Int J Theor PhysDOI 10.1007/s10773-014-2128-3

Controlled Teleportation of an Arbitrary Three-ion Statein Ion-trap Systems

Yuan-hua Li ·Li-ping Nie ·Xiao-lan Li

Received: 18 January 2014 / Accepted: 29 March 2014© Springer Science+Business Media New York 2014

Abstract We propose a seven-ion entangled channel that can be used to realize the deter-ministic controlled teleportation of an arbitrary three-ion state in ion-trap systems. Wedescribe the construction of this channel and explicitly demonstrate how the protocol works.In our scheme, it does not involve Bell-state measurement and only needs to perform thesingle-ion measurements and single ion unitary operations.

Keywords Controlled teleportation · Ion-trap systems · Arbitrary three-ion state

1 Introduction

Quantum entanglement lies at the heart of quantum mechanics plays a crucial role in quan-tum information processing, such as quantum teleportation [1], quantum cryptography [2],and quantum information splitting(QIS) [3–7]. Entangled states for two-level particles havebeen observed for photons [8], atoms in cavity QED [9], and ions in a trap [10, 11]. Sincethe first controlled teleportation protocol of single qubit was proposed [12], and then manycontrolled teleportation protocols of single qubit has been investigated [13–15]. Especially,the ion trap is considered to be one of the ideal systems for quantum information pro-cessing including QIS. In ion-trap systems, QIS of an arbitrary single-ion state can berealized by using W state [16]. In 2009, a scheme for teleportation of an unknown two-ionentangled state was proposed by using the three-ion entangled GHZ state in terms of ion-trap systems [17]. In the case of teleporting an arbitrary two-ion state, the probabilistic

Y.-h. Li (�) · L.-p. Nie · X.-l. LiDepartment of Physics, Jiangxi Normal University, Nanchang 330022, Chinae-mail: lyhua1984@163.com

Y.-h. Li · L.-p. NieKey Laboratory of Optoelectronic and Telecommunication of Jiangxi Province, Nanchang 330022,China

Int J Theor Phys

teleportation scheme [18] and the perfect teleportation scheme [19] have been carried outalmost at the same time.

In the present paper, we propose a seven-ion entangled channel that can be used to realizethe deterministic controlled teleportation of an arbitrary three-ion state in ion-trap systems.We show that the the seven-ion entangled resource can be prepared from a five-ion clusterstate and a Bell pairs by using one controlled-NOT gate. And unlike other schemes forthree-qubit teleportation, our protocol are not involving the joint Bell-state measurementsand multi-particle unitary transformation, which only requires single-ion measurements andsingle ion unitary operations that are feasible in current experiments.

2 The Model

Suppose that two two-level ions are confined in a linear trap. We simultaneously drive thetwo ions with a laser beam, tuned to ω0 − ν − δ, where ω0 is a frequency of the transition|1〉 → |0〉 and v is a frequency of the center-of-mass mode of the collective motion of theions. Under the condition of δ ≤ ν, the excitation of the stretch modes is far off-resonantand thus can be neglected. In the rotating-wave approximation, the Hamiltonian (assuming� = 1) is given by [20]

H = νa†a + ω0

∑

j=m,n

Sz,j +�∑

j=m,n

[e−i

[(ω0−ν−δ)t−η

(a+a†

)+φ]S+j +H.c.

], (1)

where the subscripts m and n denote two different ions, S+j = |1〉 ⟨0j∣∣, S−j = ∣∣0j

⟩ ⟨1j

∣∣, and

Sz,j = 12

(|1〉 〈1j | − |0j 〉〈0j |), with the ground

∣∣0j⟩

and excited∣∣1j

⟩states of the j -th ion.

While a† and a are the creation and annihilation operators for the center-of-mass mode ofthe collective motion of the two ions, and � and φ are the Rabi frequency and phase of

the laser field. η = k/√

2νMis a Lamb-Dicke parameter with the mass M of an ion, k is

wave vector. Under the conditions ofδ ≥ η�, φ = π/

2 and the vibrational frequency v ismuch larger than other characteristic frequencies, in the Lamb-Dicke regime (i.e. the spatialextension of the ionic wave function is much smaller than the wavelength of the lasers), theeffective Hamiltonian in the interaction picture can be further described as follows [21]

He = λ

⎡

⎣1

2

∑

j=m,n

(∣∣0j 〉〈0j∣∣ + ∣∣1j 〉〈1j

∣∣) + (S+mS+n + S+mS−n +H.c.

