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Testing Bell inequalities with weak measurements Shmuel Marcovitch and Benni Reznik School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel-Aviv University, Tel-Aviv 69978, Israel. Quantum theory is inconsistent with any local hidden variable model as was first shown by Bell. To test Bell inequalities two separated observers extract correlations from a common ensemble of identical systems. Since quantum theory does not allow simultaneous measurements of noncom- muting observables, on each system every party measures a single randomly chosen observable out of a given set. Here we suggest a different approach for testing Bell inequalities that is experimen- tallyrealizable bycurrentmethods. WeshowthatBellinequalitiescanbemaximally violatedeven whenallobservablesaremeasured oneachmemberoftheensemble. Thisispossiblebyusingweak 1 measurements that produce small disturbance, at the expense of accuracy. However, our approach 1 doesnotconstituteanindependenttestofquantumnonlocalitysincethelocalhiddenvariablesmay 0 correlate the noise of the measurement instruments. Nevertheless, by adding a randomly chosen 2 precise measurement at the end of every cycle of weak measurements, the parties can verify that n thehidden variables were not interfering with the noise, and thusvalidate thesuggested test. a J 8 I. INTRODUCTION 1 tA tB Quantum nonlocality [1] has been tested [2, 3] in dif- ] tB ph fseimrepnltesmyastneimfesstaatniodniissgniovwenwbyidCellyauascecre-Hptoerdn.e-SIhtsimmonoyst- q2A (cid:86)z t2A 2 q2B (cid:86)34(cid:83) - Holt (CHSH) inequality [4], in which every cycle A and t n B measure one out of two randomly chosen observables a a1,a2 1 and b1,b2 1 . In a local hidden vari- tB qu able m∈od{e±l B}≤2, where∈ {± } (cid:86)x t1A 1 (cid:86)(cid:83)4 [ q1A q1B B = E(a1b1)+E(a1b2)+E(a2b1) E(a2b2). (1) (cid:92) | − | AB 2 Above E(a b ) is the joint expectation value. The EPR v i j A B 6 state 3 1 2 ψ = ( A B + A B ), (2) 3 | i √2 |↑z↑zi |↓z↓z i FIG. 1: Sequential CHSH setup. Parties A and B measure . two observables at sequential times. The knobs control the 5 saturates the maximal bound 2√2 [5] with i.e. a1 = inaccuracyofthedevices. Intheaccuratemeasurementlimit 00 σx, a2 = σz, b1 = σπ/4, and b2 = σ3π/4, where σπ/4 = the pointer readings correspond to the possible values of the 1 1 observables( 1). Intheinaccurate(weak)limitthereadings 1 √2(σx+σz) and σ3π/4 = √2(σx−σz). are noisy,and±themicroscopic values cannot be inferred. We : In the regular setting every cycle the parties measure v requirethat thedistance between theparties d= XA XB i a single observable. In our approach to be referred to satisfies d ctA2 and d ctB2. | − | X as the sequential setting, for each member of the ensem- ≪ ≪ r ble A measures both a1 and a2 (one after another), and a similarly B measures both b1 and b2, as is schematically describedinFig. 1. Obviously,asdiscussedabove,foror- decreasingthe accuracyof measurements,one in fact re- dinary measurements the first measurement of a1 would gains the maximal violation of Bell inequalities. This randomize the result in the subsequent measurement of result is stated in theorem 1 and corollary 1. The out- a2, and no violation of Bell inequalities would be man- comes of inaccurate measurements may lie outside the ifested. However, as is well known, there is a trade-off usual range of the measured observables. In this case betweentheaccuracyofthemeasurementandthedistur- local hidden variables may interfere with the noise to bance caused to the system [6]. The limit in which indi- violate Bell inequalities. Therefore our approach