Controlled Remote Preparation of a Two-Qubit State via an Asymmetric Quantum Channel

Transcription

1 Commun. Theor. Phys. 55 (0) Vol. 55 No. February 5 0 Controlled Remote Preparation of a Two-Qubit State via an Asymmetric Quantum Channel WANG Zhang-Yin ( ) Key Laboratory of Optoelectronic Information Acquisition & Manipulation of Ministry of Education of China School of Physics & Material Science Anhui University Hefei China (Received October 8 00; revised manuscript received November 00) Abstract I present a new scheme for probabilistic remote preparation of a general two-qubit state from a sender to either of two receivers. The quantum channel is composed of a partial entangled tripartite Greenberger Horne Zeilinger (GHZ) state and a W-type state. I try to realize the remote two-qubit preparation by using the usual projective measurement and the method of positive operator-valued measure respectively. The corresponding success probabilities of the scheme with different methods as well as the total classical communication cost required in this scheme are also calculated. PACS numbers: Hk a w Key words: remote state preparation partial entangled Greenberger Horne Zeilinger (GHZ) state W-type state projective measurement positive operator-valued measure unitary operation Introduction In 993 Bennett et al. [] proposed a method for interchanging quantum resources between different places via a quantum channel with the help of some classical communication which is termed as quantum teleportation (QT). Later another interesting novel method called remote state preparation (RSP) was presented by Lo [] Pati [3] and Bennett et al. [4] which also utilizes a prior shared entanglement and some classical communication to transmit pure quantum states. In RSP the prepared state is assumed to be completely known by the sender. In contrast the teleported state is not required to be known by the sender in QT. Consequently this new communication protocol is also viewed as teleportation of a known state. Further due to the prior knowledge about the original state to some extent the classical communication and entanglement cost can be reduced in RSP process. For an example Pati [3] has shown that for a qubit chosen from equatorial or polar great circles on a Bloch sphere RSP requires only forward classical bit exactly half that of QT. However for general states RSP procedure requires as much communication cost as QT. The detailed trade-off between the classical communication cost and the required entanglement in RSP procedure can be studied distinctly in the protocol proposed by Bennett et al. [4] Up to now RSP has already attracted many attentions [5 ] e.g. low-entanglement RSP [5] higherdimensional RSP [6] optimal RSP [7] oblivious RSP [8] RSP without oblivious conditions [9] generalized RSP [0] faithful RSP [] RSP for multi-parties [] and continuous variable RSP in phase space [3 4] etc. On the other hand some RSP schemes have already been experimentally implemented [9 ] e.g. Peng et al. presented an RSP scheme with the technique of NMR (nuclear magnetic resonance) [9] Xiang et al. [0] and Peters et al. [] proposed two other RSP schemes using spontaneous parametric down-conversion. Recently also some RSP schemes are investigated by using different entangled states as quantum channel. [ 48] In terms of entanglements in quantum channels these RSP schemes can be classified into two types. One uses maximally entangled states [ ] while another utilizes nonmaximally entangled states. [ ] In the latter case usually one or more auxiliary qubits need to be incorporated and entangled with the original qubits. After this a proper measurement on qubits including the ancillas should be executed such that the original-qubit state is collapsed to one of the eligible states. Subsequently the prepared state is retrieved from the eligible state by performing appropriate unitary operations which correspond to the measurement results. Note that the so-called proper measurement is usually projective measurement (PM) [49] in the latter type of existing RSP schemes. [ ] As a matter of fact there lies another type of measurement named positive operator-valued measure (POVM) [50 5] which has already attracted many attentions and been employed in various quantum information processing. [ ] Nonetheless to my best knowledge there have been few proposals for how to probabilistically generate RSP of a general two-qubit state with the method of POVM. In view of that in this paper using a tripartite GHZ-type state and a W-type state as the quantum channel I attempt to propose a scheme to address the question raised above in which not only the usual PM but also the method of POVM are considered to implement the probabilistic remote two-qubit Supported by the Project of Anhui University under Grant No. 009QN08B Corresponding author zywang@ahu.edu.cn c 0 Chinese Physical Society and IOP Publishing Ltd

