How Much Measurement Independence Is Needed to Demonstrate Nonlocality?

Jonathan Barrett

Department of Mathematics, Royal Holloway, University of London, Egham, Surrey TW20 0EX, United Kingdom

Nicolas Gisin

Group of Applied Physics, University of Geneva, 1211 Geneva 4, Switzerland(Received 13 September 2010; published 10 March 2011)

If nonlocality is to be inferred from a violation of Bell’s inequality, an important assumption is that the

measurement settings are freely chosen by the observers, or alternatively, that they are random and

uncorrelated with the hypothetical local variables. We demonstrate a connection between models that

weaken this assumption, allowing partial correlation, and (i) models that allow classical communication

between the distant parties, (ii) models that exploit the detection loophole. Even if Bob’s choices are

completely independent, all correlations from projective measurements on a singlet can be reproduced,

with mutual information between Alice’s choice and local variables less than or equal to one bit.

DOI: 10.1103/PhysRevLett.106.100406 PACS numbers: 03.65.Ud, 03.65.Ta, 03.67.�a

Quantum nonlocality, whereby particles appear to influ-ence one another instantaneously even though they arewidely separated in space, is one of the most remarkablephenomena known to modern science [1–4]. Historically,this peculiar prediction of quantum theory triggered manydebates and even doubts about its validity. Today, it is awell established experimental fact [5].

The profound implications of quantum nonlocality forour world view remain controversial. But it is no longerconsidered as suspicious, or of marginal interest. It iscentral to our understanding of quantum physics, and, inparticular, it is essential for the powerful applications ofquantum information technologies. Ekert showed [6] howentanglement could enable distant partners to establish ashared cryptographic key. More recently, it was shown thatnonlocal correlations alone can ensure security of a secretkey, with no further assumptions about quantum theory [7].Simultaneously, it was realized that Bell inequalities arethe only entanglement witnesses that can be trusted incases where the dimensions of the relevant Hilbert spacesare unknown [8].

In an experimental demonstration of quantum nonlocal-ity, measurements are performed on separated quantumsystems in an entangled state, and it is shown that themeasurement outcomes are correlated in a manner thatcannot be accounted for by local variables. In order toconclude that nonlocality is exhibited, it is crucial for theanalysis that the choices of which measurement to performare freely made by the experimenters. Alternatively, theymust be random and uncorrelated with the hypotheticallocal variables. It is well known that if the measurementsettings are not random, but are in fact determined by thelocal variables, then arbitrary correlations can be repro-duced [9].

Here we reverse the argument. Taking for grantedthat quantum nonlocal correlations are produced, how

independent must the measurement settings be assumedto be in order to rule out an explanation in terms of localvariables [10]? We show that all correlations obtained fromprojective measurements on a singlet state can be repro-duced with the mutual information between local variablesand the measurement setting on one side less than or equalto one bit. This is not only of fundamental interest. If acryptographic protocol is relying on the nonlocality ofcorrelations for security, it is vital that there is no under-lying local mechanism that an eavesdropper may be ex-ploiting. The result shows that random number generatorsused to determine settings must be assumed to have a veryhigh degree of independence from the particle source.Bell experiments.—In an experimental test of a Bell

inequality, two experimenters—traditionally named Aliceand Bob—are spatially separated, see Fig. 1. They eachcontrol a quantum system, and the joint state of these twosystems is an entangled state. Alice and Bob each performa measurement on their quantum system, in such a way thatthe measurements are spacelike separated. Alice’s mea-surement is chosen from a finite set of possibilities, as isBob’s. Call these measurement settings the inputs and labelthem x for Alice’s choice and y for Bob’s choice. Theoutcome of Alice’s measurement (her output) is labeled a

FIG. 1 (color online). Schematic illustration of an experimentfor testing quantum nonlocality.

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and Bob’s b. By repeating this procedure many times andcollating the data, Alice and Bob can estimate conditionalprobabilities of the form Pða; bjx; yÞ.