)⎤

⎦ , (2)

where λ = 2(�η)2

δ. Then the evolution operator of two-ion system is given by

U(t) = e−iHet . (3)

From (3) one can see that the evolution operator U(t) is independent of the vibrationalquantum number. This implies that the vibrational state of the ions can be a thermal statein principle. Using the evolution operator described by (3), one can easily work out thefollowing transformations:

|0〉m| 0〉n → e−iλt (cos λt |00〉mn − i sin λt | 11〉mn) , (4)

|0〉m| 1〉n → e−iλt (cos λt |01〉mn − i sin λt | 10〉mn) , (5)

Int J Theor Phys

|1〉m| 0〉n → e−iλt (cos λt |10〉mn − i sin λt | 01〉mn) , (6)

|1〉m| 1〉n → e−iλt (cos λt |11〉mn − i sin λt | 00〉mn) , (7)

3 Seven-ion Entangled Quantum Channel

In order to facilitate the discussion of our seven-ion entangled channel and the resultsobtained in this article, we first briefly describe the properties of a quantum channel com-posed of the direct product of a five-ion cluster state |C5〉12345 = 1

2 (|00000〉 + |00111〉 +|11101〉 + |11010〉)12345 and a Bell pairs

∣∣+⟩67 = 1√

2(|00〉 + |11〉)67, i.e.

| �〉1234567 = |C5〉12345⊗|+〉67. (8)

The seven-ion entangled channel |�〉1234567 can be obtained from the direct product state|�〉1234567 in (8) by performing one controlled-NOT operation. This can be achieved byusing the on ions 1 (set as a control ion) and 7 (set as a target ion). Hence, the seven-ionquantum channel can be written as

|�〉1234567 =√

2

4(|0000000〉 + |0000011〉 + |0011100〉 + |0011111〉

+ |1110101〉 + |1110110〉 + |1101001〉 + |1101010〉) . (9)

After implementing the single-qubit operations |1〉3 → i |1〉3, |1〉4 → i |1〉4 and |1〉6 →i |1〉6 on ions 3, 4 and 6, respectively, the state |�〉1234567 becomes

∣∣�′⟩1234567 =

√2

4(|0000000〉 + i |0000011〉 − |0011100〉 − i |0011111〉

+i |1110101〉 − |1110110〉 + i |1101001〉 − |1101010〉) . (10)

Next, we implement controlled teleportation of an arbitrary three-ion state by using theabove seven-ion entangled state as quantum channel in ion-trap systems.

4 Controlled Teleportation of an Arbitrary Three-ion State

Our scheme can be described as follows. Suppose the sender Alice has an arbitrary three-ionstate, which is given by

|ψ〉ABC = a |000〉+b |001〉+c |010〉+d |011〉+e |100〉+f |101〉+g |110〉+h |111〉 , (11)

Int J Theor Phys

where |a|2 + |b|2 + |c|2 + |d|2 + |e|2 + |f |2 + |g|2 + |h|2 = 1. Now Alice wants tosend the state to Bob who is assigned to reconstruct the original state with the help ofCharlie. Hence, she prepares a seven-ion entangled state

∣∣�′⟩1234567. The ions A, B, C,

3, 4 and 6 belong to Alice, the ion 2 belongs to Charlie and the ions 1, 5 and 7 belongto Bob, respectively. It is evident that the overall ten-ion state for the arbitrary three-ion state and a seven-ion entangled state can be written in the following form of directproduct

| 〉ABC1234567 = |ψ〉ABC ⊗ ∣∣�′⟩1234567 . (12)

To achieve the purpose of controlled teleportation of an arbitrary three-ion state in ion-trapsystems, Alice first let the ions pairs (3, A) , (4, B) and (6, C) in turn interact with threedifferent laser beams by choosing the λt = π

/4. Then Alice measures her ions A, B, C,

3, 4 and 6 in the basis of {|0〉 , |1〉}, respectively. It is known that Alice may obtain one ofthe 64 kinds of possible measured results with equal probability, and the remaining ionsmay collapse into one of the 64 states

∣∣φi⟩1257 (i = 1, 2, · · · , 64) after the measurement.

We summarize the possible outcomes of the measurement and the possible joint states ofthe remaining ions in the Appendix. Next Alice informs her measured results to Bob andCharlie through a classical channel. Whether it is possible for Bob to reconstruct the orig-inal state with local operations to the state

∣∣φi⟩1257 is dependent on the controller Charlie.

If Charlie allows Bob to reconstruct the initial unknown state, he needs to carry out a sin-

gle ion measurement on the ion 2 under the basis |±〉 = (|0〉 ± |1〉)/√

2, and then tells

Bob the result. At last, by combining information from the sender and controller, Bob canreconstruct the original state |ψ〉ABCwith an appropriate unitary transformation on the ionsat hand.