2 No. Communications in Theoretical Physics 45 preparation. Although the quantum channel is asymmetric in this scheme each one of the two receivers has the chance to construct the original state. Furthermore the corresponding success probabilities of the scheme with different methods and the total classical communication cost required in this scheme are also calculated in detail. The outline of this paper is organized as follows: in Sec. a controlled remote two-qubit preparation protocol is designed via a usual PM as well as a proper POVM. The success probabilities are also calculated in the two different cases respectively. In Sec. 3 some discussions regarding the required amount of classical communication as well as the implement feasibility of the protocol are given together with the summary. Controlled RSP Scheme of a General Two- Qubit State via PM as Well as POVM Now let me present the controlled RSP scheme. Suppose Alice is the state preparer Bob and Charlie are her two remote ministrants. Alice wants to prepare remotely ξ ξ ξ 3 = ξ 4 α β γ δ ηα ηβ η γ η δ β α δ γ ηβ ηα η δ η γ a general two-qubit state in either ministrant s place. The state to be prepared is V = α 00 +β 0 +γ 0 +δ α β γ δ are arbitrary complex numbers and satisfy α + β + γ + δ =. Alice knows it exactly while Bob and Charlie do not. The quantum channel linking Alice Bob and Charlie is composed of a partial entangled tripartite GHZ state and a W-type entangled state ψ 3 = a b 3 ( a + b = ) φ 456 = c d e ( c + d + e = ) () a b c d and e are nonzero real numbers and satisfy a b c d e. Qubit pair ( 4) belongs to Alice while qubit pairs ( 5) and (3 6) to Bob and Charlie respectively. In order to realize the state preparation Alice carries out a two-qubit projective measurement (PM) on her qubit pair ( ) in a set of mutually orthonormal basis vectors { ξ ξ ξ 3 ξ 4 } which are given by () η ( γ + δ )/( α + β ). These four non-maximally entangled basis states { ξ ξ ξ 3 ξ 4 } are related to the computation basis vectors { } and form a complete orthonormal basis set in a four-dimensional Hilbert space i.e. ξ i ξ j = δ ij. Note that the joint state of the whole system can be rewritten as ψ 3 φ 456 = (ac ad ae bc bd be ) = ξ 4 Υ ξ 4 Υ ξ 3 4 Υ ξ 4 4 Υ (3) Υ 356 = α ac α ad β ae γ bc γ bd δ be Υ 356 = ηα ac ηα ad ηβ ae η γ bc η γ bd η δ be Υ = βac βad αae δbc δbd γbe Υ = ηβac ηβad ηαae η δbc η δbd η γbe After the two-qubit PM Alice broadcasts her measurement result via a classical channel (shown in Fig. (a)). From Eq. (3) one can see that Alice s measurement result should be one of the four states defined in Eq. (). Without loss of generality suppose Alice measures ξ 3 4 then the collapsed state of the qubit pairs ( 3) and (5 6) will be Υ Provided that Alice intends to restore the original state in Charlie s location then Bob acts as a controller and he collaborates with Charlie to help him construct the prepared state in his place. For this purpose Bob performs firstly a single-qubit PM on his qubit in the X bases { + = ( 0 + )/ = ( 0 )/ }. After this operation Bob then measures his another qubit 5 in the bases { 0 }. Then the collapsed state Υ can be reexpressed as Υ = + (βac βad αae δbc δbd γbe ) + (βac βad αae δbc δbd γbe ) = + [(βac 0 36 αae δbc 36 γbe 0 36 ) (βad δbd 0 36 ) 5 ]

3 46 Communications in Theoretical Physics Vol [(βac 0 36 αae δbc 36 + γbe 0 36 ) (βad δbd 0 36 ) 5 ]. (4) From Eq. (4) one can see that if Bob gets ± 5 Charlie cannot reconstruct the original state and the RSP scheme fails. Only when Bob gets ± 0 5 the two-qubit preparation may be realized in Charlie s place. Then after the two single-qubit measurements Bob tells Charlie his measurement results via a classical channel. Upon receiving Bob s classical message Charlie knows properly the collapsed state of his qubits 3 and 6. If Bob gets then Charlie s qubits 3 and 6 are left in Q 36 = (βac 0 36 αae δbc 36 γbe 0 36 ). (5) If Bob gets 0 5 the collapsed state of qubit pair (3 6) will be Q 36 = (βac 0 36 αae δbc 36 + γbe 0 36 ). (6) In this way Charlie performs a local unitary operation I 3 σ z 6 on her qubit pair (3 6) in the first case and σz 3 σz 6 in the latter one. After Charlie s performing the state of Charlie s qubit pair (3 6) will collapse to Q 36 = (αae βac γbe