Maximally Non-Local and Monogamous Quantum Correlations 0605182

7/27/2019 Maximally Non-Local and Monogamous Quantum Correlations 0605182

1/5

arXiv:quant-ph/0605182v1

21May2006

Maximally Non-Local and Monogamous Quantum Correlations

Jonathan Barrett,1, Adrian Kent,2, and Stefano Pironio3,

1Perimeter Institute for Theoretical Physics, 31 Caroline Street N, Waterloo, Ontario N2L 2Y5, Canada2Centre for Quantum Computation, DAMTP, Centre for Mathematical Sciences,

University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, U.K.3ICFO-Institut de Ciencies Fotoniques, Mediterranean Technology Park, 08860 Castelldefels (Barcelona), Spain

Institute for Quantum Information, California Institute of Technology, Pasadena, CA 91125, U.S.A.

(Dated: May 19, 2006)We introduce a version of the chained Bell inequality for an arbitrary number of measurement

outcomes, and use it to give a simple proof that the maximally entangled state of two d dimensionalquantum systems has no local component. That is, if we write its quantum correlations as amixture of local correlations and general (not necessarily quantum) correlations, the coefficient ofthe local correlations must be zero. This suggests an experimental programme to obtain as good anupper bound as possible on the fraction of local states, and provides a lower bound on the amount ofclassical communication needed to simulate a maximally entangled state in dd dimensions. We alsoprove that the quantum correlations violating the inequality are monogamous among non-signallingcorrelations, and hence can be used for quantum key distribution secure against post-quantum (butnon-signalling) eavesdroppers.

Quantum theory predicts that measurements on sep-

arated entangled systems will produce outcome correla-tions that, in Bells terminology [1], are not locally causalor, in what has become standard terminology, are non-local. In particular, if a Bell inequality is violated thenone cannot consistently assume that the outcomes ofmeasurements on each system are predetermined and in-dependent of the measurements carried out on the othersystem(s). Violation of Bell inequalities has been con-firmed in numerous experiments [2].

Violation of Bell inequalities not only tells us some-thing fundamental about Nature, but also has practi-cal applications. For example, the nonlocality of quan-

tum correlations allows communication complexity prob-lems to be solved using an amount of communicationthat is smaller than is possible classically [3]. Barrett-Hardy-Kent (BHK) recently showed that testing partic-ular nonlocal quantum correlations allows two parties todistribute a secret key securely, in such a way that thesecurity is guaranteed by the no-signalling principle alone[4] (i.e. without relying on the validity of quantum the-ory).

In this paper, we extend the chained Bell inequalityto an inequality with an arbitrary number of measure-ment outcomes, which can thus be applied to states inarbitrary dimensions. In the limit of a large number

of measurement settings, quantum mechanics predictscorrelations for a maximally entangled bipartite statethat resemble those of the tripartite Greenberger-Horne-Zeilinger (GHZ) state in that, with probability tendingto one, the predictions of any local hidden variable modelcontradict those of quantum mechanics for at least onepair of measurements. We use this to give a construc-tive proof of a result originally due to Elitzur, Popescuand Rohrlich (EPR) [5]: if the quantum correlations ofa maximally entangled state of two qubits are written as

a convex combination of local and nonlocal correlations,

then the local fraction must be zero. Our proof extendsEPRs result to maximally entangled states in any dimen-sion and also removes the need for a technical assumptionrequired for EPRs original proof [6]. Moreover, becauseour proof is constructive, it motivates an experimentalprogramme to establish the best possible upper boundon the fraction of local states in a maximally entangledstate. More generally, our proof method works, and mo-tivates experimental tests, for any example of GHZ-typecorrelations.

Next, we give a rigorous proof of the monogamy ofthe correlations obtained from a d d-dimensional max-imally entangled state. Here, monogamy means that athird party cannot get any information about the mea-surement outcomes, so long as a no-signalling conditionholds, even if quantum theory is incorrect. This propertyhas a particular significance in the context of secret keydistribution, and our results here generalise those of BHK[4], which proved monogamy in the 22-dimensional casein order to demonstrate the security of BHKs schemeagainst general non-signalling eavesdroppers.

Finally, as a corollary, we derive a lower bound onthe classical communication needed to simulate measure-ments on a maximally entangled state in dd dimensions.

