Quantum Inf Process (2017) 16:93 DOI 10.1007/s11128-017-1542-x

Deterministic remote preparation via the Brown state Song-Ya Ma1 · Cong Gao1 · Pei Zhang1 · Zhi-Guo Qu2

Received: 31 May 2016 / Accepted: 8 February 2017 / Published online: 23 February 2017 © Springer Science+Business Media New York 2017

Abstract We propose two deterministic remote state preparation (DRSP) schemes by using the Brown state as the entangled channel. Firstly, the remote preparation of an arbitrary two-qubit state is considered. It is worth mentioning that the construction of measurement bases plays a key role in our scheme. Then, the remote preparation of an arbitrary three-qubit state is investigated. The proposed schemes can be extended to controlled remote state preparation (CRSP) with unit success probabilities. At variance with the existing CRSP schemes via the Brown state, the derived schemes have no restriction on the coefficients, while the success probabilities can reach 100%. It means the success probabilities are greatly improved. Moreover, we pay attention to the DRSP in noisy environments under two important decoherence models, the amplitudedamping noise and phase-damping noise. Keywords Remote state preparation · Brown state · Complete orthogonal basis · Amplitude-damping noise · Phase-damping noise

1 Introduction Secure transmission of a quantum state from one place to another is one of the central tasks in the field of quantum information. However, the state itself should not be directly

B

Song-Ya Ma [email protected]

1

School of Mathematics and Statistics, Henan University, Kaifeng 475004, China

2

Jiangsu Engineering Center of Network Monitoring, Nanjing University of Information Science & Technology, Nanjing 210044, China

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sent because any enemy can easily replace it by a fake one without recognition by the authorized parties. The application of quantum entanglement provides some novel ways for the transmission of a quantum state. Quantum teleportation (QT) was first proposed by Bennett et al. [1] to securely transmit a qubit by utilizing priorly shared entanglement and some classical communication. In QT, the sender has the physical instance of the state, but the information of the state is unknown to her. When the state is known to the sender, Lo [2] proposed another secure method called remote state preparation (RSP), which can be done via the same quantum channel as that in QT but with simpler measurement and less classical communication cost (CCC). Bennett et al. [3] studied the trade-off between the required quantum resources and CCC in RSP. Pati [4] demonstrated that a single qubit chosen from polar great circles on a Bloch sphere can be remotely prepared with one classical bit (cbit) from the sender to the receiver if they share one EPR pair in advance, while two cbits are needed in QT. In addition to the traditional RSP with two parties, multi-party RSP [5–17] plays an important role in the general quantum network communication and quantum distributed computation. One branch of multi-party RSP is called controlled remote state preparation (CRSP) [5– 10], which transmits a quantum state in a controlled manner. In CRSP, one or several controllers are introduced in addition to the sender and the receiver. Although the controlling party has no information about the prepared state, he determines whether the state is constructed for the receiver or not. The other is named as joint remote state preparation (JRSP) [11–17]. Different from RSP that all the secret information is provided by one sender, JRSP deals with the situation that the complete classical knowledge of the state is independently shared among numbers of senders who have to cooperatively complete the task. However, most of the previous RSP protocols are achieved in ideal conditions, which does not suffer from any unwanted interactions with the realistic environments. In recent years, some RSP protocols have taken noisy influences into account. For example, Liang et al. [18] analyzed how to remotely prepare a single-qubit and a bipartite entangled states via a GHZ-class channel subject to Markovian Pauli noises. Zhang et al. [19] discussed the remote preparation of a general two-particle state through two Bell-like states in the presence of structured reservoirs. Guan et al. [20] studied the JRSP of an arbitrary two-qubit state via a sixqubit cluster state suffering from amplitude-damping noise and phase-damping noise. Li et al. [21] investigated the influence of several Markovian noises on the DRSP of an arbitrary two-qubit state by using four EPR pairs as the entangled channel. Li et al. [22] proposed a new scheme for efficient remote preparation of an arbitrary two-qubit state by introducing two auxiliary qubits and using two EPR states as the quantum channel in a non-recursive way. So far, various generalizations have been made for RSP such as oblivious RSP, low-entanglement RSP, optimal RSP, deterministic RSP, and RSP in noisy environments [5–29]. Meantime, experimental implementation of RSP has been demonstrated [30–34]. Like quantum operation sharing [35–37], various entangled resources such as EPR pair, GHZ state, W sate, cluster state or χ state have been exploited for carrying out RSP since the first appearance of RSP. Why different entangled states are explored as the quantum channel to realize RSP? This is because various entanglement maybe used in some peculiar circumstances due to the security consideration and their own features. Particularly, in some special case only some kinds of entan-

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gled states are available, one has to use them to try to fulfill tasks instead of doing nothing. In this sense, it is interesting and useful to explore the ability of different entangled states in implementing RSP tasks. In 2005, Brown et al. [38] discovered a maximally entangled five-qubit state, i.e., Brown state, through a numerical optimization procedure. This notable state exhibits the genuine multi-partite entanglement according to both negative partial transpose measure and von Neumann entropy measure. Moreover, the Brown state is more highly entangled than other five-qubit GHZ state. As a peculiar entangled resource, the Brown state has been shown to be useful for quantum information processing, such as quantum operation sharing [36,37], QT, quantum-state sharing, superdense coding [39], secure quantum communication [40], and quantum information splitting [41]. Chen et al. [8] first investigated the problem of using the Brown state as the shared quantum channel to fulfill the RSP tasks. They proposed two CRSP schemes for two- and three-qubit states in the case of complex coefficients with the success probabilities 50%. However, the measurement bases they constructed have many restrictions, which means their methods can not adapt to arbitrary two- and three-qubit states. In order to improve the application range, Gao et al. [9] reinvestigated the CRSP of arbitrary two- and three-qubit states in complex space via the Brown state. They put forward two schemes that have no requirements on the prepared state while preserving the same success probabilities 50%. To our knowledge, there are no deterministic RSP (or CRSP, JRSP) schemes via the Brown state till now. In fact, the authors in Ref. [8] also raised such a question: In the future, one may consider how a certain ensemble of quantum states with complex coefficients can be remotely prepared in a deterministic way? Success probability is one of the most important indexes to evaluate a protocol. Thus, deterministic RSP has attracted much attention and been intensively studied in recent years [6,10,16,17,29]. Of course, it would be better to consume less resource if achieving unit success probability. In 2012, Zhan et al. [29] put forward two deterministic schemes for remote preparation of arbitrary two- and three-qubit states with complex coefficients via GHZ states as the quantum channel. In this paper, we are concerned with the question “can one make a general RSP via the Brown state with unit success probability?”. In Sect. 2, we put forward a deterministic remote state preparation (DRSP) scheme of an arbitrary two-qubit state with complex spectra via the Brown state as the entangled channel. It is worth stressing that the novel set of measurement basis plays a key role in this scheme. DRSP of an arbitrary three-qubit state is considered in Sect. 3. In fact, the proposed protocols can be extended to the CRSP with unit success probabilities, which will be explained in the last section. Compared with the previous CRSP proposals [8,9], the success probabilities of the extended protocols are greatly improved. The above two DRSP schemes are achieved under ideal conditions. In general, an actual quantum system will undergo unavoidable interaction with its surrounding world. So it is necessary to consider and explore quantum transmission process in realistic environment. In Sect. 4, we pay attention to the noisy influence on the process of distributing of the channel qubits under two specific kinds of noise, the amplitude-damping noise and phase-damping noise. Discussions and conclusions are given in the last section.