δbc 36 ). (7) Then to construct the original state finally either the usual PM or the method of POVM can be employed by Charlie to complete the preparation which are designed as follows. Fig. (a) Alice Bob and Charlie share the partial entangled tripartite GHZ state and the W-type state. Alice makes a two-qubit projective measurements (PM) and informs Bob and Charlie of her measurement results ( bits); (b) Bob is assigned to make two single-qubit measurements(sm) in the different bases and then tells Charlie his measurement results ( bits). Conditioned on Alice and Bob s classical message Charlie constructs the prepared state via a usual PM. (c) Conditioned on Alice and Bob s classical message Charlie constructs the prepared state via POVM. See text for more details. Case PM is employed (shown in Fig. (b)) Charlie then introduces an auxiliary qubit A in the initial state 0 and performs a collective unitary operation Ξ on the qubits 3 6 and A under the basis vectors { A 00 36A 00 36A 0 36A 00 36A 0 36A 0 36A 36A } [49] the collective unitary operation Ξ (an eight eight matrix) may take the following form ( ) A A Ξ = (8) A A A i (i = ) is a 4 4 matrix and may be expressed respectively as A = diag 3 4 ) A = diag ( 3 4 ). i (i = 3 4 and i ) depends on the joint state of the three qubits 3 6 and A Q 3 36A = (αae βac γbe δbc 36 ) 0 A. (9) According to this state explicitly one gets ( b A = diag a be ac e ) c

4 No. Communications in Theoretical Physics 47 ( A = diag b a b e ) a c 0 e c. After Charlie s collective unitary operation on the qubits 3 6 and A the initial joint state Q 3 36A is transformed into Ξ Q 3 36A = [ be(α β γ δ 36 ) 0 A + ( a b eα a c b e β c e bδ 36 ) A ]. (0) At last Charlie measures the qubit A in the bases { 0 }. If the measurement result is 0 A it means Charlie has already constructed the original state V on her qubit pair (3 6). The success probability in this case is (be) /. Otherwise the RSP scheme fails. Case POVM is employed (shown in Fig. (c)) Charlie then introduces two auxiliary qubits m and n in the initial state 00 mn and performs two controlled-not (CNOT) operations with qubits 3 and 6 as the controlled qubits while the auxiliary qubits m and n as the target ones respectively. These two CNOT operations transform the state Q mn into the following form Q 3 36mn = (αae mn + βac 00 36mn + γbe 00 36mn + δbc 36mn ) = 4 ( K 36 H mn + K 36 H mn + K 3 36 H 3 mn + K 4 36 H 4 mn ) () K 36 = α β γ δ 36 V H mn = ae 00 mn + ac 0 mn + be 0 mn + bc mn K 36 = α β 0 36 γ 0 36 δ 36 H mn = ae 00 mn + ac 0 mn be 0 mn bc mn K 3 36 = α β γ 0 36 δ 36 H 3 mn = ae 00 mn ac 0 mn + be 0 mn bc mn K 4 36 = α β 0 36 γ δ 36 H 4 mn = ae 00 mn ac 0 mn be 0 mn + bc mn. From the Eq. () one can see that Charlie can get the states K i 36 (i = 3 4) provided that the states H i mn (i = 3 4) are distinguished via an appropriate measurement. Note that K is exactly the prepared state. Readily the prepared state can be further retrieved from K K 3 and K 4. Unfortunately the four states H i mn (i = 3 4) are not orthonormal in general. As a consequence they can not be differentiated deterministically by using a usual PM. Nevertheless the discrimination can be achieved in a probabilistic manner by making an optimal POVM measurement [50 553] on the ancillary qubits m and n as follows P = x M M P = x M M P 3 = x M 3 M 3 P 4 = x M 4 M 4 P 5 = I x 4 M i M i () i= M = ae 00 + ac 0 + be 0 + bc )mn M = ae 00 + ac 0 be 0 bc )mn M 3 = ae 00 ac 0 + be 0 bc )mn M 4 = ae 00 ac 0 be 0 + bc )mn = (ae) + (ac) + (be) + (bc) I is an identity operator x is a coefficient relating to a b c and e and should be able to assure P 5 to be a positive operator. To exactly determine x I would like to rewrite the five elements P P P 3 P 4 and P 5 in the matrix form P = x P = x P 3 = x P 4 = x P 5 = (ae) (ae) (ac) (be) bebc (ac) (ae) bcbe (bc) (be) bcbe bebc (ac) (ae) A B C D (bc) (be) bebc (ac) bcbe (bc) (be) bcbe bebc (bc)

