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Quantum teleportation of arbitrary twoqubit state via entangled cavity ﬁelds Zhongjie Wang ∗ , Xu Fang College of Physics and Electronic Information, Anhui Normal University, Wuhu 241000, China
a r t i c l e
i n f o
Article history: Received 14 October 2013 Accepted 5 May 2014 Available online xxx PACS: 03.67.Hk 03.67.Mn
a b s t r a c t We propose a new protocol for quantum teleportation of an arbitrary two qubit state via continuous variables entangling channel. In our scheme two pairs of entangled light ﬁelds are employed. An outstanding characteristic of this scheme is that arbitrary state of two atoms is transmitted deterministically and directly to another pair of atoms without the help of the other atoms. © 2014 Elsevier GmbH. All rights reserved.
Keywords: Teleportation Entanglement state Entanglement cavity ﬁeld
1. Introduction Recent years quantum teleportation has been extensively investigated both theoretically and experimentally due to its important application in quantum computation [1] and quantum communication [2–10]. Quantum teleportation, originally proposed by Bennett et al. in 1993, is such a technique as information is transferred from a sender to a receiver through a quantum entanglement channel with the help of some classical information. Since then, many various proposals were proposed through various quantum entanglement channels such as entangled Bell state [2], threeparticle entangled state [7], entangled photon pairs, etc. quantum teleportation has been demonstrated experimentally in optical system [6], nuclear magnetic resonance [11], and trapped ions [12]. The author of Ref. [13] proposed a teleportation scheme for an arbitrary two qubit state which is based on the 16 orthogonal generalized Bell states. The author of Ref. [14] used a genuine irreducible fourqubit entangled state to transport arbitrary twoqubit state. In these proposed protocols, it is required that the multiqubit entangled state is taken as quantum channel and the Bell state measurement technique for atomic states is used. In this paper, we present a new quantum teleportation scheme for an arbitrary twoqubit state. In our scheme, the entanglement
cavity ﬁelds instead of the entangled atomic states are taken as quantum channel and cavity quantum electric dynamics (QED) technique is used. This protocol needs not the Bell state measurement technique for atomic states and has 100% success probability.
2. Theoretical description Let us consider a threelevel atom inside a singlemode light cavity driven by a classical ﬁeld with frequency ω. The atomic states are denoted by g> , e >, and i>. We assume that the transition frequency between the states e> and i> is highly detuned from the cavity ﬁeld frequency. The Hamiltonian of this system is [9] ( = 1) H=
ω0 z + ωc a+ a + g(+ a + − a+ ) + ˝(+ e−iωt + − eiωt ) 2
where + = e > < g, − = g > < e, z = e > < e − g > < g, e> and g> are the excited and ground states of the atom, a+ and a are the creation and annihilation operators for the cavity mode, ω0 is the transition frequency of the atom, g is the atomcavity coupling strength, and ˝ is the Rabi frequency of the classical ﬁeld. We assume that ω0 = ω. Then the interaction Hamiltonian, in the interaction picture, is HI = e−iıt g ∗ + a + eiıt g − a+ + ˝( + + − )
∗ Corresponding author. Email address:
[email protected] (Z. Wang).
(1)
(2)
where ı = ωc − ω0 is the detuning between the atomic transition frequency and cavity ﬁeld frequency, here we set ı = 0. By making