Now, let us take an example to demonstrate the principle of this protocol. For instance,if Alice’s measurement outcome is |000000〉ABC346, then the state of the remaining ionscollapse into the state

∣∣∣φ1⟩

1257= (a |0000〉 + b |0001〉 + c |1101〉 + d |1100〉 + e |1111〉 + f |1110〉 + g |0010〉 + h |0011〉)1257

=√

2

2

[|+〉2 (a |000〉 + b |001〉 + c |101〉 + d |100〉 + e |111〉 + f |110〉 + g |010〉 + h |011〉)157

+ |−〉2 (a |000〉 + b |001〉 − c |101〉 − d |100〉 − e |111〉 − f |110〉 + g |010〉 + h |011〉)157],(13)

where{|±〉2 = 1√

2(|0〉 ± |1〉)2

}. Charlie can now make a single ion measurement on ion

2 in the basis of {|±〉2}, and then he sends the result of his measurement to Bob. If the resultof the single ion is |+〉2 or |−〉2, the ions 1, 5 and 7 will collapse into one of the followingstate, respectively,

|ϕ1〉157 = (a |000〉+b |001〉+c |101〉+d |100〉+e |111〉+f |110〉+g |010〉+h |011〉) (14)

or

|ϕ2〉157 = (a |000〉+b |001〉−c |101〉−d |100〉−e |111〉−f |110〉+g |010〉+h |011〉) (15)

Int J Theor Phys

In order to obtain the original three-ion state easily, firstly, Bob performs a correspondingsingle ion unitary operation I1 or σz

1 on the ion 1 to transfer its state into

(a |000〉 + b |001〉 + c |101〉 + d |100〉 + e |111〉 + f |110〉 + g |010〉 + h |011〉)157 . (16)

Then Bob can carry out a controlled-NOT operation on his ions with the ion 5 as controlledion and the ion 1 as target ion, then the (16) becomes

(a |000〉 + b |001〉 + c |011〉 + d |010〉 + e |101〉 + f |100〉 + g |110〉 + h |111〉)517 .

(17)Next Bob makes two controlled-NOT operations on his ions with the ions 5, 1 as controlledions and the ion 7 as target ion, then the (17) becomes

(a |000〉 + b |001〉 + c |010〉 + d |011〉 + e |100〉 + f |101〉 + g |110〉 + h |111〉)517 .

(18)After doing those operations, Bob can successfully reconstruct the original arbitrary three-ion state |ψ〉ABC .

5 Discussions and Conclusions

Now we give a brief discussion on the experimental matters. Based on the current ion-traptechniques [22], we can use one Zeeman level of the S1/2 ground state of 40Ca+ ions asthe state |0〉 and one Zeeman level of the metastable D5/2 state as the state |1〉. It is knownthat the lifetime of the metastable D5/2 state is about 1.16s and the accessible center-of-mass mode frequency ν = 1.2 MHz [23]. Under choosing the parameters δ = 0.1ν,and� = 0.1ν, thus, the condition �η ≤ δ ≤ ν can be easily realized. Therefore, the effectiveHamiltonian described in (3) may be used to describe a realistic physical system. In additionthe five-ion cluster state has been experimentally realized in ion-trap systems [24]. Thus ourschemes are realizable with the present ion-trap techniques.

In conclusion, we have demonstrated that the seven-ion entangled state can be used as thequantum channel to realize the deterministic controlled teleportation of an arbitrary three-ion state in ion-trap systems. The scheme does not involve Bell-state measurement and onlyneeds to perform the single-ion measurements and single ion unitary operations. We hopethat this controlled teleportation protocol of three-ion state can be realized experimentallywith presently available techniques in the future.

Acknowledgments This work is supported by the National Natural Science Foundation of China (GrantNo. 61265001), the Natural Science Foundation of Jiangxi Province, China (Grant No. 20122BAB202005),the Research Foundation of state key laboratory of advanced optical communication systems and networks,Shanghai Jiao Tong University, China (2011GZKF031104), and the Research Foundation of the EducationDepartment of Jiangxi Province.

Appendix

Alice’s possible results and the corresponding possible joint states for Bob and Charlie,where the normalization factors have been omitted for convenience (Table 1).

