Einstein-Podolsky-Rosen Steering in Two-sided Sequential Measurements with One Entangled Pair
Jie Zhu, Meng-Jun Hu, Guang-Can Guo, Chuan-Feng Li, Yong-Sheng Zhang
aa r X i v : . [ qu a n t - ph ] M a r Einstein-Podolsky-Rosen Steering in Two-sided Sequential Measurements with OneEntangled Pair
Jie Zhu,
1, 2
Meng-Jun Hu,
3, 1, 2, ∗ Chuan-Feng Li,
1, 2
Guang-Can Guo,
1, 2 and Yong-Sheng Zhang
1, 2, † Key Laboratory of Quantum Information, University of Science and Technology of China, CAS, Hefei, 230026, China CAS Center for Excellence in Quantum Information and Quantum Physics, Hefei, 230026, China Beijing Academy of Quantum Information Sciences, Beijing, 100089, China (Dated: March 2, 2021)Non-locality and quantum measurement are two fundamental topics in quantum theory and theirinterplay attracts intensive focuses since the discovery of Bell theorem. Recently, non-locality sharingamong multiple observers with one entangled pair has been predicted and experimentally observed bygeneralized quantum measurement – weak measurement. However, only one-sided sequential case,i.e., one Alice and multiple Bobs is widely discussed and little is known about the two-sided case.Here, we theoretically and experimentally explore the non-locality sharing in two-sided sequentialmeasurements case in which one entangled pair is distributed to multiple Alices and Bobs. Weexperimentally observed double EPR steering among four observers in a photonic system. In thecase that all observers adopt the same measurement strength of the weak measurement, it is observedthat double EPR steering can be demonstrated simultaneously. The results not only deepen ourunderstanding of relation between sequential measurements and non-locality but also may findimportant applications in many quantum information tasks, such as randomness certification.
Introduction—
Non-locality, which is the core charac-teristic of quantum theory [1], plays a fundamental rolein many quantum information tasks. Bell non-locality[2, 3] and Einstein-Podolsky-Rosen (EPR) steering [4, 5]are two extensively investigated notions that capture thequantum non-locality in which EPR steering is proved tobe a strict general form than Bell non-locality [6]. Fromthe perspective of quantum information, both Bell non-locality and EPR steering can be demonstrated by con-sidering two separated observers, Alice and Bob, thatperform local measurements on a shared quantum state ρ AB and quantum non-locality is witnessed via violationof corresponding inequalities. Recently, R. Silva et al ex-tended Bell test to include one Alice and many Bobs withintermediate Bobs performing sequential weak measure-ments and showed that Bell non-locality can be sharedamong multiple observers with one entangled pair [1].Double Bell-Clauser–Horne–Shimony–Holt (CHSH) [8]inequality violations among three observers are then ex-perimentally observed with one entangled photon pair bytwo independent groups including ours [2, 10]. Based onthis sequential scenario lots of works have been reported[11–22] and EPR steering among multiple observers isexperimentally demonstrated very recently [23].To date, however, almost all discussions are limited toone-sided sequential case, i.e., one entangled pair is dis-tributed to one Alice and multiple Bobs. As emphasizedby R. Silva et al in their last sentence in Ref. [1] that itwould be interesting to investigate the two-sided sequen-tial case, i.e., including multiple Alices in the setup. Inthis Letter, we theoretically and experimentally explorethe two-sided sequential case that one entangled pair isdistributed to multiple Alices and Bobs, in which mid-dle Alices and Bobs perform optimal weak measurementsand the last Alice and Bob perform projective measure- FIG. 1.
Theoretical sketch.
