n Comms 1416

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nATuRE CommunICATIons | 2:411 | DoI: 10.1038/c1416 | www.atre.c/atrecicati

© 2011 Macmillan Publishers Limited. All rights reserved.

Received 17 Ja 2011 | Accepted 30 J 2011 | Pblihed 2 Ag 2011 DOI: 10.1038/ncomms1416

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1 Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario, N2L 2Y5 Canada. 2 Institute for Theoretical Physics, ETH Zurich,

Zurich 8093, Switzerland. Correspondence and requests for materials should be addressed to R.C. (email: [email protected]).

n extei qat thery ca have

iprved predictive pwer

Rger Clbeck1 & Reat Reer2

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nATuRE CommunICATIons | DoI: 10.1038/c1416



Given a system and a set o initial conditions, classical mecha-nics allows us to calculate the uture evolution to arbitrary precision. Any uncertainty we might have at a given time

is caused by a lack o knowledge about the conguration. In quan-tum theory, on the other hand, certain properties—or exam-ple, position and momentum—cannot both be known precisely.Furthermore, i a quantity without a dened value is measured,quantum theory prescribes only the probabilities with whichthe various outcomes occur, and is silent about the outcomes

themselves.Tis raises the important question o whether the outcomes

could be better predicted within a theory beyond quantum mecha-nics1. An intuitive step towards its answer is to consider appendinglocal hidden variables to the theory 2. Tese are classical variablesthat allow us to determine the experimental outcomes (see lateror a precise denition). Here we ask a new, more general question:is there any extension o quantum theory (not necessarily takingthe orm o hidden variables) that would convey any additionalinormation about the outcomes o uture measurements?

We proceed by giving an illustrative example. Consider aparticle heading towards a measurement device that has a numbero possible settings, denoted by a parameter, A, corresponding tothe dierent measurements that can be chosen by the experimenter.

Te measurement generates a result, denoted X . For concreteness,one could imagine a spin-1/2 particle incident on a Stern–Gerlachapparatus. Each choice o measurement corresponds to a particu-lar orientation o the device, and the outcome is assigned depend-ing on which way the beam is deected. Within quantum theory,a description o the quantum state o the particle and o themeasurement apparatus allows us to calculate the distribution,P X | A, o the outcome, X , or each measurement choice, A. Anotherexample is described in Figure 1.

In this work, we consider the possibility that there exists urther,yet to be discovered, inormation that allows the outcome X to bebetter predicted. We do not place any restrictions on how this inor-mation is maniest, nor do we demand that it allows the outcomes tobe calculated precisely. In particular, it could be that the additional

inormation gives rise to a more accurate distribution over the out-comes. For example, in an experiment or which quantum theory predicts a uniorm distribution over the outcomes, X , there couldbe extra inormation that allows us to calculate a value, X ′, suchthat X = X ′ with probability 3/4 (in the model proposed by Leggett3,or instance, the local hidden variables provide inormation o thistype). More generally, we allow or the possibility o an extendedtheory that provides non-classical inormation. For example, itcould comprise a ‘hidden quantum system’, which, i measured inthe correct way, gives a value correlated to X .

ResultsAssumptions. o ormulate our main claim about the non-extendibility o quantum theory, we introduce a ramework within

which any arbitrary extra inormation provided by an extension o the current theory can be considered. In the ollowing, we explainthis ramework on an inormal level (see Supplementary Methodsor a ormal treatment).

Te crucial eature o our approach is that it is operational, inthe sense that we only reer to directly observable objects (such asthe outcome o an experiment), but do not assume anything aboutthe underlying structure o the theory. Note that the outcome, X , o a measurement is usually observed at a certain point in spacetime.Te coordinates o this point (with respect to a xed reerence sys-tem) can be determined operationally using clocks and measuringrods. Analogously, the measurement setting A needs to be availableat a certain spacetime point (beore the start o the experiment). omodel this, we introduce the notion o a spacetime random variable(SV), which is simply a random variable together with spacetime

coordinates (t , r 1, r 2, r 3). Operationally, a SV can be interpreted asa value that is accessible at a given spacetime point (t , r 1, r 2, r 3). Wenow model a measurement process as one that takes an input, A, toan output, X , where both X and A are SVs.

