DSpace at VNU: Deterministic Joint Remote Preparation of an Arbitrary Qubit via Einstein-Podolsky-Rosen Pairs

DSpace at VNU: Deterministic Joint Remote Preparation of an Arbitrary Qubit via Einstein-Podolsky-Rosen Pairs tài liệu,...

DOI 10.1007/s10773-012-1107-9

Deterministic Joint Remote Preparation of an Arbitrary

Qubit via Einstein-Podolsky-Rosen Pairs

Cao Thi Bich · Nung Van Don · Nguyen Ba An

Received: 11 October 2011 / Accepted: 14 February 2012 / Published online: 1 March 2012

© Springer Science+Business Media, LLC 2012

Abstract Joint remote state preparation is a secure and faithful method based on local

op-eration and classical communication to transmit quantum states without the risk of full in-formation leaking to either of the participants In this work, we propose a new deterministic protocol for two parties to remotely prepare an arbitrary single-qubit state for a third party using two Einstein-Podolsky-Rosen pairs as the nonlocal resource We figure out the advan-tages as well as the disadvanadvan-tages of this new protocol in comparison with others, showing

in general that the proposed protocol is superior to the existing ones We also describe the situation when there are more than two preparers

Keywords Joint remote state preparation· EPR pairs · Feed-forward measurements · Unit

success probability

1 Introduction

About three decades ago questions such as “Why one cannot precisely measure two conju-gate variables at the same time?”or “Why an arbitrary quantum state cannot be cloned?” or

“Why there is spooky action at distance?”, etc., were regarded as idle The loath answer at that time was likely “God only knows” Nowadays, perhaps because of “God is subtle but malicious he is not”, people have become aware that such weird features, which are laws of Nature, constitute the very necessary ingredients to guarantee absolute cryptography [1], to exponentially speed up computation [2] or to perform a global task only by means of local operation and classical communication (LOCC), etc

C.T Bich () · N.V Don · N.B An

Center for Theoretical Physics, Institute of Physics, 10 Dao Tan, Hanoi, Vietnam

e-mail: ctbich@iop.vast.ac.vn

C.T Bich

Physics Department, Hanoi University of Education No 1, 136 Xuan Thuy, Hanoi, Vietnam

N.V Don

Physics Department, Hanoi University of Science, 334 Nguyen Trai, Hanoi, Vietnam

In quantum cryptography [1] information is encoded in classical keys which are however distributed in a quantum way under the enemy’s very nose by sending qubits prepared ran-domly in one of the four states{|0, |1, |+ = (|0 + |1)/√2,|− = (|0 − |1)/√2}, with

{|0, |1} the two orthonormal states of a qubit in the computational basis Information can

also be encoded quantumly in the state of a qubit of the form

with any parameters a, b, ϕ∈R satisfying the normalization condition a2+ b2= 1

Never-theless, transmitting quantum information by physically sending such informative qubits is dangerous since any unauthorized parties can alter the qubits before they reach the intended party without detection of the latter Therefore, secure and faithful transmission of quantum information without sending the qubits themselves proves to be a great idea This implic-itly means that the transmission should be performed by LOCC, which appears realizable

if some nonlocal resource is provided between the sending and the receiving parties The first intriguing protocol of such kind of global tasks, called quantum teleportation (QT), was proposed in 1993 by Bennett et al [3] using the quantum entanglement [4] as the nonlo-cal resource In principle, by QT any unknown quantum states can be transmitted (see, e.g., [5 10] and references therein), if the transmitter owns the states Yet, transmission of known states can be done by a simpler method with the same nonlocal resource as in QT but without the need of having the states at hand This method was referred to as remote state prepara-tion (RSP) [11–14] As a per se feature of RSP, the preparer knows the full information encoded in the to-be-prepared quantum state, that may appear unwanted in some circum-stances To avoid this feature, joint remote state preparation (JRSP) protocols [15–30] have

recently been designed, in which there are N ≥ 2 preparers [19], who jointly perform the task but no one among them is able to identify the full content of the encoded information Experimental architecture of JRSP has also been dealt with in Ref [31]