The question is then, when can the correlations producedin an experiment like this be explained by underlying localvariables? Locality here implies that the outcome ofAlice’s measurement cannot be directly influenced byBob’s choice of which measurement to perform, andvice versa. More precisely, let the hypothetical underlyingvariables be denoted �, and assume a distribution Pð�Þ.Then Bell locality is the condition that

Pða; bjx; y; �Þ ¼ Pðajx; �ÞPðbjy; �Þ: (1)

If the correlations could in principle be explained as arisingfrom underlying local variables, then they can be written inthe form

Pða; bjx; yÞ ¼ X

�

Pð�ÞPðajx; �ÞPðbjy; �Þ: (2)

(Here and throughout, sums should be replaced with inte-grals when the variable is not discrete.) The correlationscannot be written in this form if and only if they violate aBell inequality.

Suppose that correlations are obtained which do violatea Bell inequality. What assumptions are necessary to con-clude that this is indicative of nonlocality? There is a largeamount of literature on this topic, and a reader could do alot worse than refer to Bell’s original papers [1]. But it isuncontroversial that the standard argument needs to as-sume that the inputs x, y are freely, or at least indepen-dently and randomly, chosen by Alice and Bob.

Nonindependent measurement choices.—We wish toanalyze the case in which measurement settings can becorrelated with local variables. Hence, in addition to a setof correlations Pða; bjx; yÞ, assume a distribution over in-puts Pðx; yÞ. A correlated-settings model is defined by avariable � and an overall joint probability distributionPCSða; b; x; y; �Þ. The correlated-settings model reprodu-ces the given correlations and input distribution ifPCSða; bjx; yÞ ¼ Pða; bjx; yÞ and PCSðx; yÞ ¼ Pðx; yÞ. Themodel is Bell-local if the distribution PCS also satisfies anequation of the form of Eq. (1). In this case

PCSða; bjx; yÞ ¼X

�

PCSð�jx; yÞPCSða; bjx; y; �Þ

¼ X

�

PCSð�jx; yÞPCSðajx; �ÞPCSðbjy; �Þ: (3)

Note the similarity with Eq. (2). The usual kind of model,in which Alice’s and Bob’s inputs are assumed independentof the local variables, is a special case with PCSðx; yj�Þ ¼PCSðx; yÞ. By Bayes’ rule, this also gives PCSð�jx; yÞ ¼PCSð�Þ, and Eq. (3) reduces to an equation of the form ofEq. (2).

Another special case is the extreme case where theinputs are completely determined by �, so that x ¼ fð�Þand y ¼ gð�Þ for functions f and g. Here, for all x, y, �,

PCSðx; yj�Þ ¼ 0 or 1. Any correlations can be produced bya Bell-local model of this form [9].In other models, x and y will be neither independent nor

determined by �. In general there are many numericalmeasures of the degree to which they are correlated. Oneparticularly natural and simple measure is the mutualinformation. Thus a correlated-settings model can be char-acterized by the mutual information between the measure-ment choices x, y, and �:

Iðx; y:�Þ ¼ Hðx; yÞ þHð�Þ �Hðx; y; �Þ; (4)

where H is the Shannon entropy. When x and y are inde-pendent of �, Iðx; y:�Þ ¼ 0. If x and y are functions of �,on the other hand, then Iðx; y:�Þ ¼ Hðx; yÞ. Since in gen-eral Iðx; y:�Þ � Hðx; yÞ, this means that the mutual infor-mation is maximal [with respect to a fixed distribution overinputs Pðx; yÞ].In the context of correlated-settings models, the question

