DSpace at VNU: Joint remote preparation of four-qubit cluster-type states revisited

Home Search Collections Journals About Contact us My IOPscience Joint remote preparation of four-qubit cluster-type states revisited This content has been downloaded from IOPscience Please scroll down to see the full text 2011 J Phys B: At Mol Opt Phys 44 135506 (http://iopscience.iop.org/0953-4075/44/13/135506) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 132.174.255.116 This content was downloaded on 02/10/2015 at 19:02 Please note that terms and conditions apply IOP PUBLISHING JOURNAL OF PHYSICS B: ATOMIC, MOLECULAR AND OPTICAL PHYSICS doi:10.1088/0953-4075/44/13/135506 J Phys B: At Mol Opt Phys 44 (2011) 135506 (6pp) Joint remote preparation of four-qubit cluster-type states revisited Nguyen Ba An1 , Cao Thi Bich1 and Nung Van Don2 Center for Theoretical Physics, Institute of Physics, 10 Dao Tan, Thu Le, Ba Dinh, Hanoi, Vietnam Physics Department, Hanoi National University, 334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam E-mail: nban@iop.vast.ac.vn Received May 2011, in final form 24 May 2011 Published 21 June 2011 Online at stacks.iop.org/JPhysB/44/135506 Abstract We revisit the protocols proposed recently (Zhan et al 2011 J Phys B: At Mol Opt Phys 44 095501) for joint remote preparation of four-qubit cluster-type states We not only point out errors in those protocols but also make considerable improvements Our protocols, for both the cases of real and complex coefficients of the state to be prepared, consume much less quantum resource as well as classical communication cost with x, y, z and w being real numbers satisfying the normalization condition x + y + z2 + w = 1, or the form Introduction Joint remote state preparation (JRSP) is a multiparty protocol for high-level secure and faithful transmission of confidential quantum information contained in a quantum state | over any distance by means of only local operation and classical communication Let S be the set of numbers that fully characterize the state | In the simplest three-party case, set S can be split, say by David, the Director of an Intelligence Bureau, into two subsets S1 and S2 , each of which does not identify | , but both of them S1 is given to Alice, S2 to Bob and the two are ordered to prepare | for a distant Charlie, who knows nothing about S1 , S2 and S Of course, David can ask either Alice or Bob to teleport [1] the state, at the inconvenience that David himself must beforehand prepare the | and then hand it over the teleporter Alternatively, David can simply let either Alice or Bob know S and ask her/him to perform a remote state preparation protocol (see, e.g., [2–4]), but by doing so all confidentiality of | is leaked out to the preparer The merit of JRSP is the maintenance of confidentiality since neither Alice nor Bob can infer S from S1 or S2 To avoid the situation in which Alice and Bob may gather together to infer S from S1 and S2 , David can secretly choose among his personnel two preparers, one does not know who is the other Since the first JRSP