Some chained Bell inequalities. Consider a standard

Bell-type experiment: two parties, Alice and Bob, sharea joint system in an entangled quantum state, and per-form measurements on their local subsystems. Eachparty may choose one out of N different measurements,and each measurement Ak of Alice and Bl of Bob(k, l = 1, . . . , N ) may have d possible outcomes: Ak, Bl =0, . . . , d1. Quantum theory predicts the joint probabil-ities PQM(Ak = a, Bl = b) that Alices and Bobs mea-surements, Ak and Bl, have respective outcomes a and b.The correlations predicted by quantum theory are local
http://arxiv.org/abs/quant-ph/0605182v1http://arxiv.org/abs/quant-ph/0605182v1http://arxiv.org/abs/quant-ph/0605182v1http://arxiv.org/abs/quant-ph/0605182v1http://arxiv.org/abs/quant-ph/0605182v1http://arxiv.org/abs/quant-ph/0605182v1http://arxiv.org/abs/quant-ph/0605182v1http://arxiv.org/abs/quant-ph/0605182v1http://arxiv.org/abs/quant-ph/0605182v1http://arxiv.org/abs/quant-ph/0605182v1http://arxiv.org/abs/quant-ph/0605182v1http://arxiv.org/abs/quant-ph/0605182v1http://arxiv.org/abs/quant-ph/0605182v1http://arxiv.org/abs/quant-ph/0605182v1http://arxiv.org/abs/quant-ph/0605182v1http://arxiv.org/abs/quant-ph/0605182v1http://arxiv.org/abs/quant-ph/0605182v1http://arxiv.org/abs/quant-ph/0605182v1http://arxiv.org/abs/quant-ph/0605182v1http://arxiv.org/abs/quant-ph/0605182v1http://arxiv.org/abs/quant-ph/0605182v1http://arxiv.org/abs/quant-ph/0605182v1http://arxiv.org/abs/quant-ph/0605182v1http://arxiv.org/abs/quant-ph/0605182v1http://arxiv.org/abs/quant-ph/0605182v1http://arxiv.org/abs/quant-ph/0605182v1http://arxiv.org/abs/quant-ph/0605182v1http://arxiv.org/abs/quant-ph/0605182v1http://arxiv.org/abs/quant-ph/0605182v1http://arxiv.org/abs/quant-ph/0605182v1http://arxiv.org/abs/quant-ph/0605182v1http://arxiv.org/abs/quant-ph/0605182v1http://arxiv.org/abs/quant-ph/0605182v1http://arxiv.org/abs/quant-ph/0605182v1


2/5

2

if and only if they can be written in the form

PQM(Ak = a, Bl = b) =

qP(Ak = a)P(Bl = b) ,

with 0 q 1 and

q = 1. Without loss of gen-erality, we can assume that the terms P(Ak = a) andP(Bl = b) are deterministic, that is, that they only takethe values 0 or 1 [7]. The correlations are thus local if andonly if they can be reproduced by a mixture of hiddenstates assigning definite values to each measurement. Vi-olation of a Bell inequality implies that the correlationscannot be written in this form.

In general one can write the correlations as

PQM(Ak = a, Bl = b) = p PL(Ak = a, Bl = b)

+ (1 p) PNL(Ak = a, Bl = b) , (1)where PL(Ak = a, Bl = b) and PNL(Ak = a, Bl = b)represent local and non-local joint distributions, respec-tively, and p and (1 p) their corresponding weights inthe mixture, with 0 p 1. Suppose now that wehave a set of quantum correlations which satisfy somerelation with certainty, and we can show that any localmodel must fail to satisfy the relation at least some ofthe time. It follows that the correlations cannot be writ-ten in the form of Eq. (1) except with p = 0. (Note thatthis is true even if the term PNL(Ak = a, Bl = b) is al-lowed to describe signalling correlations.) A particularlywell-known example of such correlations was that pro-duced by GHZ [8] and simplified by Mermin [9]; hencewe refer to correlations with these properties as GHZ-type correlations. Bipartite examples have been given byHeywood-Redhead [10] and Cabello [11]. A set of quan-

tum correlations is of GHZ-type if and only if it has anassociated Bell inequality which is violated right up tothe algebraic limit of the expression.

We now derive a Bell inequality and show that in thelimit of a large number of measurement settings, thequantum correlations from a maximally entangled stateviolate the inequality up to the algebraic limit. Thus thequantum correlations tend to GHZ-type as the numberof measurement settings becomes large. Hence, we show,maximally entangled states in any dimension have zerolocal component.

Consider first the case where Alice and Bob each havea choice between two different measurements (N = 2).

