (( A , A ′ ) , ( B , B ′ )) , (( A , A ′ ) , ( B , B ′ )) , (( A , A ′ ) , ( B , B ′ )) , (( A , A ′ ) , ( B , B ′ )) , (( A , A ′ ) , ( B , B ′ )) , (( A , A ′ ) , ( B , B ′ )) , (( A , A ′ ) , ( B , B ′ )) , (( A , A ′ ) , ( B , B ′ )) , (( A , A ′ ) , ( B , B ′ )) , (( A , A ′ ) , ( B , B ′ )) , (( A , A ′ ) , ( B , B ′ )) , (( A , A ′ ) , ( B , B ′ )) . (3)From their probabilities, one can easily deduce those of the 4 different possible joint sub-measurements,for instance the one obtained by considering sub-measurement A in association with sub-measurement B . More precisely, the outcome-probability P ( A , B ), of obtaining outcome A for A and outcome B for B , would be given by the sum: P ( A , B ) = P (( A , A ′ ) , ( B , B ′ )) + P (( A , A ′ ) , ( B , B ′ ))+ P (( A , A ′ ) , ( B , B ′ )) + P (( A , A ′ ) , ( B , B ′ )) , (4)and similarly for the other outcome-probabilities.Now, the probabilities deduced from a single measurement situation can always fit into a singleKolmogorovian probability space, and therefore be represented in terms of deterministic, non-contextualhidden-variables; see for instance the representation theorem in [8]. In other words, if all measurementsperformed by Alice and Bob are compatible, no probabilistic structure extending beyond the classicalone can be revealed. This means that in order to highlight the existence of elements of reality that cannotbe described by classical probability models, and therefore by classical hidden-variables theories, oneneeds to consider situations where an entity is not always subjected to the same experimental context,i.e., to the same measurement, however big such measurement is. Bell’s work was precisely aboutidentifying and analyzing experimental situations able to produce joint probabilities that cannot bemodeled using a single Kolmogorovian probability space, in order to test whether the type of correlationsidentified by EPR [9] did truly exist in reality.In the case of the Bell-CHSH inequality, 4 different measurements are required and represented bythe tensor product observables A ⊗ B , A ′ ⊗ B , A ⊗ B ′ and A ′ ⊗ B ′ . Using A = A ′ = B = B ′ = I ,2ne can then deduce the the 6 commutation relations:[ A ⊗ B, A ′ ⊗ B ] = [ A, A ′ ] ⊗ I , [ A ⊗ B, A ⊗ B ′ ] = I ⊗ [ B, B ′ ] , [ A ′ ⊗ B, A ′ ⊗ B ′ ] = I ⊗ [ B, B ′ ] , [ A ⊗ B ′ , A ′ ⊗ B ′ ] = [ A, A ′ ] ⊗ I , [ A ⊗ B, A ′ ⊗ B ′ ] = [ A, A ′ ] ⊗ BB ′ + A ′ A ⊗ [ B, B ′ ] , [ A ′ ⊗ B, A ⊗ B ′ ] = [ A ′ , A ] ⊗ BB ′ + AA ′ ⊗ [ B, B ′ ] . (5)It is clear from the above that for the 4 observables A ⊗ B , A ′ ⊗ B , A ⊗ B ′ and A ′ ⊗ B ′ , to describemeasurements that cannot be incorporated into a single measurement scheme, we must have [ A, A ′ ] = 0and/or [ B, B ′ ] = 0, i.e., Alice’s and Bob’s local observables must not all commute. However, thistransfer of the incompatibility requirement from the non-local to the local level of the observables, isonly the consequence of the fact that a specific representational choice has been a priori adopted: thatof describing all joint measurements between Alice and Bob as product measurements relative to a unique tensor product representation of the state space. But this is a very special situation, whichwill not necessarily apply to all experimental situations. For instance, it will certainly be invalid if inaddition to the Bell-CHSH inequality also the marginal laws (also called no-signaling conditions) areviolated, as observed in many experiments [10, 11, 12, 13, 14, 15]. But even when the marginal laws are obeyed, a single tensor product representation for all theobservables will not work if the Bell-CHSH inequality is violated beyond Tsirelson’s bound, as it is thecase for, say, the “Bertlmann wears no socks” experiment described in [18]. This is an experimentalsituation where the Bell-CHSH inequality is maximally violated even though Alice’s measurements A and A ′ , and Bob’s measurements B and B ′ , are perfectly compatible at the local level. In other words,this is a situation that cannot be described by (2), as the violation originates from an incompatibilitywhich manifests at the non-local level of the joint measurements, the reason being that the correlationsare created by the joint action of Alice and Bob, in a purely contextual way, i.e., the common causesat the origin of the correlations are contextually actualized.So, while being true that incompatibility does play a central role in the construction of a Bell-testexperiment, and in the understanding of its rationale, it is incompatibility at the global level of thejoint measurements that is fundamental to have, which only reduces to local incompatibility whenall the entanglement can be “pushed” into the state of the system. This is only possible, within thestandard Hilbertian formulation of quantum mechanics, if the Bell-CHSH inequality is violated belowTsirelson’s bound and all marginal laws are satisfied. In more general situations, entanglement needsto be allocated also at the level of measurements, as they cannot anymore be all described as productobservables relative to a same tensor product representation. This is of course a manifestation ofcontextuality: the possibility of using a tensor product representation for the observables describingjoint measurements becomes contextual, in the sense that one needs to adopt a different isomorphismfor each joint measurement, in order to allocate the entanglement resource only in the state; see [16, 17]for the details.To put the above differently, the incompatibility of the different joint measurements means that onecannot find a non-contextual hidden-variables representation for the observed outcome-probabilities,i.e., one cannot find common causes in the past explaining all the correlations that are revealed by Note that the marginal laws can be easily violated when joint measurements are performed on spatially interconnectedmacroscopic entities, as well as on conceptual entities that are connected through meaning; see [17] and the referencescited therein. can be in more or less abstractstates, with