The Uncertainty PrincipleDetermines the Nonlocalityof Quantum MechanicsJonathan Oppenheim1* and Stephanie Wehner2,3*

Two central concepts of quantum mechanics are Heisenberg’s uncertainty principle and a subtleform of nonlocality that Einstein famously called “spooky action at a distance.” These twofundamental features have thus far been distinct concepts. We show that they are inextricably andquantitatively linked: Quantum mechanics cannot be more nonlocal with measurements that respectthe uncertainty principle. In fact, the link between uncertainty and nonlocality holds for all physicaltheories. More specifically, the degree of nonlocality of any theory is determined by two factors:the strength of the uncertainty principle and the strength of a property called “steering,” whichdetermines which states can be prepared at one location given a measurement at another.

Ameasurement allows us to gain informa-tion about the state of a physical system.For example, when measuring the posi-

tion of a particle, the measurement outcomes corre-spond to possible locations. Whenever the state ofthe system is such that we can predict this positionwith certainty, then there is only onemeasurementoutcome that can occur. Heisenberg (1) observedthat quantum mechanics imposes strict restrictionson what we can hope to learn—there are incompat-ible measurements such as position andmomentumwhose results cannot be simultaneously predictedwith certainty. These restrictions are known as un-certainty relations. For example, uncertainty relationstell us that if we were able to predict the momen-tum of a particle with certainty, then, when mea-suring its position, all measurement outcomeswould occur with equal probability. That is, wewould be completely uncertain about its location.

Nonlocality can be exhibited when measure-ments on two ormore distant quantum systems areperformed; that is, the outcomes can be correlatedin a way that defies any local classical description(2). This is whywe know that quantum theorywillnever be superseded by a local classical theory.Nonetheless, even quantum correlations are re-stricted to some extent: Measurement results can-not be correlated so strongly that theywould allowsignaling between two distant systems. However,quantum correlations are still weaker than whatthe no-signaling principle demands (3).

So why are quantum correlations strongenough to be nonlocal, yet not as strong as theycould be? Is there a principle that determines thedegree of this nonlocality? Information theory(4, 5), communication complexity (6), and localquantum mechanics (7) provided us with somerationale for why limits on quantum theory may

exist. But evidence suggests that many of theseattempts provide only partial answers. Here, wetake a very different approach and relate the limi-tations of nonlocal correlations to two inherentproperties of any physical theory.

At the heart of quantum mechanics liesHeisenberg’s uncertainty principle (1). Tradition-ally, uncertainty relations have been expressed interms of commutators

DADB ≥1

2j⟨yj½A, B�jy⟩j ð1Þ

with standard deviationsDX¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi⟨yjX 2jy⟩− ⟨yjXjy⟩2

pfor X ∈ {A,B}. However, themoremodern approachis to use entropic measures. Let p(x(t)|t)s denotethe probability that we obtain outcome x(t) whenperforming a measurement labeled t when the sys-tem is prepared in the state s. In quantum theory,s is a density operator, whereas for a general phys-ical theory, we assume that s is simply an abstractrepresentation of a state. The well-known ShannonentropyH(t)s of the distribution over measurementoutcomes of measurement t on a system in state sis thus

HðtÞs :¼ −∑xðtÞ

pðxðtÞjtÞs log pðxðtÞjtÞs ð2Þ

In any uncertainty relation, we wish to compareoutcome distributions for multiple measurements.In terms of entropies, such relations are of the form

∑tpðtÞHðtÞs ≥ cT ,D ð3Þ

where p(t) is any probability distribution over theset of measurements T, and cT,D is some positiveconstant determined by T and the distributionD = {p(t)}t. To see why Eq. 3 forms an uncer-tainty relation, note that whenever cT,D > 0 wecannot predict the outcome of at least one of themeasurements t with certainty [i.e., H(t)s > 0].Such relations have the great advantage that thelower bound cT,D does not depend on the state

s (8). Instead, cT,D depends only on the mea-surements and hence quantifies their inherentincompatibility. It has been shown that for twomeasurements, entropic uncertainty relations doin fact implyHeisenberg’s uncertainty relation (9),providing uswith amore general way of capturinguncertainty [see (10) for a survey].

