Multiparty-controlled remote preparation of four-qubit cluster-type entangled states
aa r X i v : . [ qu a n t - ph ] N ov Multiparty-controlled remote preparation of four-qubit cluster-type entangled states
Dong Wang a,b,c, ∗ , Liu Ye b, † , Sabre Kais a,d, ‡ a Department of Chemistry and Birck Nanotechnology Center,Purdue University, West Lafayette, IN 47907, USA b School of Physics & Material Science, Anhui University, Hefei 230601, China c National Laboratory for Infrared Physics, Shanghai Institute of Technical Physics,Chinese Academy of Sciences, Shanghai 200083, China d Qatar Environment and Energy Research Institute, Qatar Foundation, Doha, Qatar
We present a strategy for implementing multiparty-controlled remote state preparation (MCRSP)for a family of four-qubit cluster-type states with genuine entanglements while employing,Greenberg-Horne-Zeilinger-class states as quantum channels. In this scenario, the encoded informa-tion is transmitted from the sender to a spatially separated receiver via the control of multi-party.Predicated on the collaboration of all participants, the desired state can be entirely restored withinthe receiver’s place with high success probability by application of appropriate local operations andnecessary classical communication . Moreover, this proposal for MCRSP can be faithfully achievedwith unit total success probability when the quantum channels are distilled to maximally entangledones.
PACS numbers: 03.67.-a; 03.67.Hk
I. INTRODUCTION
An important focus in the field of quantum information processing (QIP) has been the secure and faithful transmis-sion of information from one node of quantum network to another non-local node with finite classical and quantumresources. Quantum teleportation (QT) originated from the pioneering work of Bennett [1] is one application of non-local physics which may accomplish such a task. the central idea of QT is to deliver magically quantum informationwithout physically transporting any particles from the sender to the receiver by means of an established entangle-ment. Apart from QT there exists another such efficient method, the so-called remote state preparation (RSP) [2–4].RSP allows for the transfer of arbitrary known quantum states from a sender (Alice) to a spatially distant receiver(Bob), provided that the two parties share an entangled state and may communicate classically. Although both QTand RSP are able to achieve the task of information transfer [5–7], there are some subtle differences between QTand RSP which can be summarized as follow: (i) Precondition. In RSP, the sender of the states is required to becompletely knowledge about the prepared state. In contrast, neither the sender nor the receiver necessarily possessesany knowledge of the information associated with the teleported states in QT. (ii) State existence. The state to beteleported initially inhabits a physical particle in the context of QT, while this is not required in RSP. That is to say,the sender in RSP is full aware of the information regarding the desired state, without any particle in such a statewithin his possession. (iii) Resource trade-off. Bennett [4] has shown that quantum and classical resources can betraded off in RSP but cannot in QT. In standard teleportation, an unknown quantum state is sent via a quantumchannel, involving 1 ebit, and 2 cbits for communication. In contrast, if the teleported state is known to the senderprior to teleportation, the required resources can be reduced to 1 ebit and 1 cbit in RSP at the expense of lower ∗ [email protected] (D. Wang) † [email protected] (L. Ye) ‡ [email protected] (S. Kais) success probability, half of that in QT. However, Pati [3] has argued that for special ensemble states (e.g., states oneither the equator or great polar circle of the Bloch sphere) RSP requires less classical information than teleportationwith the same unitary success probability.Owing to its importance in QIP, RSP has received great attention and a large number of theoretical investigationshave been proposed [8–38]. Specifically, there have been investigations concerning: low-entanglement RSP [8], optimalRSP [9], oblivious RSP [10, 11], RSP without oblivious conditions [12], generalized RSP [13], faithful RSP [14], jointRSP (JRSP) [15–27], Multi-controlled