Local correlations satisfy the following inequality [12],

I2 = [A1 B1] + [B1 A2] + [A2 B2]+ [B2 A1 1] d 1 , (2)

where X =d1i=1 iP (X = i) is the average value of therandom variable X {0, . . . , d 1} and [X] denotes Xmodulo d. This follows from the identity

[A1 B1 + B1 A2 + A2 B2 + B2 A1 1] = d 1 ,

and the relation [X] + [Y] [X + Y]. When d = 2, itcorresponds to the CHSH inequality [13].

We can extend the above inequality to an arbitrarynumber N of measurement choices:

IN = [A1 B1] + [B1 A2] + [A2 B2] + + [AN BN] + [BN A1 1] d 1 . (3)

This extended inequality can be viewed as a chained ver-sion of inequality (2), and follows by a similar argument.It is equivalent, when d = 2, to the chained inequal-ity introduced by Pearle [14] and Braunstein and Caves[15]. The relation between the Pearle-Braunstein-Cavesinequality and GHZ-type correlations was first pointedout by Hardy [16].

We now show that if Alice and Bob share the maxi-mally entangled state

|d = 1/

dd1

q=0|qA|qB , (4)

there exist measurement settings such that for large N,IN(QM) tends to zero. The maximally entangled state|d has the property that if Alice measures an observablewith eigenvectors |r (r = 0, . . . , d1) and Bob measuresthe observable with complex conjugate eigenvectors |r,they get perfectly correlated outcomes. We define theeigenvectors characterizing Alices measurement Ak as

|rAk =1

d

d1q=0

exp

2i

dq(r k)

|qA , (5)

and those characterizing Bobs measurement Bl as

|rBl =1

d

d1q=0

exp

2i

dq(r l)

|qB , (6)

where k = (k 1/2)/N and l = l/N.A straightforward calculation shows that each term

in (3) is equal to /N2 + O(1/N3), where =

2/(4d2)d1

j=1 j/ sin2(j/d). As there are 2N such

terms in the inequality (3), it follows that

IN(QM) = 2/N + O(1/N2), (7)

which can be made arbitrarily small for sufficiently largeN.

For any particular finite value of N, the quantityIN(QM) implies an upper bound on the fraction of lo-cal states in any model that reproduces the correlations,i.e., an upper bound on the p of Eq. (1). This is be-cause although the term PNL(Ak = a, Bl = b) can vio-late the Bell inequality (3), it must satisfy IN(NL) 0,since each term in (3) is a positive quantity. Thus we


3/5

3

can write IN(QM) = p IN(L) + ( 1 p) IN(NL), and, sinceIN(L) d 1 and IN(NL) 0, IN(QM) p (d 1), or

p IN(QM)d 1 . (8)

Of course, in a real experiment, the state prepared willnot be precisely (4), the measurements will not be pre-

cisely defined by projections onto the vectors (5) and(6), and so on, and thus the experimentally determinedvalue, IN(EXP), will generally be greater than IN(QM).Nonetheless, given a value for IN(EXP), we can obtain abound of the form

p IN(EXP)d 1 . (9)

Hence we propose an experimental challenge: to obtainthe lowest possible bound on p (for any d and N) forbipartite maximally entangled states.

Our proof extends to any example of GHZ-type corre-lations; thus another natural challenge is to obtain the

lowest possible bound on p for any set of quantum corre-lations on any bipartite or multipartite entangled state.

Indeterminacy of the measurement outcomes and

Monogamy. The correlations that we just introducedhave a particular significance in the context of key distri-bution. BHK showed [4] how non-local correlations canbe used as the basis of a key distribution scheme thatis secure against non-signalling eavesdroppers. Non-localcorrelations can also be used to give at least partial secu-rity against non-signalling eavesdroppers in more prac-tical QKD schemes [17, 18]. In these discussions, it isnot assumed that such eavesdroppers are constrained bythe laws of quantum mechanics, but it is assumed that

they can only prepare systems in states whose correla-tions are non-signalling, in the following sense. Supposethat Alice and Bob share a bipartite system character-ized by correlations P(Ak = a, Bl = b). The correlationsare non-signalling if they satisfy

a

P(Ak = a, Bl = b) =a

P(Ak = a, Bl = b) , (10)

for all k, k, l , b, and a similar set of conditions obtainedby summing over Bobs input. These no-signalling con-ditions ensure that the marginal distributions P(Ak = a)and P(Bl = b) are well defined quantities. The defini-

tion of no-signalling can be extended to more than twoparties, by requiring a condition similar to (10) for everypossible grouping of the parties into two subsets.

