Local Quantum Measurement and No-Signaling Imply Quantum Correlations
LLocal quantum measurement and no-signaling imply quantum correlations
H. Barnum, S. Beigi, S. Boixo, M. B. Elliott, and S. Wehner Perimeter Institute for Theoretical Physics, 31 Caroline St. N, Waterloo, ON, N2L 2Y5 Canada Institute for Quantum Information, California Institute of Technology, Pasadena, CA 91125, USA (Dated: October 29, 2018)We show that, assuming that quantum mechanics holds locally, the finite speed of information isthe principle that limits all possible correlations between distant parties to be quantum mechanicalas well. Local quantum mechanics means that a Hilbert space is assigned to each party, and then alllocal POVM measurements are (in principle) available; however, the joint system is not necessarilydescribed by a Hilbert space. In particular, we do not assume the tensor product formalism betweenthe joint systems. Our result shows that if any experiment would give non-local correlations beyondquantum mechanics, quantum theory would be invalidated even locally.
Quantum correlations between space-like separatedsystems are, in the words of Schr¨odinger, “ the character-istic trait of quantum mechanics, the one that enforcesits entire departure from classical lines of thought”[1].Indeed, the increasing experimental support [2] for corre-lations violating Bell inequalities [3] is at odds with localrealism. Quantum correlations have been investigatedwith increasing success [4], but what is the principle thatlimits them [5]?Consider two experimenters, Alice and Bob, at twodistant locations. They share a preparation of a bipar-tite physical system, on which they locally perform oneof several measurements. This shared preparation maythereby cause the distribution over the possible two out-comes to be correlated. In nature, such non-local cor-relations cannot be arbitrary. For example, it is a con-sequence of relativity that information cannot propagatefaster than light. The existence of a finite upper boundon the speed of information is known as the principle of no-signaling . This principle implies that if the events cor-responding to Alice’s and Bob’s measurements are sepa-rated by space-like intervals, then Alice cannot send in-formation to Bob by just choosing a particular measure-ment setting. Equivalently, the probability distributionover possible outcomes on Bob’s side cannot depend onAlice’s choice of measurement setting, and vice versa.Quantum mechanics, like all modern physical theories,obeys the principle of no-signaling.But is no-signaling the only limitation for correlationsobserved in nature? Bell [3] initiated the study of theselimitations based on inequalities, such as the CHSH ex-pression [6]. It is convenient to describe this inequalityin terms of a game played by Alice and Bob. Suppose wechoose two bits x, y ∈ { , } uniformly and independentlyat random, and hand them to Alice and Bob respectively.We say that the players win, if they are able to returnanswers a, b ∈ { , } respectively, such that x · y = a + b mod 2. Alice and Bob can agree on any strategy before-hand, that is, they can choose to share any preparationpossible in a physical theory, and choose any measure-ments in that theory, but there is no further exchange ofinformation during the game. The probability that the players win is14 (cid:88) x,y ∈{ , } (cid:88) a,b ∈{ , } x · y = a + b mod 2 p ( a, b | M xA , M yB ) (1)where p ( a, b | M xA , M yB ) denotes the probability that Aliceand Bob obtain measurement outcomes a and b whenperforming the measurements M xA and M yB respectively(any pre- or post-processing can be taken as part of themeasurement operation). Classically, i.e. in any localrealistic theory, this probability is bounded by [6] p classical ≤ / . (2)Such an upper bound is called a Bell inequality.Crucially, Alice and Bob can violate this inequalityusing quantum mechanics [3]. The corresponding boundis [7] p quantum ≤
12 + 12 √ , (3)and there exists a shared quantum state and measure-ments that achieve it [6]. Further, there is now com-pelling experimental evidence that nature violates Bellinequalities and does not admit a local realistic descrip-tion [2]. Yet, there exist stronger no-signaling correla-tions (outside quantum mechanics) which achieve successprobability p nosignal = 1 [5]. So why, then, isn’t naturemore non-local [8]?Studying limitations on non-local correlations thusforms an essential element of understanding nature. Onone hand, it provides a systematic method to both theo-retically and experimentally compare candidate physicaltheories [9]. On the other hand, it crucially affects ourunderstanding of information in different settings suchas cryptography and communication complexity [10–13].For example, if nature would admit p nosignal = 1, anytwo-party communication problem could be solved us-ing only a single bit of communication, independent ofits size [11]. Also, for the special case of the CHSH in-equality, it is known that the bound (3) is a consequence a r X i v : . [ qu a n t - ph ] A p r of information theoretic constraints such as uncertaintyrelations [12] or the recently proposed principle of infor-mation causality [13]. However, characterizing generalcorrelations remains a difficult challenge [14], and it isinteresting to consider what other constraints may im-pose limits on quantum correlations. Result.