joint RSP [28], RSP for many-particle states [29–35], RSP for qutrit states [36]and continuous variable RSP in phase space [37, 38]. While, several RSP proposals by means of different physicalsystems have been experimentally demonstrated as well [39–45]. For examples, Peng et al. investigated a RSP schemeusing NMR [39], Xiang et al. [40] and Peters et al. [41] proposed other two RSP schemes using spontaneous parametricdown-conversion. Julio et al. [45] reported the remote preparation of two-qubit hybrid entangled states, including afamily of vector-polarization beams; where single-photon states are encoded in the photon spin and orbital angularmomentum, and then the desired state is reconstructed by means of spin-orbit state tomography and transversepolarization tomography.Recently, many authors proceed to focus on RSP for cluster-type state by exploring various novel methods [46–51];because cluster states are one of the most important resources in quantum information processing and can be efficientlyapplied to information processing tasks, such as: quantum teleportation [52], quantum dense coding [53, 54], quantumsecret sharing [55], quantum computation [56], and quantum correction [57]. In general, a cluster-state is expressedas | Ω N i = 12 N/ N O s =1 ( | i s Z ( s +1) + | i s ) , (1)with the conventional use of Z is a pauli operator and Z N +1 ≡
1. It has been shown that one-dimensional N -qubitcluster states are generated in arrays of N qubits mediated with an Ising-type interaction. It may easily be seenthat the state will be reduced into a Bell state for N = 2 (or 3); the cluster states are equivalent to Bell states (orGreenberger-Horne-Zeilinger (GHZ) states) respectively under stochastic local operation and classical communication(LOCC). Yet when N >
3, the cluster state and the N -qubit GHZ state cannot be converted to each other by LOCC.When N = 4, the four-qubit cluster-state is given by | Ω i = 12 ( | i + | i + | i − | i ) . (2)In this work our aim is to examine the implementations of multiparty-controlled remote state preparation (MCRSP)for a family of four-qubit cluster-type entangled states with the aid of general quantum channels [46–51].The paper is structured as follows: in the next section, we present the MCRSP scheme for four-qubit cluster-type entangled states with multi-agent control by the utilization of GHZ-class entanglements as quantum channels.The results show that the desired state can be faithfully reconstructed within Bob’s laboratory with high successprobability. Moreover, the required classical communication cost (CCC) and total success probability (TSP) will bediscussed. Finally, features of our proposed scheme are detailed followed by a conclusion section. II. MCRSP FOR FOUR-QUBIT CLUSTER-TYPE ENTANGLED STATES
Suppose there are ( m + n + 2) authorized participants, say, Alice, Bob, Charlie , · · · , Charlie n , Dick , · · · , andDick m (where m, n ≥ i andDick j are truthful agents. Now, Alice would like to assist Bob remotely in the preparation of a four-qubit cluster-typeentangled state | P i = α | i + βe iϕ | i + γe iϕ | i + δe iϕ | i , (3)with the control of the agents, where α , β , γ , δ and ϕ i are real-valued, satisfy the normalized condition α + β + γ + δ = 1, and ϕ i ∈ [0 , π ]. In order to obtain MCRSP, Alice, Bob, Charlie i and Dick j share previously generatedgenuine quantum resources – i.e., GHZ entanglements – which are given by | Υ (1) i A A B B C ··· C n = , X k a k | k i ⊗ ( n +4) A A B B C ··· C n , (4)and | Υ (2) i A A B B D ··· D m = , X l b l | l i ⊗ ( m +4) A A B B D ··· D m , (5)respectively, without loss of generality a , b ∈ R , and these bounds | a | ≥ | a | and | b | ≥ | b | are maintained. Initially,qubits A , A , A and A are sent to Alice, qubits B , B , B and B to Bob, C i to Charlie i ( i ∈ { , · · · , n } ) and D j to Dick j ( j ∈ { , · · · , m } ).For implementing MCRSP, the procedure can be divided into the following steps: Step 1 . Firstly, Alice makes a two-qubit projective measurement on her qubit pair ( A , A ) under a set of completeorthogonal basis vectors {|L ij i} composed of computational basis {| i , | i , | i , | i} , which can be written as( |L i , |L i , |L i , |L i ) T = Q ( | i , | i , | i , | i ) T , (6)where, Q = α βe − iϕ γe − iϕ δe − iϕ β − αe − iϕ δe − iϕ − γe − iϕ γ − δe − iϕ − αe − iϕ βe − iϕ δ γe − iϕ − βe − iϕ − αe − iϕ . (7)Since the total systemic state taken as quantum channels can be described as | Ψ T i = | Υ (1) i A A B B C ··· C n ⊗ | Υ (2) i A A B B D ··· D m = , X i,j |L ij i A A ⊗ |X ij i A A B B B B C ··· C n D ··· D m , (8)where the non-normalized state |X ij i ≡ A A hL ij | Ψ T i ( i, j = 0 ,