does vidualmeasurementsprovidevanishing informationgain notconstitute anindependent test ofquantumnonlocal- was firstanalyzedby Aharonovet. al. (in the contextof ity. However,byaddingasinglerandomlychosenprecise post-selection) [7] and was termed weak measurements. measurementattheendofeachcycle,thepartiescande- In this limit the measurements become inaccurate and termine whether the hidden variables interfere with the onecouldexpectthatnonlocalcorrelationswouldnotbe noise. The expected negative result implies that the in- observed. accurate measurements still provide a valid test. This is The main result of this paper is that as one gradu- manifested in corollary 2. We then proceed by quantita- ally reduces the disturbance caused to the system, by tivelyillustratingthetransitionfromstrongtoweakmea- 2 surements in the sequential setting in realistic finite en- After the measurementsthe state ofthe systemsandthe semblesandshowthatsimultaneouslydifferentBelltests pointers is given by can be attained using the same ensemble. We conclude m m by discussing implications of our approach,in particular maximal violation of the Leggett and Garg inequalities |Ψi=U(tm) φ(ql)− φ′(qk) φ(ql)Ok(tk) [8] and possible realizations of the suggested method. hYl Xk lY6=k + φ′(qk)φ′(ql) φ(qs)Ol(tl)Ok(tk) (9) k<l s=k,l X Y6 II. VIOLATION OF BELL INEQUALITIES m 1 2 3 WITH WEAK MEASUREMENTS + 2 φ′′(qk) φ(ql)Ok(tk)+o(ǫ )|ψi. Xk lY6=k i Theorem1. Letasystem ψ bemeasuredsequentially by an arbitrary number of ob|seirvables O (i = 1,...,m) Let us compute EQM(qiqj), where without loss of gener- i ality i<j. Define U(t )ψ = δ ψ , U(t )O (t )ψ awteatikmmeseatsiuwrehmereent0s,<t1 <t2 <···<tm. In the limit of = αkr|ψri, and U(jtj|)Oil(tl)POk(rt|k)|rψii = j kηrklk|ψ|rii, where ψ is any basis and k,l j. Expanding Eq. r P | i ≤ P (9) in the ψ basis up to the jth measurement yields EQM(qi)= ψ Oi(ti)ψ , (3) | ri h | | i EQM(qiqj)=Re ψ Oi(ti)Oj(tj)ψ , (4) j j h | | i Ψ = δ φ(q ) αkφ(q ) φ(q ) | i r k − r ′ k l pwohienrteerq.iTishtehoebrseeardvianbgleosfOtheariethwmriettaesnurienmtheentHaepispeanrabteurgs Xr h Yk Xk lY6=k i + φ(q )φ(q ) φ(q )ηkl representation, O (t ) U (t )O U(t ), U(t) = e iH0t, ′ k ′ l s r (10) i i † i i i − and H0 is the free Ham≡iltonian. Xk<l sY6=k,l j 1 Proof. The full Hamiltonian is + 2 φ′′(qk) φ(ql)ηrkk+o(ǫ3)|ψri. m Xk lY6=k i H =H0+ δ(t−ti)piOi, (5) To compute EQM(qiqj) we trace out the system and qk, Xi k 6= i 6= j. Using qφ2(q)dq = 0, φ(q)φ′(q)dq = 0, it can be shown that to second order operations of O do where [q ,p ] = i (we take ~ = 1). The second term in R R k i i not contribute. Therefore, Eq. (5) corresponds to the von-Neumann measurement interaction, where for simplicity instantaneous measure- 1 E (q q )= (δ ηij +δ η ij +α iαj +αiα j)+o(ǫ4) ments are assumed. In addition, we assume identical QM i j 4 r∗ r r r∗ ∗r r r ∗r initial Gaussian wavepackets for the pointers: Xr 4 =Re ψ O (t )O (t )ψ +o(ǫ ). i i j j h | | i φ(qi,t=0)= ǫ 1/4e−ǫqi2/4, (6) (11) 2π (cid:16) (cid:17) This completes the proof of theorem 1. where ǫ 1 = [∆q (t = 0)]2 σ2. The initial state of the − i ≡ system and the apparatuses Using the same considerations it can be shown that Eqs. (3,4) are valid for simultaneous measurements too. Related results for correlations of two-level system with Ψ = ψ φ(q ), (7) i | i | i⊗ continuous weak measurements have been discussed i Y in [9], the case of post-selection in [10–12] and two evolves in time according to sequential measurements in [13]. = U (t )e ipiOiU(t ). (8) Corollary 1. In the sequential setting we define i † i − i U U ≡ Yi Yi B = E(qAqB)+E(qAqB)+E(qAqB) E(qAqB). (12) S 1 1 