is no longer whether quantum correlations violate a Bellinequality. The question is, how large must Iðx; y:�Þ be ifthe correlations are to be reproduced within a Bell-localmodel? Alternatively, for a fixed degree of correlationIðx; y:�Þ, do quantum correlations violate a Bell inequalityby a sufficiently large margin that the model cannot pos-sibly be Bell-local? This depends on the precise experi-ment. For example, if the input alphabet consists of onlytwo choices, as with the CHSH inequality, then any corre-lations can be reproduced by a correlated-settings modelwith Iðx; y:�Þ � 1. But the question remains open forlarger input alphabets. Intuitively one may think that forlarge alphabets, if Alice’s choice is only mildly correlatedwith the variable �, then a sufficient violation of a suitableBell inequality should still rule out locality. But we shallsee that this intuition is wrong, at least for maximallyentangled qubits.Connection to communication cost.—One way to simu-

late quantum correlations, including nonlocal correlations,is for Alice and Bob to communicate with one another afterinputs are chosen, but before outputs are produced. Anycorrelations can be produced this way if Alice and Bobshare random data, and Alice simply informs Bob of herinput choice. Hence, if Alice is choosing from nA possibleinputs, then an obvious upper bound on howmany classicalbits need to be communicated is given by lognA. Theminimum number of bits that must be communicated, onthe other hand, provides a natural way of quantifying theamount of nonlocality inherent in a given set of correla-tions. This interesting line of research was started byMaudlin [2] and independently by Tapp and co-workersand by Steiner [11,12]. It culminated with a model of Tonerand Bacon, who proved that one single bit of communica-tion from Alice to Bob is sufficient to simulate all corre-lations obtained from arbitrary projective (i.e.,von Neumann) measurements on two qubits in amaximally entangled state [13]. It was extended to all

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two-party correlations, though ignoring the marginals,in [14].

A slightly more precise description of a communicationmodel for simulating quantum correlations is as follows.The random data shared between Alice and Bob are avariable �, with distribution PCð�Þ. Assuming that Alicecommunicates first, Alice sends a bit string c1 to Bob. Thestring c1 can depend on � and on Alice’s input x. It mayalso depend on further random data held by Alice, but thisdata can without loss of generality be absorbed into �.Hence, assume that c1 ¼ c1ð�; xÞ. Depending on the pro-tocol Bob may now send a bit string c2 to Alice, whereagain without loss of generality, we assume that c2 is afunction of �, c1, and y. This continues for a total of kmessages. The entire conversation between Alice and Bobon a particular round is the sequence m ¼ c1; c2; . . . ; ckand is a function m ¼ fðx; y;�Þ of x, y, and �. Finally,Alice outputs a, where again absorbing all randomness into�, a can be assumed to be a function a ¼ gðx;�;mÞ of herinput, the shared random data, and the conversation.Similarly, Bob outputs b, which is a function b ¼hðy;�;mÞ. The protocol then terminates.

For each pair of inputs, a communication model definesa distribution PCða; b;m;�jx; yÞ, and reproduces correla-tions Pða; bjx; yÞ if PCða; bjx; yÞ ¼ Pða; bjx; yÞ. If adistribution over inputs Pðx; yÞ is also given, the conversa-tion m has a well-defined distribution PðmÞ ¼P

x;yPCðmjx; yÞPðx; yÞ. Let the entropy of this distribution

be HðmÞ.The connection with correlated-settings models is ex-

pressed in the following.Theorem 1: Consider a fixed input distribution

Pðx; yÞ, and a communication model with total conversa-tion m. Then there exists a Bell-local correlated-settingsmodel, which reproduces the same correlations and Pðx; yÞ,with Iðx; y:�Þ � HðmÞ.