Int J Theor PhysTa

ble

1T

heou

tcom

eof

mea

sure

men

tspe

rfor

med

byA

lice

,and

the

corr

espo

ndin

gpo

ssib

lejo

ints

tate

sob

tain

edby

Bob

and

Cha

rlie

Ali

ce’s

resu

lts

Stat

esob

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Bob

and

Cha

rlie

| 0000

00〉 A

BC

346

∣ ∣ φ1⟩ 12

57=

(a| 00

00〉 +

b| 00

01〉 +

c| 11

01〉 +

d| 11

00〉 +

e| 11

11〉 +

f| 11

10〉 +

g| 00

10〉 +

h| 00

11〉 ) 1

257

−i| 00

1001〉 A

BC

346

∣ ∣ φ2⟩ 12

57=

(a| 00

00〉 −

b| 00

01〉 +

c| 11

01〉 −

d| 11

00〉 +

e| 11

11〉 −

f| 11

10〉 +

g| 00

10〉 −

h| 00

11〉 ) 1

257

| 0010

00〉 A

BC

346

∣ ∣ φ3⟩ 12

57=

(a| 00

01〉 +

b| 00

00〉 +

c| 11

00〉 +

d| 11

01〉 +

e| 11

10〉 +

f| 11

11〉 +

g| 00

11〉 +

h| 00

10〉 ) 1

257

−i| 00

0001〉 A

BC

346

∣ ∣ φ4⟩ 12

57=

(−a| 00

01〉 +

b| 00

00〉 −

c| 11

00〉 +

d| 11

01〉 −

e| 11

10〉 +

f| 11

11〉 −

g| 00

11〉 +

h| 00

10〉 ) 1

257

−i| 01

0100〉 A

BC

346

∣ ∣ φ5⟩ 12

57=

(a| 00

00〉 +

b| 00

01〉 −

c| 11

01〉 −

d| 11

00〉 +

e| 11

11〉 +

f| 11

10〉 −

g| 00

10〉 +

h| 00

11〉 ) 1

257

−| 01

1101〉 A

BC

346

∣ ∣ φ6⟩ 12

57=

(a| 00

00〉 −

b| 00

01〉 −

c| 11

01〉 +

d| 11

00〉 +

e| 11

11〉 −

f| 11

10〉 −

g| 00

10〉 +

h| 00

11〉 ) 1

257

−i| 01

1100〉 A

BC

346

∣ ∣ φ7⟩ 12

57=

(a| 00

01〉 +

b| 00

00〉 −

c| 11

00〉 −

d| 11

01〉 +

e| 11

10〉 +

f| 11

11〉 −

g| 00

11〉 −

h| 00

10〉 ) 1

257

−| 01

0101〉 A

BC

346

∣ ∣ φ8⟩ 12

57=

(−a| 00

01〉 +

b| 00

00〉 +

c| 11

00〉 −

d| 11

01〉 −

e| 11

10〉 +

f| 11

11〉 +

g| 00

11〉 −

h| 00

10〉 ) 1

257

−i| 10

0010〉 A

BC

346

∣ ∣ φ9⟩ 12

57=

(a| 00

00〉 +

b| 00

01〉 +

c| 11

01〉 +

d| 11

00〉 −

e| 11

11〉 −

f| 11

10〉 −

g| 00

10〉 −

h| 00

11〉 ) 1

257

−| 10

1011〉 A

BC

346

∣ ∣ φ10

⟩ 1257

=(a

| 0000〉 −

b| 00

01〉 +

c| 11

01〉 −

d| 11

00〉 −

e| 11

11〉 +

f| 11

10〉 −

g| 00

10〉 +

h| 00

11〉 ) 1

257

−i| 10

0010〉 A

BC

346

∣ ∣ φ11

⟩ 1257

=(a

| 0001〉 +

b| 00

00〉 +

c| 11

00〉 +

d| 11

01〉 −

e| 11

10〉 −

f| 11

11〉 −

g| 00

11〉 −

h| 00

10〉 ) 1

257

−| 10

0011〉 A

BC

346

∣ ∣ φ12

⟩ 1257

=(−

a| 00

01〉 +

b| 00

00〉 −

c| 11

00〉 +

d| 11

01〉 +

e| 11

10〉 −

f| 11

11〉 +