1 2 2 1 2 2 | − | Each operation of p yields an order of ǫ. In the limit of In the limit of weak measurements B B, coinciding weak measurements, the inaccuracy σ (ǫ 0). S → →∞ → withthatdefinedinEq. (1). Inparticular,maximalvio- Byexpanding tofirstorderinǫ,whilekeepingterms i to first order inU and tracing out the system and q , lationof2√2isobtainedwiththesameobservablesasin j j 6=i, one obtainsUEq. (3). OthBe r=egIuAlar sσeBttinagnwdiOthBO=1AI=AσxAσ⊗BIB., O2A =σzA⊗IB, To derive Eq. (4) we expand to second order 1 ⊗ π/4 2 ⊗ 3π/4 1 Corollary 2. Assume that at the end of every e−ipiOi =1−ipiOi− 2p2iOi2+o(ǫ3). cycleofweakmeasurementsthe partiesrandomlychoose 3 a single observable to measure strongly OA, OB respec- s s tively. Then E (qA,qB) = ψ OA OB ψ . This can QM s s h | s ⊗ s | i 2 2 be straightforwardly seen from the weak disturbance of 2.5 thepreviouslymeasuredobservables. Inquantumtheory 106 the weakly measured correlations equal the strongly 2 measured ones. 1.5 105 1 III. TESTING LOCAL HIDDEN VARIABLE 104 MODELS IN THE SEQUENTIAL SETTING 0.5 103 In the sequential setting the outcomes of weak mea- (cid:16)1 0 1 2 3 e e e e e surements may lie outside the usual range of the measured observables. Then the hidden variables can inter- ferewiththenoisetoviolateCHSHinequality. Therefore thesequentialsettingdoesnotconstituteanindependent FIG.2: ViolationofthesequentialCHSHtest. Solidline: BS testofquantumnonlocality. The requirementofrandom asafunctionofthemeasurementsinaccuracyσ ∆q(t=0). ≡ choice seems inevitable in any independent test of quan- BS > 2 for σ > p[ 2ln(23/4 1)]−1 1.1425. Note − − ∼ tum nonlocality. that precise sequential measurements satisfy CHSH inequal- Nevertheless,physicallyonewouldexpectadditiveran- ity. Brokenline: ensemblesizeN3(σ)thatmanifestsviolation withthreestandarddeviations. B 2.43correspondstothe dom noise n: qiA = ai +nAi , qjB = bj +nBj such that minimal ensemblesize N3 =3088.SF∼orcomparison, theregu- E(ainj)=0 and E(ninj)i6=j =0. Then lar non-sequentialprecise setting requires N3 =105. E(qAqB)=E(a b ), (13) i j i j and no violation of CHSH inequality can be shown. By corollary 2 we can use the regular test of nonlocality as a certificate for the sequential setting. That is, let each N 3 partypreciselymeasurearandomlychosenobservableat theendofeverycycle. Bell’stheoremstatesthatthecor- relations of the precise measurements would not violate CHSH inequality. If the parties observe that the weakly measuredcorrelationsdiffer fromthe preciselymeasured ones,thentheyknowthatthesequentialtestisnotvalid. However, then they also know that the hidden variables (”maliciously”) interfere with the noise. IV. TRANSITION FROM STRONG TO WEAK MEASUREMENTS FIG. 3: Simultaneous violation of several CHSH tests. The partiesperformnCHSHexperimentsbeforestartingthecur- The strict limit of weak measurements requires an in- rent CHSH sequence. BS > 2 for σ > σmin (dotted line), finite ensemble, which is not practical. However, viola- where σmin = 23/4√n (for large enough n). Also shown is theminimal valueof N3 (brokenline) and thecorresponding tion can be observed using a finite ensemble, and finite σ3(solidline)whichmanifestsviolationwith3standarddevi- measurement apparatus inaccuracy σ ∆q(t = 0), as ≡ ations. N3scalesquadraticallywithn. Inset: Expectedviola- illustrated in fig. 2. In fact, we find that BS > 2 for tiongivenσ3. BS(n)decreasesslowlywithnwhereBS 2.4. relatively mild value of σ 1.15. As the inaccuracy in- ∼ ∼ creases B 2√2. Also shown is the required ensemble S → sizeN3 bywhichviolationismanifestedwith threestan- and the corresponding inaccuracy σ3 by which violation