One may consider many entropic measuresinstead of the Shannon entropy. For example, themin-entropy

H∞ðtÞs :¼ −log maxxðtÞ

pðxðtÞjtÞs ð4Þ

used in (8) plays an important role in cryptographyand provides a lower bound on H(t)s. Entropicfunctions are, however, a rather coarse way ofmeasuring the uncertainty of a set of measure-ments, as they do not distinguish the uncertaintyinherent in obtaining any combination of outcomesx(t) for different measurements t. It is thus usefulto consider much more fine-grained uncertaintyrelations consisting of a series of inequalities,one for each combination of possible outcomes,which we write as a string x = (x(1),…, x(n) ) ∈B×n

with n = |T | (11). That is, for each x, a set ofmeasurements T , and distribution D = {p(t)}t,

Pcertðs;xÞ :¼ ∑n

t¼1pðtÞ pðxðtÞjtÞs ≤ zxðT ,DÞ

ð5ÞFor a fixed set of measurements, the set ofinequalities

U ¼ ∑n

t¼1pðtÞ pðxðtÞjtÞs ≤ zxj∀x ∈ B�n

� �

ð6Þthus forms a fine-grained uncertainty relation, as itdictates that one cannot obtain a measurement out-come with certainty for all measurements simulta-neously whenever zx < 1. The values of zx thusconfine the set of allowed probability distributions,and themeasurements have uncertainty if zx < 1 forall x. To characterize the “amount of uncertainty”in a particular physical theory, we are thus inter-ested in the values of

1DAMTP, University of Cambridge, Cambridge CB3 0WA, UK.2Institute for Quantum Information, California Institute of Tech-nology, Pasadena, CA 91125, USA. 3Centre for Quantum Tech-nologies, National University of Singapore, 117543 Singapore.

*To whom correspondence should be addressed. E-mail:jono@damtp.cam.ac.uk (J.O.); wehner@nus.edu.sg (S.W.)

Fig. 1. Any game where, for every choice of settingss and t and answer a of Alice, there is only one correctanswer b for Bob can be expressed using strings xs,asuch that b = xðtÞs,a is the correct answer for Bob (13)for s and a. The game is equivalent to Alice out-putting xs,a and Bob outputting x

ðtÞs,a.

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zx ¼ maxs∑n

t¼1pðtÞ pðxðtÞjtÞs ð7Þ

where the maximization is taken over all statesallowed on a particular system (for simplicity, weassume it can be attained in the theory considered).We will also refer to the state rx that attains themaximum as a “maximally certain state.” How-ever, we will also be interested in the degree ofuncertainty exhibited by measurements on a set ofstates S quantified by zSx defined with the maxi-mization in Eq. 7 taken over s ∈ S. Fine-graineduncertainty relations are directly related to theentropic ones, and theyhaveboth aphysical and aninformation-processing appeal (12).

As an example, consider the binary spin-½observables Z and X. If we can obtain a partic-ular outcome x(Z) given that we measured Z withcertainty—that is, p(x(Z)|Z) = 1—then the comple-mentary observable must be completely uncertain,that is, p(x(X)|X) = ½. If we choose which measure-ment to make with probability ½, then this notionof uncertainty is captured by the relations

1

2pðxðX ÞjX Þ þ 1

2pðxðZÞjZÞ ≤ zx ¼

1

2þ 1

2ffiffiffi2

p

for all x ¼ ðxðX Þ, xðZÞÞ ∈ f0,1g2

where the maximally certain states are given by theeigenstates of ðX þ ZÞ= ffiffiffi

2p

and ðX − ZÞ= ffiffiffi2

p.

We now introduce the concept of nonlocalcorrelations. Instead of considering measurementson a single system, we considermeasurements ontwo (or more) space-like separated systems tradi-tionally named Alice and Bob. We use t ∈ T tolabel Bob’s measurements, and b ∈B to label hismeasurement outcomes. For Alice, we use s ∈ Sto label her measurements, and a ∈A to label heroutcomes (11). When Alice and Bob performmeasurements on a shared state sAB, the outcomesof their measurements can be correlated. Letpða,bjs,tÞsAB be the joint probability that theyobtain outcomes a and b for measurements sand t. We can now again ask ourselves: Whatprobability distributions are allowed?