g| 00

11〉 −

h| 00

10〉 ) 1

257

−| 11

0110〉 A

BC

346

∣ ∣ φ13

⟩ 1257

=(a

| 0000〉 +

b| 00

01〉 −

c| 11

01〉 −

d| 11

00〉 −

e| 11

11〉 −

f| 11

10〉 +

g| 00

10〉 +

h| 00

11〉 ) 1

257

i| 11

1111〉 A

BC

346

∣ ∣ φ14

⟩ 1257

=(a

| 0000〉 −

b| 00

01〉 −

c| 11

01〉 +

d| 11

00〉 −

e| 11

11〉 +

f| 11

10〉 +

g| 00

10〉 −

h| 00

11〉 ) 1

257

−| 11

1110〉 A

BC

346

∣ ∣ φ15

⟩ 1257

=(a

| 0001〉 +

b| 00

00〉 −

c| 11

00〉 −

d| 11

01〉 −

e| 11

10〉 −

f| 11

11〉 +

g| 00

11〉 +

h| 00

10〉 ) 1

257

i| 11

0111〉 A

BC

346

∣ ∣ φ16

⟩ 1257

=(−

a| 00

01〉 −

b| 00

00〉 +

c| 11

00〉 −

d| 11

01〉 +

e| 11

10〉 −

f| 11

11〉 −

g| 00

11〉 +

h| 00

10〉 ) 1

257

| 0101

00〉 A

BC

346

∣ ∣ φ17

⟩ 1257

=(a

| 1101〉 +

b| 11

00〉 +

c| 00

00〉 +

d| 00

01〉 +

e| 00

10〉 +

f| 00

11〉 +

g| 11

11〉 +

h| 11

10〉 ) 1

257

−i| 01

0000〉 A

BC

346

∣ ∣ φ18

⟩ 1257

=(a

| 1101〉 −

b| 11

00〉 +

c| 00

00〉 −

d| 00

01〉 +

e| 00

10〉 −

f| 00

11〉 +

g| 11

11〉 −

h| 11

10〉 ) 1

257

| 0110

00〉 A

BC

346

∣ ∣ φ19

⟩ 1257

=(a

| 1100〉 +

b| 11

01〉 +

c| 00

01〉 +

d| 00

00〉 +

e| 00

11〉 +

f| 00

10〉 +

g| 11

10〉 +

h| 11

11〉 ) 1

257

−i| 01

0001〉 A

BC

346

∣ ∣ φ20

⟩ 1257

=(−

a| 11

00〉 +

b| 11

01〉 −

c| 00

01〉 +

d| 00

00〉 −

e| 00

11〉 +

f| 00

10〉 −

g| 11

10〉 +

h| 11

11〉 ) 1

257

−i| 00

0100〉 A

BC

346

∣ ∣ φ21

⟩ 1257

=(−

a| 11

01〉 −

b| 11

00〉 +

c| 00

00〉 +

d| 00

01〉 −

e| 00

10〉 −

f| 00

11〉 +

g| 11

11〉 +

h| 11

10〉 ) 1

257

−| 00

1101〉 A

BC

346

∣ ∣ φ22

⟩ 1257

=(−

a| 11

01〉 +

b| 11

00〉 +

c| 00

00〉 −

d| 00

01〉 −

e| 00

10〉 +

f| 00

11〉 +

g| 11

11〉 −

h| 11

10〉 ) 1

257

−i| 00

1100〉 A

BC

346

∣ ∣ φ23

⟩ 1257

=(−

a| 11

00〉 −

b| 11

01〉 +

c| 00

01〉 +

d| 00

00〉 −

e| 00

11〉 −

f| 00

10〉 +

g| 11

10〉 +

h| 11

11〉 ) 1

257

−| 00

0000〉 A

BC

346

∣ ∣ φ24

⟩ 1257

=(a

| 1100〉 −

b| 11

01〉 −

c| 00

01〉 +

d| 00

00〉 +

e| 00

11〉 −

f| 00

10〉 −

g| 11

10〉 +

h| 11

11〉 ) 1

257

−i| 11

0000〉 A

BC

346

∣ ∣ φ25

⟩ 1257

=(a

| 1101〉 +

b| 11

00〉 +

c| 00

00〉 +

d| 00

01〉 −

e| 00

10〉 −

f| 00

11〉 −

g| 11

11〉 −

h| 11

10〉 ) 1

257

Int J Theor Phys

Tabl

e1

(con

tinu

ed)