The two-sided sequential sce-nario in which one entangled pair is distributed to multiple Al-ices and Bobs. Here ~x i , ~y j represent measurement inputs and a i , b j are corresponding dichotomic measurement outcomes,respectively. ment. The relation between sequential weak measure-ments and Bell non-locality is explicitly derived under theunbiased input condition for two-sided sequential case.It is shown that no more than two Bell-CHSH inequal-ity violations can be obtained in the same method (seeSupplementary Materials). However, here the analyti-cal forms of EPR steering is obtained for two Alices andtwo Bobs case, showing that Alice1-Bob1, Alice2-Bob2can demonstrate EPR steering simultaneously. Using en-tangled photon pair, we experimentally observed doubleEPR steering simultaneously with n = 6 and n = 10 mea-surement settings in the case of two pairs of observers. Theoretical framework—
Consider a two-party state ρ AB distributed to Alices and Bobs who perform se-quential weak measurements as shown in Fig. 1. Forconvenience of calculations and experimental realiza-tion, we choose ρ AB = | Ψ − i AB h Ψ − | with the singletstate | Ψ − i AB = ( | ↑i A | ↓i B − | ↓i A | ↑i B ) / √ FIG. 2.
Experimental setup. (a)
Polarization-entangled photon pair are generated via the type I phase-matching spontaneousparametric down-conversion (SPDC) process by pumping a joint β -barium-borate (BBO) crystal with a 404 nm semiconductorlaser. Signal and idler photons are then distributed to Alices and Bobs with Alice1, Bob1 performing optimal weak measurementsand Alice2, Bob2 performing projective measurements. (b) Setup for realizing optimal weak measurement. (c)
Setup forrealizing projective measurement. BBO: β -barium-borate, QWP: quarter wave plate, HWP: half wave plate, PBS: polarizationbeam splitter, BD: beam displacer, SPD: single photon detector, FC: fiber coupler. tled as ˆ M ± | ~k = cos θ | k ± ih k ± | + sin θ | k ∓ ih k ∓ | (1)with dichromatic observable ˆ σ ~k ≡ | k + ih k + | −| k − ih k − | , h k + | k − i = 0 and parameter θ ∈ [0 , π/
4] de-termines the strength of measurement. When θ = 0,ˆ M ± | ~k reduces to the projector | k ± ih k ± | corresponding toproject measurement, while θ = π/ M ± | ~k = ˆ I/ √ F measuring the disturbance of measure-ment and information gain G and they satisfy the trade-off relation F + G ≤ F = sin2 θ, G = cos2 θ and satisfy optimal weakmeasurement condition F + G = 1 [1, 2, 25].The quantum non-locality can be witnessed via viola-tions of corresponding inequalities. Quantities measurequantum correlation need to be calculated to see whetheror not they can surpass the threshold supported by lo-cal hidden variables/states theory [8, 26]. It is shown inthe following that these quantities are deeply connectedto the quality factor F and information gain G of weakmeasurements. Bell quantity I and EPR steering quan-tity S both are determined by the two-party correlation C ( ~x,~y ) = P a,b abP ( a, b | ~x, ~y ). In order to obtain a gen-eral process of calculations, we first consider the jointconditional probability distribution of four observers intwo-sided sequential case, which is given as P ( a , a , b , b | ~x , ~x , ~y , ~y )=Tr[( ˆ H a ,a | ~x ,~x ⊗ ˆ H b ,b | ~y ,~y ) ρ AB ] , (2) where ˆ H a ,a | ~x ,~x ≡ ˆ M † a | ~x ˆΠ a | ~x ˆ M a | ~x with ˆΠ repre-sents projection operator and ˆ H b ,b | ~y ,~y is defined in thesame way. The joint conditional probability distributionof any two observers is P ( a i , b j | ~x i , ~y j )= X a i ′ b j ′ ~x i ′ ~y j ′ P ( ~x