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2 DRSP scheme for an arbitrary two-qubit state Suppose that the sender Alice wishes to help the receiver Bob remotely prepare an arbitrary two-qubit state h(a0 , a1 , a2 , a3 ) = a0 |00 + a1 |01 + a2 |10 + a3 |11,

(1)

 where a j , j = 0, . . . , 3 are complex numbers and satisfy 3j=0 |a j |2 = 1. Alice knows the coefficients a0 , . . . , a3 , while Bob does not have any knowledge of the original state at all. Assume that two participants share the Brown state |Br 12345 =

1 (|001|φ−  + |010|Φ−  + |100|φ+  + |111|Φ+ )12345 , 2

(2)

where 1 1 |φ±  = √ (|00 ± |11) , |Φ±  = √ (|01 ± |10) . 2 2

(3)

Besides the Brown state, Alice introduces an EPR pair |φ+ 67 . The particles (1, 2, 6, 3, 7) are in the possession of Alice, and the particles (4, 5) belong to Bob. Hence, the quantum channel can be rewritten as: 1 |C7  = √ |Br 12345 (|00 + |11)67 2 1 = √ (|00010|φ−  + |01000|Φ−  + |10101|φ+  + |11111|Φ+  2 2 +|00111|φ−  + |01101|Φ−  + |10000|φ+  + |11010|Φ+ )1263745 . (4) The DRSP scheme for an arbitrary two-qubit state is described as follows: S1 In order to help the receiver remotely prepare the original state with unit success probability, the sender Alice needs to perform suitable projective measurement under a set of useful measurement basis that relies on the parameters a0 , . . . , a3 of the prepared state. Denote b0 =

a1 − a2 a0 + a3 a0 − a3 a1 + a2 √ , b1 = √ , b2 = √ , b3 = √ . 2 2 2 2

(5)

 It is noticed that 3j=0 |b j |2 = 1. Since Alice knows a j , she knows b j completely. Split b j = r j eiθ j into the amplitude information r j and phase information θ j , where r j ≥ 0 and θ j ∈ [0, 2π ), j = 0, . . . , 3. Construct a set of measurement basis |ζ0 , . . . , |ζ7 , which has the following relationship to the computation basis |000, . . . , |111

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⎞ ⎛ r0 |ζ0  ⎜ |ζ1  ⎟ ⎜ −r1 ⎜ ⎜ ⎟ ⎜ |ζ2  ⎟ ⎜ −r2 ⎜ ⎟ ⎜ ⎜ |ζ3  ⎟ ⎜ −r3 ⎜ ⎟ ⎜ ⎜ |ζ4  ⎟ = ⎜ 0 ⎜ ⎟ ⎜ ⎜ |ζ5  ⎟ ⎜ 0 ⎜ ⎟ ⎜ ⎝ |ζ6  ⎠ ⎝ 0 0 |ζ7  ⎛

r1 r0 −r3 r2 0 0 0 0

r2 r3 r0 −r1 0 0 0 0

r3 −r2 r1 r0 0 0 0 0

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0 0 0 0 r0 −r1 −r2 −r3

0 0 0 0 r1 r0 −r3 r2

0 0 0 0 r2 r3 r0 −r1

⎞ ⎞⎛ |010 0 ⎟ ⎜ 0 ⎟ ⎟ ⎜ |101 ⎟ ⎜ ⎟ 0 ⎟ ⎜ |000 ⎟ ⎟ ⎟ ⎜ 0 ⎟ ⎟ ⎜ |111 ⎟ . ⎜ ⎟ r3 ⎟ ⎜ |011 ⎟ ⎟ ⎟ ⎜ −r2 ⎟ ⎟ ⎜ |100 ⎟ ⎝ ⎠ |001 ⎠ r1 |110 r0

(6)

Alice performs a three-qubit projective measurement on her particles (1, 2, 6) under the basis |ζ0 , . . . , |ζ7 . After the measurement, the entangled channel in Eq. (4) can be expanded in terms of the measurement basis as

 1 |C7  = √ |ζ0  r0 |00|Φ−  + r1 |01|φ+  + r2 |10|φ−  + r3 |11|Φ+  2 2

 + |ζ1  −r1 |00|Φ−  + r0 |01|φ+  + r3 |10|φ−  − r2 |11|Φ+ 

 + |ζ2  −r2 |00|Φ−  − r3 |01|φ+  + r0 |10|φ−  + r1 |11|Φ+ 

 + |ζ3  −r3 |00|Φ−  + r2 |01|φ+  − r1 |10|φ−  + r0 |11|Φ+ 

 + |ζ4  r0 |01|Φ−  + r1 |00|φ+  + r2 |11|φ−  + r3 |10|Φ+ 

 + |ζ5  −r1 |01|Φ−  + r0 |00|φ+  + r3 |11|φ−  − r2 |10|Φ+ 

 + |ζ6  −r2 |01|Φ−  − r3 |00|φ+  + r0 |11|φ−  + r1 |10|Φ+ 

 + |ζ7  −r3 |01|Φ−  + r2 |00|φ+  − r1 |11|φ−  + r0 |10|Φ+  1263745 . (7) S2 After the first-step measurement, Alice does not perform the second-step measurement immediately but performs a unitary operation U n 1 on her particles (3, 7) conditioned on her first step measurement result |ζn 1 , n 1 ∈ {0, . . . , 7}. Here U 0 = U 5 = I ⊗ I, U 1 = U 4 = I ⊗ X, U 2 = U 7 = X ⊗ I, U 3 = U 6 = X ⊗ X. (8) Then Alice performs a two-qubit projective measurement on her particles (3, 7) under the basis |η0 , . . . , |η3  which depends on the phase information θ0 , . . . , θ3 and is defined by ⎛

⎛ −iθ ⎞ e 0 |η0  ⎜ |η1  ⎟ 1 ⎜ −e−iθ0 ⎜ ⎜ ⎟ ⎝ |η2  ⎠ = 2 ⎝ −e−iθ0 |η3  −e−iθ0

e−iθ1 e−iθ1 −e−iθ1 e−iθ1

e−iθ2 e−iθ2 e−iθ2 −e−iθ2

⎞⎛ ⎞ e−iθ3 |00 ⎜ ⎟ −e−iθ3 ⎟ ⎟ ⎜ |01 ⎟ . −iθ 3 ⎠ ⎝ |10 ⎠ e −iθ 3 |11 e

(9)

After the two-step measurement, Alice transmits the classical message n 1 n 2 to Bob through the classical channel if the measurement result is |ζn 1 |ηn 2 , n 1 = 0, . . . , 7, n 2 = 0, . . . , 3.

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S3 The receiver Bob performs recovery unitary operations to get the prepared state depending on the received classical message. To see how our protocol works, without loss of generality, assume Alice’s first-step measurement result is |ζ1 126 . As a result, the particles (3, 7, 4, 5) collapse into − r1 |00|Φ−  + r0 |01|φ+  + r3 |10|φ−  − r2 |11|Φ+ .

(10)

In this situation, Alice performs the unitary operation U 1 = I ⊗ X on her particles (3, 7) and gets − r1 |01|Φ−  + r0 |00|φ+  + r3 |11|φ−  − r2 |10|Φ+ .

