DSpace at VNU: Deterministic joint remote state preparation

Physics Letters A 375 (2011) 3570–3573 Contents lists available at SciVerse ScienceDirect Physics Letters A www.elsevier.com/locate/pla Deterministic joint remote state preparation Nguyen Ba An a,∗ , Cao Thi Bich a,b , Nung Van Don a,c a b c Center for Theoretical Physics, Institute of Physics, 10 Dao Tan, Ba Dinh, Hanoi, Viet Nam Physics Department, University of Education No 1, 136 Xuan Thuy, Cau Giay, Hanoi, Viet Nam Physics Department, Hanoi National University, 334 Nguyen Trai, Thanh Xuan, Hanoi, Viet Nam a r t i c l e i n f o Article history: Received August 2011 Accepted 19 August 2011 Available online 25 August 2011 Communicated by V.M Agranovich a b s t r a c t We put forward a new nontrivial three-step strategy to execute joint remote state preparation via Einstein–Podolsky–Rosen pairs deterministically At variance with all existing protocols, in ours the receiver contributes actively in both preparation and reconstruction steps, although he knows nothing about the quantum state to be prepared © 2011 Elsevier B.V All rights reserved Keywords: Joint remote state preparation Einstein–Podolsky–Rosen pairs Unit success probability Introduction Secure faithful transmission of quantum states encoding quantum information is a primarily important task in quantum information processing, which can now be achieved by dual usage of local operation and classical communication without physically transmitting the states themselves, thanks to previously shared quantum resource (SQR) called entanglement [1] The first intriguing protocol for such kind of tasks is quantum teleportation (QT) [2] in which a sender Alice is able to teleport a unknown quantum state to a space-like receiver Bob In case full knowledge of the state (i.e., the complete set of parameters identifying the state) is known to Alice, the task can be done by a protocol named remote state preparation (RSP) [3] using the same SQR as in QT but with simpler local operation and reduced classical communication cost (CCC) Obviously, however, in RSP all the secret information encoded in the quantum state is leaked out to the sender Alice An idea to get rid of the leakage of information to the sender is that one more sender enters the game and the state’s parameter set is split, by some confidential rule, into two subsets, each of which is given to a sender By doing so it is impossible for neither sender alone to infer the secret from his/her own parameter subset Yet, to faithfully transmit the full secret to a remote receiver the two senders must cooperate correctly Original protocols realizing such idea were designed some years ago which were referred to as joint * Corresponding author E-mail address: nban@iop.vast.ac.vn (N.B An) 0375-9601/$ – see front matter © 2011 Elsevier B.V All rights reserved doi:10.1016/j.physleta.2011.08.045 remote state preparation (JRSP) [4] (see also Refs [5–8] for various extensions) As a rule, all JRSP protocols employ the SQR in terms of entangled states of which Greenberger–Horne–Zeilinger (GHZ) trios [9], W states [10] and Einstein–Podolsky–Rosen (EPR) pairs [11] are typical Most JRSP protocols proposed so far succeed with a finite probability which, however, is less than one That is, they are probabilistic [4–7] Very recently, a strategy has been introduced in Ref [8] to improve JRSP so that the success probability is boosted to one The SQR in Ref [8] are GHZ trios Here we shall make use only of EPR pairs as the SQR and unit success probability is also achieved by adopting a strategy which is different from that in Ref [8] Apart from easier production of the SQR, using EPR pairs has an advantage in the process of