i ′ , ~y j ′ ) P ( a i , a i ′ , b j , b j ′ | ~x i , ~x i ′ , ~y j , ~y j ′ )(3)with i, i ′ , j, j ′ ∈ { , } and i = i ′ , j = j ′ . Since Alices andBobs are independent observers P ( ~x i , ~y j ) = P ( ~x i ) P ( ~y j )and P ( ~x i ) = P ( ~y j ) = 1 /n for unbiased inputs with n isthe number of measurement settings. Define correlationobservable asˆ W ( ~x i ,~y j ) = X a ,a ,~x i ′ ,b ,b ,~y j ′ a i b j P ( ~x i ′ , ~y j ′ ) ˆ H a ,a | ~x ,~x ⊗ ˆ H b ,b | ~y ,~y , (4)we can obtain correlation C ( ~x i ,~y j ) = Tr[ ˆ W ( ~x i ,~y j ) ρ AB ] . (5)Definition of Eq.(4) can also be used for multiple ob-servers with the generalized definitionˆ H a ,...,a N | ~x ,...,~x N = ˆ M † a | ~x · · · ˆ M † a N − | ~x N − ˆΠ a N | ~x N ˆ M a N − | ~x N − · · · ˆ M a | ~x (6)and ˆ H b ,...,b N | ~y ,...,~y N is defined in the same way as above.The situation of EPR steering in two-sided sequentialcase is more complicated compared to Bell non-localitydue to asymmetry of EPR steering. As a demonstration,here the calculations are limited only to two Alices andtwo Bobs case, in which we ask whether or not Alice1-Bob1 and Alice2-Bob2 can demonstrate EPR steeringsimultaneously with Alice2, Bob2 performing projectivemeasurements. EPR steering quantity S and correspond-ing classical bound B for n measurement settings can bedefined as [26] S n = 1 n | n X m =1 C ( ~x m ,~y m ) | ,B n = max { A m } { λ max ( 1 n n X m =1 A m ˆ σ Bm ) } , (7)whereas S n > B n refutes any local hidden states theory.Here A m ∈ {− , } represents Alice’s declared result forthe m − th measurement setting of Bob’s and λ max ( ˆ O )denotes the largest eigenvalue of ˆ O . Detailed calculationsgive S A − B n = G A · G B . (8)After the measurement of Alice1 and Bob1, the statebecomes ρ A − B ~k = 1 n X ~k X i,j ∈{ + , −} ( ˆ M i~k ⊗ ˆ M j~k ) | Ψ i AB h Ψ | ( ˆ M i~k ⊗ ˆ M j~k ) † , (9)where the ~k denotes the measurement direction of Al-ice1 and Bob1. Then the detailed form of the steeringquantify between Alice2 and Bob2 can be obtained S A − B n = 1 − − F A F B ) P k,l |h k + | l + i| |h k − | l + i| n n − F A F B P k,l Re[ h l − | k + ih l + | k − ih k − | l − ih k + | l + i ] n n , (10)where P k,l denotes double summation P n k =1 P n l =1 with n , n are numbers of measurement settings for Bob1 andBob2 respectively, and {| k ± } ( {| l ± i} ) is the measurementbasis of A1-B1 (A2-B2). Since the distributed state is thesinglet state | ψ − i AB , the measurement directions of Aliceand Bob are chosen to be opposite to maximize the S n .When measurement settings are settled S A − B n is onlydetermined by F A · F B . Consider the case of n = 3 inwhich measurement settings are chosen as { X, Y, Z } andmeasurement strength θ is the same for Alice1 and Bob1we can obtain that S A − B = G , S A − B = 1 − G / G = cos2 θ . The corresponding classical bound is B = 1 / √ G ∈ (0 . , . FIG. 3.
Experimental results.
The steering quantities S A − B , S A − B and S A − B are measured with differentmeasurement strengths of Alice1 and Bob1 ( G A = cos(2 θ A )and G B = cos(2 θ B )). In the upper panel, the results,that Alice1’s and Bob1’s measurement strengths are equal,are presented. The blue line and dots are the theoreticalprediction and experimental results of S A − B , respectively.The red line represents the theoretical predictions of S A − B n with n = { , } , and the corresponding experimental resultsare denoted by the red rhombus and green triangle that al-most overlap. The two horizontal lines are the bounds of B = 0 . B = 0 . θ A = θ B = 0 .