(11)

Then Alice performs a two-qubit projective measurement on the particles (3, 7) under the basis |η0 , . . . , |η3 . The state in Eq. (11) can be rewritten as: √ 2 {|η0  [−b1 (|01 − |10) + b0 (|00 + |11) 4 + b3 (|00 − |11) − b2 (|01 + |10)] + |η1  [−b1 (|01 − |10) − b0 (|00 + |11)−b3 (|00 − |11) − b2 (|01 + |10)] + |η2  [b1 (|01 − |10) − b0 (|00 + |11) + b3 (|00 − |11) − b2 (|01 + |10)] + |η3  [−b1 (|01−|10) − b0 (|00 + |11) + b3 (|00−|11) + b2 (|01+|10)]} 1 = [|η0 (a1 |00 − a0 |01 + a3 |10 − a2 |11) + |η1 (−a1 |00 − a0 |01 2 + a3 |10 + a2 |11) + |η2 (a2 |00 + a3 |01 − a0 |10 − a1 |11) + |η3 (a2 |00 − a3 |01 + a0 |10 − a1 |11)] . (12) From the above equation, one can see whatever Alice’s second-step measurement outcome is, Bob can get the prepared state by performing appropriate unitary operation on his qubits. As for the other seven cases corresponding to Alice’s first-step measurement result, similar analysis can be made. Here, we do not depict them one by one anymore. Bob’s recovery operations conditioned on Alice’s two-step measurement results |ζn 1  and |ηn 2  (n 1 = 0, . . . , 7, n 2 = 0, . . . , 3) are listed in Table 1. Thus, our protocol for remotely preparing an arbitrary two-qubit state can be achieved deterministically.

3 DRSP scheme for an arbitrary three-qubit state In this section, we proceed to show the utility of the Brown state for remotely preparing an arbitrary three-qubit state f (x0 , x1 , x2 , x3 , x4 , x5 , x6 , x7 ) = x0 |000 + x1 |001 + x2 |010 + x3 |011 +x4 |100 + x5 |101 + x6 |110 + x7 |111, (13)

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Table 1 Relation among Alice’s two-step measurement outcomes n 1 , n 2 (n 1 = 0, . . . , 7, n 2 = 0, . . . , 3), the collapsed state (CS) of the qubits (4, 5) after the measurements, and Bob’s recovery operation (BRO) Un 1 n 2 on the collapsed state n1 , n2

CS

BRO

n1 , n2

CS

BRO

0(4), 0

h(a0 , a1 , a2 , a3 )

I4 I5

2(6), 0

h(−a2 , a3 , a0 , −a1 )

Y4 Z 5

0(4), 1

h(a0 , −a1 , −a2 , a3 )

Z4 Z5

2(6), 1

h(a2 , a3 , a0 , a1 )

X 4 I5

0(4), 2

h(−a3 , a2 , a1 , −a0 )

Y4 Y5

2(6), 2

h(a1 , a0 , a3 , a2 )

I4 X 5

0(4), 3

h(a3 , a2 , a1 , a0 )

X4 X5

2(6), 3

h(−x1 , x0 , x3 , −x2 )

Z 4 Y5

1(5), 0

h(a1 , −a0 , a3 , −a2 )

I4 Y5

3(7), 0

h(−x3 , −x2 , x1 , x0 )

Y4 X 5

1(5), 1

h(−a1 , −a0 , a3 , a2 )

Z4 X5

3(7), 1

h(−a3 , a2 , −a1 , a0 )

X 4 Y5

1(5), 2

h(a2 , a3 , −a0 , −a1 )

Y4 I5

3(7), 2

h(a0 , −a1 , a2 , −a3 )

I4 Z 5

1(5), 3

h(a2 , −a3 , a0 , −a1 )

X4 Z5

3(7), 3

h(−a0 , −a1 , a2 , a3 )

Z 4 I5

Here I, X, Y, Z denote the Pauli operations

 where x j , j = 0, . . . , 7 are complex numbers and satisfy 7j=0 |x j |2 = 1. Besides the Brown state |Br 12345 in Eq. (2), Alice and Bob share a GHZ state |G H Z 678 = √1 (|000 + |111)678 and an EPR pair |φ+ 90 as the quantum channel. 2 Alice processes the qubits (1, 2, 6, 9, 3, 7, 0) and Bob owns the particles (4, 5, 8). The combined system can be rewritten as: 1 (|0000100|φ− |0 4 + |0100000|Φ− |0 + |1000000|φ+ |0 + |1100100|Φ+ |0 + |0010110|φ− |1 + |0110010|Φ− |1 + |1010010|φ+ |1

|Br 12345 |G H Z 678 |φ+ 90 =

+ |1110110|Φ+ |1 + |0001101|φ− |0 + |0101001|Φ− |0 + |1001001|φ+ |0 + |1101101|Φ+ |0 + |0011111|φ− |1 + |0111011|Φ− |1 + |1011011|φ+ |1 + |1111111|Φ+ |1)1269370458 . (14) Next, we detail the process of the new scheme: S1 For the purpose of preparing an arbitrary three-qubit state at the remote position with unit success probability, we need to construct useful measurement basis. Denote x2 − x4 √ , 2 x0 − x6 y4 = √ , 2

y0 =

x0 + x6 √ , 2 x2 + x4 y5 = √ , 2 y1 =

x3 − x5 √ , 2 x1 − x7 y6 = √ , 2

y2 =

x1 + x7 √ , 2 x3 + x5 y7 = √ . 2 y3 =

(15)

Let y j = s j eiα j , where s j ≥ 0 and α j ∈ [0, 2π ). Real coefficients s j satisfy the 7 7   s 2j = 1 as |y j |2 = 1. normalization condition j=0

j=0

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Alice first performs a four-qubit projective measurement on her particles (1, 2, 6, 9) under the complete orthogonal basis |ξ0 , . . . , |ξ15 , which are related to the computation basis |0000, . . . , |1111 as ⎞ ⎛ ⎞ ⎛ ⎛ ⎛ ⎞ ⎞ |ξ8  |0100 |0101 |ξ0  ⎜ |1001 ⎟ ⎜ |ξ9  ⎟ ⎜ |1000 ⎟ ⎜ |ξ1  ⎟ ⎟ ⎜ ⎟ ⎜ ⎜ ⎜ ⎟ ⎟ ⎜ |0110 ⎟ ⎜ |ξ10  ⎟ ⎜ |0111 ⎟ ⎜ |ξ2  ⎟ ⎟ ⎜ ⎟ ⎜ ⎜ ⎜ ⎟ ⎟ ⎟ ⎜ ⎟ ⎜ ⎜ ⎜ |ξ3  ⎟ ⎟ ⎜ ⎟ = G ⎜ |1011 ⎟ , ⎜ |ξ11  ⎟ = G ⎜ |1010 ⎟ , (16) ⎜ |0000 ⎟ ⎜ |ξ12  ⎟ ⎜ |0001 ⎟ ⎜ |ξ4  ⎟ ⎟ ⎜ ⎟ ⎜ ⎜ ⎜ ⎟ ⎟ ⎜ |1101 ⎟ ⎜ |ξ13  ⎟ ⎜ |1100 ⎟ ⎜ |ξ5  ⎟ ⎟ ⎜ ⎟ ⎜ ⎜ ⎜ ⎟ ⎟ ⎝ |0010 ⎠ ⎝ |ξ14  ⎠ ⎝ |0011 ⎠ ⎝ |ξ6  ⎠ |1111 |1110 |ξ7  |ξ15  where