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Please cite this article in press as: Z. Wang, X. Fang, Quantum teleportation of arbitrary twoqubit state via entangled cavity ﬁelds, Optik  Int. J. Light Electron Opt. (2014), http://dx.doi.org/10.1016/j.ijleo.2014.05.038
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2 y
a unitary transformation HI = THI T + = ei /4 HI e−i obtain the transformed interaction Hamiltonian HI
y /4
, we can
1 1 = ga+ ( z − i y ) + g ∗ a( z + i y ) + ˝ z 2 2
− (aeg>12 + bee>12 + cgg>12 + dge>12 )gg>34 R3A R1B + (aeg>12 + bee>12 + cgg>12 + dge>12 )eg>34 R4A R1B
(3)
− (agg>12 + bge>12 + ceg>12 + dee>12 )ge>34 R1A R2B
Making the rotating wave approximation, which is equivalent to the transformation e−i2˝tz HI e−i2˝tz , the transformed interaction Hamiltonian HI becomes
+ (agg>12 + bge>12 + ceg>12 + dee>12 )ee>34 R2A R2B
1 1 (ga+ + g ∗ a) z − (g ∗ a − ga+ )( + ei2˝t − − e−i2˝t ) + ˝ z 2 2 (4)
− (aeg>12 + bee>12 + cgg>12 + dge>12 )ee>34 R4A R2B
Assuming that 2˝ » g, we can neglect the fast oscillating terms. Then HI reduces to
+ (age>12 + bgg>12 + cee>12 + deg>12 )eg>34 R2A R3B
HI =
HI
1 = (ga+ + g ∗ a) z + ˝ z 2
(5)
Proceeding to making the antitransformation, HI = T + HI T , we can obtain HI =
1 (ga+ + g ∗ a) x + ˝ x 2
(6)
x
+ +g ∗ a) x /2
− (age>12 + bgg>12 + cee>12 + deg>12 )gg>34 R1A R3B
+ (aee>12 + beg>12 + cge>12 + dgg>12 )gg>34 R3A R3B −(aee>12 + beg>12 + cge>12 + dgg>12 )eg>34 R4A R3B + (age>12 + bgg>12 + cee>12 + deg>12 )ge>34 R1A R4B − (age>12 + bgg>12 + cee>12 + deg>12 )ee>34 R2A R4B − (aee>12 + beg>12 + cge>12 + dgg>12 )ge>34 R3A R4B
The evolution operator for HI is represented as U = e−itHI = e−i˝t e−i(ga
+ (aeg>12 + bee>12 + cgg>12 + dge>12 )ge>34 R3A R2B
(7)
+ (aee>12 + beg>12 + cge>12 + dgg>12 )ee>34 R4A R4B
(12)
where
In next section, we will apply Eq. (7) in quantum teleportation of an arbitrary twoqubit state.
R1 = 20, 0> + i2˛, i2˛> + i2˛, 0> + 0, i2˛> + 0, −i2˛> +  − i2˛, 0> +  − i2˛, −i2˛>
3. Teleportation of arbitrary twobit state
(13)
We consider the teleportation of an unknown arbitrary twoqubit state  > 12 using continuous variables entangled state as quantum channel. Suppose the state of atoms 1 and 2 which is teleported by Alice to Bob is an unknown arbitrary twoqubit state  >12 = agg>12 + bge>12 + ceg>12 + dee>12
(8)
where a, b are unknown coefﬁcients, a2 + b2 + c2 + d2 = 1. Atoms 3 and 4 belonging to Bob are prepared in the ground state. Two pairs of entangled cavity ﬁelds (cavity 1, 2 belongs to Alice and cavity 3, 4 belongs to Bob) are used as quantum channel ϕ >=
(i˛>1 i˛>3 +  − i˛>1  − i˛>3 )(i˛>2 i˛>4 +  − i˛>2  − i˛>4 ) N
(9)
where N is normalized constant. The initial state of the whole system composed of atoms 1, 2, 3, 4 and light cavities are given by  (0) >= (agg>12 + bge>12 + ceg>12 + dee>12 ) ⊗ ϕ > ⊗gg>34
(10)
Step 1: Alice and Bob send atoms 1, 2 and 3, 4 into each singlemode cavity 1, 2 and 3, 4, respectively. The evolution operator Uk of the kth atoms and the kth cavity ﬁeld is described by Eq. (7). We have assumed that coupling parameters ˝, g between atoms and light ﬁelds are same. By selecting the interacting time t and the parameters , ˝ to satisfy the condition ˝t = m, m being a large positive even integer, we can obtain the evolution operator Uk as (11)
i˛(ai +a+ ) i
is displacement operator of light ﬁeld in where Dk (i˛) = e the kth cavity, ˛ = gt/2 (here it is assumed that g is real). According to Eq. (11), we can obtain the state of the whole system as 
>= U1 U2 U3 U4  (0) > =
1 16