34 and the viola-tion values are 0 . ± . S A − B , 0 . ± . S A − B and 0 . ± . S A − B . The errorbarscome from the Poissonian distribution of photon count thatare too small to present in the figure. In the lower panel, themore experimental results with different G A and G B arepresented. The blue and red surface denote the theoreticalvalues of S A − B and S A − B , and the blue and red dots arethe corresponding experimental results. practice larger n is needed to obtain more violations. Itshould be emphasized here that in the one-sided sequen-tial case multiple EPR steering refers to multiple Bobsaim at steering the state of one Alice, all Bobs have tochoose the same measurement settings [12, 20, 23]. In thetwo-sided sequential case, however, Bobs aim at steeringthe corresponding Alices and their choice of measurementsettings is thus independent of each other. Experimental realization—
We now describe the ex-perimental setup to observe non-locality sharing amongfour observers. As shown in Fig. 2a, a 404 nmsemiconductor laser with 100 mW power is used topump a joint β -barium-borate (BBO) crystal to pro-duce the polarization-entangled photon pairs via thetype I phase-matching spontaneous parametric down-conversion (SPDC) process [27]. By adjusting waveplates placed before the BBO crystal, the singlet state | Ψ − i = ( | H i| V i − | V i| H i ) / √ | H i and | V i refer to horizontal and vertical polarization statesrespectively. The fidelity of the entangled pair state ismeasured to be 98 . ± .
08% [28]. Each half of the entan-gled pair is coupled into different optical fibre and thendistributed to Alices and Bobs with Alice1, Bob1 per-forming optimal weak measurements and Alice2, Bob2performing projective measurements. Coincidence eventsbetween four detectors are registered by avalanche photo-diode single-photon detectors and a coincidence counter.The joint probability distributions for different measure-ment settings and outcomes are extracted from these co-incidence counts within 10 s integral time.As the core part of experimental setup, Fig. 2b realizesoptimal weak measurements described by Kraus opera-tors ˆ M ± | ~k in Eq. (1). The basic idea of the setup isfirstly to transform the measurement basis {| k + i , | k − i} into basis {| H i , | V i} via the basis converter consistsof a quarter-wave plate (QWP) and a half-wave plate(HWP). The interference between two beam displacers(BDs) then realize optimal weak measurements ˆ M ± | ~z with ˆ σ ~z ≡ | H ih H | − | V ih V | [2, 24]. At last another ba-sis converter is used to transform {| H i , | V i} back intothe measurement basis. To be specifically, an input state | φ k i = α | k + i + β | k − i passes the basis converter placedbefore BD1 becomes | φ z i = ˆ R | φ k i = α | H i + β | V i . Pho-tons with polarization | H i be deflected down after pass-ing BD, while nothing happens for | V i . BD1 can beused to entangle the path and polarization degrees offreedom of photons that ˆ U BD | φ z i = α | H i| d i + β | V i| u i with | d i , | u i represent down and up path between twoBDs respectively. With operations of HWPs the stateof photons before BD2 can be written as α (cos θ | V i +sin θ | H i ) | d i + β (cos θ | V i + sin θ | H i ) | u i . Since only mid-dle path out of BD2 is retained, the components | V i| d i and | H i| u i in state | ϕ i are post-selected and path de-gree of freedom is eliminated. With a HWP fixed at π/ α cos θ | H i + β sin θ | V i = ˆ M +1 | ~z | φ z i . With another basisconverter applied subsequently, the full setup completeˆ M +1 | ~k operation corresponding to the +1 outcome of themeasurement. By adjusting the HWP after BD1 from θ/ π/ − θ/
2, operation ˆ M − | ~k corresponding to the − n = 6 measurement settings in EPR steering scenariosuch that B n =6 = 0 . n = 6 set- ting but Alice2 and Bob2 measure with the n = 10 set-ting. In the ( θ A , θ B ) parameter space we have cho-sen different points with equal strength that θ A = θ B ∈ { π/ , π/ , . , π/ , π/ } . Especially, thedouble EPR steering simultaneously can be clearly ob-served when θ A = θ B = 0 .
34. The measured non-locality quantities support theoretical predictions witherrors mainly come from the Poisson distribution of pho-ton counting and imperfection of optical elements. It isclearly shown that while Alice1-Bob1 and Alice2-Bob2can not demonstrate Bell non-locality simultaneously(see Supplementary Materials), they can both demon-strate EPR steering with proper choice of measurementstrength. It is interesting to point out that Alice2 andBob2 can demonstrate one-way EPR steering if Bob1 per-forms no measurement and Alice1 preforms proper weakmeasurement [29].