We forge a fundamental link between localquantum theory and non-local quantum correlations. Inparticular, we show that if Alice and Bob are locally quan-tum , then relativity theory implies that their non-localcorrelations admit a quantum description. The assump-tion of being locally quantum may thus provide another“reason” why the correlations we observe in nature arerestricted by more than the principle of no-signaling it-self. Figure 1 states our result.Let us explain more formally what we mean by be-ing locally quantum (see also Figure 1). We say thatAlice is locally quantum, if her physical system can bedescribed by means of a Hilbert space H A of some fixedfinite dimension d , on which she can perform any lo-cal quantum measurement (POVM) M A = { Q a } a givenby bounded operators Q a ∈ B ( H A ). The probability p ( a | M A ) that she obtains an outcome a for measure-ment M A = { Q a } a is given by a function B ( H A ) → [0 , ρ A ∈ B ( H A ) [15], and similarlyfor Bob, where we use H B and M B = { R b } b to denotehis Hilbert space and measurements respectively. Con-ceptually, this means that quantum mechanics describesAlice and Bob’s local physical systems.However, we make no a priori assumption about thenature of the joint system held by Alice and Bob. In par-ticular, we do not assume that it is described by a tensorproduct of their local Hilbert spaces, or that their jointsystem is quantum mechanical. This means that Aliceand Bob can share any possible preparation which as-signs probabilities to local POVM measurements. Thatis, their preparation is simply a function ω such thatthe probabilities of observing outcomes a and b for mea-surements M A = { Q a } a and M B = { R b } b are given by p ( a, b | M A , M B ) = ω ( Q a , R b ). In particular, the state oftheir joint system may not be described by any densitymatrix.Nevertheless, we are able to show that just from theassumptions that Alice and Bob are locally quantum andthat the no-signaling principle is obeyed, it follows thatthere exist a Hilbert space H AB = H A ⊗ H B , a state ρ AB ∈ B ( H AB ) and measurements ˜ M A = { ˜ Q a } a and˜ M B = { ˜ R b } b for Alice and Bob, such that p ( a, b | M A , M B ) = ω ( Q a , R b ) = tr (( ˜ Q a ⊗ ˜ R b ) ρ AB ) (4)That is, all correlations can be reproduced quantum me-chanically. Implications.
Our result solves an important piece ofthe puzzle of understanding non-local correlations, and their relation to the rich local phenomena we encounterin quantum theory such as Bohr’s complementarity prin-ciple, Heisenberg uncertainty and Kochen-Specker non-contextuality. In particular, it implies that if we obeylocal quantum statistics we can never hope to surpass aTsirelson-type bound on p quantum like that of (3), rul-ing out the possibility of such striking differences withrespect to information processing as those pointed outin [11]. Indeed, if we were able to surpass such bounds,then the local systems of Alice and Bob could not bequantum.Other recent works also attempt to explain the limita-tions of quantum correlations. For example, the principleof information causality [13] starts with the assumptionthat nature demands that certain communication tasksshould be hard to solve. Together with the assumptionof the no-signaling principle, this allows one to obtainTsirelson’s bound for the special case of the CHSH in-equality. In our work, we also assume the no-signalingprinciple, but combine it with a different assumption,namely, that the world is locally quantum, that is, quan-tum mechanics correctly describes the laws of nature oflocal physical systems. Making this assumption we re-cover the quantum limit on all possible non-local cor-relations (not only the Tsirelson’s bound for the CHSHinequality). Proof.
To prove our result, we now proceed in twosteps. First, we explain a known characterization of allno-signaling probability assignments to local quantummeasurements [16–18]. Second, we use this characteri-zation to show that the resulting correlations can be ob-tained in quantum mechanics.