⎛

s0 ⎜ −s1 ⎜ ⎜ −s2 ⎜ ⎜ −s3 G=⎜ ⎜ −s4 ⎜ ⎜ −s5 ⎜ ⎝ −s6 −s7

s1 s0 −s3 s2 −s5 s4 s7 −s6

s2 s3 s0 −s1 −s6 −s7 s4 s5

s3 −s2 s1 s0 −s7 s6 −s5 s4

s4 s5 s6 s7 s0 −s1 −s2 −s3

s5 −s4 s7 −s6 s1 s0 s3 −s2

s6 −s7 −s4 s5 s2 −s3 s0 s1

⎞ s7 s6 ⎟ ⎟ −s5 ⎟ ⎟ −s4 ⎟ ⎟. s3 ⎟ ⎟ s2 ⎟ ⎟ −s1 ⎠ s0

(17)

Under the measurement basis |ξ0 , . . . , |ξ15 , the shared channel in Eq. (14) can be written as: |Br12345 |GHZ678 |φ+ 90 1

g0 (s0 , −s1 , −s2 , −s3 , −s4 , −s5 , −s6 , −s7 )|000|Φ− |0 = 4 + g0 (s1 , s0 , −s3 , s2 , −s5 , s4 , s7 , −s6 )|001|φ+ |0 + g0 (s2 , s3 , s0 , −s1 , −s6 , −s7 , s4 , s5 )|010|Φ− |1 + g0 (s3 , −s2 , s1 , s0 , −s7 , s6 , −s5 , s4 )|011|φ+ |1 + g0 (s4 , s5 , s6 , s7 , s0 , −s1 , −s2 , −s3 )|100|φ− |0 + g0 (s5 , −s4 , s7 , −s6 , s1 , s0 , s3 , −s2 )|101|Φ+ |0 + g0 (s6 , −s7 , −s4 , s5 , s2 , −s3 , s0 , s1 )|110|φ− |1 + g0 (s7 , s6 , −s5 , −s4 , s3 , s2 , −s1 , s0 )|111|Φ+ |1 + g1 (s0 , −s1 , −s2 , −s3 , −s4 , −s5 , −s6 , −s7 )|000|Φ− |0 + g1 (s1 , s0 , −s3 , s2 , −s5 , s4 , s7 , −s6 )|001|φ+ |0 + g1 (s2 , s3 , s0 , −s1 , −s6 , −s7 , s4 , s5 )|010|Φ− |1 + g1 (s3 , −s2 , s1 , s0 , −s7 , s6 , −s5 , s4 )|011|φ+ |1 + g1 (s4 , s5 , s6 , s7 , s0 , −s1 , −s2 , −s3 )|100|φ− |0 + g1 (s5 , −s4 , s7 , −s6 , s1 , s0 , s3 , −s2 )|101|Φ+ |0

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+ g1 (s6 , −s7 , −s4 , s5 , s2 , −s3 , s0 , s1 )|110|φ− |1

 + g1 (s7 , s6 , −s5 , −s4 , s3 , s2 , −s1 , s0 )|111|Φ+ |1 1269370458 ,

(18)

where g j (u 0 , u 1 , u 2 , u 3 , u 4 , u 5 , u 6 , u 7 ) =

7

u d |ξ8 j+d ,

j = 0, 1.

(19)

d=0

S2 After the measurement, Alice does not perform the second-step measurement immediately but performs a unitary operation U m 1 on her particles (3, 7, 0) conditioned on her first measurement result |ξm 1 , m 1 = 0, . . . , 15. Here U 0 = U 9 = I3 I7 I0 , U 1 = U 8 = I3 I7 X 0 , U 2 = U 11 = I3 X 7 I0 , U 3 = U 10 = I3 X 7 X 0 , U 4 = U 13 = X 3 I7 I0 , U 5 = U 12 = X 3 I7 X 0 , U 6 = U 15 = X 3 X 7 I0 , U 7 = U 14 = X 3 X 7 X 0 .

(20)

Then Alice performs a three-qubit projective measurement on the particles (3, 7, 0) under the complete orthogonal basis |τ0 , . . . , |τ7  which are defined by (|τ0 , |τ1 , |τ2 , |τ3 , |τ4 , |τ5 , |τ6 , |τ7 )T ⎛ −iα0 e e−iα1 e−iα2 e−iα3 ⎜ −e−iα0 e−iα1 e−iα2 −e−iα3 ⎜ −iα 0 −e−iα1 e−iα2 ⎜ e−iα3 √ ⎜ −e −iα −iα −iα 0 1 2 2⎜ e −e e−iα3 ⎜ −e = −iα −iα −iα ⎜ 0 1 2 −e −e −e−iα3 4 ⎜ −e ⎜ −e−iα0 e−iα1 −e−iα2 e−iα3 ⎜ ⎝ −e−iα0 e−iα1 e−iα2 −e−iα3 −e−iα0 −e−iα1 e−iα2 e−iα3 ⎞ ⎛ |000 ⎜ |001 ⎟ ⎟ ⎜ ⎜ |010 ⎟ ⎟ ⎜ ⎜ |011 ⎟ ⎟ ×⎜ ⎜ |100 ⎟ ⎟ ⎜ ⎜ |101 ⎟ ⎟ ⎜ ⎝ |110 ⎠ |111

e−iα4 e−iα4 e−iα4 e−iα4 e−iα4 −e−iα4 −e−iα4 −e−iα4

e−iα5 −e−iα5 e−iα5 −e−iα5 e−iα5 e−iα5 e−iα5 −e−iα5

e−iα6 −e−iα6 −e−iα6 e−iα6 e−iα6 −e−iα6 e−iα6 e−iα6

⎞ e−iα7 e−iα7 ⎟ ⎟ −e−iα7 ⎟ ⎟ −e−iα7 ⎟ ⎟ e−iα7 ⎟ ⎟ e−iα7 ⎟ ⎟ −e−iα7 ⎠ e−iα7

(21)

Alice transmits her measurement outcome |ξm 1 |τm 2  to Bob through classical message m 1 m 2 , where m 1 = 0, . . . , 15, m 2 = 0, . . . , 7. S3 Depending on the classical message from Alice, Bob performs proper recovery operations to obtain the initial state. In order to illustrate our protocol, let’s just take Alice’s first-step measurement result as |ξ0 1269 . Then, the state collapses into

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1

s0 |000|Φ− |0 + s1 |001|φ+ |0 + s2 |010|Φ− |1 + s3 |011|φ+ |1 4 + s4 |100|φ− |0 + s5 |101|Φ+ |0  (22) + s6 |110|φ− |1 + s7 |111|Φ+ |1 370458 , which can be rewritten in terms of the measurement basis |τ0 , . . . , |τ7  as

1

√ |τ0  y0 |Φ− |0 + y1 |φ+ |0 + y2 |Φ− |1 + y3 |φ+ |1 8 2  +y4 |φ− |0 + y5 |Φ+ |0 + y6 |φ− |1 + y7 |Φ+ |1 458