Quantum mechanics as well as classical me-chanics obeys the no-signaling principle, meaningthat information cannot travel faster than light.

This means that the probability that Bob obtainsoutcome b when performing measurement t can-not depend onwhichmeasurement Alice choosesto perform (and vice versa). More formally,∑a pða,bjs,tÞsAB ¼ pðbjtÞsB for all s, where sB =trA(sAB). Curiously, however, this is not the onlyconstraint imposed by quantum mechanics (3).Here we find that the uncertainty principleimposes the other limitation.

To do so, let us recall the concept of so-called Bell inequalities (2). These are most easilyexplained in their more modern form in terms ofa game played between Alice and Bob. Let uschoose questions s ∈ S and t ∈ T according tosome distribution p(s,t) and send them to Aliceand Bob, respectively, where we take for sim-plicity p(s,t) = p(s)p(t). The two players now re-turn answers a ∈A and b ∈B. Every game comeswith a set of rules that determines whether a andb are winning answers given questions s and t.Again for simplicity, we thereby assume that forevery s, t, and a, there exists exactly one winninganswer b for Bob (and similarly for Alice). Thatis, for every setting s and outcome a of Alice,there is a string xs,a ¼ ðxð1Þs,a, :::, x

ðnÞs,aÞ ∈ B�n of

length n = |T | that determines the correct answerb ¼ xðtÞs,a for question t for Bob (Fig. 1). We saythat s and a determine a “random-access coding”(13), meaning that Bob is not trying to learn thefull string xs,a but only the value of one entry. Thecase of non-unique games is a straightforward butcumbersome generalization.

As an example, the Clauser-Horne-Shimony-Holt (CHSH) inequality (14), one of the simplestBell inequalities whose violation implies nonlocal-ity, can be expressed as a game in which Alice andBob receive binary questions s,t ∈ {0,1} respec-tively, and similarly their answers a,b ∈ {0,1} aresingle bits. Alice and Bob win the CHSH game iftheir answers satisfy a ⊕ b = s · t. We can labelAlice’s outcomes using string xs,a, and Bob’s goalis to retrieve the tth element of this string. For s =0, Bob will always need to give the same answeras Alice in order to win, and hence we have x0,0 =(0,0), and x0,1 = (1,1). For s = 1, Bob needs to givethe same answer for t = 0, but the opposite answerif t = 1. That is, x1,0 = (0,1) and x1,1 = (1,0).

Alice andBobmay agree on any strategy aheadof time, but once the game starts, their physicaldistance prevents them from communicating. Forany physical theory, such a strategy consists of achoice of shared state sAB, as well as measure-ments, where we may without loss of generalityassume that the players perform a measurementdepending on the question they receive and returnthe outcome of said measurement as their answer.For any particular strategy, we may hence write theprobability that Alice and Bob win the game as

PgameðS,T ,sABÞ ¼∑s,tpðs,tÞ∑

apða,b ¼ xðtÞs,ajs,tÞsAB ð9Þ

To characterize what distributions are allowed,we are generally interested in the winning

probability maximized over all possible strat-egies for Alice and Bob:

Pgamemax ¼ max

S,T ,sAB

PgameðS,T ,sABÞ ð10Þ

which, in the case of quantum theory, we refer toas a Tsirelson’s type bound for the game (15). Forthe CHSH inequality (14), we have Pgame

max ¼ 3=4classically, Pgame

max ¼ 1=2 þ 1=ð2 ffiffiffi2

p Þ quantummechanically, andPgame

max ¼ 1 for a theory allowingany nonsignaling correlations. Pgame

max quantifies thestrength of nonlocality for any theory, with the un-derstanding that a theory possesses genuine non-locality when it differs from the value that can beachieved classically. However, the connection wewill demonstrate between uncertainty relations andnonlocality holds even before the optimization.