From local quantum measurements to POPT states:
Fix two finite dimensional Hilbert spaces on Alice andBob’s sides. A local quantum measurement (or POVM)consists of a pair of measurements M A and M B withoutcome labels { a } and { b } respectively on Alice andBob’s Hilbert spaces. Such POVMs are described bycomplex Hermitian matrices M A = { Q a } a , M B = { R b } b , Q a , R b ≥
0, which sum to the identity, i.e., (cid:80) a Q a = (cid:80) b R b = p ( a, b | M A , M B ) to any choice of measurements M A and M b . More precisely, it corresponds to a function ω onthe pair of POVM elements such that p ( a, b | M A , M B ) = ω ( Q a , R b ).Kl¨ay, Randall, and Foulis [17] have shown (see Ap-pendix) that assuming no-signaling, the shared prepara-tions (or equivalently the functions ω ) are in one-to-onecorrespondence with matrices W AB such that tr ( W AB ) =1 and [22] p ( a, b | M A , M B ) = tr (( Q a ⊗ R b ) W AB ) ≥ . (5)The matrices W AB are called positive on pure tensors (POPT) states. All quantum states are POPT states, ? M A M B a bAlice BobQM QM QM M B a bAlice BobQM QM M A FIG. 1: If the principle of no-signaling is obeyed and Alice and Bob are locally quantum, their non-local correlations can beobtained in quantum mechanics. Alice and Bob are locally quantum if their local systems can be described by a Hilbert spaceand they can choose to measure any local POVM M A = { Q a } a and M B = { R b } b . A shared preparation between Alice and Bobcorresponds to a function ω on the pair of POVM elements such that p ( a, b | M A , M B ) = ω ( Q a , R b ). This, with no-signaling,implies that the marginal distributions are given by Born’s rule, p ( a | M A ) = tr ( Q a ρ A ) and p ( b | M B ) = tr ( R b ρ B ), where ρ A and ρ B are quantum states (but the state of their joint system may not be quantum). We show that, in this setting, for anypreparation ω there exist a joint quantum state σ AB and a relabeling of POVM measurements { ˜ M A } and { ˜ M B } , such that ω ( Q a , R b ) = p ( a, b | M A , M B ) = tr (( ˜ Q a ⊗ ˜ R b ) σ AB ) where ˜ M A = { ˜ Q a } a and ˜ M B = { ˜ R b } b . but there are POPT states that do not correspond toquantum states [23].Note that POPT states cannot be combined arbitrar-ily [18]. For example, not all entangled measurements(measurements which are not a convex combination oftensor products Q a ⊗ R b ) of POPT states are well de-fined because they would result in negative “probabili-ties” for non-quantum POPTs. Specifically, if Alice andBob share a POPT, and Charlie and Bob share anotherone, then if Alice and Charlie come together, entangledmeasurements between their POPTs are not necessarilydefined. This does not affect our result, since we are onlyinterested in the case where we consider parties (here Al-ice and Charlie together) which are locally quantum. From POPT states to quantum correlations:
We nowshow that there exist a quantum state σ AB and a mapon POVM measurements f : { M A = { Q a } a } (cid:55)→ { ˜ M A = { ˜ Q a } a } (6)such that p ( a, b | M A , M B ) = tr (cid:16) ( ˜ Q a ⊗ R b ) σ AB (cid:17) . (7)In order to do so, we associate to each POPT state W AB a map W from matrices to matrices using the Choi-Jamio(cid:32)lkowski isomorphism. Explicitly, W AB is obtainedfrom W by acting on Bob’s side of the (projection on the)maximally entangled state | Φ (cid:105) W AB = ⊗ W ( | Φ (cid:105)(cid:104) Φ | ) . (8)Because W AB is a POPT, the associated map W is pos-itive, i.e., it sends positive matrices to positive ones, but it may not be an admissible quantum operation. Never-theless, if W still maps POVMs to POVMs we can obtainthe POPT correlations by moving the action of W fromthe maximally entangled state to the measurement ele-ments. In particular, if W is unital ( W (