In this work we concentrate on JRSP of an arbitrary qubit state of the most general form

as in (1) In fact, such a problem was studied previously [15–18] The nonlocal resource em-ployed in Refs [15,16] was a single Greenberger-Horne-Zeilinger (GHZ) trio [32], while

it was two Einstein-Podolsky-Rosen (EPR) pairs [34] in Ref [16] and one W state [33] in Ref [17] However, all those cited protocols were probabilistic, i.e., they succeed just with

a probability P suc < 1 As the overhead expenses in resources scales with P suc−1, on average,

if several protocols use the same amount/kind of resource, then the better is the one that has

a greater success probability and, of course, the best one it that having P suc= 1 Although

GHZ trios and W states are also useful (see e.g Refs [15,17]), we aim at employing EPR pairs as the nonlocal resource because these are the most elementary kind of entanglement and their production/distribution is easier than those of GHZ trios and W states For joint remote preparation of state| in (1) two EPR pairs are necessary [16,18] In Ref [16]

P suc <1, while in Ref [18] P suc= 1 Here we shall propose a different protocol which also

uses two EPR pairs as the nonlocal resource and has P suc= 1 but offers pronounced

advan-tages From an application point of view, our work would reveal diversity and flexibility in the ways a quantum task can be done given the same physical resource

In the next section we present in detail our protocol for the case of N= 2 preparers

Then, in the conclusion section, we compare it with other existing protocols to expose its merit We also provide anAppendixto deal with the generalization to any N > 2 preparers.

2 Our Protocol

Consider the simplest case of N= 2 The two preparers are Alice and Bob, while the

re-ceiver’s name is Charlie The full information contained in state|, (1), is S = {a, b, ϕ},

which can be somehow divided into S1and S2in such a way that S cannot be inferred from either S1 or S2, but can from both Let S1 be given only to Alice, S2 only to Bob, but no information to Charlie

The two EPR pairs|EPR AC |EPR BCused in Refs [16,18] have the qubits’ distribution shown in Fig.1, i.e., qubit A belongs to Alice, qubit B to Bob and qubits C, Cto Charlie Here, we adopt the notation

|EPR XY=√1

2



The protocol in Ref [16] is a two-step one The first step is called the preparation step and the second the reconstruction one In the preparation step Alice (and independently

Bob) measures her (his) own qubit in a basis determined solely by S1 (S2), then publicly broadcasts their outcome In the reconstruction step Charlie first performs a controlled-NOT gate1on her two qubits, followed by measuring one of them in the computational basis In

a lucky case when Charlie’s measurement outcome “matches” those of Alice and Bob, she will apply a proper operator on her unmeasured qubit to reconstruct it into the desired state

Although there are many ways to divide S into S1 and S2[15–17], neither one can make the protocol in Ref [16] deterministic Its maximum success probability is just P suc = 1/2.

The protocol in Ref [18] uses the same nonlocal resource with the same qubits’ distri-bution as in Fig.1(i.e., as in Ref [16]), but it is deterministic To achieve P suc= 1 it was

designed to be a three-step (not two-step) one: there are two preparation steps and one re-construction step In the first preparation step Bob does nothing, Alice does the same thing

as in Ref [16] and, at the same time, Charlie performs a controlled-NOT gate on her two qubits, followed by measuring one of them in the computational basis, but postpones ap-plication of any operators on the unmeasured qubit Then, both Alice and Charlie openly announce their measurement outcomes In the second preparation step Bob starts his action

by measuring his qubit in a basis that is judiciously determined by him, making use not only

of S2but also of the concrete outcomes announced by Alice and Charlie in the first step After the measurement, Bob publishes his outcome via a classical communication channel

as well What remains in the last step (i.e., the reconstruction step) is Charlie’s application

on the untouched qubit of the right operator conditioned on the outcomes of both Alice and