protocols for a single-qubit state [5–8], there have been extended ones for two- and three-qubit [8–13] cases Very recently, JRSP of four-qubit cluster-type states has also been proposed [14] The states considered in [14] have the form |φ = x |0000 + y |0011 + z |1100 + w |1111 , 0953-4075/11/135506+06$33.00 |ϕ = α |0000 + β |0011 + γ |1100 + δ|1111 , (2) with α, β, γ and δ being complex numbers satisfying the normalization condition |α|2 + |β|2 + |γ |2 + |δ|2 = However, the protocols in [14], which are referred to here as ZHM ones, suffered from errors and were not optimal Therefore, in this paper, we revisit the same problem as in [14] In section we outline the ZHM protocols and point out their errors Then, in section 3, we propose alternative protocols which are much more economical both in shared quantum resource and classical communication cost Finally, we conclude with some discussion in section ZHM protocols 2.1 Real coefficients For the state |φ , equation (1), S = {x, y, z, w} In [14], S1(2) = {x1(2) , y1(2), , z1(2) , w1(2) } such that x2 = x , x1 y2 = y , y1 z2 = z , z1 w2 = w w1 (3) As the shared quantum resource they use six Einstein–Podolsky–Rosen (EPR) pairs [15] √12 (|00 + |11 )A1 B1 (A2 C1 ,A3 B3 ,A4 C3 ,B2 C2 ,B4 C4 ) , of which qubits A1 , A2 , A3 , A4 belong to Alice, B1 , B2 , B3 , B4 to Bob and C1 , C2 , C3 , C4 to Charlie (see figure 1(a)) The bases they design for (1) © 2011 IOP Publishing Ltd Printed in the UK & the USA J Phys B: At Mol Opt Phys 44 (2011) 135506 N B An et al Alice and Bob to measure their qubits can be rewritten in a single equation as ⎛ ⎞ ⎛ (k) ⎞ ξ1 τ1 ⎜ ⎟ ⎜ (k) ⎟ ⎜ ξ2 ⎟ ⎜ τ2 ⎟ ⎜ ⎟ ⎜ ⎟ ⎟, ⎜ ⎟ = Uk ⎜ (4) ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ (k) ⎟ ⎝ ξ15 ⎠ ⎝τ ⎠ 15 (k) ξ16 τ16 in which k = (2) for Alice (Bob), |ξ1 = |0000 , |ξ3 = |1100 , |ξ2 = |0011 , |ξ4 = |1111 , |ξ5 = |0001 , |ξ7 = |1101 , |ξ6 = |0010 , |ξ8 = |1110 , |ξ9 = |0100 , |ξ11 = |1000 , |ξ13 = |0101 , |ξ10 = |0111 , |ξ12 = |1011 , |ξ14 = |0110 , |ξ15 = |1001 , |ξ16 = |1010 , O F (k) O O O O F (k) O ⎞ O O ⎟ ⎟, O ⎠ F (k) O is the × zero matrix and ⎛ yk xk ⎜ y −x k k F (k) = ⎜ ⎝ zk −wk wk zk zk wk −xk −yk ⎞ wk −zk ⎟ ⎟ yk ⎠ −xk ⎛ (k) F ⎜ O Uk = ⎜ ⎝ O O (5) (a) Figure The shared quantum resource for JRSP of four-qubit cluster-type states in (a) ZHM protocols and (b) our protocols Qubits are represented by dots and entanglement by a solid line connecting the entangled qubits (6) The shared quantum resource used is the same as for the case of real coefficients However, the design of measurement bases is delicate for the case of complex coefficients These are defined by equations (10) and (11) of [14] which can be rewritten as ⎛ (k) ⎞ ⎞ ⎛ μ1 |ξ1 ⎜ (k) ⎟ ⎜ |ξ2 ⎟ ⎜ μ2 ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ ⎜ ⎟ = Vk ⎜ (10) ⎜ ⎟ , ⎜ ⎟ ⎟ ⎜ ⎜ (k) ⎟ ⎝|ξ15 ⎠ ⎝ μ15 ⎠ |ξ16 μ(k) 16 (7) It is worth pointing out two obvious errors First, the transformation matrix F (k) (and so Uk ) were inappropriate because they are non-unitary and thus the states |τn(k) are not normalized This is due to the general inequality xk2 + yk2 + zk2 + wk2 = for both k = 1, In fact, the equalities 2 2 x1(2) + y1(2) + z1(2) + w1(2) = are incompatible with the normalization condition x + y + z2 + w = (see the proof of incompatibility for the single-qubit case in [6, 7]) The correct transformation should be F (k) ⇒ Fk = Nk F (k) with Nk = where and (8) G(k) For the state |ϕ , equation (2), S = {α, β, γ , δ} In [14], S1(2) = {α1(2) , β1(2), , γ1(2) , δ1(2) } such that β2 = β , β1 γ2 = γ , γ1 δ2 = δ δ1 αk∗ ⎜λk α ∗ k =⎜ ⎝ βk λk βk O G(k) O O βk∗ λk βk∗ −αk −λk αk O O G(k) O ⎞ O O ⎟ ⎟ O ⎠ G(k) γk∗ ∗ −λ−1 k γk δk −λ−1 k δk ⎞ δk∗ ∗⎟ −λ−1 k δk ⎟ −γk ⎠ λ−1 k γk (11) (12) with λk = (|γk |2 + |δk |2 )/(|αk |2 + |βk |2 ) Note that the adequate expression of λk should be λk = (|γk |2 + |δk |2 )/(|αk |2 + |βk |2 ) Even with such definition of λk , the transformation matrix G(k) does not normalize the 2 2 states |μ(k) n , because in general |αk | + |βk | + |γk | + |δk | = 2 for both k = 1, In fact, |α1(2) | +|β1(2) | +|γ1(2) | +|δ1(2) |2 = and |α|2 + |β|2 + |γ |2 + |δ|2 = are incompatible The correct transformation should be G(k) ⇒ Gk = Mk G(k) with Mk = (13) 2 |αk | + |βk | + |γk |2 + |δk |2 2.2 Complex coefficients α , α1 ⎛ (k) G ⎜ O Vk = ⎜ ⎝ O O ⎛ xk2 + yk2 + zk2 + wk2 Second, the success probability P = 1/4 obtained in [14] is not correct either Such a result was produced because of their wrong argument and the omission of the global factor of 1/8 in equation (8) of [14] With the two above-mentioned errors corrected the right total success probability should be P = N12 N22 α2 = (b) (9) J Phys B: At Mol Opt Phys 44 (2011) 135506 N B An et al = [(x1 y2 + z1 w2 ) |00 − (x1 x2 + z1 z2 ) |01 + (y1 y2 + w1 w2 ) |10 − (y1 x2 + w1 z2 ) |11 ]C1 C3 , |D12 Furthermore, the argument leading to their total success probability P = 1/16 was wrong and the global factor of 1/8 in their equation (12) was not taken into account If all of the errors mentioned above are corrected, the right total success probability should be P = M12 M22 /4 |D13 C1 C3 = [(x1 z2 − z1 x2 ) |00 − (x1 w2 − z1 y2 ) |01 + (y1 z2 + w1 x2 ) |10 − (y1 w2 − w1 y2 ) |11 ]C1 C3 , (20) C1 C3 (21) Our protocols |D14 In this section we propose our own protocols with properly defined unitary transformations between the measurement and computational bases We also present the cases of real and complex coefficients separately for clarity As the shared quantum resource is most expensive, it is of paramount importance to economize it So, instead of six shared EPR pairs as in [14], we employ only three such pairs Namely, the shared quantum channel in our protocols is (23) C1 C3 = [(y1 y2 + w1 w2 ) |00 − (y1 x2 + w1 z2 ) |01 − (x1 y2 + z1 w2 ) |10 + (x1 x2 + z1 z2 ) |11 ]C1 C3 , (24) with ‘ .’ consisting of ten more other terms Looking at the above expressions suggests choosing the coefficients of |00 C1 C3 , |01 C1 C3 , |10 C1 C3 and |11 C1 C3 in any of |Dmn C1 C3 to be x, y, z and w, respectively The choice (16) for information splitting results from setting the coefficients of |D11 C1 C3 so that x1 x2 + z1 z2 = x, x1 y2 + z1 w2 = y, y1 x2 + w1 z2 = z and w1 w2 + y1 y2 = w Substituting (16) into (18) we can figure out that the task fails if the measurement outcomes of Alice and Bob are (m, n) = (1, 3), (1, 4), (2, 3), (2, 4), (3, 1), (3, 2), (4, 1) or (4, 2) Fortunately, when (m, n) = (1, 1), (1, 2), (2, 1), (2, 2), (3, 3), (3, 4), (4, 3) or (4, 4), each happens randomly with an equal probability of N12 N22 /8, the two qubits C1 and C3 collapse into the state 3.1 Real coefficients Since Alice and Bob each hold only two qubits, we define the bases in which Alice (k = 1) and Bob (k = 2) measure their qubits as ⎛ (k) ⎞ ⎛ ⎞ |u1 |00 ⎜ (k) ⎟ ⎜|u2 ⎟ ⎜|01 ⎟ ⎜ ⎜ ⎟ ⎟ (15) ⎜ (k) ⎟ = Fk ⎝|10 ⎠ ⎝|u3 ⎠ |11 |u(k) |D11 where, as noticed before, Fk = Nk F (k) , with F (k) and Nk defined by equations (7) and (8) For the reason to be clarified later, we split {x, y, z, w) into {x1 , y1 , z1 , w1 } and {x2 , y2 , z2 , w2 } as follows We choose x1 , y1 , z1 , w arbitrarily, but determine x2 , y2 , z2 , w from x1 , y1 , z1 , w and x, y, z, w in the following way: ⎫ x2 = d1 (xw1 − zz1 ),⎪ ⎪ ⎬ y2 = d1 (yw1 − wz1 ), (16) z2 = d1 (zx1 − xy1 ),⎪ ⎪ ⎭ w2 = d1 (yy1 − wx1 ), |D12 |D21 |D22 with |D33 x1 w1 − y1 z1 (17) In terms of |u(k) n , the shared quantum channel, equation (14), can be expressed in the form |D34 C1 C3 C1 C3 C1 C3 C1 C3 C1 C3 C1 C3 u(1) m A1 A2 u(2) n B1 B2 |Dmn C1 C3 , (18) |D43 m,n=1 where |D11 C1 C3 = [(y1 x2 + w1 z2 ) |00 + (y1 y2 + w1 w2 ) |01 − (x1 x2 + z1 z2 ) |10 − (x1 y2 − z1 w2 ) |11 ]C1 C3 , |D22 of which qubits A1,2 belong to Alice, B1,2 to Bob and C1,3 to Charlie (see figure 1(b)) N1 N2 |Q = √ 2 (22) |D21 (|00 + |11 )A1 B1 (|00 + |11 )A2 C1 (|00 + |11 )B2 C3 , |Q = √ √ √ 2 (14) d1 = = [(x1 w2 − z1 y2 ) |00 + (x1 z2 − z1 x2 ) |01 + (y1 w2 − w1 y2 ) |10 + (y1 z2 − w1 x2 ) |11 ]C1 C3 , C1 C3 = [(x1 x2 + z1 z2 ) |00 + (x1 y2 + z1 w2 ) |01 + (y1 x2 + w1 z2 ) |10 + (w1 w2 + y1 y2 ) |11 ]C1 C3 , or |D44 C1 C3 (19) C1 C3 C1 C3 ⇒ φ1 C1 C3 = (x |00 + y |01 + z |10 + w |11 )C1 C3 , (25) ⇒ φ2 C1 C3 = (−x |01 + y |00 − z |11 + w |10 )C1 C3 , (26) ⇒ φ3 C1 C3 = (−x |10 − y |11 + z |00 + w |01 )C1 C3 , (27) ⇒ φ4 C1 C3 = (x |11 − y |10 − z |01 + w |00 )C1 C3 , (28) ⇒ φ5 C1 C3 = (x |00 − y |01 − z |10 + w |11 )C1 C3 , (29) ⇒ φ6 C1 C3 = (x |01 + y |00 − z |11 − w |10 )C1 C3 , (30) ⇒ φ7 C1 C3 = (x |10 − y |11 + z |00 − w |01 )C1 C3 (31) ⇒ φ8 C1 C3 = (x |11 + y |10 + z |01 + w |00 )C1 C3 , (32) J Phys B: At Mol Opt Phys 44 (2011) 135506 N B An et al 3.2 Complex coefficients Table The measurement outcomes of Alice and Bob (m, n) and the corresponding recovery operators R1 (R2 , R3 ) for the information splitting (16) ((35), (36)) m n R1 R2 R3 1 1 2 2 3 3 4 4 4 4 I ⊗I I ⊗ ZX Failure Failure ZX ⊗ I ZX ⊗ ZX Failure Failure Failure Failure Z⊗Z Z⊗X Failure Failure X⊗Z X⊗X I ⊗ ZX I ⊗I Failure Failure ZX ⊗ ZX ZX ⊗ I Failure Failure Failure Failure Z⊗X Z⊗Z Failure Failure X⊗X X⊗Z Failure Failure I ⊗I I ⊗ ZX Failure Failure ZX ⊗ I ZX ⊗ ZX Z⊗Z Z⊗X Failure Failure X⊗Z X⊗X Failure Failure For the case of complex coefficients we propose two different protocols in which the measurement bases of Alice and Bob may