1) = f : Q a (cid:55)→ ˜ Q a = W ( Q Ta ) T , (9)maps POVM measurements to POVM measurements.We then show that (7) holds with σ AB = | Φ (cid:105)(cid:104) Φ | . Let d be the local dimension of Alice and Bob. If W is unitalwe have tr (( Q a ⊗ R b ) W AB ) = tr (( Q a ⊗ R b ) ⊗ W ( | Φ (cid:105)(cid:104) Φ | ))= tr ( | Φ (cid:105)(cid:104) Φ | ( ⊗ W ∗ )( Q a ⊗ R b ))= tr ( | Φ (cid:105)(cid:104) Φ | ( Q a ⊗ W ∗ ( R b )))= 1 d tr (cid:0) Q Ta W ∗ ( R b ) (cid:1) = 1 d tr (cid:0) W ( Q Ta ) R b (cid:1) = tr (cid:16) ( ˜ Q a ⊗ R b ) | Φ (cid:105)(cid:104) Φ | (cid:17) , (10)where W ∗ denotes the adjoint of W . This establishes (7)in the unital case.In general, W can be decomposed into a unital mapand another map. This other map gives a quantum state σ AB by acting on | Φ (cid:105) . Then f is defined in terms of theunital map as before. We finish the proof by showing that σ AB is well-normalized and (7) is satisfied. For a generalpositive map, let M be the image of the identity, i.e., W (
1) = M . The matrix M is normalized, tr ( M ) /d = tr ( W AB ) = 1. We assume initially that M is invertible,and define ˜ W ( · ) = M − / W ( · ) M − / . (11)The map ˜ W is unital. Further, the quantum state σ AB = | ψ (cid:105)(cid:104) ψ | given by | ψ (cid:105) = ( M / ) T ⊗ | Φ (cid:105) (12)is well-normalized, that is, tr ( σ AB ) = tr ( M T ) /d = 1.Thus by defining f as in (9) but in terms of ˜ W we con-clude tr (( Q a ⊗ R b ) W AB ) = 1 d tr (cid:0) W ( Q Ta ) R b (cid:1) = 1 d tr (cid:16)(cid:16) M / ˜ W ( Q Ta ) M / (cid:17) R b (cid:17) = tr (cid:16) ( ˜ Q a ⊗ R b ) σ AB (cid:17) . (13)If M is not invertible, in order to define ˜ W , one canstart with the map (1 − (cid:15) ) W ( · ) + (cid:15) tr ( · ), and then takethe limit (cid:15) → Conclusion.
We have shown that being locally quan-tum is sufficient to ensure that all non-local correlationsbetween distant parties can be reproduced quantum me-chanically, if the principle of no-signaling is obeyed. Thisgives us a natural explanation of why quantum corre-lations are weaker than is required by the no-signalingprinciple alone, i.e., given that one can describe localphysics according to quantum measurements and states,then no-signaling already implies quantum correlations.It would be interesting to know whether our work canbe used to derive more efficient tests for non-local quan-tum correlations than those proposed in [14]. Finally, itis an intriguing question whether one can find new limitson our ability to perform information processing locally based on the limits of non-local correlations, which wenow know to demand local quantum behavior.This work was supported by the National ScienceFoundation under grant PHY-0803371 through the In-stitute for Quantum Information at the California In-stitute of Technology, and by the US Department of En-ergy through the LDRD program at Los Alamos NationalLaboratory. Research at Perimeter Institute is supportedby the Government of Canada through Industry Canadaand by the Province of Ontario through the Ministry ofResearch and Innovation.
Appendix
We include a derivation of the POPT states for com-pleteness. We follow the more general version in [18].The outline is the following: using no-signalling, we ap-ply Gleason’s theorem on both sides, Alice and Bob. This implies that the no-signaling POPT state is bilinear onAlice and Bob measurements, which gives its form.We denote the local POVMs by M A = { Q a } a and M B = { R b } b . The joint probability distribution is givenby a function ω acting on POVM elements p ( a, b | M A , M B ) = ω ( Q a , R b ) . (14)Notice that for any pair of POVMs (cid:88) a,b ω ( Q a , R b ) = 1 , (15)but ω is not assumed to be bilinear at this point. No-signalling implies that for all M B (cid:88) b ω ( Q a , R b ) = (cid:88) b p ( a, b | M A , M B ) (16)= p ( a | M A , M B ) = p ( a | M A ) = ω ( Q a ) . That