Our result is based on the assumption that measurement set-tings can be chosen reely (which we call assumption FR). We notethat this assumption is common in physics, but oen only madeimplicitly. It is, or example, a crucial ingredient in Bell’s theorem(see re. 4). Formulated in our ramework, assumption FR is thatthe input, A, o a measurement process can be chosen such that it is

uncorrelated with certain other SVs, namely all those whose coor-dinates lie outside the uture lightcone o the coordinates o A. Wenote that this reerence to a lightcone is only used to identiy a seto SVs and does not involve any assumptions about relativity theory (see the Supplementary Inormation). However, the motivation orassumption FR is that, when interpreted within the usual relativis-tic spacetime structure, it is equivalent to demanding that A can bechosen such that it is uncorrelated with any pre-existing values in any reerence rame. Tat said, the lack o correlation between the relevantSVs could be justied in other ways, or example, by using a notion o ‘eective reedom’ (discussed in re. 4).

We also remark that Assumption FR is consistent with a notiono relativistic causality in which an event B cannot be the cause o A i there exists a reerence rame in which A occurs beore B. In act,

our criterion or A to be a ree choice is satised whenever anythingcorrelated to A could potentially have been caused by A. However,in an alternative world with a universal (rame-independent) time,one might reject assumption FR and replace it with somethingweaker, or example, that A is ree, i it is uncorrelated with any-thing in the past with respect to this universal time. Nevertheless,since experimental observations indicate the existence o relativisticspacetime, we use a notion o ree choice consistent with this.

We additionally assume that the present quantum theory iscorrect (we call this assumption QM ). Tis assumption is naturalbecause we are asking whether quantum theory can be extended.In act, we only require that two specic aspects o quantum theory hold, and so split assumption QM into two parts. On an inormallevel, the rst is that measurement outcomes obey quantum statis-tics, and the second is that all processes within quantum theory can

a

b

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Figure 1 | Illustration o the scenario. A eareet i carried t a

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be considered as unitary evolutions, i one takes into account theenvironment (see the Supplementary Inormation or more details).We remark that the second part o this assumption need only holdor microscopic processes on short timescales and does not precludesubsequent waveunction collapse.

Main fndings. Consider a measurement that depends on a setting A and produces an output X . According to quantum theory, we canassociate a quantum state and measurement operators with this

process rom which we can compute the distribution P X | A.We ask whether there could exist an extension o quantum the-

ory that provides us with urther inormation (which we denote by Ξ) that is useul to predict the outcome. o keep the description o the inormation, Ξ, as general as possible, we do not assume that itis encoded in a classical system, but, instead, characterize it by howit behaves when observed. (Formally, we model access to Ξ analo-gously to the measurement o a quantum system, that is, as a processwhich takes an input SV and produces an output SV). We demandthat Ξ can be accessed at any time (similarly to classical or quantuminormation held in a storage device) and that it is static, that is, itsbehaviour does not depend on where or when it is observed.

Our main result is that we answer the above question in thenegative, that is, we show that, using assumptions FR and QM , the

distribution P X | A is the most accurate description o the outcomes.More precisely, or any xed (pure) state o the system, the chosenmeasurement setting, A, is the only non-trivial inormation about X , and any extra inormation, Ξ, provided by an extended theory isirrelevant. We express this through the Markov chain condition

X A↔ ↔ Ξ.

Tis condition expresses mathematically that the distribution o X given A and Ξ is the same as the distribution o X given only A5.Hence, access to Ξ does not decrease our uncertainty about X , andthere is no better way to predict measurement outcomes than by using quantum theory.