be similar or dissimilar 3.2.1 Similar measurement bases The bases in which Alice (k = 1) and Bob (k = 2) measure their qubits are defined as ⎛ (k) ⎞ ⎛ ⎞ |v1 |00 ⎜ (k) ⎟ ⎜|01 ⎟ ⎜|v2 ⎟ ⎟ (37) ⎜ (k) ⎟ = Gk ⎜ ⎝|10 ⎠ ⎝|v3 ⎠ |11 |v4(k) where, as noticed before, Gk = Mk G(k) , with G(k) and Mk given by equations (12) and (13) In terms of vn(k) , |Q = C1 C3 = (x |00 + y |01 + z |10 + w |11 )C1 C3 |T11 C1 C3 |00 C2 C4 = |φ C1 C2 C3 C4 (1) vm A1 A2 vn(2) B1 B2 |Tmn C1 C3 , (38) m,n=1 C1 C3 = [(α1 α2 + γ1 γ2 ) |00 + (α1 β2 + γ1 δ2 ) |01 + (β1 α2 + δ1 γ2 ) |10 + (δ1 δ2 + β1 β2 ) |11 ]C1 C3 , |T13 (33) C1 C3 |T31 C1 C3 |T33 C1 C3 |T12 C1 C3 (41) = [(β1∗ β2∗ + δ1∗ δ2∗ ) |00 − (β1∗ α2∗ + δ1∗ γ2∗ ) |01 − (α1∗ β2∗ + γ1∗ δ2∗ ) |10 + (α1∗ α2∗ + γ1∗ γ2∗ ) |11 ]C1 C3 , Likewise, we can set the coefficients of |D12 C1 C3 so that x1 y2 + z1 w2 = x, x1 x2 + z1 z2 = −y, y1 y2 + w1 w2 = z and y1 x2 + w1 z2 = −w, yielding ⎫ x2 = d1 (wz1 − yw1 ),⎪ ⎪ ⎬ y2 = d1 (xw1 − zz1 ), (35) z2 = d1 (yy1 − wx1 ), ⎪ ⎪ ⎭ w2 = d1 (zx1 − xy1 ) (40) = [(β1∗ α2 + δ1∗ γ2 ) |00 + (β1∗ β2 + δ1∗ δ2 ) |01 − (α1∗ α2 + γ1∗ γ2 ) |10 − (α1∗ β2 + γ1∗ δ2 ) |11 ]C1 C3 , (34) (39) = [(α1 β2∗ + γ1 δ2∗ ) |00 − (α1 α2∗ + γ1 γ2∗ ) |01 + (β1 β2∗ + δ1 δ2∗ ) |10 − (β1 α2∗ + δ1 γ2∗ ) |11 ]C1 C3 , With |φ C1 C3 at hand, Charlie proceeds by preparing two ancillary qubits C2 and C4 in the state |00 C2 C4 , and then applying two controlled-NOT gates CNOTC1 C2 CNOTC3 C4 to φ C1 C3 |00 C2 C4 , with C1,3 the control qubits and C2,4 the target ones As a result, she obtains the desired four-qubit cluster-type state |φ in equation (1), since CNOTC1 C2 CNOTC3 C4 |φ where the 16 states |Tmn have the form to which Charlie can respectively apply the recovery operators R = I ⊗I, I ⊗ZX, ZX ⊗I, ZX ⊗ZX, Z ⊗Z, Z ⊗X, X ⊗Z or X ⊗ X, with X = {{0, 1}, {1, 0}} and Z = {{1, 0}, {0, −1}} Pauli’s bit- and phase-flip operator, to obtain, up to a global phase factor, the state |φ M1 M √ 2 (42) = [(α1 α2 λ2 − γ1 γ2 λ−1 ) |00 + (α1 β2 λ2 − γ1 δ2 λ−1 ) |01 (43) + (β1 α2 λ2 − δ1 γ2 λ−1 ) |10 + (β1 β2 λ2 − δ1 δ2 λ−1 ) |11 ]C1 C3 , Or, we can set the coefficients of |D13 C1 C3 so that x1 z2 − z1 x2 = x, x1 w2 − z1 y2 = −y, y1 z2 + w1 x2 = z and y1 w2 − w1 y2 = −w, yielding ⎫ x2 = d1 (xy1 − zx1 ), ⎪ ⎪ ⎬ y2 = d1 (wx1 − yy1 ), (36) z2 = d1 (xw1 − zz1 ), ⎪ ⎪ ⎭ w2 = d1 (wz1 − yw1 ), (44) and 11 other terms containing λ1 , λ−1 or/and λ2 , λ−1 Analysing the above expressions suggests four possible choices of information splitting in which α1 , β1 , γ1 , δ1 are chosen arbitrarily but α2 , β2 , γ2 , δ2 need to be determined appropriately Choice is due to the setting {α1 α2 + γ1 γ2 , α1 β2 + γ1 δ2 , β1 α2 + δ1 γ2 = γ , δ1 δ2 + β1 β2 } = {α, β, γ , δ}, resulting in ⎫ α2 = χ1 (γ γ1 − αδ1 ), ⎪ ⎪ ⎬ β2 = χ1 (δγ1 − βδ1 ), (45) γ2 = χ1 (αβ1 − γ α1 ),⎪ ⎪ ⎭ δ2 = χ1 (ββ1 − δα1 ), and so on In table 1, the recovery operators R1 (R2 , R3 ) associated with the information splitting (16) (( 35), (36)) are listed in correspondence with Alice’s and Bob’s measurement outcomes (m, n) For each choice of information splitting, among the 16 possible outcomes yield