The final concept we need in our discussion issteerability,which determineswhat statesAlice canprepare on Bob’s system remotely. Imagine thatAlice and Bob share a state sAB, and consider thereduced state sB = trA(sAB) on Bob’s side. In quan-tum mechanics, as well as many other theories inwhich Bob’s state space is a convex set S, the statesB ∈ S can be decomposed in many different waysas a convex sum,

sB ¼∑apðajsÞss,a with ss,a ∈ S ð11Þ

corresponding to an ensemble Es = {p(a|s), ss,a}a.Schrödinger (16, 17) noted that in quantummechanics, for all s there exists a measurementon Alice’s system that allows her to prepare Es ={p(a|s), ss,a}a on Bob’s site (Fig. 2). That is, formeasurement s, Bob’s system will be in the statess,a with probability p(a|s). Schrödinger calledthis steering to the ensemble Es, and it does notviolate the no-signaling principle, because foreach of Alice’s settings, the state of Bob’s systemis the same once we average over Alice’s mea-surement outcomes.

Even more, he observed that for any set ofensembles {Es}s that respect the no-signaling con-straint (i.e., for which there exists a state sB suchthat Eq. 11 holds), we can in fact find a bipartitequantum state sAB and measurements that al-low Alice to steer to such ensembles. We can im-agine theories in which our ability to steer is morerestricted, or perhaps even less restricted (someamount of signaling is permitted). Our notion ofsteering thus allows forms of steering not consid-ered in quantum mechanics (16–19) or other re-stricted classes of theories (20). Our ability to steer,however, is a property of the set of ensembles weconsider, and not a property of one single ensemble.

We are now in a position to derive the relationbetween how nonlocal a theory is and how un-certain it is. We may express the probability (Eq.9) that Alice and Bob win the game by using aparticular strategy (Fig. 1) as

PgameðS,T ,sABÞ ¼

∑spðsÞ∑

apðajsÞPcertðss,a; xs,aÞ ð12Þ

(8)

Fig. 2. When Alice performs a measurement la-beled s and obtains outcome a with probabilityp(a|s), she effectively prepares the state ss,a onBob’s system. This is known as steering.

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where ss,a is the reduced state of Bob’s system forsetting s and outcome a of Alice, and p(t) inPcert(⋅)is the probability distribution over Bob’s questionsT in the game. This immediately suggests thatthere is indeed a close connection between gamesand our fine-grained uncertainty relations. In par-ticular, every game can be understood as givingrise to a set of uncertainty relations, and vice versa.It is also clear from Eq. 5 for Bob’s choice of mea-surements T and distribution D over T that thestrength of the uncertainty relations imposes anupper bound on the winning probability,

PgameðS,T ,sABÞ ≤ ∑spðsÞ∑

apðajsÞ zxs,aðT ,DÞ

≤ maxxs,a

zxs,aðT ,DÞ ð13Þ

where we have made explicit the functional de-pendence of zxs,a on the set of measurements.This seems curious given that we know from (21)that any two incompatible observables lead to aviolation of the CHSH inequality, and that toachieve Tsirelson’s bound Bob must measuremaximally incompatible observables (22) thatyield stringent uncertainty relations. However,from Eq. 13 we may be tempted to conclude thatfor any theory it would be in Bob’s best interest toperform measurements that are very compatibleand have weak uncertainty relations in the sensethat the values zxs,a

can be very large. But canAlice prepare states ss,a that attain zxs,a

for anychoice of Bob’s measurements?

The other important ingredient in under-standing nonlocal games is thus steerability. Wecan think of Alice’s part of the strategy S, sAB aspreparing the ensemble {p(a|s), ss,a}a on Bob’ssystem whenever she receives question s. Thus,when considering the optimal strategy for nonlocalgames, wewant tomaximizePgame(S,T, sAB) overall ensembles Es = {p(a|s), ss,a}a that Alice cansteer to, and use the optimal measurements Toptfor Bob. That is,

Pgamemax ¼ max

fE sgs∑spðsÞ∑

apðajsÞzss,a

xs,aðTopt,DÞ

ð14Þand hence the probability that Alice and Bob winthe game depends only on the strength of the un-certainty relations with respect to the sets ofsteerable states. To achieve the upper boundgiven by Eq. 13, Alice needs to be able to preparethe ensemble fpðajsÞ,rxs,a

gaof maximally cer-

tain states on Bob’s system. In general, it is notclear that the maximally certain states for themeasurements that are optimal for Bob in thegame can be steered to.