In the Methods, we sketch the proo o this (the ull proo isdeerred to the Supplementary Inormation).

DiscussionWe now discuss experimental aspects related to our result. Note thatat the ormal level, we present a theorem about certain denedconcepts on the basis o certain assumptions, hence what remainsis to connect our denitions to observations in the real world, andexperimentally conrm the assumptions, where possible. Assump-tion FR reers to the ability to make ree choices and—while we cannever rule out that the universe is deterministic and that ree willis an illusion—this is in principle alsiable, or example, by adevice capable o guessing an experimentalist’s choices beore they are made. (See also re. 6 where the possibility o weakening thisassumption is discussed.)

Te validity o assumption QM could be argued or on the basiso experimental tests o quantum theory. However, the existence o the particular correlations we use in the second part o our proo is quantum-theory independent, so worth establishing separately.Because o experimental inefciencies, these correlations cannotbe veried to arbitrary precision. Figure 2 bounds our ability toexperimentally establish (1) depending on the quality o the setupused (characterized here by the visibility). For more details, seethe Methods.

We proceed by discussing previous work on extensions o quan-tum theory. o the best o our knowledge, all such extensions thathave been excluded to date can also be excluded using our result.

Te question asked by Einstein, Podolsky and Rosen1 waswhether quantum mechanics could be considered complete. Tey appealed to intuition to argue that an extended theory should exist

(1)(1)

and one might then have hoped or a deterministic completion,that is, one that would uniquely determine the measurement out-comes—contrast this with our (more general) notion, where theextended theory may only give partial inormation. Bell2 amously showed that a deterministic completion is not possible when thetheory is supplemented by local hidden variables. (o relate thisback to our result, this corresponds to the special case where the

urther inormation, Ξ, is a classical value specied by the local hid-den variables. A short discussion on the term ‘local’ can be ound inthe Supplementary Inormation.) Recently, a conclusion7 similar toBell’s has been reached using the Kochen–Specker theorem8. Teseresults have been extended to arbitrary (that is, not necessarily local)hidden variables9,10, under the assumption o relativistic covariance(see also re. 11, as well as re. 12, where a condition slightly weakerthan locality is used to derive a theorem similar to Bell’s).

Te aorementioned papers le open the question o whetherthere could exist an extended theory that provides urther inor-mation about the outcomes without determining them completely.(Note that, in his later works, Bell uses denitions that potentially allow probabilistic models13. However, as explained in the Supple-mentary Inormation, non-deterministic models are not compatible

with Bell’s other assumptions.) In the case that the urther inorma-tion takes the orm o local hidden variables, an answer to the abovequestion can be ound in res 3,14,15, and the strongest result is thatany local hidden variables are necessarily uncorrelated with the out-comes o measurements on Bell states15. (We remark that the modelin re. 3 also included non-local hidden variables. However, we havenot reerred to these in this paragraph, because, as mentioned belowin the context o de Broglie–Bohm theory, the presence o non-localhidden variables contradicts assumption FR).

In the present work, we have taken this idea urther and excludedthe possibility that any extension o quantum theory (not necessarily in the orm o local hidden variables) can help predict the outcomeso any measurement on any quantum state. In this sense, we showthe ollowing: under the assumption that measurement settings canbe chosen reely, quantum theory really is complete.

0.90 0.92 0.94 0.96 0.98 1.000.0

0.1

0.2

0.3

0.4

0.5

0.6

0

10

20

30

40

50

Visibility

I N N

Figure 2 | Achievable values o I N depending on the experimental

visibility. Thi fgre relate t the eareet etp ed r tetig

the accracy apti QM a decribed i the methd. The etp

ivlve tw partie ad i paraeterized by the ber pible

eareet chice available t each party, N. The plt give the

ii In achievable depedig the viibility (red lie), which

deterie the allet pper bd the variatial ditace r

the perect markv chai cditi (1) that cld be btaied with that

viibility (ee eqati (8)). It al hw the ptial vale N which

achieve thi (ble lie). Fr cpari, the vale achievable ig N= 2,

which crrepd t the CHsH eareet26 (yellw lie), ad the

cae N= 8, which i ptial r viibility 0.98 (gree lie), are hw.