a success; thus, the total success probability is P = N12 N22 with χ1 = (β1 γ1 − α1 δ1 )−1 J Phys B: At Mol Opt Phys 44 (2011) 135506 N B An et al Choice is due to the setting {α1 β2∗ +γ1 δ2∗ , −α1 α2∗ −γ1 γ2∗ , + δ1 δ2∗ , −β1 α2∗ − δ1 γ2∗ } = {α, β, γ , δ}, resulting in ⎫ α2∗ = χ1 (βδ1 − δγ1 ), ⎪ ⎪ ⎬ β2∗ = χ1 (γ γ1 − αδ1 ), (46) ∗ γ2 = χ1 (δα1 − ββ1 ),⎪ ⎪ ⎭ ∗ δ2 = χ1 (γ α1 − αβ1 ) ⎞ |00 ⎟ ⎟ ⎜ ⎟ = G2 ⎜|01 ⎟ , ⎠ ⎝|10 ⎠ |11 (49) N1 M2 √ 2 |vn B1 B2 (51) |Lmn C1 C3 , = [(w1 α2 − y1 γ2 ) |00 + (w1 β2 − y1 δ2 ) |01 + (z1 α2 − x1 γ2 ) |10 + (z1 β2 − x1 δ2 ) |11 ]C1 C3 , (59) C1 C3 C1 C3 |L43 C1 C3 |L12 C1 C3 = [(x1 α2 λ2 − z1 γ2 λ−1 ) |00 We have carefully reinvestigated the protocols in [14] for JRSP of four-qubit cluster-type states Apart from pointing out their errors, we have improved them considerably In our protocols only three (not six as in [14]) EPR pairs are used as the shared quantum channel So, Alice and Bob only need to perform measurements in Hilbert spaces of dimension (not 16 as in [14]), at the expense of Charlie’s two ancillary qubits and two CNOTs, which are, however, local resource and local operation As for classical communication cost, it was bits (4 per preparer) in [14] In our protocols, it is at most just bits (2 per preparer), which, however, can be managed rationally (52) m,n=1 = [(x1 α2 + z1 γ2 ) |00 + (x1 β2 + z1 δ2 ) |01 + (y1 α2 + w1 γ2 ) |10 + (w1 δ2 + y1 β2 ) |11 ]C1 C3 , (58) C1 C3 Conclusion where |L11 = [(z1 β2∗ − x1 δ2∗ ) |00 − (z1 α2∗ − x1 γ2∗ ) |01 − (w1 β2∗ − y1 δ2∗ ) |10 + (w1 α2∗ − y1 γ2∗ ) |11 ]C1 C3 , C1 C3 and seven other terms of the structure such as |L12 C1 C3 , i.e containing λ2 and λ−1 Simple ways of splitting S → {S1 , S2 } can be realized by making one of the states (53)–(60) to be ϕ C1 C3 = (α |00 + β |01 + γ |10 + δ |11 )C1 C3 For example, if it is |L11 C1 C3 (|L13 C1 C3 , |L31 C1 C3 or |L33 C1 C3 ), then (m, n) = (1, 1) ((1, 3), (3, 1) or (3, 3)) yields |ϕ C1 C3 and (m, n) = (2, 1) ((2, 3), (4, 1) or (4, 3)) yields XZ ⊗ I ϕ C1 C3 , while all the other outcomes lead to a failure This means that, for any of the ways of choosing information splitting just mentioned, the total success probability of this protocol is P = N12 M22 /4 ⎛ A1 A2 (57) C1 C3 −1 + (x1 β2 λ2 − z1 δ2 λ−1 ) |01 + (y1 α2 λ2 − w1 γ2 λ2 ) |10 + (y1 β2 λ2 − w1 δ2 λ−1 (61) ) |11 ]C1 C3 , |um = [(z1 α2 − x1 γ2 ) |00 + (z1 β2 − x1 δ2 ) |01 + (y1 γ2 − w1 α2 ) |10 + (y1 δ2 − w1 β2 ) |11 ]C1 C3 , C1 C3 = [(w1 β2∗ − y1 δ2∗ ) |00 − (w1 α2∗ − y1 γ2∗ ) |01 (60) − (x1 δ2∗ − z1 β2∗ ) |10 + (x1 γ2∗ − z1 α2∗ ) |11 ]C1 C3 , with F1 = N1 F (k=1) and G2 = M2 G(k=2) Correspondingly, S1 = {x1 , y1 , z1 , w1 } and S2 = {α2 , β2, , γ2 , δ2 } In terms of |um and |vn , |Q = (56) |L41 3.2.2 Dissimilar measurement bases The design of measurement bases is not unique In the preceding subsection Alice and Bob use similar measurement bases Now we consider the case when their measurement bases are dissimilar More concretely, let Alice’s basis be ⎞ ⎛ ⎞ ⎛ |00 |u1 ⎜|01 ⎟ ⎜|u2 ⎟ ⎟ ⎜ ⎟ ⎜ (50) ⎝|u3 ⎠ = F1 ⎝|10 ⎠ , |11 |u4 ⎞ = [(y1 β2∗ + w1 δ2∗ ) |00 − (y1 