dard deviations. The minimal ensemble is about thirty is manifested. times larger the one required to manifest B > 2 in the We proceed by quantifying the required inaccuracy usual setting. of the measurements as depicted in figures 2 and 3. Furthermore, simultaneous violation of (possibly dif- By explicitly expanding the joint state of the system ferent) Bell inequalities can be observed using the same and the measurement apparatuses it can be shown that ensemble. Forexample,assumethatthetwopartiesper- E(qAqB) = 1 , E(qAqB) = E(qAqB) = y and formnCHSHsequencesoneachpairoftheensemblewith 1 1 √2 2 1 1 2 √2 a fixed inaccuracy σ before starting the current CHSH E(q2Aq2B) = √y22, where y ≡ e−ǫ/2 = e−2σ12. For exam- test. In fig. 3 we show the minimal ensemble size N3 ple, the mutual state of the system and the measuring 4 devices after party A measures both σA and then σA Leggett and Garg (LG) [8]. LG discuss measurements x z and party B measures σB is onalocalsystemwhichareperformedatdifferenttimes. π/4 They analyze the scenario in which at each cycle the ex- z π/4 φ(q2A 1)φ(q1B 1) αφ(q1A 1) β(q1A+1) perimentalist chooses two out of m > 2 measurements |↑ ↑ i − − − − and compute their mutual expectation value. Given a +|↓z↑π/4iφ(q2A+1)φ(q1B−1)(cid:0)αφ(q1A−1)+β(q1A+1)(cid:1) realistic model and non-demolition measurements, cor- + z π/4 φ(q2A 1)φ(q1B+1)(cid:0)βφ(q1A 1)+α(q1A+1)(cid:1) relations satisfy the LG inequalities, which have a sim- |↑ ↓ i − − ilar structure to Bell inequalities. Interestingly, it was +|↓z↓π/4iφ(q2A+1)φ(q1B+1)(cid:0)βφ(q1A−1)−α(q1A+1)(cid:1), shown that quantum theory violates the LG inequali- (cid:0) (cid:1) ties using weak measurements [14–16] even if all observ- 1 1 ables are measured on each member of the ensemble. In where α= [cos(π/8)+sin(π/8)], β = [sin(π/8) 2√2 2√2 − this setting precise measurements do not violate LG in- cos(π/8)]. By tracing out the system and q1A, one can equalities by definition. The addition of noise, however, compute E(q2Aq1B) given above. Therefore BS = √12(1+ distinguishesmodelsincorporatingLGassumptionsfrom y)2, where B denotes the sequential setting. quantumtheory,sincenon-demolitionmeasurementsim- S By the central limit theorem the size of the ensemble ply that the noise and the measured observable are in- which is requiredto manifest violation of Bell inequality dependent [14]. This is violated in quantum theory even with z standard deviations is determined by thoughtheeffectofweakmeasurementsonthesystemis small. An immediate consequence of Eq. (4) is that LG inequalitiesaremaximallyviolatedbyquantumtheoryin V(B) z <B 2, (14) the limit of weak measurements. Note that the realism s N3 − assumption is crucial since one can write a joint proba- bilitydistributiontothe pointersofweakmeasurements, where V(B) is the variance of B. In the regular non- which do not correspond to 1 values. sequentialsetting B is the sum of four independent vari- ± ables V(B) V(qAqB) = 4(1 + σ2)2 2. In ≡ i,j i j − case of sequential measurementsthe experiments are dependent, thus V(PB ) CV(qAqB,qAqB), where S ≡ i,j,k,l i j k l CV(x,y) = E(xy) E(x)E(y) is the mutual covari- − P VI. CONCLUSIONS ance. By a straightforward computation it can be shown that for i = k and j = l, CV(qAaB,qAaB) = 6 6 i j i l CV(qiAqjB,qkAqlB)=0, and V(qiAqjB)=(1+σ2)2. There- We propose a new approach to test quantum nonlo- fore V(B ) = 4(1+σ2)2 1(1+y)4. These results are cality, in which all observables are measured on each S 2 − illustrated in fig. 2. system without the need to select randomly a single Now assume that the two parties perform n CHSH observable. By enlarging the sequence