the less abstract ones (i.e., the more concrete ones) being precisely those associated withthe condition of “being in space.”A few additional remarks are in order, to better understand what a Bell-test can tell us aboutthe reality of a composite micro-system. We mentioned that a single tensor product representationcan only be used when the Bell-CHSH inequality is violated below Tsirelson’s bound and all marginallaws are satisfied. This should lead one to reflect on the practice of mathematically representing jointmeasurements by product operators, and then have superpositions of product states to describe thepresence of entanglement.If we accept the idea that density operators can also represent genuine states, then the situation isnot in conflict with the general physical principle saying that a composite system exists, and thereforeis in a well-defined state, only if its sub-systems also exist, and therefore are also in well-defined states[21]. On the other hand, if we believe that only vector-states (i.e., pure states) can represent genuinestates, then the choice of representing the state of a composite system as a superposition of productstates, not allowing then to attach vector-states to the sub-systems, becomes questionable, as physicallyunintelligible (how could the sub-systems exist if they are not in well-defined states?).A different possibility would be to drop the requirement of describing entanglement as a property ofthe state, i.e., to use a product state to describe the state of the system and then non-product operatorsto describe the joint measurements. In other words, when confronted with a violation of the Bell-CHSHinequality, there is no a priori requisite to mathematically model the experimental situation in termsof entangled states: entangled (non-product) measurements can also be used, and in fact must be used Just as electromagnetic waves and acoustic waves share the same undulatory nature, but remain completely distinctentities, the same would apply to human conceptual entities and the microscopic entities: they would share a sameconceptual nature, while remaining completely distinct entities.
4f the marginal laws are also disobeyed.But even this situation is not full satisfactory, as is clear that when a system evolves, a productstate will generally transform into a non-product state, hence the interpretational problem remains andthe use of density operators to describe genuine states seems to be inescapable [21].The above issue can be better understood if one observes, as one of us did many decades ago, thatin the Hilbert space formalism separated entities cannot be consistently modeled. The reason for thisis that the Hilbertian formalism is, structurally speaking, too specific, as it satisfies an axiom calledthe ‘covering law’ [22, 23, 24, 7]. This means that non-locality, which in ultimate analysis means non-separability, is intrinsically part of the quantum formalism, so much so that locality/separability cannotbe even properly expressed within it (not for a lack of states, but for a lack of properties). In otherwords, the very decision of using a Hilbert space to model a physical system already and unavoidablyintroduces non-locality/non-separability in its description.Note that the Bell-CHSH inequality, being expressed only in terms of probabilities, is independentof the mathematical formalism used to model an experimental situation. Also, Bell was primarilyinterested in separability, and not whether the probability structure of a composite system would beKolmogorovian or non-Kolmogorovian, and in particular if the probabilities of the different joint mea-surements could be described as the marginals of a unique joint probability distribution. In particular,Bell wasn’t focused on the marginal laws being satisfied or not. This question only came about later,with the analysis of Fine [26], Pitowsky [27] and more recently Dzhafarov and Kujala [28].Note also that the special attention placed on the marginal laws only resulted from their interpreta-tion as ‘no-signaling conditions’ [30, 29]. Indeed, it has been argued that if violated they could be usedto achieve faster than light communication. However, a more attentive analysis of the situation showsthat this is not necessarily the case, for at least two reasons: it is not clear what are the times involved,in order to handle a large enough statistical ensemble of identically prepared systems, and if, when theyare properly accounted for, they would still allow for an effective supraluminal communication. Also,and more importantly, the existence of correlations separated by space-like intervals does not per seimply that they result from underlying phenomena propagating in space faster than the speed of light,as the numerous models investigated by our group have clearly shown; see for instance the discussionsin [31, 17, 7, 18]Our digression on marginal laws allows us to “close the circle” of our analysis. We mentioned thatthe description of all the observables associated with the different joint measurements in a Bell-testexperiment, as product observables relative to a single tensor product representation, is what createsthe illusion that Bell-tests would only be about local incompatibility. There is however no a prioriphysical justification to believe that such a peculiar representation would correspond to the generalcase. Certainly, it is not forced upon us by the available empirical data, which in fact tell us a ratherdifferent story, considering that the marginal laws are typically violated.The proof of the marginal laws also relies on the existence of a single tensor product representation,which however, again, is not imposed by the quantum formalism (see in particular the discussion in[32]), hence should be justified by the data.To conclude, Bell-tests are not just about local incompatibility: taking into account the availabledata, and until proven to the contrary, they also are about non-local incompatibility, hence about non-locality. 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