α2∗ + w1 γ2∗ ) |01 − (x1 β2∗ + z1 δ2∗ ) |10 + (x1 α2∗ + z1 γ2∗ ) |11 ]C1 C3 , |L33 from which Charlie is able to obtain the desired state |ϕ C1 C2 C3 C4 = (α |0000 + β |0011 + γ |1100 + δ |1111 )C1 C2 C3 C4 , equation (2), as described above with the help of the ancillary two-qubit state |00 C2 C4 and two CNOT gates CNOTC1 C2 CNOTC3 C4 The success probability is thus P = M12 M22 /8 while Bob’s basis is ⎛ |v1 ⎜|v2 ⎜ ⎝|v3 |v4 (55) |L31 Evidently, if choice (2, 3, 4) applies, then only when the measurement outcome is (m, n) = (1, 1) ((1, 3), (3, 1), (3, 3)) Charlie’s qubits collapse into the state = (α |00 + β |01 + γ |10 + δ |11 )C1 C3 , = [(y1 α2 + w1 γ2 ) |00 + (y1 β2 + w1 δ2 ) |01 − (x1 α2 + z1 γ2 ) |10 − (x1 β2 + z1 δ2 ) |11 ]C1 C3 , |L23 Choice is due to the setting {β1∗ β2∗ +δ1∗ δ2∗ , −β1∗ α2∗ −δ1∗ γ2∗ , ∗ ∗ −α1 β2 − γ1∗ δ2∗ , α1∗ α2∗ + γ1∗ γ2∗ } = {α, β, γ , δ}, resulting in ⎫ α2∗ = −χ1∗ (βγ1∗ + δδ1∗ ),⎪ ⎪ ⎬ β2∗ = χ1∗ (αγ1∗ + γ δ1∗ ), (48) ∗ ∗ ∗ ∗ γ2 = χ1 (βα1 + δβ1 ), ⎪ ⎪ ⎭ δ2∗ = −χ1∗ (αα1∗ + γβ1∗ ) C1 C3 (54) |L21 Choice is due to the setting {β1∗ α2 + δ1∗ γ2 , β1∗ β2 + δ1∗ δ2 , ∗ −α1 α2 − γ1∗ γ2 , −α1∗ β2 − γ1∗ δ2 } = {α, β, γ , δ}, resulting in ⎫ α2 = χ1∗ (αγ1∗ + γ δ1∗ ), ⎪ ⎪ ⎬ β2 = χ1∗ (βγ1∗ + δδ1∗ ), (47) γ2 = −χ1∗ (αα1∗ + γβ1∗ ),⎪ ⎪ ⎭ δ2 = −χ1∗ (βα1∗ + δβ1∗ ) |ϕ = [(x1 β2∗ + z1 δ2∗ ) |00 − (x1 α2∗ + z1 γ2∗ ) |01 + (y1 β2∗ + w1 δ2∗ ) |10 − (y1 α2∗ + w1 γ2∗ ) |11 ]C1 C3 , |L13 β1 β2∗ C1 C3 (53) J Phys B: At Mol Opt Phys 44 (2011) 135506 N B An et al or to reduce the actual cost depending on the case In fact, the case of real coefficients does require bits (see table 1), but the case of complex coefficients does not When Alice and Bob measure in similar bases (subsection 3.2.1), only bits (1 per preparer) are consumed upon an agreed choice of information splitting Precisely, if choice (see equation (45)) is effective, then Alice (Bob) just announces ‘1’ when m (n) = and ‘0’ when m (n) = If choice (see equation (46)) is effective, then Alice (Bob) just announces ‘1’ when m = (n = 3) and ‘0’ when m = (n = 3) If choice (see equation (47)) is effective, then Alice (Bob) just announces ‘1’ when m = (n = 1) and ‘0’ when m = (n = 1) And, if choice (see equation (48)) is effective, then Alice (Bob) just announces ‘1’ when m (n) = and ‘0’ when m (n) = The protocol succeeds if m + n = and fails if m + n = When Alice and Bob measure in dissimilar bases (subsection 3.2.2), bit from Bob suffices in the worst situation and more bits from Alice are needed in the other situations For example, if |L11 C1 C3 is made to be |ϕ C1 C3 , then Bob announces ‘0’ when his n = 1, implying failure, or ‘1’ when n = 1, in which case Alice announces ‘0’ or ‘1’ when her m = or 2, implying the collapsed state |ϕ C1 C3 or XZ ⊗ I |ϕ C1 C3 at Charlie’s It is worth noting that the improvement made in our paper is based on the observation that the four-qubit clustertype states (1) or (2) can be locally constructed from the corresponding two-qubit states (33) or (49) by means of two ancillary qubits and two CNOTs Thus, the problem essentially reduces to JRSP of a