Bob Note that the division of S into S1 and S2in Ref [18] should be

and

After a condensed summary of the protocols in Refs [16,18], our purpose here is to propose a new deterministic protocol that also uses two EPR pairs but relaxes the function

of Charlie The state of nonlocal resource employed in the new protocol is |Q ABAC=

|EPR AB |EPR ACwith the qubits’ distribution shown in Fig.2which is remarkably different from that in Refs [16,18] Now, Alice holds two qubits A and A, while Bob and Charlie

each holds just a qubit (B and C) We also adopt the division S ⇒ {S1, S2} as in (3), (4) and the new protocol is a three-step one as well However, it proceeds differently as follows

1Controlled-NOT gate denoted by CN OT XY is a quantum gate acting on two qubits X (control qubit) and Y (target qubit) as CN OT |i |j = |i |i ⊕ j , where i, j ∈ {0, 1} and ⊕ stands for an addition mod 2.

Fig 1 The qubits’ distribution

for JRSP of the most general

single-qubit state via two EPR

pairs following the protocols

proposed in Refs [ 16 , 18 ].

Qubits are represented by dots

and entangled qubits are

connected by solid lines

Fig 2 The qubits’ distribution

for JRSP of the most general

single-qubit state via two EPR

pairs following the protocol

proposed in this work Qubits are

represented by dots and

entangled qubits are connected

by a solid line

In the first preparation step only Alice plays a role She measures qubits A and Ain the

basis determined solely by S1 = {a, b} as

⎛

⎜

⎝

|u00AA

|u01AA

|u10AA

|u  

⎞

⎟

⎠ = U(a, b)

⎛

⎜

⎝

|00AA

|01AA

|10AA

|11 

⎞

⎟

U (a, b)=

⎛

⎜

⎝

⎞

⎟

and publicly publishes kl (k, l ∈ {0, 1}), if the outcome |u klAA is found Since the

transfor-mation U (a, b) is unitary, the states {|u klAA} constitute an orthonormal complete set in a

4D Hilbert space Inversing (5) yields

|00AA= a|u00AA+ b|u01AA, (7)

|01AA= a|u10AA− b|u11AA, (8)

|10AA= b|u10AA+ a|u11AA (9) and

|11AA= b|u00AA− a|u01AA. (10)

As is well known, thanks to the effect of entanglement swapping, after Alice completed her

measurement the two qubits B and C are projected onto an entangled state, despite they are

far apart and still untouched This is verified mathematically by substituting (7)–(10) into the expression of the two EPR pairs in use|Q ABAC = |EPR AB |EPR AC, which can then

be rewritten in the form

|Q ABAC=1

2 1

l=0 1

k=0

|u klAA|L klBC (11) where

|L00BC = a|00 BC + b|11 BC , (12)

|L01BC = b|00 BC − a|11 BC , (13)

|L10BC = a|01 BC + b|10 BC (14) and

|L11BC = −b|01 BC + a|10 BC (15)

In the second preparation step only Bob plays a role He measures the qubit B but the choice of measurement basis is delicate To achieve P suc= 1, Bob must make use not only

of S2 = ϕ, which he is supposed to know, but also of the Alice’s outcome kl, which he hears

from the public media It turns out, however, that only ϕ and l suffice Explicitly, if l= 0,

Bob chooses the measurement basis as

|v0B

|v1B

=√1

2

1 e −iϕ

|0B

|1B

whereas if l= 1, he chooses the measurement basis as

|v0B

|v1B

=√1

2

1 e iϕ

|0B

|1B

Combining (16) and (17), Bob’s measurement bases are determined by ϕ and l in the

fol-lowing fashion:

|v (l)

0 B

|v (l)

= V (l) (ϕ)

|0B

|1B

Table 1 The collapsed state| klmC of Charlie’s qubit C and her reconstruction operator R klm, depending

on the outcomes kl and m of Alice and Bob, respectively I is the identity operator, X = {{0, 1}, {1, 0}} the bit-flip operator and Z = {{1, 0}, {0, −1}} the phase-flip one

where

V (l) (ϕ)=√1

2

1 e −(−1) l iϕ

e ( −1) l iϕ −1



Obviously, for a given l, the states {|v (l)

mB ; m = 0, 1} constitute an orthonormal

com-plete set in a 2D Hilbert space because V (l) (ϕ) is a unitary transformation In terms of