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We remark that several other attempts to extend quantum theory have been presented in the literature, the de Broglie–Bohm theory 16,17 being a prominent example (this model recreates the quantum cor-relations in a deterministic way but uses non-local hidden variables,see re. 18 or a summary). Our result implies that such theoriesnecessarily come at the expense o violating assumption FR.

Another way to generate candidate extended theories is via modelsthat simulate quantum correlations. We discuss the implications o ourresult in light o such models in the Supplementary Inormation. In

addition, we remark that a claim in the same spirit as ours has recently been obtained on the basis o the assumption o non-contextuality 19.

Randomness is central to quantum theory, and with it comes arange o philosophical implications. In this article, we have shownthat the randomness is inherent: any attempt to better explain theoutcomes o quantum measurements is destined to ail. Not only is the universe not deterministic, but quantum theory provides theultimate bound on how unpredictable it is. Aside rom these un-damental implications, there are also practical ones. In quantumcryptography, or example, the unpredictability o measurementoutcomes can be quantied and used to restrict the knowledge o anadversary. Most security proos implicitly assume that quantum the-ory cannot be extended (although there are exceptions, the rst o which was given in re. 20). However, in this work, we show that this

ollows, i the theory is correct.

MethodsSketch o the proo . Our main result is the ollowing theorem whose proo wesketch here (see the Supplementary Inormation or the ormal treatment).

Teorem 1: For any quantum measurement with input SV A and output SV X and or any additional inormation,Ξ, under assumptions QM and FR, the Markovchain condition (1) holds.

Te proo is divided into three parts. Te rst two are related to a Bell-type set-ting, involving measurements on a maximally entangled state. In Part I, we show thatassumption FR necessarily enorces that Ξ is non-signalling (in the sense denedbelow). In Part II, we show that or a particular set o bipartite correlations, i Ξ isnon-signalling, it cannot be o use to predict the outcomes. Tese correlations occurin quantum theory (c. the rst part o assumption QM ) when measuring a maximally entangled state, and hence we conclude that no Ξ can help predict the outcomes o measurements on one-hal o such a state. Finally, in Part III, we use the second part o

assumption QM to argue that this conclusion also applies to all measurements on anarbitrary (pure) quantum state. ogether, these establish our claim.Te bipartite scenario used or the rst two parts o the proo involves two

quantum measurements, with inputs A and B and respective outputs X and Y .Te setup is such that the two measurements are spacelike separated in the sensethat the coordinates o A are spacelike separated with the coordinates o Y , and,likewise, those o B are spacelike separated with those o X .

As mentioned in the main text, we model the inormation provided by theextended theory, Ξ, by its behaviour under observation. We introduce a SV, C,which can be thought o as the choice o what to observe, and another SV, Z , whichrepresents the outcome o this observation (all the SVs used in the setup are shownin Supplementary Fig. S1). In terms o these variables, our main result, equation (1),can be restated that or all values o a, c and x, we have

P P Z acx Z ac| | .=

(Note that we use lower case to denote specic values o the corresponding upper

case SVs).

Proo part I. Te entire setup described above (including the extra inormationΞ, accessed by choosing an observable, C, and obtaining an outcome, Z ) gives riseto a joint distribution P XYZ | ABC. Te purpose o this part o the proo is to show thatassumption FR implies that P XYZ | ABC must satisy particular constraints, called non-signalling constraints, which characterize situations where operations on dierentisolated systems cannot aect each other. Formally, these are

P P YZ ABC YZ BC | |=

P P XZ ABC XZ AC | |=

P P XY ABC XY AB| |=

We remark that the obser vation that the assumption o ree choice gives rise tocertain non-signalling constraints has been made already in re. 11, and a similar

(2)(2)

(3)(3)

(4)(4)

(5)(5)

argument has been presented by Gisin9 and Blood10. (Note that the arguments inres 9,10 implicitly assume that measurements can be chosen reely).