It turns out that in quantum mechanics, thiscan be achieved in cases where we know the op-timal strategy. For all XOR games (23)—that is,correlation inequalities for two outcome observ-ables (which include CHSH as a special case)—aswell as other games where the optimal measure-ments are known, we find that the states that aremaximally certain can be steered to (12). The

uncertainty relations for Bob’s optimal measure-ments thus give a tight bound,

Pgamemax ¼∑

spðsÞ∑

apðajsÞ zxs,aðTopt,DÞ ð15Þ

where we recall that zxs,a is the bound given by themaximization over the full set of allowed states onBob’s system. It is an open question whether thisholds for all games in quantum mechanics.

An important consequence of this is that anytheory that allows Alice and Bob to win with aprobability exceeding Pgame

max requires measure-ments that do not respect the fine-grained uncer-tainty relations given by zxs, a for themeasurementsused by Bob (the same argument can be made forAlice). Even more, it can lead to a violation of thecorresponding min-entropic uncertainty relations(12). For example, if quantum mechanics were toviolate CHSH using the same measurements forBob, it would need to violate the min-entropicuncertainty relations (8). This relation holds evenif Alice and Bob were to perform altogetherdifferent measurements when winning the gamewith a probability exceeding Pgame

max : For these newmeasurements, there exist analogous uncertaintyrelations on the set S of steerable states, and ahigher winning probability thus always leads to aviolation of such an uncertainty relation. Converse-ly, if a theory allows any states violating even oneof these fine-grained uncertainty relations forBob’s (or Alice’s) optimal measurements on thesets of steerable states, thenAlice andBob are ableto violate the Tsirelson’s type bound for the game.

Although the connection between nonlocalityand uncertainty is more general, we examine theexample of the CHSH inequality in detail to gainsome intuition on how uncertainty relations ofvarious theories determine the extent to which thetheory can violate Tsirelson’s bound (12). To brieflysummarize, in quantum theory, Bob’s optimal mea-surements are X and Z, which have uncertaintyrelations given byzxs,a

¼ 1=2þ 1=ð2 ffiffiffi2

p Þ of Eq.8. Thus, if Alice could steer to the maximallycertain states for these measurements, they wouldbe able to achieve a winning probability given byPgamemax ¼ zxs,a

—that is, the degree of nonlocalitywould be determined by the uncertainty relation.This is the case. If Alice and Bob share the singletstate, then Alice can steer to the maximally cer-tain states by measuring in the basis given by theeigenstates of ðX þ ZÞ= ffiffiffi

2p

or of ðX − ZÞ= ffiffiffi2

p.

For quantum mechanics, our ability to steer isonly limited by the no-signaling principle, but weencounter strong uncertainty relations. On theother hand, for a local hidden variable theory, wecan also have perfect steering, but only with anuncertainty relation given byzxs,a

¼ 3=4, and thuswe also have the degree of nonlocality given byPgamemax ¼ 3=4. This value of nonlocality is the

same as deterministic classical mechanics, wherewe have no uncertainty relations on the full set ofdeterministic states but our abilities to steer tothem are severely limited. In the other directionare theories that are maximally nonlocal, yet re-

main no-signaling (3). These have no uncertainty(i.e., zxs,a¼1), but unlike in the classical world,we still have perfect steering, so they win theCHSH game with probability 1.

For any physical theory, we can thus considerthe strength of nonlocal correlations to be a tradeoffbetween two aspects: steerability and uncertainty. Inturn, the strength of nonlocality can determine thestrength of uncertainty in measurements. However,it does not determine the strength of complemen-tarity of measurements (12). The concepts of un-certainty and complementarity are usually linked,but we find that it is possible to have theories thatare just as nonlocal and uncertain as quantum me-chanics, but that have less complementarity. Thissuggests a rich structure relating these quantities.