{|v (l)

0 B , |v (l)

1 B}, we have

|0B=√1

2



|v (l)

0 B + e −(−1) l iϕ |v (l)

1 B



(20) and

|1B=√1

2



e ( −1) l iϕ |v (l)

0 B − |v (l)

1 B



In measuring the quit B, if Bob finds it in state |v (l)

mB , he reveals m ∈ {0, 1} publicly Using

(20) and (21) in (12)–(15) we see that, conditioned on the outcomes klm, Charlie’s qubit C

would collapse, up to a global phase factor, into a certain state labeled| klmC, all of which are collected in Table1

In the last step only Charlie plays a role She makes use of the outcomes klm announced

by Alice and Bob to decide the right operator R klm to be applied on her qubit C to cast it to

the target state| C The concrete operators R klmare also listed in Table1 It is interesting

that we can express the dependence of R klm on klm by a single compact formula as

where⊕ denotes an addition mod 2

The probability for Alice finding state|u klAAand Bob finding state|v (l)

mB is P klm = 1/8,

independent of concrete values of k, l and m Since there exists a reconstruction operator for

each of the eight possible outcomes, our protocol succeeds with the total probability

P suc=

1

m=0 1

l=0 1

k=0

P klm= 8 ×1

i.e., it is deterministic

3 Conclusion

We have reconsidered the problem of joint remote preparation of an arbitrary qubit Al-though this problem was dealt with previously, the present investigation exhibits enlight-ening aspects The nonlocal resource in terms of a GHZ trio and a W state was employed

for the task in Ref [15] and Ref [17], respectively In this work we instead concentrate on two EPR pairs as the nonlocal resource, which were already employed in Refs [16,18] The nice feature of the protocol in Ref [16] is that the receiver participates only in the last step and his function is just to reconstruct the target state without the need of classical

com-munication It however suffers a bad feature of being probabilistic (i.e., P suc < 1) As for

the protocol in Ref [18], the nice feature is that it is deterministic (i.e., P suc= 1) On the

other hand, its bad feature is that the receiver’s function is more involved: she participates not only in the last step but also in the first preparation step in which she is required to per-form a controlled-NOT gate and to carry out a measurement as well as to communicate his measurement outcome As described in detail in the preceding section, the present protocol, though being executed also via two EPR pairs, incorporates the nice features of both the protocols in Refs [16,18] That is, in the new protocol both P suc= 1 and simple function

of the receiver are achieved The only price to pay is the first preparer’s ability to carry out

a two-qubit measurement instead of just a single-qubit one as in Refs [16,18] This price is regarded cheaper than that spent by the receiver in Ref [18] Moreover, an additional advan-tage of the present protocols over those in Refs [16,18] rests in the symmetry between Bob and Charlie (compare Figs.1and2) Hence, this new protocol allows, in case of need, to exchange the role of Bob and Charlie, while the other protocols do not Concerning classical communication cost, it is two bits in the probabilistic protocol [16] As for those in Ref [18] and here, an extra bit arises, but this is worth to make them deterministic

To conclude, we emphasize that to achieve unit success probability, P suc= 1, the protocol

should consist of three steps (two preparation steps and a reconstruction step) and, more importantly, feed-forward measurements should be done in the two first steps Namely, the basis for the measurement in the second step should be decided by the measurement outcome

in the first step

Last but not least we would like to mention two meaningful issues The first one con-cerns the question “What if the quantum channel initially consists of partially entangled qubit-pairs?” and the second one asks “How about the situation of more than two prepar-ers?” Of course partially entangled quantum channels can also do the job (usually with auxiliary qubits and measurements on them), but only in a probabilistic manner Maximum entanglement is therefore a prerequisite for determinism of JRSP protocols If it were not so,

a priori local filtering [35] or distillation process [36] is required to supply the participants

with (maximally entangled) EPR pairs As for the situation of any N > 2 preparers, the

pro-cedure is somewhat nontrivial, very different from Ref [19] and, thus, deserves a separate description in theAppendix

Acknowledgements We thank the two anonymous referees for their suggestions that improved the manuscript This work is supported by the Vietnam Foundation for Science and Technology Development (NAFOSTED) through a project no 103.99-2011.26.

