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Teleportation of an arbitrary mixture of diagonal states of multiqubits v ia classical correlation and classical communication

An Min Wang

Department of Modern Physics, University of Science and Technology of China, Hefei, 230026, P.R.China We propose a protocol in which the faithful and deterministic teleportation of an arbitrary mixture of diagonal states is completed via classical correlation and classical communication. Our scheme can be generalized straightforwardly to the case of N -qubits by using N copies of classical correlated pairs and classical communication. Moreover, a varying scheme by using the generalized classical correlated state within a multiqubit space is also presented. In addition, the arbitrary mixed state whose set of eigenvectors is known are a direct application of our protocol.

arXiv:quant-ph/0509164v3 25 Jan 2006

PACS numbers: 03.67.Hk, 03.67.-a, 03.65.Ud

I.

INTRODUCTION

Since Bennett, Brassard, Cr? epeau, Jozsa, Peres, and Wootters (BBCJPW) proposed quantum teleportation in 1993 [1], many theoretical protocols were suggested and some experimental implementations were proposed. Recently, the teleportation of multiqubits has been well studied [2]. Quantum teleportation has become the one of the most important and the most in?uential achievements in quantum theory, in particular, in the end of twenty century. Quantum teleportation transports an unknown quantum state from Alice (sender) to Bob (receiver) via quantum correlation (Einstein-Podolsky-Rosen’s pair) and classical communication. When transporting a known quantum state, one can used the remote state preparation (RSP) which was proposed by Lo [3] and Pati [4]. As to the protocol of transporting a partially known quantum state, sometimes it is called as teleportation and sometimes it is called the remote state preparation either. Whatever if the quantum state one wants to transport is known, partially known or unknown, it is very interesting and important to know what resources will be costed at least and which resources can be replaced by the other ones or be traded o? among the used resources, as well as how new resources can be exploited. This is just our main motivation. In addition, we would like to show the action of classical correlation in quantum information processing and further help for understanding the nature of quantum and classical correlations produced in quantum theory. In this paper, we propose a protocol in which the faithful and deterministic teleportation of an arbitrary mixture of diagonal states is completed via classical correlation and classical communication. Here, a classical correlated pair of two qubits can be written as

p CAB =

1 (|00 2

AB

00| + |11

AB

11|)

(1)

Its name is from that it does not violate local hidden variable (LHV) theory [5]. In fact, it is a separable mixed state of two qubits and then there is no any quantum entanglement. In transporting and distributing quantum state, it has played a substituting and active role. For example, Cubitt etal. [6] used it to distribute entanglement. Toner and Bacon [7] constructed a protocol showing that the teleportation of a single qubit admits a local hidden variable theory. Ghosh et.al [8] provided an alternative simple proof of the necessary of entanglement in quantum teleportation and tried to show that it is su?cient to have a classical correlated channel in order to teleport any commuting qubits. Our scheme can be generalized straightforwardly to the case of multiqubits. Teleporting an arbitrary mixture of diagonal states of N -qubits needs to using N copies of classical correlated pairs and classical communication. Actually, we also can use the generalized classical correlated state within a multiqubit space to carry out our teleportation. In addition, we discuss the application of our protocol to the arbitrary mixed state whose set of eigenvectors is known.

II. ONE QUBIT

First, let us describe how to teleport a mixture of diagonal states of one qubit via a classical correlation and classical communication [8, 9]. As a class of mixed state, such a state can be written as ρd X (1) = a0 |0

X

0| + a1 |1

X

1|

(2)

where for a density matrix, a0 and a1 ought to be real and positive, as well as their summation is 1.

2 Initially, Alice and bob shared a classical correlated pair. Thus, the joint system of them is just

p ρini (1) = ρd X (1) ? CAB

(3)

where the ?rst two qubits (denoted by the subscripts XA) belong to Alice and the third one (denoted by the subscript B ) belongs to Bob. Our protocol can be divided into two steps. Step one: Alice performs the operation [11]

X B B OAlice (1) = (σ0 ? H A ? σ0 ).(CN OT0 (X, A) ? σ0 )

(4)

where σ0 is 2 × 2 unit matrix, H is a Hadamard transformation and CN OT0 (X, A) is a controlled not of two neighbor qubits in which A is a control qubit lying at the latter. The de?nitions of these operations are respectively 1 H= √ 2 1 1 1 ?1 ,

