JOURNAL OF OPTICS APPLICATIONS – Oct. 2012, Vol. 1, Iss. 2 14

Schemes of Teleportation with Six-qubit

Cluster State Qian Lan*1, Xinwei Zha2, Jing Wei3

School of Science, Xi’an University of Posts and Telecommunications, Xi’an 710061, China

*1lanqian198711@163.com; 2zhxw@xupt.edu.cn; 3weijing129@126.com

Abstract

In this paper, we have proposed a quantum state

teleportation of an arbitrary two-qubit state using

non-maximally entangled six-qubit Cluster state as channel.

The first scheme is described as: the sender Alice introduced

an auxiliary particle and operators orthogonal complete

bases measurement, and the controller Bob performed

non-Bell measurements on his particles in the quantum

teleportation process. At last, the receiver Charlie can get the

arbitrary two-qubit state only by operating appropriate

unitary transformation. The second scheme is: the sender

Alice operated joint Bell state measurement, and the

controller Bob performed non-Bell measurements on his

particles, the receiver introduced an auxiliary particle and

operated appropriate unitary transformation, the quantum

teleportation will be successful.

Keywords

Quantum Teleportation; Non-Bell Measurements; Joint Bell State

Measurement; Unitary Transformation

Introduction

Quantum communication is a popular topic in the

communication field. The quantum communication

includes many sides, such as quantum teleportation

[1-14], quantum key distribution [15], quantum secure

direct communication [16], dense coding [17-19],

quantum secret sharing [20], remote state preparation

[21-22] and Quantum state sharing [23-24] which is

also named QSTS. QSTS, which plays an important

role in quantum communication, is the scheme of

sharing an unknown state among some agents.

Recently Cluster state has been discussed a lot not only

in quantum information but also in quantum

communication. In this paper we compared two

schemes of teleporting an arbitrary two-qubit state

which used a non-maximally entangled six-qubit

Cluster state as channel. In the first scheme, the sender

Alice first introduces an auxiliary particle and

performs an orthogonal complete bases measurement;

the second one is the receiver introduces an auxiliary

particle and performs an appropriate unitary

transformation, the unknown particle state can be

teleported. And both of the two schemes’ probability

of success is proved to be2

4 a 。

The First Telepertation Scheme of the

Arbitrary Two-Qubit State

Suppose that the sender Alice plans to teleportate the

following arbitrary two-qubit entangled state to

Charlie

1 2

1 2

0 1

2 3

| ( 00 01

10 11 )

a a

a a

x x

x x

(1)

Where 0 1 2 3, , ,x x x x are arbitrary complex numbers,

and it is assumed that the wave function satisfies the

normalization condition 23

01ii

x

.The six-qubit

cluster state can be written as

1 2 1 2 1 2

1 2 1 2 1 2

6 ( 000000 010101

101010 111111 )

A A B B C C

A A B B C C

C a b

c d

(2)

A n d 2 2 2 2

, 1a b c d a b c d . N o w w e

consider that Alice, Bob, and Charlie hold particles

(1A ,

2A ), (1B ,

2B ), (1C ,

2C ) respectively. In order to

teleportate the state 1 2a a

in this scheme, Alice first

introduces an auxiliary particle 0A

.The system state

of particles becomes

15 JOURNAL OF OPTICS APPLICATIONS – Oct. 2012, Vol. 1, Iss. 2

1 2 1 2 1 2 1 2

0

1

2

3

| 0

[ ( 000000000 000010101

001001010 001011111 )

( 010000000 010010101

011001010 011011111 )

( 100000000 100010101

101001010 101011111 )

( 110000000 110010101

1

a as A A B B C C AC

x a b

c d

x a b

c d

x a b

c d

x a b

c

1 2 1 2 1 2 1 2

11001010 111011111 )]a a A AA B B C Cd

(3)

step1: In order to realize the teleportation, Alice can

make a measurement on the five particles ‘1 2 1 2a a A AA ’

and convey her results to Bob via classical

communication. If Alice operates a measurement using

the states 1 2 1 2

, 1,2,3,...30,31,32i a a A AAg i , which are

given as follow:

1 2 1 2

2

1

2 2

1( 00000 01001 1 01011 10100

2

1 10110 11101 1 11111 )a a A AA

a a ag

b b c

a a a

c d d

(4.1)

1 2 1 2

2

2

2 2

1( 00000 01001 1 01011 10100

2

1 10110 11101 1 11111 )a a A AA

a a ag

b b c

a a a

c d d

(4.2)

