Both qubits of the singlet state can be steered simultaneously by multiple independent observers via sequential measurement
aa r X i v : . [ qu a n t - ph ] F e b Both qubits of the singlet state can be steered simultaneously bymultiple independent observers via sequential measurement
Kun Liu , , , Tongjun Liu , , , Wei Fang , , , Jian Li , , , ∗ and Qin Wang , , † Institute of quantum information and technology,Nanjing University of Posts and Telecommunications, Nanjing 210003, China. “Broadband Wireless Communication and SensorNetwork Technology” Key Lab of Ministry of Education,NJUPT, Nanjing 210003, China. and “Telecommunication and Networks” National EngineeringResearch Center, NJUPT, Nanjing 210003, China. (Dated: February 25, 2021) Abstract
Quantum correlation is a fundamental property which distinguishes quantum systems from clas-sical ones, and it is also a fragile resource under projective measurement. Recently, it has beenshown that a subsystem in entangled pairs can share nonlocality with multiple observers in se-quence. Here we present a new steering scenario where both subsystems are accessible by multipleobservers. And it is found that the two qubits in singlet state can be simultaneously steered bytwo sequential observers, respectively.PACS number(s): 03.67.Dd, 03.67.Hk,42.65.Lm ∗ [email protected] † [email protected] . INTRODUCTION The correlation between the entangled distant quantum systems can be beyond any classi-cal systems and plays an important role in both fundamental quantum physics and quantuminformation science. Quantum correlation is also a powerful resource in quantum informa-tion processes, it is widely applied in protocols such as teleportation, secure key distribution,quantum computation and etc. Generally, quantum correlations can be descripted as a hier-archy with three inequivalent concepts, quantum entanglement [1], Einstein-Podolsky-Rosen(EPR) steering [2], and Bell nonlocality [3]. For bipartite systems, the weakest quantumcorrelation, entanglement, means that the state density cann’t be decomposed into any sep-arable one. Bell nonlocality, the strongest form, allows the violation of Bell inequality basedon the joint probability distribution of local measurements by two spatially separated play-ers. As the intermediate form, EPR steering, which was first introduced by Schr¨odinger,and then formally defined by Wiseman et al. [4, 5], is verified if one of the two spatiallyseparated parties can steer, via local measurements, the set of conditional quantum state ofthe subsystem on the other player’s side.For the certification of these correlations [6], the violation of Bell inequalities can verifynonlocality without any assumption of the measurement devices on both sides, which isconsidered as a device-independent scenario [7]. Quantum entanglement can be certifiedvia quantum state tomography with fully characterized measurement devices on both side,which is a device-dependent scenario. And for EPR steering, the measurement devices onone side need to be fully characterized, which is also called one-side device-independentscenario, or a semi-device-independent one [8].Usually, quantum correlation is investigated for the n particles in entanglement, eachof which is accessible by one and only one space-separated observer for projective mea-surements [9]. Some studies use different degree of freedom of parties. In Ref. [10], thesteerability of three entangled qubits is demonstrated with two photons with polarizationand spatial degree of freedom. However, it is interesting to consider whether multiple ob-servers sequentially measuring the same subsystem can share different quantum correlation.Generally, for a sharp measurement, a system would collapse into one of the eigenstates ofthe projector, where the system would be separable from other systems. However, unsharp,or weak measurement can preserve some entanglement in the post-measurement state. Weak2easurement refers to the measurement with intermediate coupling strength between thesystem and the probe. In contrast to strong projective measurement, weak