Appendix

In thisAppendixwe generalize the situation with two preparers (Alice and Bob) described

in the main text to that with an arbitrary N > 2 preparers (Alice, Bob 1, Bob 2, and Bob

N − 1) The pre-shared quantum channel consists of N EPR pairs distributed among the

N + 1 participants (N preparers plus a receiver) as shown in Fig.3

For clarity, let us first consider N = 3 in detail The three EPR pairs of the quantum

channel are|EPR A1B1|EPR A2B2|EPR A3C = |Q A1B1A2B2A3C , with qubits A1 , A2, A3 hold

by Alice, B1 by Bob 1, B2 by Bob 2 and C by the receiver Charlie While Alice is allowed

Fig 3 The qubits’ distribution

for JRSP of the most general

single-qubit state via N EPR

pairs for the situation of N

preparers (Alice, Bob 1, Bob 2,

and Bob N− 1) Qubits are

represented by dots and

entangled qubits are connected

by a solid line

to know{a, b} as in the case of N = 2, Bob 1 and Bob 2 now share the knowledge of ϕ in

the following way: Bob 1 knows ϕ1 and Bob 2 knows ϕ2 where ϕ1 and ϕ2 sum up to ϕ.

First, Alice measures her three qubits in the basis{|u klmA1A2A3; k, l, m ∈ {0, 1}}:

⎛

⎜

⎜

⎜

⎜

⎜

⎝

|u000A1A2A3

|u001A1A2A3

|u010A1A2A3

|u011A1A2A3

|u100A1A2A3

|u101A1A2A3

|u110A1A2A3

|u111A1A2A3

⎞

⎟

⎟

⎟

⎟

⎟

⎠

=

⎛

⎜

⎜

⎜

⎜

⎜

⎝

⎞

⎟

⎟

⎟

⎟

⎟

⎠

⎛

⎜

⎜

⎜

⎜

⎜

⎝

|000A1A2A3

|001A1A2A3

|010A1A2A3

|011A1A2A3

|100A1A2A3

|101A1A2A3

|110A1A2A3

|111A1A2A3

⎞

⎟

⎟

⎟

⎟

⎟

⎠

(24)

and publicly broadcasts klm, if she finds state |u klmA1A2A3, projecting qubits B1, B2and

C onto an entangled state |L klmB1B2C Expressing the quantum channel |Q A1B1A2B2A3C

through{|u klmA1A2A3},

|Q A1B1A2B2A3C= 1

2√

2 1

m=0 1

l=0 1

k=0

|u klmA1A2A3|L klmB1B2C , (25)

we derive|L klmB1B2Cin the form

|L000B1B2C = a|000 B1B2C + b|111 B1B2C , (26)

|L001B1B2C = −a|111 B1B2C + b|000 B1B2C , (27)

|L010B1B2C = a|001 B1B2C + b|110 B1B2C , (28)

|L011B1B2C = −a|110 B1B2C + b|001 B1B2C , (29)

|L100B1B2C = a|010 B1B2C + b|101 B1B2C , (30)

|L101B1B2C = −a|101 B1B2C + b|010 B1B2C , (31)

|L110B B C = a|011 B B C + b|100 B B C (32)

|L111B1B2C = a|100 B1B2C − b|011 B1B2C (33) Next, Bob 1 and Bob 2 independently measure their qubits in a basis conditioned not

only on ϕ1and ϕ2but also on Alice’s outcome Concretely, when klm= 000 or 010, each

