Observers can always generate nonlocal correlations without aligning measurements by covering all their bases
OObservers can always generate nonlocal correlations without aligning measurementsby covering all their bases
Joel J. Wallman and Stephen D. Bartlett
School of Physics, The University of Sydney, Sydney, New South Wales 2006, Australia (Dated: 6 February 2012)Quantum theory allows for correlations between the outcomes of distant measurements that areinconsistent with any locally causal model, as demonstrated by the violation of a Bell inequality.Typical demonstrations of these correlations require careful alignment between the measurements,which requires distant parties to share a reference frame. Here, we prove, following a numericalobservation by Shadbolt et al., that if two parties share a Bell state and each party preformsmeasurements along three perpendicular directions on the Bloch sphere, then the parties will alwaysviolate a Bell inequality. Furthermore, we prove that this probability is highly robust againstlocal depolarizing noise, in that small levels of noise only decrease the probability of violating aBell inequality by a small amount. We also show that generalizing to N parties can increase therobustness against noise. These results improve on previous ones that only allowed a high probabilityof violating a Bell inequality for large numbers of parties. PACS numbers: 03.65.Ta, 03.65.Ud, 03.67.-a
One of the most fascinating and useful features ofquantum theory is that the correlations between the out-comes of spatially separated measurements can be nonlo-cal, i.e., inconsistent with any locally causal model [1, 2].Typically, to obtain nonlocal correlations experimentally,great care is taken to choose measurements that give thestrongest nonlocal correlations possible, which requiresdistant parties to share a reference frame [3, 4]. Whilethere are proposals for violating a Bell inequality withoutthe need for a prior shared reference frame [4, 5], theseproposals add substantial complexity to the simple formof a standard Bell test.Distant parties could attempt to violate a Bell in-equality without aligning reference frames by perform-ing measurements in random directions [6–8], and recentresults prove that such a method can demonstrate a vi-olation with some nonzero probability. Specifically, for N spatially-separated parties who share a Greenberger-Horne-Zeilinger (GHZ) state [9], almost all choices of twomeasurements at each site lead to nonlocal correlationsbetween measurement outcomes if the number of par-ties N is large [7]. Therefore distant parties that do notshare a reference frame can randomly choose measure-ments that violate some Bell inequality with a probabil-ity that approaches 1 as N increases. If the parties alsoshare a single direction on the Bloch sphere (as can bethe case in, e.g., photon polarization encodings [4]), thenthey can always violate one of two Bell inequalities by anamount that is exponential in N [8].These results are the weakest for the scenario most rel-evant to experiments, namely, the bipartite N = 2 case:the probability of violating a Bell inequality by choosingtwo mutually unbiased measurements randomly in thebipartite case is ∼
42% [7]. In this paper, we show thatif the two parties each choose three measurements cor-responding to the x , y and z components of their local Cartesian reference frame (hereafter referred to as a triadof measurements ), then they will always violate a Bell in-equality. That this scheme always results in a violationof a Bell inequality was communicated to us by the au-thors of [10] as a conjecture. In this paper, we prove thisconjecture; we note that Ref. [10] presents an indepen-dent proof of the same. We also prove that this form ofa Bell test is robust against noise, in that small levels ofnoise only slightly decrease the probability of violating aBell