Ali

ce’s

resu

lts

Stat

esob

tain

edby

Bob

and

Cha

rlie

−| 11

1011〉 A

BC

346

∣ ∣ φ26

⟩ 1257

=(a

| 1101〉 −

b| 11

00〉 +

c| 00

00〉 −

d| 00

01〉 −

e| 00

10〉 +

f| 00

11〉 −

g| 11

11〉 +

h| 11

10〉 ) 1

257

−i| 11

1010〉 A

BC

346

∣ ∣ φ27

⟩ 1257

=(a

| 1100〉 +

b| 11

01〉 +

c| 00

01〉 +

d| 00

00〉 −

e| 00

11〉 −

f| 00

10〉 −

g| 11

10〉 −

h| 11

11〉 ) 1

257

−| 11

0011〉 A

BC

346

∣ ∣ φ28

⟩ 1257

=(−

a| 11

00〉 +

b| 11

01〉 −

c| 00

01〉 +

d| 00

00〉 +

e| 00

11〉 −

f| 00

10〉 +

g| 11

10〉 −

h| 11

11〉 ) 1

257

−| 10

0110〉 A

BC

346

∣ ∣ φ29

⟩ 1257

=(−

a| 11

01〉 −

b| 11

00〉 +

c| 00

00〉 +

d| 00

01〉 +

e| 00

10〉 +

f| 00

11〉 −

g| 11

11〉 −

h| 11

10〉 ) 1

257

i| 10

1111〉 A

BC

346

∣ ∣ φ30

⟩ 1257

=(−

a| 11

01〉 +

b| 11

00〉 +

c| 00

00〉 −

d| 00

01〉 +

e| 00

10〉 −

f| 00

11〉 −

g| 11

11〉 +

h| 11

10〉 ) 1

257

−| 10

1110〉 A

BC

346

∣ ∣ φ31

⟩ 1257

=(−

a| 11

00〉 −

b| 11

01〉 +

c| 00

01〉 +

d| 00

00〉 +

e| 00

11〉 +

f| 00

10〉 −

g| 11

10〉 −

h| 11

11〉 ) 1

257

i| 10

0111〉 A

BC

346

∣ ∣ φ32

⟩ 1257

=(a

| 1100〉 −

b| 11

01〉 −

c| 00

01〉 +

d| 00

00〉 −

e| 00

11〉 +

f| 00

10〉 +

g| 11

10〉 −

h| 11

11〉 ) 1

257

| 1000

00〉 A

BC

346

∣ ∣ φ33

⟩ 1257

=(a

| 1111〉 +

b| 11

10〉 +

c| 00

10〉 +

d| 00

11〉 +

e| 00

00〉 +

f| 00

01〉 +

g| 11

01〉 +

h| 11

00〉 ) 1

257

−i| 10

1001〉 A

BC

346

∣ ∣ φ34

⟩ 1257

=(a

| 1111〉 −

b| 11

10〉 +

c| 00

10〉 −

d| 00

11〉 +

e| 00

00〉 −

f| 00

01〉 +

g| 11

01〉 −

h| 11

00〉 ) 1

257

| 1010

00〉 A

BC

346

∣ ∣ φ35

⟩ 1257

=(a

| 1110〉 +

b| 11

11〉 +

c| 00

11〉 +

d| 00

10〉 +

e| 00

01〉 +

f| 00

00〉 +

g| 11

00〉 +

h| 11

01〉 ) 1

257

−i| 10

0001〉 A

BC

346

∣ ∣ φ36

⟩ 1257

=(−

a| 11

10〉 +

b| 11

11〉 −

c| 00

11〉 +

d| 00

10〉 −

e| 00

01〉 +

f| 00

00〉 −

g| 11

00〉 +

h| 11

01〉 ) 1

257

−i| 11

0100〉 A

BC

346

∣ ∣ φ37

⟩ 1257

=(a

| 1111〉 +

b| 11

10〉 −

c| 00

10〉 −

d| 00

11〉 +

e| 00

00〉 +

f| 00

01〉 −

g| 11

01〉 −

h| 11

00〉 ) 1

257

−| 11

1101〉 A

BC

346

∣ ∣ φ38

⟩ 1257

=(a

| 1111〉 −

b| 11

10〉 −