distributing entanglement among the participants before the execution of the JRSP protocol Namely, the presence of unavoidable malicious eavesdroppers necessitates a prior careful checking procedure that sacrifices a large number of entangled states Use of SQR in terms GHZ trios as in Ref [8] requires the checking to be carried out simultaneously between the three participants, while use of SQR in terms of EPR pairs as in our protocols requires only pair-wise checking This not only simplifies the checking operation but also economizes the overhead expenses In this Letter we put forward a new nontrivial strategy for the three remote parties to deterministically prepare the most general single- and two-qubit states without leakage of full information encoded in the state to either of them, using only EPR pairs (not GHZ trios) as the SQR The case of single-qubit states is considered in Section and that of two-qubit ones in Section Section is the conclusion N.B An et al / Physics Letters A 375 (2011) 3570–3573 Deterministic joint remote preparation of an arbitrary single-qubit state 3571 Table The reconstruction operator rln in Eq (13) versus l and n I is the identity operator, σx the bit-flip operator and σz the phase-flip one Note that rln can generally be formulated as rln = σzl⊕n σxl Let us name the two senders Alice and Alice 2, while the receiver remains Bob The essence of a JRSP protocol can be grasped as follows The complete set S of parameters identifying the state to be prepared is split into two subsets S and S The subset S is given only to Alice 1, while S only to Alice 2, so neither Alice nor Alice is able to learn the complete set S The prerequisite is that somehow Alice 1, Alice and Bob must a priori share some kind of entanglement Usually, the participants execute a two-step protocol in the following manner [4–7] In the first step both Alice j ( j = and 2) independently carry out measurements in the bases determined by S j and publicly broadcast their outcomes In the second step Bob performs proper operations depending on the broadcasted outcomes to obtain the desired state The success probability of such customary two-step protocols is always less than one [4–7] To achieve unit success probability a strategy composed of three steps has been chalked out in Ref [8] which applies when GHZ trios are served as the SQR More concretely, in the two first steps Alice and Alice carry out their measurements sequentially in such a way that the measurement basis of Alice is defined not only by her parameter subset S but also by the outcome of Alice It is somewhat surprised to recognize that such a strategy is not suitable when EPR pairs are used as the SQR We shall therefore propose a different strategy to cope with EPR pairs Our strategy also undergoes three steps but the content of each step is more delicate In the first step Alice and Bob independently carry out their measurements The measurement basis of Alice is determined by S , whereas that of Bob is the computational one As for Alice 2, she starts her job only in the second step making use of S as well as of the outcomes broadcasted by Alice and Bob in the first step The target state is then obtained in the third step by Bob’s actions which are sensitive to the outcomes of both senders In this section we detail our new strategy for JRSP of an arbitrary single-qubit state whose most general form reads k= |φ = a|0 + be i ϕ |1 , The unitary transformations v (k) in Eq (9) depends also on S = ϕ : (1) with a, b, ϕ ∈ R and a + b = The three parties share two EPR pairs, |Q A1 B A2 B where |q = |q XY = A B |q A B , √1 (2) (|00 + |11 ) X Y Each Alice holds a qubit (Al- ice holds qubit A , Alice qubit A ), while Bob holds