measurementcan be less destructive and also remain some original features of measured system. It hasbeen proved to be a good method for signal amplification [11], state tomography and solvingquantum paradoxes. For a pair of entangled particles, it is possible to reserve some quan-tum correlation after measurement at intermediate strength [12]. Silva et al. [13] showedthat multiple observers can share nonlocality on a pair of entangled qubits by using op-timal weak measurement in a Bell scenario. One qubit is accessible by a single observer,Alice, while the other one is accessible by several observers, Bobs, sequentially. Withoutany communication between Bobs, the CHSH-Bell inequality between Alice and two Bobs,respectively, can be violated simultaneously. The violation of twice CHSH-Bell inequality isdemonstrated [14] using a pair of polarization-entangled photons. Brown and Colbeck [15]shown that arbitrary number of Bobs can share the CHSH nonlocality with a single Alicewith a single maximally entangled qubit pair, if, for each Bob, one measurement is sharpwhile the other is optimal weak. It brings a further understand of the fundamental limitof entanglement applications in device-independent scenarios. Foletto et al. restudied thecorrelation between Alice and individual Bobs [16]. that a system can still sustain entan-glement and violate CHSH inequality if arbitrary number of observers (Bobs) implementsequential unsharp measurements on one subsystem and the other observer, Alice, performsimplement sharp measurement (Alice). The measurements on the subsystem are based onthe history of previously performed measurements and observed outcomes, so when appro-priate measurement strength is choosed, the path of a tree-like structure (the protocol) willextend infinitely.Shneoy et al. [17] theoretically investigated the EPR steering with multi-observer. it isfound that with the increase of dimension in system, if measurement strength η i all haveappropriate values, the number of that Bobs can steer Alice is found to be N Bob ∼ d/ log d .For the case of qubits d = 2, each Bob knows his respective position in the sequence, as heperforms a measurement with a well chosen strength η i , the maximal sequence of Bob thatcan steer Alice is Five. And Choi et al. [18] experimentally demonstrated this process byaccomplishing a sequential steering for three Bobs using photonic system.To date, most discussions of quantum correlation for multiple observers with single entan-gled qubit pair focus on that one qubit is accessible by several Bobs sequentially, while the3ther one is accessible by one Alice. In this letter, we exploit the EPR steerability of a maxi-mally entangled qubit pair, where each qubit is accessible by multiple observers sequentially.It is worth mentioning a similar scene in revealing of hidden nonlocality [19, 20], where bothqubits are locally filtered before the final sharp measurements. The local filter on each qubitcan be considered as a special positive operator-valued measurement (POVM), where onlyone output is delivered to the successive observer, while the weak measurements above areone type of POVM with two outputs, both of which is accessible for all successive observers.The EPR steering scenario and linear steering inequality are presented as the basic toolin section II. A detailed steering protocol for multiple observers using weak measurementis given in section III. The violation of steering inequality for different Alice-Bob pairs isdiscussed followed by the conclusion in last section. II. EPR STEERING AND STEERING INEQUALITYA. EPR Steering scenario