1) is obtained with probability of 1 / N ij , where N ij corresponds to the normalized parameter of state |X ij i . Step 2 . According to her own measurement outcome |L ij i , Alice makes an appropriate unitary operation ˆ U ( ij ) A A on her remaining qubit pair ( A , A ) under the ordering basis {| i , | i , | i , | i} , which is accordingly one ofˆ U (00) A A = diag(1 , , , , (9)ˆ U (01) A A = diag( e iϕ , − e − iϕ , e i ( ϕ − ϕ ) , − e i ( ϕ − ϕ ) ) , (10)ˆ U (10) A A = diag( e iϕ , − e i ( ϕ − ϕ ) , − e − iϕ , e i ( ϕ − ϕ ) ) , (11)and ˆ U (11) A A = diag( e iϕ , e i ( ϕ − ϕ ) , − e i ( ϕ − ϕ ) , − e − iϕ ) . (12)Subsequently, Alice measures her qubits A and A under the a set of complete orthogonal basis vectors {|±i := √ ( | i ± | i ) } , and broadcasts her measured outcomes via a classical channel. Incidentally, all of the authorized (S1) (S2) (S3)(S5)(S5)(S4) B B A A D C C n-1 C n C D D n-1 D n A A B B D C C n-1 C n C D D n-1 D n A A D C C n-1 C n C D D n-1 D n A A A A A A B B B B }} C b it s C b it s C b it s TQPM ( ) ˆ ijA A U SQPM D C C n-1 C n C D D n-1 D n A A A A ˆ B B B B U ( ) ˆ A ijB B B V B B B B B B B B B B B B B A B B B B B A SQPM B B B B B A AliceBob Alice AliceBob BobBobBobBobBobAlice
FIG. 1: Schematic diagram for MCRSP implementation. The procedure is explicitly decomposed as above Figures (S1) ∼ (S5).The ellipse represents two-qubit projective measurement (TQPM) under the set of basis vectors {|L ij i} ; the square representssingle-qubit projective measurement (SQPM) under the set of basis vectors {|±i} ; rectangle represents operating a bipartitecollective unitary transformation ˆ U ( ij ) A A ;the triangle represents SQPM under the set of basis vectors {| i , | i} ; the sexanglerepresents performing single-qubit unitary transformations ˆ U B B B B on Bob’s qubits; the circle represents making a tripletcollective unitary operation ˆ V ( ij ) B B B A ; Cbits represents classical information communication. participators make an agreement in advance that cbits ( i, j ) correspond to the outcome |L ij i A A , and cbits ( p, q )relate to the measuring outcome of qubits A and A , respectively. For simplicity, we denote p, q = ( , if | + i is probed1 , if |−i is probed . Step 3 . The agents proceed to carry out single-qubit measurements under the set of vector basis {|±i} on thequbits respectively, and later inform Bob of the results via classical channels. We assume that the cbit x i correspondsto the outcome of the agents C i , and y j corresponds to the outcome of the agents D j , where the values of x i and y j have been previously denoted as p and q , respectively. And we have g = Σ nx =1 x i , mod ⊕ h = Σ my =1 y j , mod ⊕ g = 0 and h = 0; II) g = 0 and h = 1; III) g = 1 and h = 0; andIV) g = 1 and h = 1. Step 4 . In response to the different measuring outcomes of the sender and agents, Bob operates on his qubits B , B , B and B with an appropriate unitary transformation ˆ U B B B B . Step 5 . Finally, Bob introduces one auxiliary qubit B A with initial state of | i . And then he makes triplet collectiveunitary transformation ˆ V ( ij ) B B B A on his qubits B , B and B A under a set of ordering basis vector {| i , | i , | i , | i , | i , | i , | i , | i} , which is given byˆ V ( ij ) B B B A = W ij U ij U ij −W ij ! , (13)where W ij and U ij are 4 × W = diag( a b a b , a a , b b , , (14) U = diag( r − ( a b a b ) , r − ( a a ) , r − ( b b ) , , (15) TABLE I: ijpqgh denotes the corresponding measurement outcomes from the authorized participants, ˆ U B B B B denotesunitary operations what Bob needs to perform on qubits B , B , B and B , respectively. ijpqgh ˆ U B B B B ijpqgh ˆ U B B B B ijpqgh ˆ U B B B B ijpqgh ˆ U B B B B I B I B I B I B I B I B X B X B X B X B I B I B X B X B X B X B I B I B Z B I B I B I B X B Z B X B X B X B Z B I B X B X B X B Z B X B Z B I B I B I B Z B I B X B X B X B Z B X B I B I B X B Z B X B X