X CN OT0 (X, A) = σ0 ? |0 A X 0| + σ1 ? |1 A

1|

(5)

where σ1 is Pauli matrix. The transformation skill (4) is actually from M. A. Nielsen and I. L. Chuang’s idea [10] in order to change Bell’s basis measurement into the computation basis measurement. Here, our aim is to make that Alice can execute the measurement in the computation basis. It is easy to obtain the transformed result as the following: 1 (|00 XA 00| + |01 XA 01|) ? (a0 |0 B 0| + a1 |1 B 4 1 + (|10 XA 10| + |11 XA 11|) ? (a1 |0 B 0| + a0 |1 B 4 1 + (|00 XA 01| + |01 XA 00|) ? (a0 |0 B 0| ? a1 |1 B 4 1 + (|10 XA 11| + |11 XA 10|) ? (a1 |0 B 0| ? a0 |1 B 4 1|) 1|) 1|) 1|) (6)

Step Two: Alice executes the measurement in the computation basis: {|00 XA 00|, |01 XA 01|, |10 XA 10|, |11 XA 11|} and she has the probability 0.25 of reducing to one of them. For the ?rst two basis, she can send a cbit 0 to Bob and so Bob has a0 |0 B 0| + a1 |1 B 1| as required. While for the later two basis, she can send a cbit 1 to Bob and then Bob has to apply a σ1 (not) transformation to obtain a0 |0 B 0| + a1 |1 B 1| as required.

III. TOW QUBITS

Now, let us consider the case of an arbitrary mixture of diagonal states of two quits. The state to be teleport reads 1 4x

1

ρd X1 X2 (2) =

1 ,x2 =0

ax1 x2 |x1 x2

X1 X2

x1 x2 |

(7)

Alice and Bob have to share two classical correlated pairs, that is, the joint system of them can be written as

p p ρini (2) = ρd X1 X2 (2) ? CA1 B1 ? CA2 B2

(8)

where four qubits X1 X2 and A1 A2 belong to Alice, and two qubits B1 B2 belong to Bob. Similar to the case of one qubit, we use two steps to teleport ρd X1 X2 (2) from Alice to Bob. Step one: Alice performs the operation OAlice (2) =

X1 X2 B1 B2 σ0 ? σ0 ? H A1 ? σ0 ? H A2 ? σ0 B1 A2 B2 . CN OT1 (X1 , A1 ) ? σ0 ? σ0 ? σ0 X1 B2 . σ0 ? CN OT2 (X2 , A2 ) ? σ0

(9)

where the control not of parting n qubits is de?ned by

Xi ?n CN OTn (Xi , Ai ) = σ0 ? σ0 ? |0 Ai Xi ?n 0| + σ1 ? σ0 ? |1 Ai

1|,

(i = 1, 2; n ≥ 1)

(10)

3 It is not di?cult to calculate out the transformed state ρAO ρAO = 1 16 x

1 1 1 1

1 ,x2 =0

′ β ,β =1 α1 ,α′ 1 =1 α2 ,α2 =1 1 2

aβ1 β2 |x1 x2 x1 x2 | (11)

′ ′ ?|α1 α′ 1 | ? [(1 ? α1 )σ0 + α1 σ3 ] σx1 |β1 β1 |σx1 [(1 ? α1 )σ0 + α1 σ3 ] ′ ′ ?|α2 α2 | ? [(1 ? α2 )σ0 + α2 σ3 ] σx2 |β2 β2 |σx2 [(1 ? α2 )σ0 + α′ 2 σ3 ]

It has 256 terms, however, only 64 diagonal terms are important because the non-diagonal terms will not appear after measurement. Obviously, the diagonal terms with the forms as the following: 1 |x1 x2 x1 x2 | ? 16

1

β1 ,β2 =0

aβ1 β2 |α1 α1 | ? σx1 |β1

B1

β1 | ? σx1 ? |α2

A2

α2 | ? σx2 |β2

B2

β2 |σx2

(12)

where x1 , x2 , α1 , α2 = 0, 1. B1 ? Step Two: Alice executes the measurement in the 16 computation basis: {|x1 x2 X1 X2 x1 x2 | ? |α1 A1 α1 | ? σ0 B2 |α2 A2 α2 | ? σ0 (x1 , x2 , α1 , α2 = 0, 1) and she has the probability 1/16 of reducing to one of them. It is clear that all of computation basis can be divided in to four groups respectively corresponding to {x1 = 0, x2 = 0}, {x1 = 0, x2 = 1}, {x1 = 1, x2 = 0} and {x1 = 1, x2 = 1}. For each group basis, Alice can send a relevant cbit to Bob and then Bob has to apply a corresponding transformation listed in the following table to obtain the teleported state ρd X1 X2 (2). Alice’s measurement B1 |00 X1 X2 00| ? |α1 A1 α1 | ? σ0 B1 |01 X1 X2 01| ? |α1 A1 α1 | ? σ0 B1 |10 X1 X2 10| ? |α1 A1 α1 | ? σ0 B1 |11 X1 X2 11| ? |α1 A1 α1 | ? σ0

IV.