+|τ1  −y0 |Φ− |0 + y1 |φ+ |0 + y2 |Φ− |1 − y3 |φ+ |1  +y4 |φ− |0 − y5 |Φ+ |0 − y6 |φ− |1 + y7 |Φ+ |1 458

+|τ2  −y0 |Φ− |0 − y1 |φ+ |0 + y2 |Φ− |1 + y3 |φ+ |1  +y4 |φ− |0 + y5 |Φ+ |0 − y6 |φ− |1 − y7 |Φ+ |1 458

+|τ3  −y0 |Φ− |0 + y1 |φ+ |0 − y2 |Φ− |1 + y3 |φ+ |1  +y4 |φ− |0 − y5 |Φ+ |0 + y6 |φ− |1 − y7 |Φ+ |1 458

+|τ4  −y0 |Φ− |0 − y1 |φ+ |0 − y2 |Φ− |1 − y3 |φ+ |1  +y4 |φ− |0 + y5 |Φ+ |0 + y6 |φ− |1 + y7 |Φ+ |1 458

+|τ5  −y0 |Φ− |0 + y1 |φ+ |0 − y2 |Φ− |1 + y3 |φ+ |1  −y4 |φ− |0 + y5 |Φ+ |0 − y6 |φ− |1 + y7 |Φ+ |1 458

+|τ6  −y0 |Φ− |0 + y1 |φ+ |0 + y2 |Φ− |1 − y3 |φ+ |1  −y4 |φ− |0 + y5 |Φ+ |0 + y6 |φ− |1 − y7 |Φ+ |1 458

+|τ7  −y0 |Φ− |0 − y1 |φ+ |0 + y2 |Φ− |1 + y3 |φ+ |1  −y4 |φ− |0 − y5 |Φ+ |0 + y6 |φ− |1 + y7 |Φ+ |1 458 .

(23)

Equation (23) equals to 1 √ [|τ0  f (x0 , x1 , x2 , x3 , x4 , x5 , x6 , x7 ) 8 2 + |τ1  f (x0 , −x1 , −x2 , x3 , −x4 , x5 , x6 , −x7 ) + |τ2  f (−x6 , x7 , x4 , −x5 , x2 , −x3 , −x0 , x1 ) + |τ3  f (x0 , x1 , −x2 , −x3 , −x4 , −x5 , x6 , x7 ) + |τ4  f (−x6 , −x7 , x4 , x5 , x2 , x3 , −x0 , −x1 ) + |τ5  f (x6 , x7 , x4 , x5 , x2 , x3 , x0 , x1 ) + |τ6  f (x6 , −x7 , x4 , −x5 , x2 , −x3 , x0 , −x1 ) + |τ7  f (−x0 , x1 , −x2 , x3 , −x4 , x5 , −x6 , x7 )] .

(24)

It is shown that whatever Alice’s second-step measurement outcome is, Bob can always obtain the desired state by performing appropriate unitary operations on his collapsed particles.

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Table 2 Relation between the Alice’s two-step measurement outcome m 1 m 2 and Bob’s recovery operation (BRO) Um 1 m 2 on the collapsed state, where m 1 = 0, . . . , 15, m 2 = 0, . . . , 7 m1, m2

BRO

m1, m2

BRO

m1, m2

BRO

m1, m2

BRO

0(8), 0

I4 I5 I8

1(9),0

I4 Y5 Z 8

2(10),0

Y4 Y5 Y8

3(11),0

I4 Y5 X 8

0(8), 1

Z4 Z5 Z8

1(9),1

Z 4 X 5 I8

2(10),1

X4 X5 X8

3(11), 1

Z 4 X 5 Y8

0(8), 2

Y4 Y5 Z 8

1(9),2

Y4 I5 I8

2(10),2

I4 I5 X 8

3(11), 2

Y4 I5 Y8

0(8), 3

Z 4 Z 5 I8

1(9),3

Z4 X5 Z8

2(10),3

X 4 X 5 Y8

3(11), 3

Z4 X5 X8

0(8), 4

Y4 Y5 I8

1(9),4

Y4 I5 Z 8

2(10),4

I4 I5 Y8

3(11),4

Y4 I5 X 8

0(8), 5

X 4 X 5 I8

1(9),5

X4 Z5 Z8

2(10),5

Z 4 Z 5 Y8

3(11),5

X4 Z5 X8

0(8), 6

X4 X5 Z8

1(9),6

X 4 Z 5 I8

2(10),6

Z4 Z5 X8

3(11),6

X 4 Z 5 Y8

0(8), 7

I4 I5 Z 8

1(9),7

I4 Y5 I8

2(10), 7

Y4 Y5 X 8

3(11),7

I4 Y5 Y8

4(12), 0

Y4 Z 5 I8

5(13), 0

Y4 X 5 I8

6(14), 0

X 4 I5 Y8

7(15), 0

Z 4 I5 Y8

4(12), 1

X 4 I5 Z 8

5(13), 1

X 4 Y5 Z 8

6(14), 1

Y4 Z 5 X 8

7(15), 1

I4 Z 5 X 8

4(12), 2

I4 X 5 Z 8

5(13), 2

I4 Z 5 Z 8

6(14), 2

Z 4 Y5 X 8

7(15), 2

X 4 Y5 X 8

4(12), 3

X 4 I5 I8

5(13), 3

X 4 Y5 I8

6(14),3

Y4 Z 5 Y8

7(15),3

I4 Z 5 Y8

4(12), 4

I4 X 5 I8

5(13),4

I4 Z 5 I8

6(14),4

Z 4 Y5 Y8

7(15),4

X 4 Y5 Y8

4(12), 5

Z 4 Y5 I8

5(13), 5

Z 4 I5 I8

6(14), 5

I4 X 5 Y8

7(15), 5

Y4 X 5 Y8

4(12), 6

Z 4 Y5 Z 8

5(13),6

Z 4 I5 Z 8

6(14),6

I4 X 5 X 8

7(15), 6

Y4 X 5 X 8

4(12), 7

Y4 Z 5 Z 8

5(13), 7

Y4 X 5 Z 8

6(14), 7

X 4 I5 X 8

7(15), 7

Z 4 I5 X 8

As for the other fifteen cases corresponding to Alice’s first-step measurement result, similar analysis process can be made. As a summary, Bob’s recovery operations conditioned on Alice’s two-step measurement results |ξm 1  and |τm 2  (m 1 = 0, . . . , 15, m 2 = 0, . . . , 7) are described in Table 2. Therefore, the success probability for preparing an arbitrary three-qubit state is unit.