Proof: Given Pðx; yÞ and a communication model asabove, construct a correlated-settings model as follows.Define � as the pair � ¼ ð�;mÞ. In the communicationmodel,m is the conversation between Alice and Bob, but inthe correlated-settings model there is no communication,and � represents the underlying variable. Define thecorrelated-settings model by PCSða; b; x; y; �Þ ¼Pðx; yÞPCða; b;�;mjx; yÞ. It is easy to see that the resultingmodel reproduces Pðx; yÞ and the same correlations as thecommunication model. Bell locality is satisfied since

PCSða; bjx; y; �Þ ¼ PCða; bjx; y;�;mÞ¼ PCðajx;�;mÞPCðbjy; �;mÞ¼ PCSðajx; �ÞPCSðbjy; �Þ:

The mutual information between � and ðx; yÞ isIð�:x; yÞ ¼ Ið�:x; yÞ þ Iðm:x; yj�Þ (5)

¼ 0þHðmj�Þ �Hðmj�; x; yÞ (6)

¼ Hðmj�Þ (7)

� HðmÞ: (8)

The first line uses the chain rule [15], the second linefollows from the fact that � is assumed to be independentof the inputs x, y, and the third line holds because m is afunction of � and x; y. This concludes the proof.Intuitively, � ¼ ð�;mÞ restricts Alice’s and Bob’s

choices to inputs (x; y) such that m ¼ fðx; y; �Þ. We stressthe generality of this result: the communication model mayhave involved two-way communication, and may havebeen such that the communication on a given run is un-bounded. As long as HðmÞ is finite, the bound is useful.We now apply the theorem with reference to the Toner-

Bacon communication model. Here, x and y are arbitraryprojective measurements. The shared random data � takethe form of two vectors on the Bloch sphere, and thedistribution Pð�Þ is such that the two vectors are indepen-dent, with each uniformly distributed. For our purposesmost of the details of the model do not matter, and we referthe reader to [13] for a description and proof that it repro-duces quantum correlations for projective measurementson a singlet. It is sufficient to know that at each round, theconversation m is a single bit communicated from Alice toBob. The bit m ¼ fðx;�Þ is a function of Alice’s input xand �.The resulting correlated-settings model also reproduces

all correlations from projective measurements on a singlet.In general, the actual value of Iðx; y:�Þ depends on thedistribution over inputs Pðx; yÞ. The theorem tells us thatfor any such distribution, Iðx; y:�Þ � HðmÞ � 1, sincem isa single bit. In particular, if Alice’s and Bob’s inputs areindependent, with each chosen from a uniform distributionover all possible directions, it is easy to verify thatIðx; y:�Þ � 0:85.In a typical Bell-type experiment with a singlet, Alice

and Bob will be choosing from finite sets of measurements.The Toner-Bacon model can still be applied, with thedistribution Pðx; yÞ having support only on these finitesets. In this case too, Iðx; y:�Þ � 1. The Bell-localcorrelated-settings model reproduces the quantum correla-tions no matter how large the input alphabets, and nomatter what Bell inequality is being tested.Finally, in the Toner-Bacon model, Bob’s setting is of

course independent from the communication he receivesfromAlice. It follows that in the derived correlated-settingsmodel, Bob’s setting is independent from �, i.e.,Iðy:�Þ ¼ 0.Detection loophole.—In a real Bell experiment, detec-

tion is inefficient. A typical analysis of the experimentestimates the correlations Pða; bjx; yÞ from those runs onwhich both Alice’s and Bob’s detectors clicked, and simplyignores all the other runs. Such an analysis is valid, as longas it is assumed that the probability of a detector clicking isindependent of the hypothetical local variables. This issometimes called the ‘‘fair sampling assumption.’’

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It is possible to reproduce nonlocal correlations with amodel that is Bell-local, but which violates the fair sam-pling assumption [16,17]. Denote the event that Alice’s(Bob’s) detector clicks by DA (DB). A detection-efficiencymodel is defined by a variable �, with distribution PDEð�Þ,independent of x; y, and for each x; y a distributionPDEða; b;DA;DBjx; y; �Þ. Bell locality is the condi-tion that PDEða; b;DA;DBjx; y; �Þ ¼ PDEða;DAjx; �ÞPDEðb;DBjy; �Þ. The model reproduces correlationsPða; bjx; yÞ if

Pða; bjx; yÞ ¼ PDEða; bjx; y;DA;DBÞ: (9)

The efficiencies of Alice’s detectors in this model aregiven by PDEðDAjxÞ ¼ P

�PDEð�ÞPDEðDAjx; �Þ, and simi-larly for Bob’s. If the detection efficiencies in a realexperiment are high enough, and if a Bell inequality isviolated by a large enough margin, then a Bell-localdetection-efficiency model reproducing the correlationscan be ruled out.