general two-qubit state Although protocols for JRSP of a general two-qubit state have been studied previously by a number of authors [8–10], ours deserve their own merit Namely, up to now either two GHZ trios [16] or two W states [17] or four EPR pairs have been served as the shared quantum channel for JRSP of a general two-qubit state, but here we employed only three EPR pairs We believe that three EPR pairs as the quantum channel are most economical for such a joint task Such simplification is realizable by judicious ways of information splitting (see, e.g., equations (16), (35), (45)–(48)), which differ from the ‘conventional’ ones (see, e.g., equations (3) and (9)) Despite more complicated forms, the splittings (16), (35), (45)– (48) cause no mathematical difficulties since they are easily calculated by any classical computer Finally, we would like to attract the reader’s attention to the fact that with the same shared quantum resource (i.e three EPR pairs) JRSP of an (M + N )-qubit cluster-type state of the forms |ω = (x |0 .00 .0 + y |0 .01 .1 + z |1 .10 .0 + w |1 .11 .1 )1 MM+1 M+N , | = (α |0 .00 .0 + β |0 .01 .1 + γ |1 .10 .0 + δ |1 .11 .1 )1 MM+1 M+N , (63) with arbitrary M, N > 2, can also be performed in the same manner as described in this paper Just at the receiver’s site Charlie needs to use M + N − ancillary qubits, all in state |0 , and apply M + N − CNOTs properly Counterparts of the states (62) and (63) in terms of (M + N )-mode cluster-type entangled coherent states have also been studied recently with some applications (see, e.g., [18–20]) Acknowledgments This work is supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) References [1] Bennett C H, Brassard G, Crepeau C, Jozsa R, Peres A and Wootters W K 1993 Phys Rev Lett 70 1895 [2] Lo H K 2000 Phys Rev A 62 012313 [3] Pati A K 2000 Phys Rev A 63 014302 [4] Bennett C H, DiVincenzo D P, Shor P W, Smolin J A, Terhal B M and Wootters W K 2001 Phys Rev Lett 87 077902 [5] Xia Y, Song J and Song H S 2007 J Phys B: At Mol Opt Phys 40 3719 [6] An N B and Kim J 2008 J Phys B: At Mol Opt Phys 41 095501 [7] An N B and Kim J 2008 Int J Quantum Inf 1051 [8] An N B 2010 Opt Commun 283 4113 [9] An N B 2009 J Phys B: At Mol Opt Phys 42 125501 [10] Xiao X O, Liu J M and Zeng G 2011 J Phys B: At Mol Opt Phys 44 075501 [11] Chen Q Q, Xia Y, Song J and An N B 2010 Phys Lett A 374 4483 [12] Luo M X, Chen X B, Ma S Y, Niu X X and Yang Y X 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OF PHYSICS B: ATOMIC, MOLECULAR AND OPTICAL PHYSICS doi:10.1088/0953-4075/44/13/135506 J Phys B: At Mol Opt Phys 44 (2011) 135506 (6pp) Joint remote preparation of four-qubit cluster-type states. .. Online at stacks.iop.org/JPhysB/44/135506 Abstract We revisit the protocols proposed recently (Zhan et al 2011 J Phys B: At Mol Opt Phys 44 095501) for joint remote preparation of four-qubit cluster-type. .. classical communication cost with x, y, z and w being real numbers satisfying the normalization condition x + y + z2 + w = 1, or the form Introduction Joint remote state preparation (JRSP) is a