Discussion and conclusion—
In summary, we haveexplored theoretically and experimentally non-localitysharing in the two-sided sequential case with one en-tangled pair is distributed to multiple Alices and Bobs.We obtain the explicit formula that relates sequentialoptimal weak measurements and Bell quantity includ-ing one-sided sequential case as a special situation. Forone-sided sequential case, it has been shown there existsmeasurement protocols to demonstrate arbitrary manyBell-CHSH inequality violations with biased inputs [1]or unequal sharpness measurement to various Bobs [22].It would be interesting to investigate whether or notsuch measurement protocols exist in two-sided sequentialcase. Due to asymmetry of Alice and Bob and freedom ofchoosing measurement settings, it remains an open ques-tion that whether or not there exists an elegant analyticalformula for EPR steering. Specifically, it would be inter-esting to investigate whether or not more than two pairsof Alice-Bob can demonstrate EPR steering simultane-ously in two-sided sequential case. Using entangled pho-ton pair, we experimentally verify the case of two Alicesand two Bobs in which Alice1, Bob1 performing optimalweak measurements and Alice2, Bob2 performing pro-jective measurements. For Alice1, Bob1 adopt the samemeasurement strength, we observed double EPR steer-ing simultaneously while it is shown that double Bell-CHSH inequality violations cannot be obtained. The re-sults present here not only shed new light on the under-standing of interplay between quantum measurement andnon-locality, but also may have important applicationssuch as unbounded randomness certification [11, 30–32],randomness access code [33, 34] and one-sided device in-dependent quantum key distribution [35–38].
Acknowledgements—
M.-J. Hu acknowledges H. M.Wiseman, Eric Calvencanti and Michael J. W. Hall forvaluable discussions. Part of the theoretical work weredone during his visiting in Griffith University. J. Zhu ac-knowledges X.-J. Ye, Y. Xiao, S. Cheng, and R. Wangfor helpful discussions. This work is funded by theNational Natural Science Foundation of China (GrantsNos. 11674306 and 92065113) and Anhui Initiative inQuantum Information Technologies.
Note:
A more genernal conclusion about CHSH-Bellinequality in sequential measurement structure has beennoted in a recent work [39].
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SUPPLEMENTARY MATERIALSA. OPTIMAL WEAK MEASUREMENTSWeak measurement
For a projective measurement {| k + i , | k − i} in a two-level system with h k + | k − i = 0, the projective measurementoperators are P + = | k + ih k + | , P − = | k − ih k − | and σ k = P + − P − = | k + ih k + | − | k − ih k − | .For a weak measurement, the positive-operator valued measurement (POVM) can be written as Kraus operatorsˆ M ± = cos( θ ) | k ± ih k ± | + sin( θ ) | k ∓ ih k ∓ | . (S1)For an initial state | ψ i , the result of the weak measurement is ρ ± = ˆ M ± | ψ ih ψ | ˆ M †± / Tr( ˆ M ± | ψ ih ψ | ˆ M †± )= F | ψ ih ψ | + (1 − F )( P + | ψ ih ψ | P + + P − | ψ ih ψ | P − ) , (S2)with probability P ( ± ) = Tr( ˆ M ± | ψ ih ψ | ˆ M †± ) = G h ψ | P ± | ψ i + (1 − G ). The meanings of factors F and G are describedin Ref. [1] and will be illustrated in the following. Realization of weak measurement
For an initial two-level state | ψ i = α | i + β | i , a projective measurement can be realized by coupling a pointerstate that can be written as ( α | i + β | i ) | i pointer → α | i| + i pointer + β | i|−i pointer (S3)with h + |−i = 0. However, for a weak measurement, the evolution of the pointer state is | ψ i = ( α | i + β | i ) | i pointer →| ψ ′ i = α | i| + ′ i pointer + β | i|− ′ i pointer , (S4)where | + ′ i = cos( θ ) | i + sin( θ ) | i and |− ′ i = sin( θ ) | i + cos( θ ) | i , with h + ′ |− ′ i = sin(2 θ ) indicating the strength ofthe weak measurement. Then a projective measurement with the basis {| i , | i} on the pointer state is performed.Hereto a completed weak measurement of ˆ M + = cos( θ ) | ih | + sin( θ ) | ih | and ˆ M − = cos( θ ) | ih | + sin( θ ) | ih | isfullfiled. The probability of two outcomes are P (+) = |h | ψ ′ i| and P ( − ) = |h | ψ ′ i| . Factors of F and G in the weak measurement In our experiment, the factors F and G is defined as the same as Ref. [1, 2]. F denotes the disturbance of themeasurement, that can be written as F = h + ′ |− ′ i . G denotes the information gain of the measurement, that can bewritten as G = 1 − |h |− ′ i| − |h | + ′ i| , where the |h |− ′ i| and |h | + ′ i| are error rates of the weak measurement.Here the optimal condition, F + G = 1, is satisfied.upplemental Material –2/3 B. BELL NON-LOCALITY IN TWO-SIDED SEQUENTIAL MEASUREMENTS CASE
FIG. S1.