is, the marginal distribution is well defined.For any POVM element Q a on Alice’s side we can de-fine a corresponding function ω a which acts on Bob’sPOVM elements. The function ω a is defined by its actionon any POVM element R b with the equation ω a ( R b ) = ω ( Q a , R b ) . (17)Notice that, for every POVM M B on Bob’s side, no-signalling from Bob to Alice implies that (cid:88) b ω a ( R b ) = (cid:88) b ω ( Q a , R b ) = ω ( Q a ) . (18)Because ω a adds to the constant value ω ( Q a ) when itis summed over any POVM, we can use Gleason’s theo-rem [19–21] to identify ω a with an unnormalized quan-tum state ˜ σ a on Bob’s side. Specifically, for any POVMelement R b , we have ω a ( R b ) = ω ( Q a , R b ) = tr (˜ σ a R b ) . (19)The previous equation allows us to define, for any givenPOPT ω , a map ˆ ω from POVM elements Q a on Alice’sside to unnormalized quantum states on Bob’s sideˆ ω ( Q a ) = ˜ σ a . (20)Now choose an informationally complete POVM M B = { R b } on Bob’s side. Then ˆ ω is given by the functions ω b defined by ω b ( Q a ) = ω ( Q a , R b ) = tr (˜ σ a R b ) . (21)We use no-signalling from Alice to Bob to apply Glea-son’s theorem to each function ω b from the information-ally complete POVM, as we did before with no-signallingin the other direction. The action of ω b is then given byan unnormalized quantum state, which implies that it islinear. This proves that ˆ ω is linear.Once we have established the linearity of ˆ ω we canidentify it with the operator W introduced in the textaccording to ˆ ω ( Q a ) = 1 d W ( Q Ta ) . (22)Finally, we can write ω ( Q a , R b ) = tr (ˆ ω ( Q a ) R b ) = 1 d tr ( W ( Q Ta ) R b ) (23)= tr (( Q a ⊗ R b ) W AB ) . [1] E. Schr¨odinger, in P. Camb. Philos. Soc. (1935), vol. 31,pp. 555–563.[2] W. Tittel et al., Phys. Rev. Lett. , 3563 (1998).G. Weihs et al., Phys. Rev. Lett. , 5039 (1998). J. Panet al., Nature , 515 (2000). M. A. Rowe et al., Na-ture , 791 (2001). M. Ansmann et al., Nature ,504 (2009).[3] J. S. Bell, Physics , 195 (1965).[4] B. Julsgaard, A. Kozhekin, and E. S. Polzik, Nature , 400 (2001). L. Duan et al., Nature , 413 (2001).H. Haffner et al., Nature , 643 (2005). C. W. Chou etal.,Nature , 828 (2005). D. L. Moehring et al., Nature , 68 (2007).[5] S. Popescu and D. Rohrlich, Found. Phys. , 379 (1994).[6] J. Clauser et al., Phys. Rev. Lett. , 880 (1969).[7] B. Tsirelson, Letters , 93 (1980).[8] S. Popescu, Nat. Phys. , 507 (2006).[9] A. Leggett, Found. Phys. , 1469 (2003). S. Groblacheret al., Nature , 871 (2007). C. Branciard et al., Nat.Phys. , 681 (2008). [10] S. Wolf and J. Wullschleger (2005), arXiv:quant-ph/0508233. G. Brassard et al., Phys. Rev. Lett. ,250401 (2006). H. Buhrman et a., P. Roy. Soc. A ,1919 (2006). A. J. Short, N. Gisin, and S. Popescu, Quan-tum Inf. Process. , 1573 (2006). H. Barnum et al., Phys.Rev. Lett. , 240501 (2007). H. Barnum et al., in P. ofIEEE ITW (2008), pp. 386–390. E. H¨anggi, R. Renner,and S. Wolf (2009), arXiv:0906.4760.[11] W. van Dam (2005), arXiv:quant-ph/0501159.[12] G. V. Steeg and S. Wehner, QIC , 801 (2009).[13] M. Pawlowski et al., Nature , 1101 (2009).[14] S. Wehner, Phys. Rev. A , 022110 (2006).M. Navascu´es, S. Pironio, and A. Acin (2008),arXiv:0803.4290. A. Doherty et al., in P. of the 23rdIEEE CCC (2008), pp. 199–210.[15] Which also holds for dimension d = 2. A. Gleason, J.Math. Mech. , 885 (1957). P. Busch, Phys. Rev. Lett. , 120403 (2003). C. Caves et al., Found. Phys. , 193(2004).[16] D. Foulis and C. Randall, Interpretations and Founda-tions of Quantum Theory , 920 (1979).[17] M. Kl¨ay, C. Randall, and D. Foulis, Int. J. of Th. Phys. , 199 (1987).[18] H. Barnum et al., arXiv:quant-ph/0507108 (2005).[19] A. Gleason, J. Math. Mech. , 885 (1957).[20] P. Busch, Phys. Rev. Lett. , 120403 (2003).[21] C. Caves, C. Fuchs, K. Manne, and J. Renes, Found.Phys.34