? |α2 ? |α2 ? |α2 ? |α2

A2 A2 A2 A2

B2 α2 | ? σ0 B2 α2 | ? σ0 B2 α2 | ? σ0 B2 α2 | ? σ0

Bob’s operation σ0 ? σ0 σ0 ? σ1 σ1 ? σ0 σ1 ? σ1

N -QUBITS

The generalization to the case of N -qubits is straightforward but signi?cant. In order to simplify our notions, we do not write obviously the symbol of direct product and unit matrix. Suppose the state to be teleported is ρd X1 X2 ···XN (N ) 1 = N 2

1

x1 ,··· ,xN =0

ax1 ,x2 ,··· ,xN |x1 , x2 , · · · , xN x1 , x2 , · · · , xN |

(13)

Initially, the joint system of Alice and Bob reads

N

ρini (N ) = ρd X1 X2 ···XN (N )

i=1

C p (Ai , Bi )

(14)

where 2N qubits X1 , X2 , · · · , XN ; A1 , A2 , · · · , AN belong to Alice, and N qubits B1 , B2 · · · , BN belong to Bob. Note p that N copies of classical correlated pairs N i=1 C (Ai , Bi ) are shared by Alice and Bob. Step one: Alice performs the operation

N N

OAlice (N ) =

i=1

H Ai

i=1

CN OT (Xi , Ai )

(15)

That is, taking each Ai as the control qubit, Xi as the corresponding target qubit, Alice ?rst performs N cnot operations; then for every Ai qubit, Alice always applies a Hadamard transformation H Ai . The diagonal terms of transformed state becomes 1 |x1 x2 · · · xN x1 x2 · · · xN | 4N

1 N

aβ1 β2 ···βN

β1 ,β2 ,··· ,βN =0 i=1

(|αi αi |) (σxi |βi βi |σxi )

(16)

Step two: Alice executes the measurements in the 4N computation basis |x1 x2 · · · xn x1 x2 · · · xn | ? |α1 α2 · · · αN α1 α2 · · · αN | (x1 , x2 , · · · , xN = 0, 1; α1 , α2 , · · · , αN = 0, 1) and send a relevant cbit to Bob. Bob perN forms the corresponding operation i=1 σxi and obtains the teleported state ρd X1 X2 ···XN (N ). Based on our previous scheme, it is easy to draw out its quantum circuit:

4

; ; ; 1 $ % $ % $ 1 % 1

+

+

+

2XWSXW ILQDOVWDWH %RE?V 2SHUDWLRQ ? ?[

?

L

L

FIG. 1: Quantum circuit of teleportation of an arbitrary mixture of diagonal states of N -qubits

V.

A VARYING SCHEME

It is worthy of pointing out that there is a varying scheme of our above protocol by using the generalized classical correlated state which is constructed by moving all of odd positions of N -copies of classical correlated pairs to the front. In fact, such a transformation can be completed by a serial of swapping operations. Introducing a swapping transformation of two neighbor qubits (2 × 2 matrix) de?ned by ? ? 1 0 0 0 ?0 0 1 0? (17) S (X, Y ) = ? 0 1 0 0? 0 0 0 1 Its action is S (X, Y )|αX βY = |βY αX , Thus,

N N

S (X, Y )(QX ? QY )S (X, Y ) = QY ? QX ?

N N

(18)

S where

k=1

|αi βi

=

i=1

|αi

? ?

N

j =1

|βj ? ,

N ?j

S

QAi QBi

k=1

S=

Q Ai

i=1

? ?

N

j =1

QBj ?

?

(19)

N ?1

S=

S (Bi , AN ?j +1 )

j =1 i=1

(20)

Note that we have used the fact S = S ?1 . Therefore, a generalized classical correlated state of N -qubit space can be written as

N s (N ) = S CA 1 A2 ···AN B1 B2 ···BN p CA i Bi i=1

S

(21)

In the experimental implementation, it is di?erent from the N -copies of classical correlation pairs, and so it is only not a problem of notion. Just as one considered a generalized four particle entanglement di?erent from two pairs of

5 Bell states [2]. The simplest case is one with two classical correlated pairs, and the generalized correlated state in the space of four qubits is just

s (2) = CX 1 X2 A1 A2 B2 A1 . ? S (B1 , A2 ) ? σ0 σ0

1 |00 2 11|

A1 B1

00| + |11

A1 B1

11|

? =

1 |00 2

A2 B2

00| + |11

A2 B2

A1 B2 . σ0 ? S (B1 , A2 ) ? σ0

1 |00 A1 A2 00| ? |00 B1 B2 00| + |01 A1 A2 01| ? |01 B1 B2 01| 4 +|10 A1 A2 10| ? |10 B1 B2 10| + |11 A1 A2 11| ? |11 B1 B2 00|

(22)

Suppose that Alice and Bob initially share the generalized state of classical correlation in the space of N -qubits. Thus, the joint system of Alice and Bob is

s d ρs ini (N ) = ρX1 X2 ,··· ,XN (N ) ? CA1 A2 ···AN B1 B2 ···BN (N )