4 DRSP subject to noisy environments In ideal situation, two schemes are proposed to realize deterministic remote preparation of arbitrary two- and three-qubit states in Sects. 2 and 3. In real situations, however, quantum noise may have an influence on quantum channel. Here, we take two types of noise, amplitude-damping and phase-damping noise, as the model. Take the noisy influence on the DRSP scheme in Sect. 2 as an example. In this scheme for remotely preparing an arbitrary two-qubit state, the two participants share the state |C7  = √1 |Br (|00 + |11) before communication. For example, suppose 2 Alice has a quantum source generator to generate the quantum channel |C7  in her laboratory. Then, she sends the particles (4, 5) to Bob via the noisy environment. Due to the interaction with the environment, the state |C7  will have some changes. We use Kraus operations [42] to describe the noisy effect and analyze two specific kinds of noises, the amplitude-damping noise and phase-damping noise. And the fidelity is used to show how close the output state and the original state are. Moreover, we will

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make a discussion between the two cases to show that in which noisy environment more information will be lost. 4.1 In the amplitude-damping noisy environment In this subsection, we consider the amplitude-damping noise. Amplitude-damping noise is one of the most important decoherence noises that can be used to describe the energy dissipation effects due to loss of energy from a quantum system. The action of amplitude-damping noise is shown by a set of Kraus operators [42]    √  1√ 0 0 ηa a a , (25) , E1 = E0 = 0 0 0 1 − ηa where ηa (0 < ηa < 1) is the decoherence rate to indicate the error probability when the quantum states pass through the amplitude-damping noisy environment. The effect of amplitude-damping noise on the shared quantum channel |C7  is εa (ρ) =

1

j1 , j2 =0

4

5

4†

5†

E aj1 E aj2 ρ E aj1 E aj2 ,

(26)

where the subscripts j1 , j2 represent which Kraus operator will be chosen to preform, the superscripts (4, 5) represent the operator E act on which qubit, and ρ = |C7 C7 | is the density matrix of shared state |C7 . Also, ε denotes a quantum operation which maps from ρ to ε(ρ) due to the noise. In this case, the entangled quantum channel ρ is replaced by εa (ρ) =

1 {[(|00010 + |00111)(|00 − (1 − ηa )|11) + (|10101 16  + |10000)(|00 + (1 − ηa )|11) + 1 − ηa (|01000   + |01101)(|01 − |10) + 1 − ηa (|11111 + |11010)(|01 + |10) × [(00010| + 00111|)(00| − (1 − ηa )11|)

 + (10101| + 10000|)(00| + (1 − ηa )11|) + 1 − ηa (01000   + |01101)(01| − 10|) + 1 − ηa (11111| + 11010|)(01| + 10|) + ηa2 (−|0001000 + |1010100 − |0011100 + |1000000) ×(−0001000| + 1010100| − 0011100| + 1000000|)    + ηa − 1 − ηa |0001010 + |0100000 + 1 − ηa |1010110 + |1111100    − 1 − ηa |0011110 + |0110100 + 1 − ηa |1000010 + |1101000    × − 1 − ηa 0001010| + 0100000| + 1 − ηa 1010110| + 1111100|    − 1 − ηa 0011110| + 0110100| + 1 − ηa 1000010| + 1101000|    + ηa − 1 − ηa |0001001 − |0100000 + 1 − ηa |1010101 + |1111100

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   − 1 − ηa |0011101 − |0110100 + 1 − ηa |1000001 + |1101000    × − 1 − ηa 0001001| − 0100000| + 1 − ηa 1010101| + 1111100|   − 1 − ηa 0011101| − 0110100| + 1 − ηa 1000001| + 1101000|)} .

(27)

S1 Alice selects M A1 = {|ζn ζn |; n = 0, 1, . . . , 7} as measurement operators for measuring her qubits (1, 2, 6). For simple but without loss of generality, assume Alice’s first-step measurement result is M0A1 = |ζ0 ζ0 |. Thus, the system of qubits (3, 7, 4, 5) becomes ⎛ ⎞ † M0A1 εa (ρ)M0A1 ⎠ ρ1 = tr126 ⎝  A1 † A1 tr M0 M0 εa (ρ)   1 r0 1 − ηa |00(|01 − |10) + r1 |01(|00 + (1 − ηa )|11) = a 16P0   + r2 |10(|00 − (1 − ηa )|11) + r3 1 − ηa |11(|01 + |10)   × r0 1 − ηa 00|(01| − 10|) + r1 01|(00| + (1 − ηa )11|)   + r2 10|(00| − (1 − ηa )11|) + r3 1 − ηa 11|(01| + 10|) + ηa2 (r1 |0100 − r2 |1000)(r1 0100| − r2 1000|)     + ηa r0 |0000 + r1 1 − ηa |0110 − r2 1 − ηa |1010 + r3 |1100     × r0 0000| + r1 1 − ηa 0110| − r2 1 − ηa 1010| + r3 1100|     + ηa −r0 |0000 + r1 1 − ηa |0101 − r2 1 − ηa |1001 + r3 |1100     × −r0 0000| + r1 1 − ηa 0101| − r2 1 − ηa 1001| + r3 1100| , (28) where

  † P0a = tr M0A1 M0A1 εa (ρ)     1  2 r1 + r22 1 + ηa2 + (1 − ηa )2 + 2ηa (1 − ηa ) + 2 r02 + r32 = 16 1 (29) = 8

is the probability that Alice gets the measurement result M0A1 . S2 Alice performs a unitary operation I3 I7 conditioned on her first-step measurement result M0A1 . Then Alice measures her qubits (3, 7) by using M A2 = {|ηn ηn |; n = 0, 1, 2, 3} as measurement operators. If she gets the measurement result M0A2 = |η0 η0 |, the qubits (4, 5) collapse into

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⎛

⎞ A† M0A2 ρ1 M0 2 ⎠ ρ2 = tr37 ⎝  † tr M0A2 M0A2 ρ1 ) 1   b0 1 − ηa (|01 − |10) + b1 (|00 + (1 − ηa )|11) = 16P0a   + b2 (|00 − (1 − ηa )|11) + b3 1 − ηa (|01 + |10)   × b0 1 − ηa (01| − 10|) + b1 (00| + (1 − ηa )11|)   + b2 (00| − (1 − ηa )11|) + b3 1 − ηa (01| + 10|) + ηa2 |b1 − b2 |2 |0000|      + ηa (b0 + b3 )|00 + (b1 − b2 ) 1 − ηa |10 b0∗ + b3∗ 00|       + b1∗ − b2∗ 1 − ηa 10| + ηa (b3 − b0 )|00 + (b1 − b2 ) 1 − ηa |01      × b3∗ − b0∗ 00| + b1∗ − b2∗ 1 − ηa 01| . (30) S3 According to Alice’s two-step measurement result |ζ0 |η0 , Bob performs recovery unitary operation U00 = I4 I5 and gets the output state † = ρ2 . ρout = U00 ρ2 U00

(31)

As for the other 31 cases corresponding to Alice’s two-step measurement results, similar analysis process can be made. The state to be prepared is h(a0 , a1 , a2 , a3 ) = a0 |00 + a1 |01 + a2 |10 + a3 |11 which equals to |T  = b0 |Φ−  + b1 |ϕ+  + b2 |ϕ−  + b3 |Φ+ 

(32)

due to Eq. (5). Thus, the fidelity can be expressed as: Fa = T |ρout |T  2 2     1 1 2 2 2 2 2 1 − ηa − 1 − ηa = ηa r1 + r2 + 1 − ηa + r1 + r2 4 2       2 2 2 2 r1 + r2 (2 − ηa ) 1 − ηa + ηa ηa cos(θ2 − θ1 ) + r1 r2 r0 + r3   − ηa 1 − ηa r02 r12 cos(2θ0 − 2θ1 ) + r22 r32 cos(2θ3 − 2θ2 )  − r12 r32 cos(2θ3 − 2θ1 ) − r02 r22 cos(2θ0 − 2θ2 )   1 2 r0 + r32 r12 + r22 ηa (2 − ηa ). + (33) 2 The fidelity of the output state is related not only to the amplitude parameter r j and noise parameter ηa , but also to the phase parameter θ j . In noisy environment, more