In the protocols of Refs. [4] and [17, 18], maximally en-tangled states are prepared by a source, which is situatedbetween Alice and Bob and assumed to be under controlof the eavesdropper Eve. On each pair of particles, Al-ice and Bob perform measurements Ak and Bl, chosenindependently and randomly, and use the correspondingmeasurement outcomes to establish a shared secret key.

If the correlations used to distribute the key admita model with fractions p and 1 p of local and non-signalling non-local states respectively, then for a frac-tion p of pairs, Eve could prepare a deterministic localstate that would give her complete information aboutAlices and Bobs measurement outcomes. This strategyis clearly not significantly useful to Eve if the quantumcorrelations imply p

0, which is the case in BHKs

protocol [4], as noted above.

However, a zero fraction of local states does not neces-sarily imply that Eves information is zero, as it does notexclude the possibility that she could prepare a non-localstate that has definite values for a non-empty proper sub-set of the measurement inputs. For instance, in the caseof Cabellos example [11], there is a model reproducingthe quantum correlations with the following properties:i) every hidden state is non-signalling, ii) for every hid-den state, at least one measurement has a predeterminedoutcome, iii) each measurement has a predetermined out-come for at least some finite fraction of the hidden states.

A non-signalling Eve exploiting this model could obtainsome knowledge of Alices and Bobs measurement out-comes. Similar remarks apply to the GHZ and Merminexamples.

The correlations defined by (4), (5) and (6), however,are stronger in the sense that any non-signalling modelthat reproduces the correlations has the following prop-erty: for every hidden state, the outcomes of any mea-surement are completely undetermined and occur all withthe same probability 1/d. This property is implied by thefollowing theorem.Theorem. Any no-signalling distribution for which

IN IN satisfies

P(Ak = a) 1d

+d

4IN and P(Bl = b)

1

d+

d

4IN

for all measurements Ak and Bl and for all outcomes aand b.

In the limit IN 0, Eve cannot therefore gainany knowledge about Alice and Bobs measurement out-comes. In other words, any tripartite distribution de-

scribing the joint systems of Alice, Bob and Eve is of theform P(Ak = a, Bl = b) P(Em = e). We say that Al-ices and Bobs correlations are monogamous, in obviousanalogy with the familiar monogamy of entanglement.

Proof of the Theorem. Suppose that P(Ak = a) >1/d + d/4 IN for some measurement Ak and outcome a.We will then show that IN > IN. (The same argumentapplies if we suppose that P(Bl = b) > 1/d + d/4 IN forsome measurement Bl and outcome b.)


4/5

4

Defining AN+1 = A1 + 1 (modulo d), we can write

IN =

Nj=1

([Aj Bj ] + [Bj Aj+1])

2N Nj=1

(P(Aj = Bj) + P(Aj+1 = Bj)) ,

(11)

since [X] = d1i=1 i P([X] = i) 1 P([X] = 0). Now

P(Ai = Bj) =

d1r=0

P(Ai = r, Bj = r)

min(P(Ai = q), P(Bj = q))+ min(1 P(Ai = q), 1 P(Bj = q))

= 1 |P(Ai = q) P(Bj = q)| ,

for any q {0, . . . , d 1}. Using this expression in theabove inequality for IN and defining N arbitrary differentvalues qj , we get

IN Nj=1

(|P(Aj = qj) P(Bj = qj)|

+ |P(Aj+1 = qj) P(Bj = qj)|)

Nj=1

|P(Aj = qj) P(Aj+1 = qj)| ,

(12)

where the second line follows from the triangle inequality.The hypothesis P(Ak = a) > 1/d + d/4 IN implies that

|P(Ak = a) P(Ak = a + 1)| > INfor some a. If we define the values qj used in (12) byq1 = . . . = qk1 = a and qk = . . . = qN = a + 1 (modulod), we obtain the inequality