two qubits B and B Clearly, the complete parameter set identifying |φ is S = {a, b, ϕ } There are many ways to split S into two subsets S and S We find out that to achieve unit success probability the proper splitting is S = {a, b} and S = ϕ In the first step, Alice and Bob independently perform the following actions Alice measures her qubit A in the basis {|ψ0 A , |ψ1 A } determined just by S as |ψ0 |ψ1 A1 A1 = a b −b a |0 |1 A1 A1 , (3) whereas Bob first applies CNOT B B on his two qubits B and B and then measures qubit B in its computational basis {|0 B , |1 B } After those actions the state | Q A B A B of the SQR, Eq (2), becomes | Q A B A B , which can be rewritten as CNOT X Y denotes a controlled-NOT gate acting on two qubits X (control qubit) and Y (target qubit) as CNOT X Y |i X | j Y = |i X |i ⊕ j Y , where i , j ∈ {0, 1} and ⊕ stands for an addition mod l n rln 0 I Q A1 B A2 B = 1 σz σ z σx σx 1 1 |ψl A |m B |λlm A B , (4) m =0 l =0 where |λ00 A2 B = a|00 + b|11 A2 B |λ01 A2 B = a|10 + b|01 A2 B |λ10 A2 B = a|11 − b|00 A2 B |λ11 A2 B = a|01 − b|10 A2 B , (5) , (6) , (7) , (8) with ⊕ an addition mod As is easily verified from Eq (4), when Alice finds qubit A in state |ψl A and Bob finds qubit B in state |m B , the unmeasured qubits A and B are automatically projected onto an entangled state |λlm A B , with a probability plm = 1/4 for any pair of possible l and m Afterwards, Alice and Bob announce their outcomes by virtue of the public classical media In the second step, Alice starts measuring her qubit A in a basis that she needs to correctly determine using the parameter subset S = ϕ available to her as well as the outcomes (l, m) announced by Alice and Bob There are two choices for the measurement basis labeled by k = 0, as (k) |θ0 |θ1(k) A2 = v (k) A2 |0 |1 A2 A2 (9) , with k = l ⊕ m, i.e., if l = m = or l = m = 1, if l = 0, m = or l = 1, m = (10) 1 e −i ϕ v ( 0) = √ −i ϕ , −e e −i ϕ v (1 ) = √ −i ϕ −e (11) (12) To specify the value of k in states |λlm (k) perindex “(k)” to them as |λlm (k) {|θn A2 ; n A2 B A2 B , we assign a su- and express them through = 0, 1} in the form 1 (k) λlm A2 B (k) =√ n =0 θn r + |φ A ln (13) B1 , with rln an operator which is in fact independent of k as listed in Table Transparently, with an equal probability pn = 1/2, Bob will (k) obtain a state |θn A with n = 0, or 2, which he needs also to announce publicly In the third step, depending on the outcomes l and n, Bob’s job is simply to apply rln on his qubit B to obtain the desired state |φ B Since Bob is always able to reconstruct |φ B from |φ B = + rln |φ B , our JRSP protocol is deterministic Mathematically, its total success probability p T is 1 pT = plm pn = × n =0 m =0 l =0 × = (14) 3572 N.B An et al / Physics Letters A 375 (2011) 3570–3573 ⎛ In passing we notice the following relationship (1 ) σx ⊗ I λlm ( 0) A2 B = λl,m⊕1 A2 B (15) (0) This allows Alice to use the same basis {|θn A } to measure qubit A for both l ⊕ m = and l ⊕ m = 1, provided that when l ⊕ m = she applies σx on A before measuring it Deterministic joint remote preparation of an arbitrary two-qubit state Several authors [6] dealt with JRSP of particular two-qubit states of the form α |00 + β|11 , with α , β ∈ C and |α |2 + |β|2 = However, such tasks can be reduced to JRSP of single-qubit states α |0 + β|1 with an addition application of a CNOT on the collapsed qubit and an ancillary qubit in state |0 at the end by Bob Other authors [7] dealt with JRSP of general two-qubit states of the form a j eiϕ j | j , |Φ = (16) j =0 where {a j , ϕ j } ∈ R, j =0 a j = 1, while |0 ≡ |00 , |1 ≡ |01 , |2 ≡ |10 and |3 ≡ |11 are shorthands for the four two-qubit orthonormal states in their computational basis, but the