5 48 Communications in Theoretical Physics Vol A = x(ae) B = 4 x(ac) (3) 4 C = x(be) D = 4 x(bc). Evidently to let P 5 be a positive operator the coefficient x should be chosen such that all the diagonal elements A B C D are nonnegative. So it should be an appropriate value within the scope x 4 as is strongly relative to a b c and e. After this POVM operation Charlie can positively conclude the states H i mn (i = 3 4) of qubits m and n in terms of the POVM s values. The probability in each case is p = 36mn Q 3 P i Q 3 36mn = mn H i P i H i mn /3 = (i = 3 4). (4) x However if Charlie gets P 5 s value (such probability is 4 p = /x) he can not infer which state the qubits m and n are in. Once Charlie determines the states H i mn (i = 3 4) it means he also knows exactly the state of his qubit 3 and 6. As a consequence Charlie can construct the prepared state V on his qubit pair (3 6) by performing an appropriate unitary operation. To be specifical if Charlie knows the state of his qubit pair (3 6) is K 36 K 36 K 3 36 or K 4 36 ) he needs only to perform the corresponding unitary operation I 3 I 6 σz 3 I 6 I 3 σz 6 or σ3 z σ6 z on his qubit pair (3 6) respectively. Thus the total success probability of the RSP scheme in this case is 4 p = x = [ x (ae) + (ac) + (be) + ] (bc) = x ( b )b ( d e )e d. (5) Above the RSP protocol is considered in the case that the state to be prepared is retrieved in Charlie s place. While Bob is assigned to construct the prepared state at his side Charlie acts as the controller. In this case Charlie will perform the two single-qubit measurements in the X bases { + = ( 0 + )/ = ( 0 )/ } and the bases { 0 } on his qubits 3 and 6 respectively and then transmits Bob his measurement results via a classical channel. After having received the classical message from Charlie applying the same analysis method as that proposed just above Bob can also fulfill the preparation on his qubits and 5 by using either the usual PM or the method of POVM. As depicted previously it is possible that Alice measures ξ 4 ξ 4 or ξ 4 4 with a certain probability. In these three cases the collapsed state of the qubit pairs ( 5) and (3 6) according to Eq. (3) will be Υ 356 Υ 356 and Υ respectively. Since neither Bob nor Charlie has knowledge of the four coefficients α β γ and δ the original state V can not be constructed via certain local unitary operations due to the involvement of an antiunitary operation. [ ] As a result the RSP scheme fails in the latter three cases. Nonetheless it should be noted that the coefficients α β γ and δ are assumed to be complex in the beginning. As depicted in Refs. [8 9] the four coefficients α β γ and δ may be some special values: (i) α β γ and δ are real; (ii) α β γ and δ satisfy η = ; (iii) α β γ δ are real and satisfy η = ; (iv) α = β = γ = δ = / and αγ = βδ; (v) α = β = γ = δ = / and αβ = δγ. While the state to be prepared is chosen from one of the above ensembles applying the same analysis method the RSP scheme can be achieved with higher probability. Here I do not depict them anymore. By above analysis one can see I have already shown the tripartite controlled RSP scheme of an arbitrary two-qubit state with PM and POVM respectively. The total success probability is (be) (PM is employed) or (4/x) [( b )b ( d e )e ]/( d ) (POVM is employed). If a = b = / c = d = e = / 3 and one can choose x = such that P 5 is a zero operator i.e. the quantum channel consists of a maximally entangled three-qubit GHZ state and a maximally entangled three-qubit W state and the so-called POVM becomes the common PM then the success probability in my RSP protocol will be /6 which is equivalent for each receiver. 3 Discussion and Summary The classical message plays an important role in RSP processes. [ ] As is shown in the above section in this scheme two kinds of classical information transmitted processes are involved. One is the classical information sent from the sender Alice to the receiver which Alice performs a two-qubit PM on her qubit pair (4) and informs the receiver of her measurement result. Another is the classical information from the controller to the receiver which the controller performs the single-qubit measurement on his two qubits respectively and then informs the receiver of the measurement results. In this section it is interested to know how many bits of classical information are required in the RSP process with three parties by two partial entangled tripartite quantum states? and my discussion is taken in the case that the original state is constructed at Charlie s side. As to the first classical communication process in this protocol based on Eq. (3) it can be noticed that after Alice s two-qubit PM she can obtain one of the four probabilistic results ξ i 356 (i = 3 4). As proposed before if Alice s measurement is ξ 4 ξ 4 or ξ 4 4 the RSP scheme fails due to Charlie s unawareness of the four coefficients α β γ and δ. In this way Alice needs not to send any classical bits to Charlie in above three cases. Only when ξ 3 4 is obtained which happens with the probability p 3 = β a c + β a d + α a e + δ b c + δ b d + γ b e the RSP scheme can be realized. Thus the amount of the classical information needed in this transmitted process is S = p 3 log p 3 bits.