Bob j (j = 1, 2) uses a measurement basis determined by

|v0B j

|v1B j

= V ( 0) (ϕ j )

|0B j

|1B j

with V ( 0) (ϕ j )given in (19) However, for klm = 001 or 011, the basis for each Bob j is

|v0B j

|v1Bj

= V ( 1) (ϕ j )

|0B j

|1B j

with V ( 1) (ϕ j )given in (19) In case klm= 100 or 110, Bob 1 uses the basis

|v0B1

|v1B1

= V ( 0) (ϕ1)

|0B1

|1B1

but the basis for Bob 2 is

|v0B2

|v1B2

= V ( 1) (ϕ2)

|0B2

|1B2

Finally, if klm= 101 or 111, the bases for Bob 1 and Bob 2 are differently defined as

|v0B1

|v1B1

= V ( 1) (ϕ1)

|0B1

|1B1

(38) and

|v0B2

|v1B2

= V ( 0) (ϕ2)

|0B2

|1B2

respectively Then the states|L klmB1B2C in (25) can be, up to an unimportant global phase factor, decomposed in terms of{|v nB1, |v sB2} as follows

|L klmB1B2C=1

2 1

s=0 1

n=0

|v nB1|v sB2R+klmns | C , (40)

with R klmnseither the identity operator or a Pauli one This implies that after Alice, Bob 1 and Bob 2 announce their measurement outcomes, Charlie will be able to convert the qubit

C to be in the desired state| C by applying R klmns on C The reconstruction operators

R klmns that Charlie needs in the last step are shown in Table2 Because each of the 25=

32 possible outcomes klmns is associated with a reconstruction operator R klmns, the total success probability is obviously 1

As inferred from the cases of N = 2 and N = 3, for any N > 3 the quantum

chan-nel should consist of N EPR pairs, |Q A1B1A2B2 A N−1B N−1A N C = |EPR A1B1|EPR A2B2 .

|EPR A N−1B N−1|EPR A N C(see Fig.3) The information is split in such a way that Alice still knows{a, b}, but Bob j (j = 1, 2, , N − 1) just knows ϕ j in such a way that the con-straintN−1

j=1 ϕ j = ϕ is satisfied The basis for Alice to measure N qubits A1, A2, , A N

is spanned by 2N orthonormal states{(−1) i1a |i1i2 i NA1A2 A N + b|i1i2 i NA1A2 A N;

i n ∈ {0, 1}; i n = 1 − i n } As for Bob j, each of them independently measures qubit B j in

the basis V ( 0) (ϕ j ).{|0B j ,|1B j } or V ( 1) (ϕ j ).{|0B j ,|1B j}, depending on Alice’s outcome

Such generalized JRSP protocol works deterministically since for each of the 22N−1

possi-ble measurement outcomes of the N preparers there exists a corresponding reconstruction

operator for the receiver to obtain the target state|

Table 2 The reconstruction operator R klmns , conditioned on the measurement outcomes klm, n and s of Alice, Bob 1 and Bob 2, respectively I is the identity operator, X = {{0, 1}, {1, 0}} the bit-flip operator and

Z = {{1, 0}, {0, −1}} the phase-flip one

1–8 00000, 00011, 10000, 10011, 01101, 01110, 11101 or 11110 I

9–16 01000, 01011, 11000, 11011, 00101, 00110, 10101 or 10110 X

17–24 00100, 00111, 10100, 10111, 01001, 01010, 11001 or 11010 ZX

25–32 01100, 01111, 11100, 11111, 00001, 00010, 10001 or 10010 Z

References

1 Bennett, C.H., Brassard, G.: In: Proceedings of IEEE International Conference on Computers, Systems and Signal Processing, p 175 IEEE, New York (1984)

2 Shor, P.W.: In: Goldwasser, S (ed.) Proceedings of the 35th Annual Symposium on the Foundation of Computer Science, p 124 IEEE Computer Society Press, Los Alamitos (1994)

3 Bennett, C.H., Brassard, G., Crepeau, C., Jozsa, R., Peres, A., Wootters, W.K.: Phys Rev Lett 70, 1895

(1993)