success probability in Ref [7] remains less than In this section we are concerned with the same most general two-qubit state as defined in Eq (16), but we propose JRSP protocols that succeed with unit probability This really matters because success probability is of paramount importance for a quantum protocol The SQR we exploit consists of four EPR pairs, |Q A1 B A1 B A2 B A2 B with |q XY = √1 = |q A j B j |q j =1 AjBj (17) , (|0 + |3 ) X Y in the abbreviated notations Qubits A and A ( A and A ) belong to Alice (Alice 2), while qubits B , B , B and B to Bob To perform JRSP of the state (16) deterministically we split its complete parameter set S = {a j , ϕ j } into S = {a j } and S = {ϕ j }, where without any loss of generality we set ϕ0 = Firstly, Alice and Bob independently act as follows Alice measures qubits A and A in the basis defined by S as ⎛ |Ψ ⎞ ⎛ a0 ⎜ |Ψ1 A A ⎟ ⎜ −a1 ⎝ |Ψ ⎠=⎝ −a2 A1 A |Ψ3 A A −a3 A1 A a1 a0 a3 −a2 a2 −a3 a0 a1 while Bob applies CNOT B B CNOT B ⎞⎛ |0 a3 a ⎟ ⎜ |1 ⎠ ⎝ |2 −a1 |3 a0 B2 A1 A1 A1 A1 A1 A1 (18) ⎛ on his qubits followed ⎜ (K ) ⎜ |Θ1 ⎜ ⎜ |Θ ( K ) ⎝ |Θ3( K ) = |Ψl A A |m B B |Λlm A A B B , ⎛ ⎞ ⎛ |Λ00 |00 ⎜ |Λ01 ⎟ ⎜ |10 ⎝ ⎠=⎝ |Λ02 |20 |Λ03 |30 V ( 0) = V (1 ) = V (2 ) = To |22 |32 |02 |12 ⎞⎛ ⎞ |33 a0 |23 ⎟ ⎜ a1 ⎟ ⎠⎝ ⎠, |13 a2 |03 a3 ⎛ |0 ⎟ ⎟ A2 A2 ⎟ ( K ) ⎜ |1 ⎟ = V ⎝ |2 A2 A2 ⎠ |3 A2 A2 ⎞ a0 ⎟ ⎜ a1 ⎟ ⎠⎝ ⎠, a2 a3 ⎞⎛ ⎞ a0 ⎟ ⎜ a1 ⎟ ⎠⎝ ⎠, a2 a3 ⎞⎛ ⎞ a0 ⎟ ⎜ a1 ⎟ ⎠⎝ ⎠ a2 a3 (21) (22) (23) A2 A2 A2 A2 A2 A2 ⎞ ⎟ ⎠, (24) A2 A2 (25) (K ) ⎞ A2 A2 B B (K ) Λlm (26) (27) (28) (29) the choices of measurement bases we write (K ) as |Λlm A A B B , which can be expressed through |Θn ; n = 0, 1, 2, (20) ⎞⎛ e − i ϕ1 e − i ϕ2 e − i ϕ3 − i ϕ − i ϕ ⎜ −e e −e −i ϕ3 ⎟ ⎝ ⎠, − i ϕ1 − i ϕ2 −e e − i ϕ3 −e e − i ϕ1 −e −i ϕ2 −e −i ϕ3 ⎛ − i ϕ1 ⎞ e e − i ϕ3 e − i ϕ2 − i ϕ − i ϕ − i ϕ ⎜ −e 1 −e e ⎟ ⎝ −i ϕ ⎠, −i ϕ −e −i ϕ2 −e 1 e − i ϕ1 − i ϕ3 − i ϕ2 e −e −e ⎛ − i ϕ2 ⎞ e e − i ϕ3 e − i ϕ1 − i ϕ3 − i ϕ1 ⎜ e − i ϕ2 −e −e ⎟ ⎝ −i ϕ ⎠, e − i ϕ3 −e −i ϕ1 −e − i ϕ2 − i ϕ3 − i ϕ1 −e −e e ⎛ − i ϕ3 ⎞ e e − i ϕ2 e − i ϕ1 − i ϕ − i ϕ − i ϕ 1 ⎜ −e e −e 1⎟ ⎝ − i ϕ3 ⎠ e −e −i ϕ2 −e −i ϕ1 −e −i ϕ3 −e −i ϕ2 e −i ϕ1 specify |Λlm |11 |01 |31 |21 ⎞ A2 A2 ⎛ (19) where −|22 −|32 −|02 −|12 |11 |01 |31 |21 −|00 −|10 −|20 −|30 More explicitly, K = if {(l, m)} = {(0, 0), (1, 1), (2, 2) or (3, 3)}, K = if {(l.m)} = {(0, 1), (1, 0), (2, 3) or (3, 2)}, K = if {(l.m)} = {(0, 2), (1, 3), (2, 0) or (3, 1)}, and K = if {(l, m)} = {(0, 3), (1, 2), (2, 1) or (3, 0)} As for the unitary transformations V ( K ) in Eq (24), they are of the form m =0 l =0 |33 |23 |13 |03 −|00 −|10 −|20 −|30 −|11 −|01 −|31 −|21 ⎧ if l − m = 0, ⎪ ⎨ if l + m = or 5, K= ⎪ ⎩ if |l − m| = 2, if l + m = V (3 ) = A1 B A1 B A2 B A2 B −|00 −|10 −|20 −|30 −|33 −|23 −|13 −|03 |22 |32 |02 |12 with |Q |11 ⎟ ⎜ |01 ⎠=⎝ |31 |21 ⎞ ⎛ |22 ⎟ ⎜ |32 ⎠=⎝ |02 |12 ⎞ ⎛ |33 ⎟ ⎜ |23 ⎠=⎝ |13 |03 (K ) |Θ0 A1 A1 by measuring qubits B and B in their computational bases {|0 , |1 , |2 , |3 } B B As a consequence, the SQR state becomes ⎛ Clearly, the qubits A , A , B and B collapse, with a probability P lm = 1/16 for any possible outcomes l, m ∈ {0, 1, 2, 3}, into an entangled state |Λlm A A B B , if Alice finds |Ψl A A and Bob 1 finds |m B B After the measurements Alice and Bob both pub2 lish their outcomes l and m through a classical communication channel Secondly, upon hearing the published outcomes, Alice measures her qubits A and A in an appropriate basis determined not only by S but also by the outcomes (l, m) There are four choices for measurement bases determined by ⎞ ⎟ ⎠, ⎞ |Λ10 ⎜ |Λ11 ⎝ |Λ12 |Λ13 ⎛ |Λ20 ⎜ |Λ21 ⎝ |Λ22 |Λ23 ⎛ |Λ30 ⎜ |Λ31 ⎝ |Λ32 |Λ33 A2 A2 B B = A2 A2 as (K ) Θn n =0 ( K )+ A2 A2 R ln |Φ B1 B1 (30) N.B An et al / Physics Letters A 375 (2011) 3570–3573 3573 Table (0) The reconstruction operator R ln in Eq (30) versus l and n I is the identity operator and the phase-flip operator l n (0) R ln 0(1, 2, 3) 0(1, 2, 3) 0(1, 2, 3) 1(0, 3, 2) 0(1, 2, 3) 2(3, 0, 1) 0(1, 2, 3) 3(2, 1, 0) I⊗I I ⊗ σz σz ⊗ σz σz ⊗ I Table (1,2,3) The reconstruction operator R ln in Eq (30) versus l and n I is the identity operator and l n σz σx (σz ) the bit-flip (phase-flip) one 0 1 1 R ln I⊗I I ⊗ σz σz ⊗ σz σz ⊗ I I ⊗ σ z σx I ⊗ σx σ z ⊗ σx σ z ⊗ σ z σx l n 2 2 3 3 3 σ z σx ⊗ σ z σ z σx ⊗ I σx ⊗ I σx ⊗ σ z σ z σx ⊗ σx σ z σx ⊗ σ z σx σx ⊗ σ z σx σx ⊗ σx (1,2,3) (1,2,3) R ln (K ) with R ln listed in Tables and After her measurement Alice broadcasts the outcome n = 0, 1, or publicly Transparently, the (K ) probability that Alice obtains the state |Θn A A is P n = 1/4 independent of n Finally, as seen from Eq (30), the desired state (16) can be re(K ) constructed by Bob’s application of R ln on his qubits B and B (0) (1,2,3) (see Since for a fixed pair of (l, n), R ln may differ from R ln Tables and 3), to determine the right reconstruction operator, Bob should take into account not only the outcomes of Alice and Alice 2, but also that of himself Precisely, Bob needs first to calculate the value of K from the values of l and m in accordance with the rule (25), then he looks for the right operator (K ) R ln from Tables and To simplify Bob’s action, Alice can (0) use the same basis {|Θn A A } to measure qubits A and A for all possible outcomes (l, m), provided that when l + m = or (|l − m| = 2, l + m = 3) she needs applying I ⊗ σx (σx ⊗ I σx ⊗ σx ) on A and A before measuring them By doing so, Table appears superfluous and Bob will need only the phase-flip operator (i.e., the bit-flip operator is not required), in accordance with Table Such a simplification comes out from the observation that (1 ) I ⊗ σx ⊗ I ⊗ I Λlm ( 0) A2 A2 B B = Λlm A2 A2 B B , (31) σx ⊗ I ⊗ I ⊗ I Λlm ( 0) A2 A2 B B = Λlm A2 A2 B B , m − m = 2, (32) (3 ) σx ⊗ σx ⊗ I ⊗ I Λlm ( 0) A2 A2 B B = Λlm A2 A2 B B m + m = , (33) Similar to the case of the single-qubit state |φ , our JRSP protocol for the two-qubit state |Φ is also deterministic since its total success probability P T is 3 PT = P lm P n = 64 × n =0 m =0 l =0 16 × = Acknowledgements This work is supported by the Vietnam Foundation for Science and Technology Development (NAFOSTED) References m + m = or 5, (2 ) plays a passive role in the sense that he participates only in the final step for reconstructing the target state, in our protocols Bob’s role turns out quite active Namely, here Bob not only performs the reconstruction operation at the end, but also participates along with the first sender, Alice 1, at the very beginning of the protocol and his measurement outcome is equally meaningful as that of the first sender for the second sender (Alice 2) to choose the right measurement basis Because of Bob’s active action and the usefulness of his outcome in the first step, an extra classical communication is needed from him The total CCC of our deterministic protocols for 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Deterministic joint remote preparation of an arbitrary single-qubit state 3571 Table The reconstruction operator rln in Eq (13) versus l and n I is the identity operator, σx the bit-flip operator... applies σx on A before measuring it Deterministic joint remote preparation of an arbitrary two-qubit state Several authors [6] dealt with JRSP of particular two-qubit states of the form α |00 + β|11... states α |0 + β|1 with an addition application of a CNOT on the collapsed qubit and an ancillary qubit in state |0 at the end by Bob Other authors [7] dealt with JRSP of general two-qubit states