Given a distribution Pðx; yÞ, and a detection-efficiencymodel reproducing correlations Pða; bjx; yÞ, it is easy toconstruct a correlated-settings model which also reprodu-ces Pðx; yÞ and Pða; bjx; yÞ. Simply define

PCSða; b; x; y; �Þ ¼ Pðx; yÞPDEða; b; �jx; y;DA;DBÞ: (10)

It is easy to show, using Eq. (9), that the correlated-settingsmodel reproduces Pðx; yÞ and Pða; bjx; yÞ.

The construction can be applied, for example, to theGisin–Gisin detection-efficiency model [18], which repro-duces correlations from arbitrary projective measurementson a singlet. The efficiencies are independent of x and yand are given by PðDAÞ ¼ 1=2, PðDBÞ ¼ 1. In this model� is a vector on the Bloch sphere, andPDEð�jx; y;DA;DBÞ ¼ j�: ~xj=2�, where ~x denotes theBloch vector representing the measurement x. In the re-sulting correlated-settings model, Iðx; y:�Þ depends onPðx; yÞ. If x and y are independently and uniformly distrib-uted, it is easy to show that Iðx; y:�Þ ¼ Iðx:�Þ � 0:28. Thisis an even lower value than was obtained above using theToner-Bacon model.

Finally, note that in the case of communication models,there was a connection between Iðx; y:�Þ and the amount ofcommunication (as measured by the entropy of the con-versation’s distribution). With detection-efficiency models,we expect there to be a connection between Iðx; y:�Þ andthe detection efficiencies. Exploring this connection is aninteresting direction for future work.

Conclusion.—It is well known that in order to derivenonlocality from violation of a Bell inequality, one has toassume that there is no correlation between the hypotheti-cal local variables � and the experimenters’ measurementchoices x and y. We have investigated how much correla-tion could be allowed if quantum predictions are to remainincompatible with local variables. Surprisingly, no matterhow large the alphabet size of the inputs, correlations fromprojective measurements on a singlet can be reproduced,

with the mutual information between Alice’s input andlocal variables not more than one bit, and Bob’s inputcompletely independent. This deserves further investiga-tion. Future work should analyze more general scenarios,involving generalized measurements, partial entanglement[19], higher dimensional Hilbert spaces, and more than twoparties. It would also be instructive to consider measures ofcorrelation other than the mutual information.Finally, let us emphasize the change in paradigm since

the old EPR paper [4]. If, contrary to EPR, one acceptsnonlocality as a fact, then not only can one develop power-ful applications in quantum information science [7,8], butmoreover one can upper bound the lack of free choice ofthe players.J. B. is supported by EPSRC. N. G. acknowledges sup-

port from the ERC advanced grant QORE and the SwissNCCR-QP. We thank M. J.W. Hall and C. Branciard forconstructive discussions.Note added.—After completion of this work, M. J.W.

Hall published a related paper [20].

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[15] Th.M. Cover and J. A. Thomas, Elements of InformationTheory (Wiley, New York, 1991).

[16] P. Pearle, Phys. Rev. D 2, 1418 (1970).[17] E. Santos, Phys. Rev. A 46, 3646 (1992).[18] B. Gisin and N. Gisin, Phys. Lett. A 260, 323 (1999).[19] For two partially entangled qubits there is a 2 bits com-

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[20] M. J.W. Hall, Phys. Rev. Lett. 105, 250404 (2010).

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