Considering a pair quantum subsystems, A and B , is shared by two spatially-separatedobservers, Alice and Bob, where each subsystem is accessible by one and only one observer.Generally said, steering describes the ability that one observer, Alice (Bob), via the mea-surement on A ( B ), can nonlocally steer the other’s subsystem B ( A ). FIG. 1. General scenario in quantum steering. Alice and Bob are two spatially separated ob-servers. The measurement device of Alice can be represented by a black box, with the classicalinput x (choice of measurements) and output a . Bob performs tomography and records his ownmeasurement choice y and the result b . To be specific, quantum steering can be described as a scenario with two spatially-4eparated players, Alice and Bob, sharing a bipartite quantum state ρ AB . Alice implementsa measurement with the inputs x on her subsystem A and outcomes a . Both x and a areclassical variables. Bob can fully characterize the state in his subsystem with trusted de-vices. Conditioned on Alice’s declare of ( x, a ) with probability p ( a | x ), Bob would found hissubsytem B collapsed into the state ˜ σ B a | x (unnormalized) [21]:˜ σ B a | x = tr A (cid:0) M a | x ⊗ I B · ρ AB (cid:1) , (1)The normalized state is σ B a | x = ˜ σ B ax p , where p = Tr (cid:16) ˜ σ B a | x (cid:17) . The set of Bob’s conditioned statesfor all measurement choice and results of Alice forms an assemblage. The assemblage obeya local hidden state (LHS) model, if each state from the assemblage can be written as, σ Ba | x = Z f ( λ ) P A ( a | x , λ ) σ λ d λ (2)where, λ is a random variable with the distribution f ( λ ) as a strategy of Alice. ThenAlice cannot steer Bob’s subsystem as the state conditioned on ( x, a ) can be chosen fromthe pre-existing ensemble { σ λ } with the distribution f ( λ ) by Alice’s declaration probabilityP A ( a | x, λ ) [17]. Otherwise, if the assemblage cannot adpot any LHS model, Bob can beconvinced that his subsystem can be steered by Alice remotely. And steerability could beone-directional, where only one player (Alice) can steer the subsystem accessible by theother one (Bob) [22]. Here, we focus on the bidirectional steering scenario and show thatdue to the existence of sequential and weak measurement, both Alices and Bobs can steerthe remote subsystem. B. Steering Inequality for qubits
The EPR steerability can be verified by the violation of linear steering inequalities [23],just like the Bell inequality for Bell nonlocality. In the scenario, Alice declare her result a k ∈{− , +1 } for each choice of measurement k ∈ { , , . . . , n } , and Bob makes the correspondingdichotomic measurement ˆ B k on his subsystem. The inequality reads:Sn = 1n n X k=1 h a k ˆ B k i ≤ Cn (3) C n is the upper bound of inequality allowed by the LHS model,Cn = max { a k } ( λ max n X k =1 a k B k !) (4)5here λ max (G) is the maximum eigenvalue of operator G . To verify the steerability, bothplayers can perform dichotomic quantum measurement, { ˆ A k } and { ˆ B k } on the qubit A and B respectively, can calculate the expected value of steerability parameter S n from thestatistics. If the inequality is violated (satisfied S n exceed C n ), it means that Alice cansteer Bob and Bob can also steer Alice, vice versa. And the LHS bound C n changes withthe number of measurements allowed, C2 = 1 / √ ≈ . / √ ≈ . ≈ . ≈ . III. MULTI-OBSERVER STEERING VIA SEQUENTIAL MEASUREMENT
In most study on multi-observer nonlocality with single entangled pair, one subsystem ismeasured sequentially by different observers, while the other one is accessible by only oneobserver. Here we call it the 1 vs n nonlocality. The model is shown in figure 2. For multiplesteering protocol, it is the one directional steering on the same subsystem. In Ref. [17], itis shown that for optimal weak measurements of all Alices, the number of that Alices cansteer Bob is found to be N Alice ∼ d/ log d . For qubit system, d = 2, the maximal number ofAlices is 5. FIG. 2. The scenario of 1 vs n nonlocality (steering) verification. State ρ is a shared state, onefor all n Alices and the other only for Bob.
Alice ∼ Alice n − implement weak measurement withstrength η ∼ η n − , Alice n and Bob implement strong measurement. Bob, half black and half white,represents two modes in different conditions. In Bell-nonlocality scenario, black box denotes nopreassumption on the measurement, which is device-dependent. However, in steering scenario,white box allows Bob to perform trusted quantum measurement or state tomography, which isdevice-independent.