1 2 1 2

2

3

2 2

1( 00000 01001 1 01011 10100

2

1 10110 11101 1 11111 )a a A AA

a a ag

b b c

a a a

c d d

(4.3)

1 2 1 2

2

4

2 2

1( 00000 01001 1 01011 10100

2

1 10110 11101 1 11111 )a a A AA

a a ag

b b c

a a a

c d d

(4.4)

1 2 1 2

2

5

2 2

1( 00001 1 00011 01000 10101

2

1 10111 11100 1 11110 )a a A AA

a a ag

b b d

a a a

d c c

(4.5)

1 2 1 2

2

6

2 2

1( 00001 1 00011 01000 10101

2

1 10111 11100 1 11110 )a a A AA

a a ag

b b d

a a a

d c c

(4.6)

1 2 1 2

2

7

2 2

1( 00001 1 00011 01000 10101

2

1 10111 11100 1 11110 )a a A AA

a a ag

b b d

a a a

d c c

(4.7)

1 2 1 2

2

8

2 2

1( 00001 1 00011 01000 10101

2

1 10111 11100 1 11110 )a a A AA

a a ag

b b d

a a a

d c c

(4.8)

1 2 1 2

2 2

9

2

1( 00100 1 00110 01101 1 01111

2

10000 11001 1 11011 )a a A AA

a a a ag

c c d d

a a

b b

(4.9)

1 2 1 2

2 2

10

2

1( 00100 1 00110 01101 1 01111

2

10000 11001 1 11011 )a a A AA

a a a ag

c c d d

a a

b b

(4.10)

1 2 1 2

2 2

11

2

1( 00100 1 00110 01101 1 01111

2

10000 11001 1 11011 )a a A AA

a a a ag

c c d d

a a

b b

(4.11)

1 2 1 2

2 2

12

2

1( 00100 1 00110 01101 1 01111

2

10000 11001 1 11011 )a a A AA

a a a ag

c c d d

a a

b b

(4.12)

1 2 1 2

2 2

13

2

1( 00101 1 00111 01100 1 01110

2

10001 1 10011 11000 )a a A AA

a a a ag

d d c c

a a

b b

(4.13)

1 2 1 2

2 2

14

2

1( 00101 1 00111 01100 1 01110

2

10001 1 10011 11000 )a a A AA

a a a ag

d d c c

a a

b b

(4.14)

1 2 1 2

2 2

15

2

1( 00101 1 00111 01100 1 01110

2

10001 1 10011 11000 )a a A AA

a a a ag

d d c c

a a

b b

(4.15)

1 2 1 2

2 2

16

2

1( 00101 1 00111 01100 1 01110

2

10001 1 10011 11000 )a a A AA

a a a ag

d d c c

a a

b b

(4.16)

17

1 2 1 2

2 21

( 00010 1 01001 01011 1 101002

210110 1 11101 11111 )a a A AA

a a ag

b b c

a a a

c d d

(4.17)

1 2 1 2

18

2 21

( 00010 1 01001 01011 1 101002

210110 1 11101 11111 )a a A AA

a a ag

b b c

a a a

c d d

(4.18)

JOURNAL OF OPTICS APPLICATIONS – Oct. 2012, Vol. 1, Iss. 2 16

1 2 1 2

19

2 21

( 00010 1 01001 01011 1 101002

210110 1 11101 11111 )a a A AA

a a ag

b b c

a a a

c d d

(4.19)

1 2 1 2

20

2 21

( 00010 1 01001 01011 1 101002

210110 1 11101 11111 )a a A AA

a a ag

b b c

a a a

c d d

(4.20)

1 2 1 2

21

2 21

( 1 00001 00011 01010 1 101012

210111 1 11100 11110 )a a A AA

a a ag

b b d

a a a

d c c

(4.21)

1 2 1 2

22

2 21

( 1 00001 00011 01010 1 101012

210111 1 11100 11110 )a a A AA

a a ag

b b d

a a a

d c c

(4.22)

1 2 1 2

23

2 21

( 1 00001 00011 01010 1 101012

210111 1 11100 11110 )a a A AA

a a ag

b b d

a a a

d c c

(4.23)

1 2 1 2

24

2 21

( 1 00001 00011 01010 1 101012

210111 1 11100 11110 )a a A AA

a a ag

b b d

a a a

d c c

(4.24)

1 2 1 2

2 2

25

2

1( 1 00100 00110 1 01101 01111

2

10010 1 11001 11011 )a a A AA

a a a ag

c c d d

a a

b b

(4.25)

1 2 1 2

2 2

26

2

1( 1 00100 00110 1 01101 01111

2

10010 1 11001 11011 )a a A AA

a a a ag

c c d d

a a

b b

(4.26)