inequality.For the multipartite case, we numerically estimate theprobability of N parties who share an N -partite GHZstate violating a Bell inequality as a function of thelevel of local depolarizing noise when the N parties eachchoose a triad of measurements. In the absence of noise,we find that the parties will always violate a Bell inequal-ity (except for N = 3, where the numerical probability is ∼ . N .An intuitive way of understanding the success rate ofthis scheme is as follows. When the parties each choosea triad of measurements, one of each parties’ three mea-surements must necessarily be within an angle π of the z -axis of the reference frame in which the entangled statewas created. Although the parties do not know which oftheir measurement directions are closest to the z -axis, bychoosing a triad of measurements they have covered allpossibilities and so can simply test each possibility usingthe method of Ref. [8] to always obtain nonlocal cor-relations. It has previously been observed that partiesthat do not share a reference frame can obtain nonlocalcorrelations by trying all possible combinations of localmeasurement directions [6]. While this is evidently true,it is also experimentally infeasible. However, our resultsshow that the parties only need to try combinations of afinite (and relatively small) number of measurements ateach site in order to always obtain nonlocal correlations. a r X i v : . [ qu a n t - ph ] F e b The scenario. —A verifier prepares many copies of an N qubit state ρ and distributes one qubit to each of N parties. The n -th party chooses a triad of measure-ments, which can be written as O s n n = Ω s n n · (cid:126)σ , where (cid:126)σ = ( σ x , σ y , σ z ) is the vector of Pauli matrices (rel-ative to the reference frame in which ρ was created)and { Ω n , Ω n , Ω n } are orthonormal vectors in the Blochsphere. The qubits are distributed over channels thatintroduce a level γ ∈ [0 ,
1] of local depolarizing noise,where γ = 1 corresponds to no noise. Regardless of itsphysical origin, local depolarizing noise can be modeledby reducing the visibility of the measurements at eachsite as γ Ω [11]. We do not consider colored noise, such aslocal dephasing noise, as such noise models could allowthe parties to establish some common direction. This inturn would allow the parties to use the method in Ref. [8]to obtain a greater violation of a Bell inequality with asmaller number of measurement settings and inequalities.For each copy of ρ , each party randomly chooses andperforms one of their three measurements on their qubit.The parties then send the verifier a list of the mea-surement choice, s n ∈ Z , and corresponding outcome, o s n n ∈ Z , for each copy of ρ . The verifier uses the lists todetermine if the measurement outcomes are inconsistentwith a locally causal model.In general, the verifier will need to use the full jointprobability distributions p ( (cid:126)o | (cid:126)s ) to determine if the rela-tion between the measurement outcomes (cid:126)o = ( o , . . . , o N )and measurement settings (cid:126)s = ( s , . . . , s N ) is inconsis-tent with a locally causal model. However, for the sce-nario we consider, the verifier only needs to calculate theprobabilities p ( a | (cid:126)s ) (as relative frequencies) that the out-comes satisfy (cid:76) Nn =1 o s n n = a for a = 0 , E ( (cid:126)s ) = p (0 | (cid:126)s ) − p (1 | (cid:126)s ) , (1)and determine if the correlation functions are inconsis-tent with any locally causal model by checking if theyviolate some Bell inequality.The Bell inequalities we consider are the Mermin-Ardehali-Belinskii-Klyshko (MABK) Bell inequali-ties [12], which only depend on two measurement settingsat each site. For N = 2, the MABK Bell inequalitiesreduce to the famous Clauser-Horne-Shimony-Holt(CHSH) [13] Bell inequalities.To use Bell inequalities that only depend on two