4 Schrödinger, E.: Naturwissenschaften 23, 807 (1935)

5 Karlsson, A., Bourennane, M.: Phys Rev A 58, 4394 (1998)

6 van Loock, P., Braunstein, S.L.: Phys Rev Lett 84, 3482 (2000)

7 An, N.B.: Phys Rev A 68, 022321 (2003)

8 An, N.B.: Phys Lett A 341, 9 (2005)

9 An, N.B., Mahler, G.: Phys Lett A 365, 70 (2007)

10 An, N.B., Kim, J.: Phys Rev A 80, 042316 (2009)

11 Lo, H.K.: Phys Rev A 62, 012313 (2000)

12 Pati, A.K.: Phys Rev A 63, 014302 (2000)

13 Bennett, C.H., DiVincenzo, D.P., Shor, P.W., Smolin, J.A., Terhal, B.M., Wootters, W.K.: Phys Rev.

Lett 87, 077902 (2001)

14 An, N.B., Bich, C.T., Don, N.V., Kim, J.: Adv Nat Sci., Nanosci Nanotechnol 2, 035009 (2011)

15 Xia, Y., Song, J., Song, H.S.: J Phys B, At Mol Opt Phys 40, 3719 (2007)

16 An, N.B., Kim, J.: J Phys B, At Mol Opt Phys 41, 095501 (2008)

17 An, N.B.: Opt Commun 283, 4113 (2010)

18 An, N.B., Bich, C.T., Don, N.V.: Phys Lett A 375, 3570 (2011)

19 An, N.B., Kim, J.: Int J Quantum Inf 6, 1051 (2008)

20 Hou, K., Wang, J., Lu, Y.L, Shi, S.H.: Int J Theor Phys 48, 2005 (2009)

21 An, N.B.: J Phys B, At Mol Opt Phys 42, 125501 (2009)

22 Zhan, Y.B., Zhang, Q.Y, Shi, J.: Chin Phys B 19, 080310 (2010)

23 Luo, M.X., Chen, X.B., Ma, S.Y, Niu, X.X., Yang, Y.X.: Opt Commun 283, 4796 (2010)

24 Chen, Q.Q., Xia, Y., Song, J., An, N.B.: Phys Lett A 374, 4483 (2010)

25 Chen, Q.Q., Xia, Y., An, N.B.: Opt Commun 284, 2617 (2011)

26 Zhan, Y.B., Hu, B.L., Ma, P.C.: J Phys B, At Mol Opt Phys 44, 095501 (2011)

27 An, N.B., Bich, C.T., Don, N.V.: J Phys B, At Mol Opt Phys 44, 135506 (2011)

28 Wang, Z.Y.: Int J Quantum Inf 9, 809 (2011)

29 Hou, K., Li, Y.B., Liu, G.H., Sheng, S.Q.: J Phys A, Math Theor 44, 255304 (2011)

30 Xiao, X.O., Liu, J.M., Zeng, G.: J Phys B, At Mol Opt Phys 44, 075501 (2011)

31 Luo, M.X., Chen, X.B., Yang, Y.X., Niu, X.X.: Quantum inf process (2011) doi: 10.1007/s11128-011-0283-5

32 Greenberger, D.M., Horne, M.A., Zeilinger, A.: In: Bell’s Theorem, Quantum Theory and Conceptions

of the Universe Kluwer, Dordrecht (1989)

33 Dur, W., Vidal, G., Cirac, J.I.: Phys Rev A 62, 062314 (2000)

34 Einstein, A., Podolsky, B., Rosen, N.: Phys Rev 47, 777 (1935)

35 Verstraete, F., Dehaene, J., DeMoor, B.: Phys Rev A 64, 010101(R) (2001)

36 Bennett, C.H., Bernstein, H.J., Popescu, S., Schumacher, B.: Phys Rev A 53, 2046 (1996)

i.e., it is deterministic

3 Conclusion

We have reconsidered the problem of joint remote preparation of an arbitrary qubit Al-though this problem was... than two prepar-ers?” Of course partially entangled quantum channels can also the job (usually with auxiliary qubits and measurements on them), but only in a probabilistic manner Maximum entanglement