(23)

Our protocol is then changed as the following: Step one: Alice performs the operation

N N

OAlice (N ) =

i=1

H Ai

i=1

CN OT (Xi , Ai ) S

(24)

That is, only adding a moving transformation than the previous scheme. In particular, in the case of two qubits

X1 X2 B1 B2 OAlice (2) = (σ0 ? H A1 σ0 ? H A2 ? σ0 ? σ0 )

X1 A2 B1 B2 (σ0 ? S (X2 , A1 ) ? σ0 ? σ0 ? σ0 )

B1 B2 (CN OT (A1 , X1 ) ? CN OT (A2 , X2 ) ? σ0 ? σ0 )

(25)

The diagonal terms of transformed state becomes 1 4N

N 1 N

i=1

|xi αi xi αi |

aβ1 β2 ···βN

β1 ,β2 ,··· ,βN =0 j =1

σxj |βj βj |σxj

(26)

For example, in the case of two qubits it has the form 1 |x1 α1 x2 α2 x1 α1 x2 α2 | ? 16

B2 B2 B1 B1 ? σx σx .ρd B1 B2 (2). σx1 ? σx2 2 1 ?

(27)

where x1 , x2 , α1 , α2 = 0, 1. Step two: Alice executes the measurements in 4N computation basis N i=1 |xi αi xi αi | (x1 , x2 , · · · , xN = 0, 1; N α1 , α2 , · · · , αN = 0, 1) and send a relevant cbit to Bob. Bob performs a corresponding operation j =1 σxj and obtain the teleported state ρd X1 X2 ···XN (N ).

VI. DISCUSSIONS AND CONCLUSIONS

Actually, our protocol has more applications. For example, it is applicable to the teleportation of a general mixed state whose set of eigenvectors is known (the eigenvalues remain unknown). Alice can teleport it by ?rst performing a diagonalized unitary transformation which can be constructed by the set of eigenvectors of this mixed state, then do our above protocol, and ?nally, Bob also has to performs this diagonalized unitary transformation in order to complete this teleportation. We have proposed a protocol in which the faithful and deterministic teleportation of an arbitrary mixture of diagonal states is completed via classical correlation and classical communication, however, without quantum entanglement. We show that the teleportation of an arbitrary mixture of diagonal states of N -qubits needs N -copies of classical correlated pairs. It must be emphasized that the teleported state in our protocol is not fully unknown but partially known. This is a reason why our protocol does not need any quantum entanglement. It reminds that there is an extreme case of trade-o? between quantum correlation and classical correlation under some preconditon. In our point

6 of view, quantum entanglement is still necessary for a quantum teleportation of a fully unknown state. A complete quantum teleportation needs both quantum entanglement and quantum measurement. Our protocol is, at most, a kind of partially quantum teleportaion since only using quantum measurement. Just because our protocol depends on quantum measurement, it is not a classical teleportation one. Of course, the results presented here o?er an intriguing glimpse into the nature of correlations produced in quantum theory, and show that classical correlation also is an important and useful resource in quantum information processing.

Acknowledgement

We particularly thank Wan Qing Niu for his earlier work about an arbitrary mixture of diagonal states of one qubit, and Liang Qiu for his surveying many related references. We are grateful all of collaborators of our quantum theory group in the institute for theoretical physics of our university. This work was founded by the National Fundamental Research Program of China with No. 2001CB309310, partially supported by the National Natural Science Foundation of China under Grant No. 60573008.

C.H. Bennett, G. Brassard, C. Cr? epeau, R. Jozsa, A. Peres, and W. K. Wootters, Phys. Rev. Lett. 70, 1895 (1993) G. Rigolin, Phys. Rev. A 71, 032303 (2005) H. K. Lo, Phys. Rev. A 62, 012313 (2000) A. K. Pati, Phys. Rev. A 63, 014302 (2001) R. F. Werner, Phys. Rev. A 40, 4277 (1989) T. S. Cubitt, F. Verstraete, W. D¨ ur, and J. I. Cirac, Phys. Rev. Lett. 91, 037902 (2003) B. F. Toner and D. Bacon, Phys. Rev. Lett. 91, 187904 (2003) S. Ghosh, G. Kar, A. Roy and U. Sen, Phys. Rev. A 65, 032309 (2002) Wan Qing Niu, PhD thesis (2005.4) unpublished. M. A. Nielsen and I. L. Chuang, “Quantum Computation and Quantum Information” (Cambridge University Press, Combridge, England, 2000) [11] In our notion, we always keep the structure of Hilbert space and the reduced space to Alice or to Bob is easy to obtained.

[1] [2] [3] [4] [5] [6] [7] [8] [9] [10]

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