c| 00

10〉 +

d| 00

11〉 +

e| 00

00〉 −

f| 00

01〉 −

g| 11

01〉 +

h| 11

00〉 ) 1

257

−i| 11

1100〉 A

BC

346

∣ ∣ φ39

⟩ 1257

=(a

| 1110〉 +

b| 11

11〉 −

c| 00

11〉 −

d| 00

10〉 +

e| 00

01〉 +

f| 00

00〉 −

g| 11

00〉 −

h| 11

01〉 ) 1

257

−| 11

0101〉 A

BC

346

∣ ∣ φ40

⟩ 1257

=(−

a| 11

10〉 +

b| 11

11〉 +

c| 00

11〉 −

d| 00

10〉 −

e| 00

01〉 +

f| 00

00〉 +

g| 11

00〉 −

h| 11

01〉 ) 1

257

−i| 00

0010〉 A

BC

346

∣ ∣ φ41

⟩ 1257

=(−

a| 11

11〉 −

b| 11

10〉 −

c| 00

10〉 −

d| 00

11〉 +

e| 00

00〉 +

f| 00

01〉 +

g| 11

01〉 +

h| 11

00〉 ) 1

257

−| 00

1011〉 A

BC

346

∣ ∣ φ42

⟩ 1257

=(−

a| 11

11〉 +

b| 11

10〉 −

c| 00

10〉 +

d| 00

11〉 +

e| 00

00〉 −

f| 00

01〉 +

g| 11

01〉 −

h| 11

00〉 ) 1

257

−i| 00

1010〉 A

BC

346

∣ ∣ φ43

⟩ 1257

=(−

a| 11

10〉 −

b| 11

11〉 −

c| 00

11〉 −

d| 00

10〉 +

e| 00

01〉 +

f| 00

00〉 +

g| 11

00〉 +

h| 11

01〉 ) 1

257

−| 00

0011〉 A

BC

346

∣ ∣ φ44

⟩ 1257

=(a

| 1110〉 −

b| 11

11〉 +

c| 00

11〉 −

d| 00

10〉 −

e| 00

01〉 +

f| 00

00〉 −

g| 11

00〉 +

h| 11

01〉 ) 1

257

−| 01

0110〉 A

BC

346

∣ ∣ φ45

⟩ 1257

=(−

a| 11

11〉 −

b| 11

10〉 +

c| 00

10〉 +

d| 00

11〉 +

e| 00

00〉 +

f| 00

01〉 −

g| 11

01〉 −

h| 11

00〉 ) 1

257

i| 01

1111〉 A

BC

346

∣ ∣ φ46

⟩ 1257

=(−

a| 11

11〉 +

b| 11

10〉 +

c| 00

10〉 −

d| 00

11〉 +

e| 00

00〉 −

f| 00

01〉 −

g| 11

01〉 +

h| 11

00〉 ) 1

257

−| 01

1110〉 A

BC

346

∣ ∣ φ47

⟩ 1257

=(−

a| 11

10〉 −

b| 11

11〉 +

c| 00

11〉 +

d| 00

10〉 +

e| 00

01〉 +

f| 00

00〉 −

g| 11

00〉 −

h| 11

01〉 ) 1

257

i| 01

0111〉 A

BC

346

∣ ∣ φ48

⟩ 1257

=(a

| 1110〉 −

b| 11

11〉 −

c| 00

11〉 +

d| 00

10〉 −

e| 00

01〉 +

f| 00

00〉 +

g| 11

00〉 −

h| 11

01〉 ) 1

257

| 1100

00〉 A

BC

346

∣ ∣ φ49

⟩ 1257

=(a

| 0010〉 +

b| 00

11〉 +

c| 11

11〉 +

d| 11

10〉 +

e| 11

01〉 +

f| 11

00〉 +

g| 00

00〉 +

h| 00

01〉 ) 1

257

−i| 11

1001〉 A

BC

346

∣ ∣ φ50

⟩ 1257

=(a

| 0010〉 −

b| 00

11〉 +

c| 11

11〉 −

d| 11

10〉 +

e| 11

01〉 −

f| 11

00〉 +

g| 00

00〉 −

h| 00

01〉 ) 1

257

Int J Theor Phys

Tabl

e1

(con

tinu

ed)