6n our protocol, the maximally entangled two subsystems A and B , are sequentiallyaccessible by two observers, ( Alice , Alice ) and ( Bob , Bob ), respectively. On the oneside with subsystem A , Alice randomly selects its input k ∈ { , , . . . , n } , performs aweak measurement A ( k, η A ) with the sharpness η A ∈ [0 , a anddelivers the post-measurement subsystem A to Alice . Here, Alice have no knowledgeof the measurement choice or the outcome of Alice . Then with the randomly choice of j ∈ { , , . . . , n } , Alice performs her measurement A ( j ). As there is no further observer of A , the measurements by Alice always sharp ones. The process is similar on the other sidewith subsystem B , where Bob performs weak measurements and Bob performs the sharpones (see Fig.3). FIG. 3. A new type of quantum network in this paper. Two subsystem share state ρ , Alice and Bob implement weak measurement and the parameter strengths are η A and η B . Alice and Bob implement strong measurement. The box with black and white shows the observer can steer otherand also can be steered by others. All the dichotomic measurements implements by four the players can be represented byPOVMs with two Kraus operators. Being a frequently-used operator in this task, a set ofKrause operators can be defined as:K ±| k = 1 √ (cid:20)p ± η · (cid:18) I + m k · σ (cid:19) + p ∓ η · (cid:18) I − m k · σ (cid:19)(cid:21) (5)Here, k is k -th measurement, ± are the outcomes of dichotomic measurement, and η is thestrength of the measurement strength for Alice s or
Bob s. The observable of POVM can bewritten as [17]: ˆE k = M + | k − M −| k = η m k · σ (6)7here M ±| k = K †±| k · K ±| k (7)Here, σ represents three Pauli matrixes ( σ x , σ y , σ z , ), and m k denotes a unit vectoron Bloch sphere. In other word, the set all observables can be represent by n vectors { m k } k ∈{ , ,...,n } and the measurement strength η , for each player. The measurement is sharpfor η = 1. And for η = 0, both Kraus operators are proportional to the identity, where thesystem is unchanged and no information can be inferred from the measurement.To verify the steerability, the choice of measurement set should be optimised. Saunders et al. [23] have shown that, for the singlet state | Ψ − i = √ ( | i − | i , the optimal measure-ments can be represented by the vectors { m k , − m k } related to the Platonic-solid dependingon n , the number of measurements allowed for each player. Measurement settings corre-spond to different solids in Bloch space, octahedron ( n = 3), cube ( n = 4), icosahedron( n = 6) and dodecahedron ( n = 10), except the case of n = 2, where the four vectors forma square in a plane. Antipodal pairs of face centres or vertices of a Platonic solid definemeasurement vectors { m k } . As for n = 2 , ,
4, the vectors are from the origin of coordinateto the face centres, while for n = 6 ,
10, to the vertices of solid [24].Let’s consider the steering scenario. The two qubits in state ρ i ′ j ′ is delivered to a pairof observers, A to Alice i and B to Bob j , respectively. Alice i performs the measurementaccording the random input k with the Kraus operators { K η A + | k , K η A −| k } and gets an output a ∈ { + , −} . And Bob j performs the measurement according the random input l withthe Kraus operators { K η B + | l , K η B −| l } and gets an output b ∈ { + , −} . The (unnormalized)conditional post-measurement two-qubit state would be expressed as, ρ ija | k,b | l = K η A a | k ⊗ K η B b | l ρ i ′ j ′ K η A † a | k ⊗ K η B † b | l . (8)with the conditional probability p ( a, b | k, l ) = Tr ρ ija | k,b | l . So the steerability parameter for Alice i and Bob j is written as,ˆ S ijn = 1 n n X k =1 X a,b ∈{ + , −} ( − a + b p ( a, b | k, k ) . (9)And the average post-measurement state after Alice i ’s and Bob j ’s measurements is ρ ij = 1 n n X k,l =1 X a,b ∈{ + , −} ρ ija | k,b | l . (10)8 (cid:3) (cid:3) (cid:3) a b cd e FIG. 4. The bullet symbols show the orientations of pure states in optimal cheating ensembles fortwo-qubit states[23]. From a to e, we can get different pair of coordinates from n = 2 , , , , which can be delivered to subsequential observers for further investigation. In our multi-observer steering scenario, the initial state is in singlet one, ρ = | Ψ − ih Ψ − | . If S ijn > C n ,the bidirectional steerability can be declared between Alice i and Bob j from state ρ i ′ j ′ , where i ′ = i − j ′ = j −