6 No. Communications in Theoretical Physics 49 With regard to the second classical communication process it has already been proposed that when Alice s measurement result is ξ 356 ξ 356 or ξ the scheme fails. So in these cases Bob needs not to implement two single-qubit measurements. Further after Bob s single-qubit measurements there are four possible measurement results the states and 5. As depicted above if Bob gets + 5 or 5 the RSP scheme fails too. So it is also unnecessary for Bob to send any classical bits to Charlie in the latter two cases. Only when Bob measures or 0 5 the RSP scheme can be realized in a probabilistic manner. Both the probabilities for Bob s measurement results and 0 5 in the case that Alice measures ξ 3 4 are given in Table. Table The probability for Bob s two single-qubit measurement results corresponding to Alice s two-qubit measurement result ξ 3 4. See text for more details. Alice s measurement result Bob s single-qubit measurement result The corresponding probability ξ P ξ P P = β a c + α a e + δ b c + γ b e β a c + β a d + α a e + δ b c + δ b d + γ b e. By above analysis the amount of the classical information required in the second transmitted process is S = p 3 [ ] P log P p 3 [ ] P log P = p 3 P ( log P) bits. (6) Therefore the total classical communication cost required in this probabilistic remote two-qubit preparation with three-party will be S = S + S = p 3 P ( log P) p 3 log p 3 bits. (7) Above the classical communication cost is discussed in the case that the original state is constructed at Charlie s side. While Bob is assigned to retrieve the prepared state with Charlie s help the total classical communication cost can also be obtained with the similar analysis method. On the other hand as for feasibility it is known for remote preparing a quantum state the quantum source has to be an entangled quantum system so that the transmission of quantum information can be completed based on entanglement swapping. In this protocol to realize the remote two-qubit preparation a tripartite entangled GHZ-type state and a W-type entangled state are taken as the quantum channel. To my best knowledge so far threephoton entanglement has been observed and used to verify quantum nonlocality [55 57] even four-particle entanglement has been demonstrated in ion traps and preparation of four five-photon entangled states has already been achieved in experiment. [58 59] Therefore I believe that this RSP protocol with three-qubit entanglements may be realized in the realm of current experimental technology. To summarize I have presented a tripartite scheme for remotely preparing an arbitrary two-qubit state via an asymmetric quantum channel which is composed of a partial entangled tripartite GHZ-type state and a W-type entangled state. By the two ministrants collaboration it is shown the remote two-qubit preparation can be realized in a probabilistic manner by using either the usual PM or the method of POVM. Moreover the successful probability of the scheme is calculated as well as the total classical communication cost required in this scheme. Comparing with the previous two-qubit RSP schemes [ ] the present one has the following advantages. First the quantum channels are different in forms. In this present scheme I exploit a GHZ-type state and a W-type state as the shared quantum channel as the quantum channels in the previous schemes were composed of Einstein Podolsky Rosen(EPR) pairs GHZ states or their modified versions. [ ] Second to realize the probabilistic RSP of a general two-qubit state in this protocol not only the usual PM is adopted but also the method of POVM is considered. while the previous RSP schemes considered only one case i.e. the RSP of a twoqubit state with PM [ ] or POVM. [37] Third I have minutely calculated the total classical communication of probabilistic RSP scheme with three parties which not only involves the transmitted communication from a sender to the receiver but also contains one between two receivers while the earlier RSP schemes only just considered the transmitted communication from a sender to a receiver or even not. [ ] Thus from the point of view of communication cost this scheme may be useful not only in understanding the essence of the classical communication in RSP process but also expanding the applied field of classical information science from the point of view of communication cost. Incidentally the present scheme can also be directly generalized to remotely prepare an arbitrary two-qubit entangled state with multiparty by using a W-type state and a multi-qubit GHZ-type state. Acknowledgments The author would like to thank the anonymous referee for his very useful and detailed suggestions.

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