Assumption FR allows us to make A a ree choice and hence we have

P P A BCYZ A| =

(the setup is such that the measurements specied by A and B are spacelikeseparated and, urthermore, Ξ is static; so, we can consider the case where itsobservation is also spacelike separated rom the measurements specied by A andB). Furthermore, using the denition o conditional probability (P Q|R: = P QR/P R), wecan write

P P P P P YZA BC YZ BC A BCYZ A YZ BC | | | | ,= × = ×

where we inserted (6) to obtain the second equality. Similarly, we have

P P P P P YZA BC A BC YZ ABC A YZ ABC | | | | .= × = ×

Comparing these two expressions or P YZA|BC yields the desired non-signallingcondition (3). By a similar argument, the other non-signalling conditions can beinerred rom assumption FR.

Proo part II. For the second part o the proo, we consider the distribution P XY | AB resulting rom certain appropriately chosen measurements on a maximally entan-gled state. We show that any enlargement o this distribution (through a systemΞ that is accessed in a process with input SV C and output SV Z ) to a distributionP XYZ | ABC that satises the above non-signalling conditions is necessarily trivial inthe sense that Ξ is uncorrelated to the rest. For this, we draw on ideas rom non-

signalling cryptography 20

, which are related to the idea o basing security on the violation o Bell inequalities21. echnically, we employ a lemma (see Lemma 1in the Supplementary Inormation), whose proo is based on chained Bellinequalities22,23 and generalizes results o res 15,24.

Consider any bipartite measurement with inputs A∈{0, 2…, 2N − 2} andB∈{1, 3…, 2N − 1}, or some positive integer N , and binary outcomes, X and Y . Tecorrelations o the outcomes can be quantied by

I P X Y A B N P X Y A a B bN

a b

a b

: ( | , ) ( | , ).,

| |

= = = = − + ≠ = =

− =

∑0 2 1

1

Our lemma then asserts that, under the non-signalling conditions derived inPart I,

D P P I Z abcx Z abc N ( , )| | ≤

or all a, b, c and x, where D is the variational distance, dened by D P Q P z Q z Z Z z Z Z ( , ) : | ( ) ( ) |= −

12 Σ . Te variational distance has the ollowing

operational interpretation: i two distributions have variational distance atmost δ , then the probability that we ever notice a dierence between them isat most δ .

Te argument up to here is ormally independent o quantum theory. However,as we describe below (see the Experimental verication section), or any xedorthogonal rank-one measurement on a two-level subsystem, one can construct2N − 1 other measurements such that, according to quantum theory, applying thesemeasurements to maximally entangled states leads to correlations that satisy I N

N ∝1/ . It ollows that, in the limit o large N , an arbitrarily small bound on

D(P Z |abcx, P Z |abc) can be obtained. We thus conclude that P Z |abcx= P Z |abc, which, by thenon-signalling condition (4), also implies (2). We have thereore shown that therelation (1) holds or the outcome X o any orthogonal rank-one measurement ona system that is maximally entangled with another one (our claim can be readily

extended to systems o dimension 2t

or positive integer t by applying the resultto t two-level systems).We also remark that Markov chains are reversible, that is, P Z |abcx=P Z |abc implies

P X |abcz =P X |abc, which together with the non-signalling conditions gives P X |abcz = P X |a.Tis establishes that, or any choices o B and C, learning Z does not allow animprovement on the quantum predictions, P X |a.