1, and delivering the post-measurement state ρ ij . To investigatethe steerability between Alice i and an observer after Bob j , Alice i has to postpone hermeasurement. Equivalently, she can perform the weak measurements with the sharpness η a = 0 and keep the post-measurement subsystem A for further measurements. IV. RESULT AND CONCLUSION
For our multi-observer steering scenario, each subsystem is accessible by two players re-spectively. So the steering should be investigated under four observer-pairs, S , S , S , and S . Let the two qubit initially in the singlet state, we numerically calculate the steerabilityparameters.Considering that both Alice and Bob perform independent measurement with the same9 ABLE I. Measurement directions for different settings. (a=(1+ √
5) / 2)Measurement setting n (X,Y,Z)2 (1,0,1),(1,0,-1)3 (1,0,0),(0,1,0),(0,0,1)4 (1,1,1),(1,-1,-1),(1,1,-1),(1,-1,1)6 (0,1,a),(0,1,-a),(1,a,0),(1,-a,0),(a,0,1),(a,0,-1)10 (0,1/a,a),(0,1/a,-a), (1/a,a,0),(1/a,-a,0)(a,0,1/a),(a,0,-1/a), (1,1,1),(1,-1,-1),(1,1,-1),(1,-1,1) sharpness, η A = η B = η . The theoretical predictions of EPR steering correlation S n ( S ij )and the violation of steering inequalities are showed in Fig. 5. The steerability parametersare calculated with a varying strength η ∈ [0 ,
1] using the measurement set based on thePlatonic solid. It is interesting that the steerability parameters are independent of n , thesize of the measurement set. And due to the symmetry between Alice s and
Bob s, it isfound that S = S . When both Alice and Bob perform the sharp measurement, onlythe observer-pair ( Alice , Bob ) can steer each other. When both Alice and Bob performthe zero-sharpness measurement, only the observer-pair ( Alice , Bob ) can steer each other.Around η ∼ .
76, it is found that all the four observer-pairs can violate against the linearsteering inequalities for n = 3 , , ,
10. That means, every player can steer the state of theremote subsystem. The steerability for the observer-pair (
Alice , Bob ) and ( Alice , Bob )is a trade-off with η , while intermediate between them for ( Alice , Bob ) and ( Alice , Bob ).In Fig. 6, we fixed the sharpness of Bob ’s measurements η B = 0 . Bob s is very close, while being a trade-off between two
Alice s with
Alice ’s measurement strength η A .In conclusion, a new type of steering scenario with a single pair of entangled subsystemis presented in this article, each of which is accessible by multiple observers. We alsoshowed that, by using weak measurements under the Platonic-solid configure, all the fourindependent observers, two sequentially measuring on the subsystems respectively, can steerthe remote subsystem. Besides, when fixing Bob ’s measurement strength as above, itwould be more interesting to study the asymmetry of the steerability originating from the10 A B S t eer i n g P a r a m e t er S i j FIG. 5. Theoretical predictions of EPR steering correlation S n ( S ij ) changing with the sharpness η , where we set η A = η B = η . Steering parameters S ij is function of η , η ∈ (0 ,
1) denotes the equalstrength of measurement in Alice and Bob. The blue solid curve represents S11 (
Alice & Bob ) intheoretical predictions, because S12=S21 is still true, black (green) solid curve represents S21/S12( Alice & Bob / Alice & Bob ), and the red one represents S22 ( Alice & Bob ); The magenta dashedhorizontal line represents steering bound C2, and the black one shows C3 (C4), red dashed linerepresents C6, blue dashed line represents C10. entangled state and other measurement sets. In such scenario, it’s also worth studying howmany observers can steer the remote subsystem if high-dimensional entanglement is shared.Furthermore, recently, it is found that the sequential multi-party quantum random accesscode can witness the quantum channel [25], verify the unsharp measurement [26, 27], and begeneralized to characterise the correlations of quantum network in prepare-and-measurementscenarios [28]. We believe that our steering scenario can also be implemented for quantumcommunication in similar network. 11 A S i j FIG. 6. The variation of S ij with η A , conditional on fixed η B . Value S ij is the function of variable η A . Only one variable affects the violations of different measurement setting n. Herein, S11, bluesolid line; S21, black solid line; S12, magenta solid line; S22, red solid line; C2, magenta dashedline; C3/C4 black dashed; C6, red dashed line; C10, blue dashed line. ACKNOWLEDGMENTS
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