Ali

ce’s

resu

lts

Stat

esob

tain

edby

Bob

and

Cha

rlie

| 1110

00〉 A

BC

346

∣ ∣ φ51

⟩ 1257

=(a

| 0011〉 +

b| 00

10〉 +

c| 11

10〉 +

d| 11

11〉 +

e| 11

00〉 +

f| 11

01〉 +

g| 00

01〉 +

h| 00

00〉 ) 1

257

−i| 11

0001〉 A

BC

346

∣ ∣ φ52

⟩ 1257

=(−

a| 00

11〉 +

b| 00

10〉 −

c| 11

10〉 +

d| 11

11〉 −

e| 11

00〉 +

f| 11

01〉 −

g| 00

01〉 +

h| 00

00〉 ) 1

257

−i| 10

0100〉 A

BC

346

∣ ∣ φ53

⟩ 1257

=(−

a| 00

10〉 −

b| 00

11〉 +

c| 11

11〉 +

d| 11

10〉 −

e| 11

01〉 −

f| 11

00〉 +

g| 00

00〉 +

h| 00

01〉 ) 1

257

−| 10

1101〉 A

BC

346

∣ ∣ φ54

⟩ 1257

=(−

a| 00

10〉 +

b| 00

11〉 +

c| 11

11〉 −

d| 11

10〉 −

e| 11

01〉 +

f| 11

00〉 +

g| 00

00〉 −

h| 00

01〉 ) 1

257

−i| 10

1100〉 A

BC

346

∣ ∣ φ55

⟩ 1257

=(−

a| 00

11〉 −

b| 00

10〉 +

c| 11

10〉 +

d| 11

11〉 −

e| 11

00〉 −

f| 11

01〉 +

g| 00

01〉 +

h| 00

00〉 ) 1

257

−| 10

0101〉 A

BC

346

∣ ∣ φ56

⟩ 1257

=(a

| 0011〉 −

b| 00

10〉 −

c| 11

10〉 +

d| 11

11〉 +

e| 11

00〉 −

f| 11

01〉 −

g| 00

01〉 +

h| 00

00〉 ) 1

257

−i| 01

0010〉 A

BC

346

∣ ∣ φ57

⟩ 1257

=(−

a| 00

10〉 −

b| 00

11〉 −

c| 11

11〉 −

d| 11

10〉 +

e| 11

01〉 +

f| 11

00〉 +

g| 00

00〉 +

h| 00

01〉 ) 1

257

−| 01

1011〉 A

BC

346

∣ ∣ φ58

⟩ 1257

=(−

a| 00

10〉 +

b| 00

11〉 −

c| 11

11〉 +

d| 11

10〉 +

e| 11

01〉 −

f| 11

00〉 +

g| 00

00〉 −

h| 00

01〉 ) 1

257

−i| 01

1010〉 A

BC

346

∣ ∣ φ59

⟩ 1257

=(−

a| 00

11〉 −

b| 00

10〉 −

c| 11

10〉 −

d| 11

11〉 +

e| 11

00〉 +

f| 11

01〉 +

g| 00

01〉 +

h| 00

00〉 ) 1

257

−| 01

0011〉 A

BC

346

∣ ∣ φ60

⟩ 1257

=(a

| 0011〉 −

b| 00

10〉 +

c| 11

10〉 −

d| 11

11〉 −

e| 11

00〉 +

f| 11

01〉 −

g| 00

01〉 +

h| 00

00〉 ) 1

257

−| 00

0110〉 A

BC

346

∣ ∣ φ61

⟩ 1257

=(a

| 0010〉 +

b| 00

11〉 −

c| 11

11〉 −

d| 11

10〉 −

e| 11

01〉 −

f| 11

00〉 +

g| 00

00〉 +

h| 00

01〉 ) 1

257

i| 00

1111〉 A

BC

346

∣ ∣ φ62

⟩ 1257

=(a

| 0010〉 −

b| 00

11〉 −

c| 11

11〉 +

d| 11

10〉 −

e| 11

01〉 +

f| 11

00〉 +

g| 00

00〉 −

h| 00

01〉 ) 1

257

−| 00

1110〉 A

BC

346

∣ ∣ φ63

⟩ 1257

=(a

| 0011〉 +

b| 00

10〉 −

c| 11

10〉 −

d| 11

11〉 −

e| 11

00〉 −

f| 11

01〉 +

g| 00

01〉 +

h| 00

00〉 ) 1

257

i| 00

0111〉 A

BC

346

∣ ∣ φ64

⟩ 1257

=(−

a| 00

11〉 +

b| 00

10〉 +

c| 11

10〉 −

d| 11

11〉 +

e| 11

00〉 −

f| 11

01〉 −

g| 00

01〉 +

h| 00

00〉 ) 1

257

Int J Theor Phys

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2. Huang, Y.B., Li, S.S., Nie, Y.Y.: Int. J. Theor. Phys. 48, 95 (2009)3. Muralidharan, S., Panigrahi, P.K.: Phys. Rev. A 78, 062333 (2008)4. Deng, F.G., Li, X.H., Li, C.Y., Zhou, P., Zhou, H.Y.: Phys. Rev. A 72, 044301 (2005)5. Luo, M.X., Deng, Y.: Quantum. Inf. Process. 12, 773 (2013)6. Pan, G.X., Liu, Y.M., Yin, X.F., Zhang, W., Zhang, Z.J.: Int. J. Quanum. Inform. 6, 1033 (2008)7. Nie, Y.Y., Li, Y.H., Liu, J.C., Sang, M.H.: Quantum. Inf. Process. 10, 297 (2011)8. Pan, J.-W., Daniell, M., Gasparoni, S., Weihs, G., Zeilinger, A.: Phys. Rev. Lett. 86, 4435 (2001)9. Osnaghi, S., et al.: Phys. Rev. Lett. 87, 0379021 (2001)

10. Lin, X., Su, W.J.: Int. J. Quanum. Inform. 9, 1299 (2011)11. Shi, Y.L., Mei, F., Yu, Y.F., Feng, X.L., Zhang, Z.M.: Quantum. Inf. Process. 11, 229 (2012)12. Karlsson, A., Bourennane, M.: Phys. Rev. A 58, 4394 (1998)13. Yan, F.L., Wang, D.: Phys. Lett. A 316, 297 (2003)14. Yang, C.P., Chu, S.I., Han, S.: Phys. Rev. A 70, 022329 (2004)15. Man, Z.X., Xia, Y.J., An, N.B.: Phys. Rev. A 75, 052306 (2007)16. Zheng, S.B.: Phys. Rev. A 74, 054303 (2006)17. Ai, L.Y., Du, G., Zhu, S.L., Zhang, Z.M.: Chin. Phys. Lett. 26, 014210 (2009)18. Lian, S.M., Yan, F.L.: Chin. Phys. Lett. 27, 040308 (2010)19. Lin, X.: Commun. Theor. Phys. 53, 1062 (2010)20. Zheng, S.B.: Phys. Rev. A 70, 045804 (2004)21. Wang, X.W., Yang, G.J.: Phys. Rev. A 78, 024301 (2008)22. Haffner, H., et al.: Nature (London) 438, 643 (2005)23. Riebe, M., et al.: Nature (London) 429, 734 (2004)24. Zheng, S.B.: Phys. Rev. A 73, 065802 (2006)