IN |P(A1 = a) P(Ak = a)+ P(Ak = a

+ 1) P(AN+1 = a + 1)|= |P(Ak = a) P(Ak = a + 1)|> IN ,

(13)

since AN+1 = A1 + 1 (modulo d), by definition. Classical simulation of the correlations. Finally, we

note that the bound (8) can be interpreted, in the

spirit of [19], as a bound on the average communica-tion necessary to simulate classically non-local correla-tions. The fact that every hidden state must be non-local in any model that reproduces measurements on amaximally entangled state implies that in any run of acommunication-assisted classical protocol some commu-nication must be exchanged between the parties. Sincethe minimal amount of communication that the partiescan exchange in any such run is one bit, it follows that atleast one bit of communication is necessary to simulate

these correlations. For qubits, i.e, when d = 2, one bitis known to be sufficient [20]. For other values of d, itis possible to extend our analysis by letting the range ofmeasuring settings go from 1 to (d 1) N, while keepingthe measurements as defined in (5) and (6). Using in-equalities similar to (3), one can then show that at leastlog2 d bits of communication are necessary. This boundis not optimal asymptotically since for large d a bound

ofO(d) bits is known [21]. It may, however, be useful forsmall d.

Summary and conclusions. The quantum correla-tions introduced imply that maximally entangled quan-tum states in arbitrary dimensions have zero local com-ponent. They motivate experiments that could boundthe weight of the local component as close to zero aspossible. In a non-signalling context, the correlations arealso provably monogamous, which gives them an immedi-ate application in key distribution. It would be interest-ing to characterise the sets of quantum correlations thatare monogamous in this sense; in particular, it would beinteresting to know if there are monogamous quantumcorrelations that can be obtained in an experiment withonly a finite number of measurement settings at each site,rather than in the limit in which the number of settingstends to infinity.

Acknowledgments. AK thanks Jonathan Oppenheimfor helpful conversations. SP acknowledges support bythe David and Alice Van Buuren fellowship of the BelgianAmerican Educational Foundation, by the National Sci-ence Foundation under Grant No. EIA-0086038, and bythe European Commission under the Integrated ProjectQubit Applications (QAP) funded by the IST directorateas Contract Number 015848.

[email protected] [email protected] [email protected]

[1] J. S. Bell, Speakable and unspeakable in quantum mechan-ics (Cambridge University Press, Cambridge, 1987).

[2] W. Tittel and G. Weihs, Quant. Inf. Comp. 1, 3 (2001).[3] G. Brassard, Found. Phys. 33, 1593 (2003).[4] J. Barrett, L. Hardy, and A. Kent, Phys. Rev. Lett. 95,

010503 (2005).[5] A. Elitzur, S. Popescu, and D. Rohrlich, Phys. Lett. A

162, 25 (1992).

[6] The assumption made in Ref. [5] is that the correlationfunction EL() introduced in the paper has an expansionin for small, positive .

[7] R. F. Werner and M. M. Wolf, Quant. Inf. Comp. 1, 1(2001).

[8] D. M. Greenberger, M. A. Horne, and A. Zeilinger, inBells Theorem, Quantum Theory, and Conceptions ofthe Universe, edited by M. Kafatos (Kluwer AcademicPublishers, Dordrecht, 1989), p. 69.

[9] N. D. Mermin, Phys. Rev. Lett. 65, 3373 (1990).[10] P. Heywood and M. Redhead, Found. Phys. 13, 481
mailto:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]


5/5

5

(1983).[11] A. Cabello, Phys. Rev. Lett. 87, 010403 (2001).[12] D. Collins, N. Gisin, N. Linden, S. Massar, and

S. Popescu, Phys. Rev. Lett. 88, 040404 (2002). The no-

tation is taken from A. Acin, R. D. Gill, and N. Gisin,Phys. Rev. Lett. 95, 210402 (2005).

[13] J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt,Phys. Rev. Lett. 23, 880 (1969).

[14] P. A. Pearle, Phys. Rev. D 2, 1418 (1970).

[15] S. L. Braunstein and C. M. Caves, Ann. Phys. 202, 22(1990).

[16] L. Hardy, Phys. Lett. A 161, 21 (1991).[17] J. Barrett, L. Hardy, and A. Kent, unpublished; A. Kent,

Perimeter Institute seminar on April 2, 2003, archived atwww.perimeterinstitute.ca.

[18] A. Acin, N. Gisin, and L. Masanes, quant-ph/0510094.[19] S. Pironio, Phys. Rev. A 68, 062102 (2003).[20] B. F. Toner and D. Bacon, Phys. Rev. Lett. 91, 187904

(2003).[21] G. Brassard, R. Cleve, and A. Tapp, Phys. Rev. Lett.

83, 1874 (1999).

Maximally Non-Local and Monogamous Quantum Correlations 0605182

Documents

Transcript of Maximally Non-Local and Monogamous Quantum Correlations 0605182