B X B Z B I B Z B I B Z B I B X B Z B X B X B Z B X B Z B I B X B Z B X B X B Z B X B I B I B Z B I B I B I B X B Z B X B X B X B Z B I B X B X B X B Z B X B I B I B I B I B I B I B X B X B X B X B I B I B X B X B X B X B Z B I B Z B I B Z B I B X B Z B X B X B Z B X B Z B I B X B Z B X B X B Z B X B I B I B Z B I B Z B I B X B X B X B Z B X B I B I B X B Z B X B X B X B Z B I B I B I B I B I B X B Z B X B X B Z B X B I B I B X B Z B X B X B X B Z B I B Z B I B Z B I B X B Z B X B X B Z B X B Z B I B X B Z B X B X B Z B X B I B I B Z B I B I B I B X B X B X B X B I B I B X B X B X B X B I B I B I B I B I B I B X B Z B X B X B X B Z B I B X B X B X B Z B X B Z B I B Z B I B Z B I B X B Z B X B X B Z B X B Z B I B X B Z B X B X B Z B X B Z B I B I B I B Z B I B X B X B X B Z B X B I B I B X B Z B X B X B X B I B I B I B I B I B X B Z B X B X B X B Z B I B X B X B X B Z B X B I B I B I B I B I B I B X B X B X B X B I B I B X B X B X B X B W = diag( a a , a b a b , , b b ) , (16) U = diag( r − ( a a ) , r − | a b a b ) , , r − ( b b ) ) , (17) W = diag( b b , , a b a b , a a ) , (18) U = diag( r − | b b | , , r − ( a b a b ) , r − ( a a ) ) , (19) W = diag(1 , b b , a a , a b a b ) , (20)and U = diag(0 , r − ( b b ) , r − ( a a ) , r − ( a b a b ) ) . (21)Then, Bob measures his auxiliary qubit, B A , under a set of measuring basis vectors {| i , | i} . If state | i is measured,his remaining qubits will collapse into the trivial state, and the MJRSP fails in this situation; otherwise, | i is probed,and the qubits’ state will transform into the desired state, that is, our MCRSP is successful in this case.Based on the above five-step protocol, it has been shown that the MCRSP for a family of cluster-type states can befaithfully performed with predictable probability. The steps can be decomposed into a schematic diagram shown inFig. 1. As a summary, we list Bob’s required local single-qubit transformations according to the sender’s and agents’different measurement outcomes in Table I.From the above analysis, one can see that the prepared state can be faithfully reconstructed with specified successprobabilities. P TQPM ( ) ˆ ijA A U ˆ B B B B U n m n m } A A A A C i D j B B B B ( ) ˆ A ijB B B V B A FIG. 2: Quantum circuit for implementing the MCRSP scheme. TQPM denotes two-qubit projective measurement under a setof complete orthogonal basis vectors {|L ij i} ; ˆ U ( ij ) A A denotes Alice’s appropriate bipartite collective unitary transformation onqubit pair ( A , A ); ˆ U B B B B denotes Bob’s appropriate single-qubit unitary transformations on his qubits B , B , B and B , respectively, and ˆ V ( ij ) A A B A denotes Bob’s triplet collective unitary transformation on his qubits B , B and B A . Now, let us turn to calculate the TSP and CCC. Alice’s measurement outcome, |L ij i , has an occurrence probabilityof P |L ij i = 1 N ij . (22)Furthermore, in considering the capture of the state | i B A , the probability should be P | i BA = |N ij a b | . (23)Thus, the success probability of MCRSP for the measurement outcome ( i, j ) should be given by P ( i,j ) = P |L ij i × P | i BA = | a b | . (24)In terms of P ( i,j ) , one can easily obtain that the TSP sums to P P , i,j ( i,j ) = , X i,j P ( i,j ) = 4 | a b | . (25)Moreover, one can show that the required CCC should be (2 + 2 + m + n ) = ( m + n + 4) cbits totally.Herein, we had described our proposal of MCRSP for a family of four-qubit cluster-type entangled states. We haveproved that our scheme can be realized faithfully with TSP of 4 | a b | and CCC of ( m + n + 4) via the control ofmulti-agent in a quantum network. For clarity, the quantum circuit for our MCRSP protocol is displayed in Fig. 2. III. DISCUSSIONS
We have found several remarkable features with respect to the scheme presented above and these features aresummarized as follows: (1) To the best of our knowledge, this is the first time one has exploited such a scenarioconcerning MCRSP for four-qubit cluster-type entangled states via control of ( m + n ) − party. Information conveyanceonly takes place between the sender and the receiver, i.e., 1 → | a b | . Moreover, when the state | a | = | b | = 1 / √ |a ||b | T SP FIG. 3: The relation between TSP and the smaller coefficients of entanglements severed as quantum channels.