Experimental results of sequential Bell test.
The results verified Eq. (S6) in which G A = G B = 1 due toAlice2, Bob2 performing projective measurements. Double Bell-CHSH inequality violations is observed only when Alice1 orBob1 performing almost no measurement such that the situation is equivalent to one-sided sequential case. When Alice1, Bob1adopt the same measurement strength , double Bell-CHSH inequality violations can not be obtained. Bell quantity I in the Bell test scenario is defined as I = | C ( ~x ,~y ) + C ( ~x ,~y ) + C ( ~x ,~y ) − C ( ~x ,~y ) | , (S5)whereas I > ~x = Z, ~x = X, ~y = ( X − Z ) / √ , ~y = − ( X + Z ) / √ √
2. In the case of two Alices and twoBobs with optimal weak measurements are performed by observers calculations based on the method of Ref. [1] give I A − B = 2 √ G A · G B ,I A − B = √
22 (1 + F A ) G A · (1 + F B ) G B ,I A ( B ) − B ( A ) = √ G A ( B ) · (1 + F B ( A ) ) G B ( A ) (S6)Due to symmetry configuration, the Bell quantity of Alice1-Bob2 and Bob1-Alice2 have the same form. The resultsare compatible with one-sided sequential case obtained by R. Silva et al if Alice1 performs no measurement with F A = 1 , G A = 0 and Alice2, Bob2 perform projective measurements with G A = G B = 1. It can be shown fromthe above equations that double Bell-CHSH inequality violations happens only when Alice1 or Bob1 performs almostno measurement and situation is very close to one-sided case. Furthermore, Alice1-Bob1 and Alice2-Bob2 can notdemonstrate Bell non-locality simultaneously. For the more general case with arbitrary N Alices and M Bobs, theexplicit analytical form of Bell quantity for arbitrary Alice and Bob is derived. It is concluded that no more thandouble Bell-CHSH inequality violations can be obtained in this scenario under the unbiased input condition.Similarly, in the two-sided sequential case in which one entangled pair is distributed to arbitrarily many N Alicesand M Bobs for sequential optimal weak measurements, the Bell quantity for arbitrary Alice and Bob, under theunbiased input condition, satisfies I A r − B s = 2 √ ( r − · ( s − (1 + F A ) · · · (1 + F A r − ) · G A r × (1 + F B ) · · · (1 + F B s − ) · G B s (S7)with r ≤ N, s ≤ M and G A N = G B M = 1 if the last Alice and Bob performing projective measurements. For N = 1,it naturally reduces to one-sided sequential case with one Alice and multiple Bobs. ∗ [email protected] † [email protected][1] R. Silva, N. Gisin, Y. Guryanova and S. Popescu, Multiple Observers Can Share the Nonlocality of Half of an EntangledPair by Using Optimal Weak Measurements , Phys. Rev. Lett. , 250401 (2015).[2] M.-J. Hu, Z.-Y. Zhou, X.-M. Hu, C.-F. Li, G.-C. Guo and Y.-S. Zhang,
Observation of non-locality sharing among threeobservers with one entangled pair via optimal weak measurement , npj Quantum Inf4