mea-surement settings at each site, the verifier can simplychoose one setting t n for each site and ignore any copyof ρ where the n -th party performed the measurementcorresponding to t n for any value of n = 1 , . . . , N . Math-ematically, this can be represented by the verifier choos-ing integers r n ∈ Z and injective functions τ n : Z → Z for n = 1 , . . . , N . The measurement settings that theverifier checks for the n -th site are s n ∈ { τ n (0) , τ n (1) } . Denoting by τ ( (cid:126)r ) = ( τ ( r ) , . . . , τ N ( r N )) the set of mea-surement settings corresponding to a specific choice of (cid:126)r = ( r , . . . , r N ) ∈ Z N , all MABK Bell inequalities canbe obtained from the inequality (cid:12)(cid:12)(cid:12) (cid:88) (cid:126)r ∈ Z N cos (cid:0) Rπ (cid:1) E ( τ ( (cid:126)r )) (cid:12)(cid:12)(cid:12) ≤ N − , (2)where R = N + 1 − (cid:80) Nk =1 r n , by varying over the 6 N functions τ [8]. Note that as presented, this is a formof post-selection, but it does not introduce a communi-cation loop-hole as it can also be viewed as a form ofpre-selection. For example, with each qubit, the verifiercould also send an integer corresponding to a setting thatthe parties cannot use to measure that qubit.The different functions τ correspond to the differentlabelings of the measurement directions, Ω s n n . We canexploit these labelings to restrict the relative orienta-tions of the triads of measurements { Ω n , Ω n , Ω n } . It isimportant to note that the verifier can only relabel themeasurements in the following manner if they know theorientation between the parties’ reference frames. With-out such knowledge, the verifier would still have to testa variety of labelings in order to identify which measure-ments violate a Bell inequality.With respect to the verifier’s reference frame (in whichthe state ρ is prepared), the n -th party’s measurementdirections can be written asΩ n = x (cid:48) n cos χ n + y (cid:48) n sin χ n Ω n = − x (cid:48) n sin χ n + y (cid:48) n cos χ n Ω n = (sin θ n cos φ n , sin θ n sin φ n , cos θ n ) , (3)where x (cid:48) n = (sin φ n , − cos φ n , y (cid:48) n = (cos θ n cos φ n , cos θ n sin φ n , − sin θ n ) , (4) θ n ∈ [0 , π ] and φ n , χ n ∈ [ − π, π ]. For each n , one of themeasurement directions Ω in must be within an angle of π / of either the ± z axis. We relabel the n -th party’smeasurements so that this direction is Ω n and swap thesign of Ω n if necessary (which corresponds to relabelingthe measurement outcomes of the measurement O s n n ), sothat θ n is in the interval [0 , π / ]. Adding multiples of π / to χ n simply permutes {± Ω n , ± Ω n } , so we can also set χ n to be in the interval [ − π / , π / ] for all n . The bipartite case. —As we now prove, two parties whoshare the singlet state, | Ψ − (cid:105) = 2 − / ( | (cid:105)| (cid:105) − | (cid:105)| (cid:105) ) , (5)where | (cid:105) and | (cid:105) are the computational basis states inthe verifier’s reference frame, in the above scenario willalways violate a Bell inequality. Note that this proofholds for any maximally entangled two-qubit state due FIG. 1. (Color online) Illustration of the labeling of the twoparties’ measurements in Eq. (7) and (8). to local equivalence, but we choose the singlet state forclarity. We also prove that this result is robust againstlocal depolarizing noise.For a singlet state distributed over channels that in-troduce local depolarizing noise parametrized by γ , thecorrelation functions are E ( s , s ) = (cid:104) Ψ − | (cid:0) γ O s ⊗ γ O s (cid:1) | Ψ − (cid:105) = − γ Ω s · Ω s . (6)The singlet state is invariant under arbitrary joint rota-tions of the two parties’ Bloch spheres, which allows us toreduce the problem to one with only three parameters.To do this, first note that we can rotate both parties’measurements so that Ω is the z axis, i.e.,Ω = (sin χ , − cos χ , = (cos χ , sin χ , = (0 , , , (7)where