Proo part III. o complete our claim, it remains to show that the Markov chaincondition (1) holds or measurements on arbitrary states (not only or those on onepart o a maximally entangled state shared between two sites). Te proo o thisproceeds in two steps. Te rst is to append an extra measurement with outcome X ′, chosen such that the pair ( X , X ′) is uniormly distributed. In the second step, wesplit the measurement into two conceptually distinct parts, where, in the rst, themeasurement apparatus becomes entangled with the system to be measured (and,possibly the environment) and, in the second, this entangled state is measuredgiving outcomes ( X , X ′). As these outcomes are uniormly distributed, the statebeore the measurement can be considered maximally entangled, so that (1) holds

(6)(6)

(7)(7)

(8)(8)

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with X replaced by ( X , X ′). Tis implies (1) and hence completes the proo o Teorem 1.

Experimental verifcation. As explained above, the validity o parts o assumptionQM can be established by a direct experiment. In particular, to veriy the existenceo the correlations required or Part II o the proo, that is, those with small I N ,one should generate a large number (much larger than N ) o maximally entangledparticles and distribute them between the measurement devices. At spacelike sepa-ration, a two-level subsystem (or example, a spin degree o reedom) should thenbe measured, the measurement being picked at random rom those specied below,and the results recorded. Tis is repeated or all o the particles. Te measurement

choices and results are then collected and used to estimate the terms in I N usingstandard statistical techniques.For an arbitrary orthogonal basis {|0⟩,|1⟩}, the required measurements can be

constructed in the ollowing way. Recall that the choice o measurement on oneside takes values A∈{0, 2…, 2N − 2} and similarly, B∈{1, 3…, 2N − 1}. We dene a

set o angles q p j

N j=

2

and states

{| ,| } cos | in | , sin | cos |q q q q q q

+ −⟩ ⟩ = ⟩ + ⟩ ⟩ − ⟩

j j j j j j

20

21

20

21s

.

Te required measurement operators are then Ea a a± ± ±= ⟩⟨| |q q and F

b b b± ± ±= ⟩⟨| |q q .

Although quantum theory predicts that arbitrarily small values o I N can beobtained or large N , due to imperections and errors in the devices, it will not bepossible to experimentally achieve this. In re. 25, a discussion o the achievable

values o I N with imperect visibilities was given. For visibilities less than 1, it is

not optimal to take N as large as possible to minimize the observed I N . Tus, to getincreasingly small bounds on the variational distance in (8), one must increase theexperimentally obtained visibilities as well as the number o measurement settings(see Fig. 2).

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description o physical reality be considered complete? Phys. Rev. 47, 777–780(1935).

2. Bell, J. S. On the Einstein-Podolsky-Rosen paradox: in Speakable and Unspeakable in Quantum Mechanics, chap. 2 (Cambridge University Press,1987).

3. Leggett, A. J. Nonlocal hidden-variable theories and quantum mechanics: anincompatibility theorem. Foundations of Physics 33, 1469–1493 (2003).

4. Bell, J. S. Free variables and local causality: in Speakable and Unspeakable inQuantum Mechanics, chap. 12 (Cambridge University Press, 1987).

5. Cover, . M. & Tomas, J. A. Elements of Information Teory, 2nd edn, section2.8. (John Wiley and Sons, 2006).

6. Colbeck, R. & Renner, R. Free randomness amplication. e-printarXiv:1105.3195 (2011).

7. Conway, J. & Kochen, S. Te ree will theorem. Foundations of Physics 36, 1441–1473 (2006).

8. Kochen, S. & Specker, E. P. Te problem o hidden variables in quantummechanics. Journal of Mathematics and Mechanics 17, 59–87 (1967).

9. Gisin, N. On the impossibility o covariant nonlocal ‘hidden’ variables inquantum physics. e-print arXiv:1002.1390 (2010).

10. Blood, C. Derivation o Bell’s locality condition rom the relativity o simultaneity. e-print arXiv:1005.1656 (2010).

11. Colbeck, R. & Renner, R. Dening the local part o a hidden variable model: acomment. e-print arXiv:0907.4967 (2009).

12. resser, C. Bell’s theory with no locality assumption. Te European Physical Journal D 58, 385–396 (2010).