Consequently, that indicates our scheme becomes a deterministic one in this case. Additionally, it should be notedthat the parameters a and b relate to the Shannon entropies of the employed quantum channels, H ( f ) = −| f | log | f | − (1 − | f | )log(1 − | f | ) , (26)where f ∈ { a , b } and a , b ∈ [ −√ / , √ / m + n )controllers to manipulate or switch the preparation procedure. If both m and n are chosen to be 0, there are noauthorized controllers during the process of the preparation, it has been found that our scheme is smoothly reducedto a scheme resembling RSP for four-qubit cluster-type states with TSP of 4 | a b | . In this case, the measurementsmade by the controllers and the communication between controllers and receivers are unnecessary, as is the auxiliaryqubit. Now, we can compare our reduced scheme with other previous schemes [46–51]; we do this with respect to RSPand JRSP for such states in view of the resource consumption and quantum operation complexity as shown in Table II.From Table II, one can directly note that the TSP of our scheme is capable of unity, and the intrinsic efficiency( η ) achieves 33 . η = N s N q + N c × T SP , (27)where N s weights the amount of qubits of the prepared states, N q weights the amount of quantum resource con-sumption, and N c weights the amount of CCC in quantum computation. Additionally, Ref. [50] can be realized witha TSP of 100%; however, there are several crucial differences between our methods and the previous, they are asfollows: (i) Quantum resource consumption. In [50], 12 qubits are indispensable in the course of RSP for four-qubitcluster-type states, while 8 qubits are sufficient to implement RSP for such states in our reduced scheme. Implyingour scheme is more economic. (ii) Operation complexity. Two four-qubit projective measurements in [50] are requirefor their procedure, while two-qubit projection measurements are required in our scheme. Experimental realization TABLE II: Comparison between our scheme and the previous ones in the case of maximally entangled channels. ET representsentanglement; SQ represents single-qubit; ASQ represents auxiliary single-qubit; CNOT represents controlled-not gates; PMrepresents projective measurement; SQPM represents single-qubit projective measurement and TSP represents total successprobability. Schemes Required qubits Quantum operations CCC TSP η Ref. [46] six 2-qubit ETs two 4-qubit PMs 8 −0.6 −0.4 −0.2 0 0.2 0.4 0.600.20.40.60.81 f E n t r opy s u m FIG. 4: The entropic diagram with variation of the parameter of quantum channels. of four-qubit projective measurement is much more difficult than that for two-qubit. Thus, in principal our scheme iseasier to experimentally realize than the previous method.
IV. CONCLUSION
Herein we have derived a novel strategy for implementing MCRSP scheme for a family of four-qubit cluster-typeentangled states by taking advantage of robust GHZ-class states as quantum channels. With the aid of suitableLOCC, our scheme can be realized with high success probability. Remarkably, our scheme has several nontrivialfeatures, including high success probability, security and reducibility. Particularly, the TSP of MCRSP can reachunity when the quantum channels are distilled to maximally entangled ones; that is, our scheme can be performeddeterministically at this limit. We argue the current MCRSP proposal might open up a new way for long-distancecommunication in prospective multi-node quantum networks.
Acknowledgments
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