1 2 1 2

2 2

27

2

1( 1 00100 00110 1 01101 01111

2

10010 1 11001 11011 )a a A AA

a a a ag

c c d d

a a

b b

(4.27)

1 2 1 2

2 2

28

2

1( 1 00100 00110 1 01101 01111

2

10010 1 11001 11011 )a a A AA

a a a ag

c c d d

a a

b b

(4.28)

1 2 1 2

2 2

29

2

1( 1 00101 00111 1 01100 01110

2

1 10001 10011 11010 )a a A AA

a a a ag

d d c c

a a

b b

(4.29)

1 2 1 2

2 2

30

2

1( 1 00101 00111 1 01100 01110

2

1 10001 10011 11010 )a a A AA

a a a ag

d d c c

a a

b b

(4.30)

1 2 1 2

2 2

31

2

1( 1 00101 00111 1 01100 01110

2

1 10001 10011 11010 )a a A AA

a a a ag

d d c c

a a

b b

(4.31)

1 2 1 2

2 2

32

2

1( 1 00101 00111 1 01100 01110

2

1 10001 10011 11010 )a a A AA

a a a ag

d d c c

a a

b b

(4.32)

If Alice’s measurement outcomes are:

{1 2 1 2

, 1,2,3,...14,15,16i a a A AAg i }.

The unknown particles entangled state can be

teleportated, if Alice’s measurement outcomes are:

1 2 1 2

, 17,18,19,...30,31,32i a a A AAg i .

The unknown particles entangled state can not be

teleportated. Therefore, if Alice’s measurement

outcome is{1 2 1 2

, 1,2,3,...14,15,16i a a A AAg i }.

Alice can inform Bob of the measurement outcomes via

a classical channel.

Step2: If Alice’s measurement outcome is 1 2 1 2

1 a a A AAg ,

the state of particle1 2 1 2B B C C will collapse into the

following states:

1 2 1 2

1 2 1 2

0 1

2 3

( 0000 01012

1010 1111 )

B B C C

B B C C

ax x

x x

(5)

Then Bob takes a joint X basis measurement

1 2

ˆ ˆB BX X on his two particles

1B and2B , which is

Von Neumann measurement under the condition of

1

0 12

X . After measuring, Bob announces

his result. If Bob's result is1 2B B

X X , the

collapsed states are written as:

17 JOURNAL OF OPTICS APPLICATIONS – Oct. 2012, Vol. 1, Iss. 2

1 21 20 1 2 3

1( 00 01 10 11 )

4C CC C

a x x x x (6)

Step3: After Alice and Bob announce their

measurement results publicly, Charlie performs the

joint unitary operation2U on his particles

1 2C C .

2

1

1

1

1

U

(7)

the resulting state of Charlie’s particles will be the

original state of 1 2

| a a . Thus Charlie can get the

original state with a successful probability 2

4 a .

The Second Scheme of the Arbitrary

Two-Qubit State Telepertation

The system state of particles is

1 2 1 2 1 2 1 2

0

1

2

3

|

[ ( 00000000 00010101

00101010 00111111 )

( 01000000 01010101

01101010 01111111 )

( 10000000 10010101

10101010 10111111 )

( 11000000 11010101

11101010 11111111

a as A A B B C CC

x a b

c d

x a b

c d

x a b

c d

x a b

c d

1 2 1 2 1 2 1 2

)]a a A A B B C C

(8)

We consider that Alice, Bob, and Charlie hold particles

(1a ,

2a1A ,

2A ),(1B ,

2B ) (1C ,

2C ) respectively.

Step1: Alice performs joint Bell measurement on her 4

qubits (1a ,

2a1A ,

2A )respectively and then she

announces her result publicly. The measurement basis

is as follow:

1 2 1 21

1( 0000 0101 1010 1111 )

2a a A Ag (9.1)

1 2 1 22

1( 0000 0101 1010 1111 )

2a a A Ag (9.2)

1 2 1 23

1( 0000 0101 1010 1111 )

2a a A Ag (9.3)

1 2 1 24

1( 0000 0101 1010 1111 )

2a a A Ag (9.4)

1 2 1 25

1( 0001 0100 1011 1110 )

2a a A Ag (9.5)

1 2 1 26

1( 0001 0100 1011 1110 )

2a a A Ag (9.6)

1 2 1 27

1( 0001 0100 1011 1110 )

2a a A Ag (9.7)

1 2 1 28

1( 0001 0100 1011 1110 )

2a a A Ag (9.8)

1 2 1 29

1( 0010 0111 1000 1101 )