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information will be lost if Fa becomes smaller. In view of the maximally entangled state to be prepared, i.e., a0 = a1 = a2 = a3 = 21 (r0 = r2 = 0, r1 = r3 = √1 , θ0 = 2 √ θ1 = θ2 = θ3 = 0), Fa∗ = 21 − 41 ηa + 21 1 − ηa . Fa∗ is plotted in Fig. 1 with noise parameter ηa in the condition that the prepared state is maximally entangled state. 4.2 In the phase-damping noise environment In this subsection, we repeat the calculation of the preceding subsection while replacing the amplitude-damping noise by phase-damping noise. Compared with the amplitudedamping noise, the phase-damping noise describes more about the loss of quantum information but without the energy dissipation. The Kraus operators [42] of phasedamping noise are p E0

=



1 − η p I,

p E1



√ = ηp

 10 , 00

p E2

√ = ηp



 00 , 01

(34)

where η p (0 < η p < 1) is the decoherence rate of phase-damping noise. Now we first describe the noisy effects on the quantum channel |C7 . In this case, the shared state |C7  becomes ε p (ρ) =

2

j1 , j2 =0

p4

p5

p4

†

†

p5

E j1 E j2 ρ E j1 E j2 ,

(35)

where the subscripts j1 , j2 denote which Karus operator will be carried out, and the superscripts (4, 5) represent the operator E will act on which qubit. In this case, the ε p (ρ) becomes 1 amplitude−damping noise phase−damping noise

0.9

Fidelity

0.8 0.7 0.6 0.5 0.4 0.3 0.2

0

0.2

0.4

0.6

0.8

1

η a(η p )

Fig. 1 Fidelities under the action of amplitude-damping noise and phase-damping noise in the condition that the prepared two-qubit state is maximally entangled state

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1  (1 − η p )2 [(|00010 + |00111)(|00 − |11) + (|10101 16 +|10000)(|00 + |11) + (|01000 + |01101)(|01 − |10) + (|11111 + |11010)(|01 + |10)] × [(00010| + 00111|)(00| − 11|) + (10101| + 10000|)(00| + 11|) + (01000| + 01101|)(01| − 10|) + (11111| + 11010|)(01| + 10|)] + η2p (|0001000 + |1010100 + |0011100 + |1000000) ×(0001000| + 1010100| + 0011100| + 1000000|) + η2p (−|0001011 + |1010111 − |0011111 + |1000011) ×(−0001011| + 1010111| − 0011111| + 1000011|) + η p (1 − η p )(|0001000 − |0100010 + |1010100 + |1111110 + |0011100 − |0110110 + |1000000 + |1101010) ×(0001000| − 0100010| + 1010100| + 1111110| + 0011100| − 0110110| + 1000000| + 1101010|) + η p (1 − η p )(|0001000 + |0100001 + |1010100 + |1111101 + |0011100 + |0110101 + |1000000 + |1101001) ×(0001000| + 0100001| + 1010100| + 1111101| + 0011100| + 0110101| + 1000000| + 1101001|) + η p (1 − η p )(−|0001011 + |0100001 + |1010111 + |1111101 − |0011111 + |0110101 + |1000011 + |1101001) ×(−0001011| + 0100001| + 1010111| + 1111101| − 0011111| + 0110101| + 1000011| + 1101001|) + η p (1 − η p )(−|0001011 − |0100010 + |1010111 + |1111110 − |0011111 − |0110110 + |1000011 + |1101010) ×(−0001011| − 0100010| + 1010111| + 1111110| − 0011111| − 0110110| + 1000011| + 1101010|) + η2p (|0100001 + |1111101 + |0110101 + |1101001) ×(0100001| + 1111101| + 0110101| + 1101001|) + η2p (−|0100010 + |1111110 − |0110110 + |1101010) ×(− 0100010| + 1111110| − 0110110| + 1101010|)} .

(36)

Then, Alice will remotely prepare the state through the channel ε p (ρ) similarly as the ideal DRSP protocol. And this process is described in Sect. 2. Suppose Alice’s first measurement result under the basis {|ζ0 , . . . , |ζ7 } on her qubits (1, 2, 6) is M0A1 = |ζ0 ζ0 |, the system of qubits (3, 7, 4, 5) becomes

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Deterministic remote preparation via the Brown state

⎞

⎛ ⎜ ρ1 = tr126 ⎜ ⎝ =

Page 17 of 22 93

A†1

⎟ M A1 ε p (ρ)M0  0 † ⎟ ⎠ A tr M0 1 M0A1 ε p (ρ)

1  2 p (1 − η p ) [r0 |00(|01 − |10) + r1 |01(|00 + |11) 16P0 + r2 |10(|00 − |11) + r3 |11(|01 + |10)] × [r0 00|(01| − 10|) + r1 01|(00| + 11|) + r2 10|(00| − 11|) + r3 11|(01| + 10|)] + η2p (r1 |0100 + r2 |1000)(r1 0100| + r2 1000|) + η2p (r1 |0111 − r2 |1011)(r1 0111| − r2 1011|) + η p (1 − η p )(−r0 |0010 + r1 |0100 + r2 |1000 + r3 |1110) ×(−r0 0010| + r1 0100| + r2 1000| + r3 1110|) + η p (1 − η p )(r2 |1000 + r0 |0001 + r1 |0100 + r3 |1101) ×(r2 1000| + r0 0001| + r1 0100| + r3 1101|) + η p (1 − η p )(r0 |0001 + r1 |0111 − r2 |1011 + r3 |1101) ×(r0 0001| + r1 0111| − r2 1011| + r3 1101|) + η p (1 − η p )(−r0 |0010 + r1 |0111 − r2 |1011 + r3 |1110) ×(−r0 0010| + r1 0111| − r2 1011| + r3 1110|) + η2p (r0 |0001 + r3 |1101)(r0 0001| + r3 1101|)

 + η2p (−r0 |0010 + r3 |1110)(−r0 0010| + r3 1110|) ,

(37)

where p

P0 =

     1 1 (1 − η p )2 + r12 + r22 η2p + r02 + r32 η2p + 2η p (1 − η p ) = (38) 8 8

is the probability that Alice gets the measurement result M0A1 . Alice performs unitary operation I3 I7 and then performs the second-step measurement under the basis |η0 , . . . , |η3  on her qubits (3, 7). If she gets the measurement M0A2 = |η0 η0 |, the system of qubits (4, 5) becomes ⎞

⎛ ⎜ ρ2 = tr37 ⎜ ⎝ =

A† M0A2 ρ1 M0 2 

tr

A†2

M0 M0A2 ρ1

⎟ ⎟ ⎠

1  2 p (1 − η p ) [(b1 + b2 )|00 + (b0 + b3 )|01 + (b3 − b0 )|10 16P0   

 +(b1 − b2 )|11] × b1∗ + b2∗ 00| + b0∗ + b3∗ 01|     + b3∗ − b0∗ 10| + (b1∗ − b2∗ )11| + η2p |b1 + b2 |2 |0000|

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+|b1 − b2 |2 |1111| + |b0 + b3 |2 |0101| + |b3 − b0 |2 |1010|



+η p (1 − η p )[(b1 + b2 )|00 + (b3 − b0 )|10]

     × b1∗ + b2∗ 00| + b3∗ − b0∗ 10| +η p (1 − η p )[(b1 + b2 )|00 + (b0 + b3 )|01

     × b1∗ + b2∗ 00| + b0∗ + b3∗ 01| +η p (1 − η p )[(b0 + b3 )|01 + (b1 − b2 )|11]    

 × b0∗ + b3∗ 01| + b1∗ − b2∗ 11| +η p (1 − η p )[(b3 − b0 )|10 + (b1 − b2 )|11]

     × b3∗ − b0∗ 10| + b1∗ − b2∗ 11| .