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13. Bell, J. S. La nouvelle cuisine: in Speakable and Unspeakable in Quantum Mechanics, 2nd edn, chap. 24 (Cambridge University Press, 2004).

14. Branciard, C. et al. esting quantum correlations versus single-particleproperties within Leggett’s model and beyond. Nature Physics 4, 681–685(2008).

15. Colbeck, R. & Renner, R. Hidden variable models or quantum theory cannothave any local part. Phys. Rev. Lett. 101, 050403 (2008).

16. de Broglie, L. La mécanique ondulatoire et la structure atomique de la matièreet du rayonnement. Journal de Physique, Serie VI VIII, 225–241 (1927).

17. Bohm, D. A suggested interpretation o the quantum theory in terms o ‘hidden’ variables. I. Phys. Rev. 85, 166–179 (1952).

18. Bell, J. S. de Broglie-Bohm, delayed choice double-slit experiment, and density matrix: in Speakable and Unspeakable in Quantum Mechanics, chap. 14(Cambridge University Press, 1987).

19. Chen, Z. & Montina, A. Measurement contextuality is implied by macroscopicrealism. e-print arXiv:1012.2122 (2010).

20. Barrett, J., Hardy, L. & Kent, A. No signalling and quantum key distribution.Phys. Rev. Lett. 95, 010503 (2005).

21. Ekert, A. K. Quantum cryptography based on Bell’s theorem. Phys. Rev. Lett. 67, 661–663 (1991).

22. Pearle, P. M. Hidden-variable example based upon data rejection. Physical Review D 2, 1418–1425 (1970).

23. Braunstein, S. L. & Caves, C. M. Wringing out better Bell inequalities. Annals of Physics 202, 22–56 (1990).

24. Barrett, J., Kent, A. & Pironio, S. Maximally non-local and monogamousquantum correlations. Phys. Rev. Lett. 97, 170409 (2006).

25. Suarez, A. Why aren’t quantum correlations maximally nonlocal? Biasedlocal randomness as essential eature o quantum mechanics. e-print

arXiv:0902.2451 (2009).26. Clauser, J. F., Horne, M. A., Shimony, A. & Holt, R. A. Proposed experiment totest local hidden-variable theories. Phys. Rev. Lett. 23, 880–884 (1969).

AcknowledgementsWe are grateul to Časlav Brukner, Adán Cabello, Jerry Finkelstein, Jürg Fröhlich,Nicolas Gisin, Gian Michele Gra, Adrian Kent, Jan-Åke Larsson, Lluís Masanes, NicolasMenicucci, Ognyan Oreshkov, Steano Pironio, Rainer Plaga, Soa Wechsler and AntonZeilinger or discussions on this work and thank Lídia del Rio or the illustrations. Tiswork was supported by the Swiss National Science Foundation (grant Nos. 200021-119868 and 200020-135048) and the European Research Council (grant No. 258932).Research at Perimeter Institute is supported by the Government o Canada throughIndustry Canada and by the Province o Ontario through the Ministry o Research andInnovation.

Author contributions

Both authors contributed equally to all areas o this work.

Additional inormationSupplementary Inormation accompanies this paper at http://www.nature.com/naturecommunications

Competing fnancial interests: Te authors declare no competing nancial interests.

Reprints and permission inormation is available online at http://npg.nature.com/reprintsandpermissions/

How to cite this article: Colbeck, R. & Renner, R. No extension o quantum theory canhave improved predictive power. Nat. Commun. 2:411 doi: 10.1038/ncomms1416 (2011).

License: Tis work is licensed under a Creative Commons Attribution-NonCommercial-Share Alike 3.0 Unported License. o view a copy o this license, visit http://creativecommons.org/licenses/by-nc-sa/3.0/

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