2a a A Ag (9.9)

1 2 1 210

1( 0010 0111 1000 1101 )

2a a A Ag (9.10)

1 2 1 211

1( 0010 0111 1000 1101 )

2a a A Ag (9.11)

1 2 1 212

1( 0010 0111 1000 1101 )

2a a A Ag (9.12)

1 2 1 213

1( 0011 0110 1001 1100 )

2a a A Ag (9.13)

1 2 1 214

1( 0011 0110 1001 1100 )

2a a A Ag (9.14)

1 2 1 215

1( 0011 0110 1001 1100 )

2a a A Ag (9.15)

1 2 1 216

1( 0011 0110 1001 1100 )

2a a A Ag (9.16)

Assuming that her measurement result is1g , the

composite system of particles 1 2 1 2B B C C becomes

1 2 1 2

0 1

2 3

1( 0000 0101

2

1010 1111 )

sub

B B C C

ax bx

cx dx

(10)

Step2: Bob takes a joint X basis measurement

1 2

ˆ ˆB BX X on his two particles

1b and 2b , which is Von

Neumann measurement under the condition of

1( 0 1 )

2X . After measuring, Bob announces

his result. If Bob's result is1 2B B

X X , the

collapsed states can be written as:

1 2

1 2

0 1

2 3

1( 00 01

4

10 11 )

C C

C C

ax bx

cx dx

(11)

Step3: Charile introduces an auxiliary two-state

particle C with the initial state 0C

to reincarnate

1 2C C under the basis as follows:

1 2 1 2 1 2 1 2

1 2 1 2 1 2 1 2

{ 000 , 010 , 100 , 110 ,

001 , 011 , 101 , 111 }

C C C C C C C C C C C C

C C C C C C C C C C C C,

1 2

1 2

0 1

2 3

1( 000 010

4

100 110 )

C C C

C C C

ax bx

cx dx

(12)

A collective unitary transformation 1U on particles

1 2, ,c c c may take the form of the following 8 8

JOURNAL OF OPTICS APPLICATIONS – Oct. 2012, Vol. 1, Iss. 2 18

matrix, namely

1 2

1

2 1

A AU

A A

(13)

Where 1A and

2A are 4 4 matrixs and may be

written as 1 0 1 2 3, , ,A diag m m m m

and 2 2 2 2

2 0 1 2 3{ 1 , 1 , 1 , 1 }A diag m m m m if 0 1,m

1 2 3, ,a a am m mb c d

, the unitary transformation

1U will transform the state 1 2C C C

as follow:

1 2

1 2

0 1

2 2

1 2

2 2

2 3

2 2

3

1( 000 010

4

011 100

101 110

111 )

C C C

C C C

ax ax

x b a ax

x c a ax

x d a

(14)

Then Charile measures particle c on the basis of

{ 0 , 1 } . If his measured outcome is 1 , the quantum

state teleportation will be failed . If 0 is obtained,

the teleportation will be successful. The state will be

collapsed into

1 21 20 1 2 3( 00 01 10 11 )

4C CC C

ax x x x (15)

After Alice and Bob announce their measurement

results publicly, Charlie performs the joint unitary

operation2U .

2

1

1

1

1

U

(16)

On his particles 1 2C C , the resulting state of Charlie’s

particles will be the original state of 1 2

| a a .

Conclusions

In summary, we have proposed two quantum state

teleportation scheme of an arbitrary two-qubit state

using a non-maximally entangled six-qubit Cluster

state as channel. The receiver Charlie recovers the

original state with a certain probability, the two

scheme’s probability of success depends on the

coefficient of the non-maximally entangled six-qubit

Cluster state, the probability is 2

2116 4 4

4a a

,

if 1

2a b c d , the probability will be 1.

Comparing the two quantum state teleportation

schemes, it is known that the first one is better than the

second one. In our first scheme, the receiver Charlie

does not need introduce an auxiliary particle and

operator unitary transformation. Only the sender Alice

introduces an auxiliary particle and makes a

measurement on her particles and the auxiliary

particle, the quantum state teleportation can be

successfully realized with the maximal probability.

The first scheme is more convenient for the receiver

Charlie.

Acknowledgements

This work is supported by the National Natural

Science Foundation of China (Grant No. 10902083) and

Shaanxi Natural Science Foundation under Contract

(No. 2009JM1007).

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Qian Lan was born On November 27

1987 in Tianjin. she will get a

master's degree in 2013. Now she is

studying in Xi’an University of

Posts and Telecommunications,

Xi’an, Shanxi China. Her major field

of study is quantum communication.

Email: lanqian198711@163.com