we have incorporated φ into χ . We then relabelthe second party’s measurements so that Ω is within π / of Ω (i.e., the z axis). Finally, we can rotate bothparties’ measurements around the z axis so that Ω isin the xz plane, i.e., the second party’s measurementdirections can be written asΩ = (sin χ cos θ, − cos χ , − sin χ sin θ )Ω = (cos χ cos θ, sin χ , − cos χ sin θ )Ω = (sin θ, , cos θ ) , (8)for some θ ∈ [0 , π / ] and χ ∈ [ − π / , π / ] and the firstparty’s measurement directions are as in Eq. (7) for somenew value of χ which we can set to be in the interval[ − π / , π / ] by relabeling the first party’s measurements.An example of a set of measurements labeled in this man-ner is shown in Fig. 1.When the measurements are labeled in this way, themeasurement statistics only need to be tested againsta small number of the 36 Bell inequalities in order todemonstrate nonlocal correlations. In particular, we only need the 3 inequalities obtained from Eq. (2) for τ =( r , − r ), (1 − r , r ) and (2 − r , r ), which can bewritten ascos ( θ / ) | sin( χ − ± π / ) | ≤ − / γ − , (9a) | a − b + 2 ab | ≤ γ − , (9b)where χ ± = χ ± χ and a = cos( χ − / ) cos( θ / ) ,b = cos( χ + / ) sin( θ / ) . (10)By adding multiples of π / to χ and/or χ and changingthe sign of χ ± (which will only permute the two inequal-ities in Eq. (9a)), we can further restrict the parametersto the region V = { θ, χ ± | θ ∈ [0 , π / ] , χ − ∈ [0 , π / ] , χ + ∈ [0 , π / ] } (11)and ignore the “ − ” inequality in Eq. (9a) as it is notviolated in V . Therefore at a given noise level γ , theprobability of the observers choosing measurements thatresult in nonlocal correlations is lower-bounded by p ( γ ) = |V| − (cid:90) V dθdχ − dχ + sin θf ( θ, χ ± , γ ) (12)where |V| is the volume of the region V and f ( θ, χ ± , γ ) =1 if ( θ, χ ± , γ ) violate Eq. (9a) or (9b) and 0 otherwise.In the absence of noise ( γ = 1), the only measurementsthat do not violate a Bell inequality are when the twoparties’ measurements are perfectly aligned, i.e., θ = 0and χ = χ . As this is a set of measure zero, the partiesalways violate a Bell inequality.As γ decreases from 1, small perturbations from thisperfect alignment also do not violate a Bell inequality.For mathematical convenience, we only consider noiselevels γ ≥ / √ ∼ .
97 analytically. Numerical data forthe full range of noise levels that allow violations of aBell inequality (determined from Tsirelson’s bound [14])is plotted in Fig. 2.For fixed γ ∈ [ / √ , θ, χ ± ) ∈ V when χ − > sin − (cid:16) − / γ − cos − ( θ / ) (cid:17) − π / := L ( θ, γ ) (13)or a − (cid:112) a − γ − − − / sin( θ / ) < , (14)respectively, where we have used a > b > − / sin( θ / )everywhere in V . The left-hand side of Eq. (14) is convexin θ and nonincreasing in χ − everywhere in V . ThereforeEq. (14) will be satisfied for all θ ∈ [ x ( γ ) , π / ] and χ − ∈ [0 , L ( θ, γ )] if it is satisfied for θ = x ( γ ), χ − = L ( x ( γ ) , γ )and θ = π / , χ − = L ( π / , γ ).Choosing x ( γ ) = cos − γ / , these conditions are sat-isfied, so a Bell inequality is always violated unless θ ≤ cos − γ / and χ − ≤ L ( θ, γ ). Therefore, for fixed γ ∈ [ / √ , − p ( γ ) ≤ / π (cid:90) cos − γ / dθ sin θL ( θ, γ ) ≤ (1 − γ / ) / . (15)For γ ≥ / √ , the probability of violating a Bell in-equality is at least 99 . The multipartite case. —We now consider N partieswho implement the same scheme using the N -partiteGHZ state, | Ψ N GHZ (cid:105) = 2 − / ( | (cid:126) N (cid:105) + | (cid:126) N (cid:105) ) , (16)where | (cid:126)i N (cid:105) denotes the state in which N qubits are pre-pared in the state | i (cid:105) . For the N -partite GHZ state withlocal depolarizing noise and measurements parametrizedas in Eq. (3), the correlation functions are E ( (cid:126)s ) = Tr (cid:0) ρ ( λ ) N (cid:79) j =1 γ O js j (cid:1) = γ N δ N N (cid:89) j =1 (Ω js j ) z + γ N Re N (cid:89) j =1 [(Ω js j ) x + i (Ω js j ) y ] , (17)where δ N ≡ − N (mod 2).To obtain a numerical estimate of the probability ofthe N parties violating a MABK Bell inequality withoutsharing a reference frame, we randomly sample 10 setsof measurements according to the uniform Haar measureon the surface of the sphere and find the fraction of mea-surements that violate an MABK Bell inequality. Theresults are plotted in Fig. 2 as a function of γ .For N (cid:54) = 3, all 10 sets of measurements led to a viola-tion of a MABK inequality, and so the numerical evidencesuggests that the parties will always violate an MABKinequality and the robustness to noise increases with N .The exceptional case of N = 3, for which the numeri-cal probability of violating an MABK inequality in theabsence of noise is ∼ . z -component ofthe measurements for odd N (indicated by the δ N termin Eq. (17)). Summary. —We have proven that two parties whoshare a maximally entangled bipartite state will alwaysviolate a Bell inequality by choosing random measure-ments from a triad of measurements (corresponding tothe x , y and z directions on their local Bloch sphere).We have also provided numerical evidence that N par-ties who share a maximally entangled state will alwaysviolate a Bell inequality (unless N = 3) with this mea-surement choice. Moreover, this scheme is robust against γ p M ABK ( % ) N=2N=3N=4N=5N=6
FIG. 2. (Color online) Plot of the probability of violatingone of the MABK Bell inequalities, p MABK , for N = 2 , . . . , γ for local depo-larizing noise. local depolarizing noise in that small levels of noise willonly slightly decrease the probability of violating a Bellinequality.We note that local depolarizing noise models a vari-ety of relevant experimental noise sources and imperfec-tions. For example, local depolarizing noise can be usedto model imperfect detectors (i.e., detectors that only de-tect a fraction γ of events) or the non-ideal preparation ofa resource state through such processes as spontaneousparametric down conversion. Local depolarizing noisealso provides a worst-case bound for other noise models.Finally, the singlet state with local depolarizing noise isequivalent to a mixed Werner state, so our results canalso be interpreted as giving a probability of violating aBell inequality for Werner states.JJW thanks Yeong-Cherng Liang for communicatingthe numerical observation in the bipartite scenario, whichinspired the results presented here and was also consid-ered independently in Ref. [10]. This research is sup-ported by the Australian Research Council. [1] J. S. Bell, Speakable and Unspeakable in Quantum Me-chanics (Cambridge University Press, Cambridge, 2004).[2] J. Barrett, N. Linden, S. Massar, S. Pironio, S. Popescuand D. Roberts, Phys. Rev. A , 022101 (2005).[3] A. Aspect, “Quantum [Un]speakables - From Bell toQuantum information”, edited by R. A. Bertlmann andA. Zeilinger, Springer (2002).[4] S. D. Bartlett, T. Rudolph, R. W. Spekkens, Rev. Mod.Phys. , 555 (2007).[5] A. Cabello, Phys. Rev. A , 042104 (2003); Phys. Rev. Lett. , 230403 (2003).[6] S. Ashhab, K. Maruyama and F. Nori, Phys. Rev. A ,052113 (2007).[7] Y.-C. Liang, N. Harrigan, S. D. Bartlett and T. Rudolph,Phys. Rev. Lett. , 050401 (2010).[8] J. J. Wallman, Y.-C. Liang and S. D. Bartlett, Phys.Rev. A , 022110 (2011).[9] D. M. Greenberger, M. A. Horne and A. Zeilinger, Bell’s Theorem, Quantum Theory, and Conceptions ofthe Universe , M. Kafatos (Ed.), Kluwer, Dordrecht, 69- 72 (1989).[10] P. Shadbolt, C. Branciard, N. Brunner, Y.-C. Liang, J. L.O’Brien and T. V´ertesi, e-print arXiv:1111.1853 [quant-ph].[11] W. Laskowski, T. Paterek, ˇC. Brukner and M. ˙Zukowski,Phys. Rev. A , 042101 (2010)[12] N. D. Mermin, Phys. Rev. Lett. , 1838 (1990); M.Ardehali, Phys. Rev. A , 5375 (1992);[13] J. F. Clauser, M. A. Horne, A. Shimony and R. Holt,Phys. Rev. Lett. , 880 (1969).[14] B. S. Cirel’son, Lett. Math. Phys.4