References and Notes1. W. Heisenberg, Z. Phys. 43, 172 (1927).2. J. S. Bell, Physics 1, 195 (1965).3. S. Popescu, D. Rohrlich, Found. Phys. 24, 379 (1994).4. G. V. Steeg, S. Wehner, QIC 9, 801 (2009).5. M. Pawłowski et al., Nature 461, 1101 (2009).6. W. van Dam, http://arxiv.org/abs/quant-ph/0501159 (2005).7. H. Barnum, S. Beigi, S. Boixo, M. B. Elliott, S. Wehner,

Phys. Rev. Lett. 104, 140401 (2010).8. D. Deutsch, Phys. Rev. Lett. 50, 631 (1983).9. I. Białynicki-Birula, J. Mycielski, Commun. Math. Phys.

44, 129 (1975).10. S. Wehner, A. Winter, N. J. Phys. 12, 025009 (2010).11. Without loss of generality, we assume that each

measurement has the same set of possible outcomes,because we may simply add additional outcomes thatnever occur.

12. See supporting material on Science Online.13. S. Wehner, M. Christandl, A. C. Doherty, Phys. Rev. A

78, 062112 (2008).14. J. Clauser, M. Horne, A. Shimony, R. Holt, Phys. Rev.

Lett. 23, 880 (1969).15. B. Cirel’son (Tsirelson), Lett. Math. Phys. 4, 93 (1980).16. E. Schrödinger, M. Born, Proc. Camb. Philos. Soc. 31,

555 (1935).17. E. Schrödinger, P. A. M. Dirac, Proc. Camb. Philos. Soc.

32, 446 (1936).18. H. M. Wiseman, S. J. Jones, A. C. Doherty, Phys. Rev.

Lett. 98, 140402 (2007).19. S. J. Jones, H. M. Wiseman, A. C. Doherty, Phys. Rev. A

76, 052116 (2007).20. H. Barnum, C. Gaebler, A. Wilce, http://arxiv.org/abs/

0912.5532 (2009).21. M. M. Wolf, D. Perez-Garcia, C. Fernandez, Phys. Rev.

Lett. 103, 230402 (2009).22. A. Peres, Quantum Theory: Concepts and Methods

(Kluwer Academic, Dordrecht, Netherlands, 1993).23. In an XOR game, the answers a ∈ {0,1} and

b ∈ {0,1} of Alice and Bob, respectively, are singlebits; whether Alice and Bob win or not depends onlyon the XOR of the answers a ⊕ b = a + b mod 2.

24. The retrieval game used was discovered in collaborationwith A. Doherty. Supported by the Royal Society ( J.O.)and by NSF grants PHY-04056720 and PHY-0803371,the National Research Foundation of Singapore,and the Singapore Ministry of Education (S.W.). Partof this work was done while J.O. was visiting CaliforniaInstitute of Technology, and while J.O. and S.W. werevisiting the Kavli Institute for Theoretical Physics(Santa Barbara, CA), which is funded by NSF grantPHY-0551164.

Supporting Online Materialwww.sciencemag.org/cgi/content/full/330/6007/1072/DC1SOM TextFigs. S1 to S4References

10 May 2010; accepted 1 October 201010.1126/science.1192065

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(6007), 1072-1074. [doi: 10.1126/science.1192065]330Science Jonathan Oppenheim and Stephanie Wehner (November 18, 2010) Quantum MechanicsThe Uncertainty Principle Determines the Nonlocality of

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   uncertainty holds true for other physical theories besides quantum mechanics.actually determined by the uncertainty principle. The unexpected connection between nonlocality and

(p. 1072) show that the degree of nonlocality in quantum mechanics isOppenheim and WehnerHeisenberg's uncertainty principle, which is one of the fundamental aspects of quantum mechanics. measuring the distant system. This form of ''action-at-a-distance,'' or nonlocality, seemingly contradicts

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