of measurements sequences before starting the currentCHSH experiment, it is also possible to manifest a simultaneous violation wherethe ithobservableofeachpartyismaximallynon- of more than one Bell inequality. Our proposal can commuting with the i 1 one for any i. Then it can be be demonstrated in all systems in which both Bell − shown that state preparation and sequential weak measurements are realizable, for example atomic physics devices [17], y2n B (n)= (1+y)2 (15) photons [18, 19], and solid-state qubits [20]. S √2 and V B (n) =4(1+σ2)2 y2n(1+y)4. These results S − 2 are illustrated in fig. 3. It can be numerically shown (cid:0) (cid:1) that Eq. (15) yields a typical value for randomly chosen n=2m measurements. Acknowledgments V. MAXIMAL VIOLATION OF LEGGETT-GARG INEQUALITIES WethankP.Skrzypczyk,N.Klinghoffer,J.Kupferman andG.Ben-Porathforhelpfuldiscussions. Thisworkhas Our suggestion to manifest violation of Bell inequali- been supported by the Israel Science Foundation grant ties is related to the time inequalities first suggested by numbers 784/06,920/09. [1] BellJS(1964)OntheEinstein-Podolsky-Rosenparadox. [2] AspectA,DalibardJ,RogerG(1982)Experimentaltest Physics (Long Island City,N.Y.) 1:195-200. 5 of Bell’s inequalities using time-varyinganalyzers. Phys. Weak Measurement on a Photon Pair as a Probe of Rev. Lett. 49:1804-1807. Hardys Paradox. Phys. Rev. Lett. 102:020404-020407. [3] WeihsG,JenneweinT,SimonC,WeinfurterH,Zeilinger [13] Johansen LM, Mello PA (2008) Quantum mechanics of A (1998) Violation of Bell’s Inequality underStrict Ein- successive measurements with arbitrary meter coupling. steinLocalityConditions.Phys. Rev. Lett.81:5039-5043. Physics Lett. A 372:5760-5764. [4] ClauserJF,HorneMA,ShimonyA,HoltRA(1969)Pro- [14] Ruskov R, Korotkov AN, Mizel A (2006) Signatures of posed Experiment to Test Local Hidden-Variable Theo- quantum behavior in single-qubit weak measurements. ries. Phys. Rev. Lett. 23:880-884. Phys. Rev. Lett. 96:200404-200407. [5] Cirel’son BS (1980) Quantum generalizations of Bell’s [15] Jordan AN, Korotkov AN, Bu¨ttiker M (2006) Leggett- inequality.Lett. Math. Phys. 4:93-100. Garginequalitywithakickedquantumpump.Phys.Rev. [6] von Neumann J (1955) in Mathematical Foundations of Lett. 97:026805-026808. QuantumMechanics(PrincetonUniversityPress,Prince- [16] Williams NS, Jordan AN (2008) Weak Values and the ton). Leggett-Garg Inequality in Solid-State Qubits. Phys. [7] Aharonov Y, Albert DZ, Casher A, Vaidman L (1987) Rev. Lett. 100:026804-026807. Surprisingquantum effects. Phys. Lett. A 124:199-203. [17] Mølmer K (2001) Counterfactual statements and weak [8] Leggett AJ, Garg A (1985) Quantum mechanics ver- measurements: an experimental proposal. Phys. Lett. A sus macroscopic realism: Is the flux there when nobody 292:151-155. looks? Phys. Rev. Lett. 54:857-860. [18] Hosten O, Kwiat P (2008) Observation of the Spin Hall [9] Korotkov AN, Averin DV (2001) Continuous weak mea- EffectofLightviaWeakMeasurements.Science319:787- surementofquantumcoherentoscillations. Phys. Rev. B 790. 64:165310-165314. [19] Goggin ME et. al. (2009) Violation of the Leggett-Garg [10] Resch KJ, Steinberg AM (2004) Extracting Joint Weak inequality with weak measurements of photons. e-print Values with Local, Single-Particle Measurements. Phys. arXiv: 0907.1679. Rev. Lett. 92:130402-130405. [20] Palacios-Laloy A. et. al. (2010) Experimental violation [11] Mitchison G, Jozsa R and Popescu S (2007) Sequential of a Bell’s inequality in time with weak measurement. weak measurement. Phys. Rev. A 76:062105-062113. Nature Phys. 6:442-447. [12] Lundeen JS, Steinberg AM (2009) Experimental Joint