(39)

According to Alice’s two-step measurement results, Bob needs to perform the recovery unitary operator U00 = I ⊗I to get the prepared the state. As the measurement † = ρ2 . As for the other 31 cases of Alice’s result is |ζ0 |η0 , that is ρout = U00 ρ2 U00 two-step measurement results, similar analysis process can be made. In order to depict how much information is lost through the phase-damping channel, it is quite useful to calculate the fidelity between ρout and the prepared state. The fidelity can be described as:  2  2  1 2  2 2 2 2 F p = T |ρout |T  = 1 − η p + η p r1 + r2 + r0 + r3 2   2 2 +η p (2 − η p ) r0 r3 [1 + cos(2θ3 − 2θ0 )] + r12 r22 [1 + cos(2θ2 − 2θ1 )] . (40) The fidelity F p∗ = 1−η p + 14 η2p of the output state in phase-damping noisy environment is also plotted in Fig. 1 with noise parameter η p in the condition that the prepared state is maximally entangled state, i.e., a0 = a1 = a2 = a3 = 21 (r0 = r2 = 0, r1 = r3 = √1 2 and θ0 = θ1 = θ2 = θ3 = 0). As is shown, Fa∗ is larger than F p∗ in this case. Similar analysis can be made in noisy environment about the DRSP scheme for an arbitrary three-qubit state. Here, we merely show the situation where the original 1 (s0 = s2 = s4 = state is maximally entangled state, i.e., x0 = x1 = · · · = x7 = √ 2 2

s6 = 0, s1 = s3 = s5 = s7 = 21 and α0 = · · · = α7 = 0). Under the influence of the amplitude-damping noise and the phase-damping noise, the fidelities are Fa∗ =

   1 4 − 3ηa − ηa 1 − ηa + 4 1 − ηa , 8

(41)

3 11 1 F p∗ = 1 − η p + η2p − η3p , 2 16 16

(42)

and

respectively. At this time, Fa∗ (F p∗ ) with noise parameter ηa (η p ) is plotted in Fig. 2.

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1 amplitude−damping noise phase−damping noise

0.9 0.8

Fidelity

0.7 0.6 0.5 0.4 0.3 0.2 0.1

0

0.2

0.4

0.6

0.8

1

η a(η p ) Fig. 2 Fidelities under the action of amplitude-damping noise and phase-damping noise in the condition that the prepared three-qubit state is maximally entangled state

5 Discussion and conclusions Two deterministic schemes are put forward to realize the remote preparation of arbitrary two- and three-qubit states via the Brown state as the entangled channel. In the first scheme for two-qubit state, the CCC is 5 cbits, while 7 cbits are required to transmit from the sender to the receiver in the RSP protocol for three-qubit state. There are several advantages of our schemes: Firstly, the coefficients of the prepared state are arbitrary complex numbers, which does not have any restrictions. Secondly, the success probabilities are unit. Finally, the two schemes require classical communication, local operations such as single-qubit Pauli unitary transformations, two-, threeand four-qubit projective measurements. These are all feasible under the present technologies [26,27], which means the proposed schemes are feasible with respect to the current experimental conditions. Comparing with the previous DRSP protocols for arbitrary two- and three-qubit states via the GHZ states as the quantum channel [29], the local unitary operations used in our schemes are only single-qubit Pauli operation without the use of Hadamard operations and other multi-qubit unitary operations. In fact, the proposed schemes can derive the corresponding CRSP schemes. Take CRSP scheme of an arbitrary two-qubit state with one controller Charlie as an example. Suppose the quantum channel is a Brown state and a GHZ state |G H Z 67C = √1 (|000 + |111)67C . The particles (1, 2, 6, 3, 7) are in the possession of Alice, the 2 particles (4,5) belong to Bob, the particle C belongs to Charlie. Alice measures her particles (1, 2, 6) under the basis |ζ1 , . . . , |ζ7  defined in Eq. (6). After the first-step measurement, Alice performs the unitary operation according to Eq. (8) on her particles (3, 7). Then she performs a two-qubit projective measurement on the particles (3, 7) under the basis |η0 , . . . , |η3  described by Eq. (9). If the controller Charlie consents to help the receiver Bob, he performs the measurement on his particle C under the Hadamard basis {|+, |−}. Alice and Charlie send the measurement results to Bob via the classical channel. At last, Bob can get the prepared two-qubit state by performing appropriate unitary operation on his collapsed particles. Similar discussion

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can be made for the CRSP of an arbitrary three-qubit state. Chen et al. [8] presented programs via the Brown state to prepare arbitrary two- and three-qubit states with one controller. When the coefficients of the prepared state are complex, they have to satisfy many restricted conditions in order to obtain high probabilities 50%. Relative to the whole complex space, the states that meet the required conditions are too limited. In other words, their protocols do not apply to all the states with arbitrary complex spectrum. To solve such a problem, Gao et al. [9] proposed two general CRSP schemes via the Brown state with no restrictions on the prepared state while keeping the same success probabilities 50%. However, the success probabilities are only one half of unit. Comparing with the previous protocols [8,9], the success probabilities of the derived schemes are improved to 100%. Apart from the fore-mentioned merits, there are also some defects about our schemes. More quantum resources and two-step projective measurement are required. To achieve this goal, some additional operations are also needed. In spite of this, our solutions are still valuable in handling realistic problems as the success probability is an important index for evaluating a protocol. It is worth mentioning that the proposed schemes use the information splitting and divide the related complex coefficients a + bi = r eiθ of the prepared state into the amplitude information r and the phase information θ before constructing the measurement basis. If r and θ are independently shared between two senders Alice1 and Alice2, the two senders cooperatively perform the corresponding measurement. Thus, the proposed schemes can be reduced to deterministic joint remote preparation schemes of arbitrary two- and three-qubit states via the Brown state as the entangled channel. At last, we investigate two proposed DRSP schemes under the influence of two important decoherence noises, the amplitude-dumping noise and the phase-damping noise. The effects of the two noises are shown by the fidelity. In view of preparing an arbitrary two-qubit state, results show that the fidelity Fa (F p ) between the output state and the original state under the amplitude(phase)-damping noise is related not only to the noise parameter ηa (η p ) and the amplitude parameter r j , but also to the phase parameter θ j . Meanwhile, the fidelity will more close to 1 when the decoherence rate ηa (η p ) gets more close to 0. Moreover, in some special conditions such as the prepared state is maximally entangled, the fidelities under amplitude-damping noise is always larger than that in phase-damping noise, which implies that the effect of phase-damping noise on the transmission is relatively stronger than that of amplitudedamping noise. Similar analysis can be made for the DRSP scheme of an arbitrary three-qubit state. Acknowledgements This work is supported by the National Natural Science Foundation of China (Nos. 61201253, 61373131, 61572246), PAPD and CICAEET funds.

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