(agg>12 + bge>12 + ceg>12 + dee>12 ) gg>34 R1A R1B
− (agg>12 + bge>12 + ceg>12 + dee>12 )eg>34 R2A R1B
0> −  − i2˛, −i2˛> R3
(14)
= i2˛, i2˛> + i2˛, 0> − 0, i2˛> + 0, −i2˛> − − i2˛,
0> −  − i2˛, −i2˛> R4
(15)
= 20, 0> + i2˛, i2˛> − i2˛, 0> − 0, i2˛> − 0,
− i2˛> −  − i2˛, 0> +  − i2˛, −i2˛>
(16)
In the above formulae, we have used symbols = A, B, A = (1, 3), B = (2, 4) and abbreviation ˛, ˇ > A = ˛ > 1 ˇ > 3 , ˛, ˇ > B = ˛ > 2 ˇ > 4 . We deﬁne the even and odd coherence states for the kth cavity ﬁeld as ± >k =
The entire teleportation protocol is described as follows:
1 Uk = {[Dk (i˛) + Dk+ (i˛)] − [Dk (i˛) − Dk+ (i˛)]kx } 2
R2 = i2˛, i2˛> − i2˛, 0> + 0, i2˛> − 0, −i2˛> + −i2˛,
(i2˛>k ±  − i2˛>k )
N1
(k = 1, 2)
(17)
where N1 is normalized constant. By using the even and odd coherence states, Eqs. (13)–(16) can be represented as
R1 = 20, 0> + +
N1 (+ , 0> + 0, + > )
R2 = +
N1 (0, + > − + , 0> )
(19)
N1 (+ , − > + − , + > ) 2
N1 (+ , 0> − 0, + > )
R4 = 20, 0> + −
(18)
N1 (+ , − > + − , + > ) 2
R3 = +
N1 (+ , + > + − , − > ) 2
(20)
N1 (+ , + > + − , − > ) 2
N1 (+ , 0> + 0, + > )
(21)
Please cite this article in press as: Z. Wang, X. Fang, Quantum teleportation of arbitrary twoqubit state via entangled cavity ﬁelds, Optik  Int. J. Light Electron Opt. (2014), http://dx.doi.org/10.1016/j.ijleo.2014.05.038
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Table 1 In teleportation process a variety of measurement results and the transformations made by Bob (measurement results for light ﬁelds is in set of (R1A R1B , R1A R4B , R4A R1B , R4A R4B )). Measurement results on light ﬁelds made by Alice and Bob
Measurement results on atoms 1, 2 made by Alice
States of atoms 3 and 4
Transformation operated by Bob
00 > A 00 > B 00 > A + + > B 00 > A − − > B + + > A 00 > B − − > A 00 > B + + > A + + > B + + > A − − > B − − > A + + > B − − > A − − > B
gg > 12
agg > 34 + bge > 34 + ceg > 34 + dee > 34
I
ge > 12
bgg > 34 + age > 34 + deg > 34 + cee > 34
4x
eg > 12
cgg > 34 + dge > 34 + aeg > 34 + bee > 34
3x
ee > 12
dgg > 34 + cge > 34 + beg > 34 + aee > 34
3x 4x
00 > A + 0 > B 00 > A 0+ > B + + > A + 0 > B + + > A 0+ > B − − > A + 0 > B − − > A 0+ > B
gg > 12 ge > 12 eg > 12 ee > 12
agg > 34 − bge > 34 + ceg > 34 − dee > 34 bgg > 34 − age > 34 + deg > 34 − cee > 34 cgg > 34 − dge > 34 + aeg > 34 − bee > 34 dgg > 34 − cge > 34 + beg > 34 − aee > 34
4z 4z 4x 4z 3x 4z 3x 4x
+ 0 > A 00 > B 0+ > A 00 > B + 0 > A + + > B 0+ > A + + > B + 0 > A − − > B 0+ > A − − > B
gg > 12 ge > 12 eg > 12 ee > 12
agg > 34 + bge > 34 − ceg > 34 − dee > 34 bgg > 34 + age > 34 − deg > 34 − cee > 34 cgg > 34 + dge > 34 − aeg > 34 − bee > 34 dgg > 34 + cge > 34 − beg > 34 − aee > 34
3z 4x 3z 3x 3z 3x 3z 4x
+ 0 > A + 0 > B + 0 > A 0+ > B 0+ > A + 0 > B 0+ > A 0+ > B
gg > 12 ge > 12 eg > 12 ee > 12
agg > 34 − bge > 34 − ceg > 34 + dee > 34 bgg > 34 − age > 34 − deg > 34 + cee > 34 cgg > 34 − dge > 34 − aeg > 34 + bee > 34 dgg > 34 − cge > 34 − beg > 34 + aee > 34
3x 4z 3z 4x 4z 3x 3z 4z 3x 3z 4x 4z
Step 2: Alice and Bob perform a Belllike measurement operation on the state of their cavity ﬁelds and inform the results each other. The Belllike measurement for ﬁeld states has been studied in Ref. [15]. After measurement, two pairs of atoms are located in the entangled states. Step 3: After Alice receiving the information about the operation accomplishment, Alice then performs a Von Neumann measurement operation on the state of atoms 1, 2, and informs the results to Bob. Step 4: According to Alice’s measurement results, Bob then decides whether to perform a unitary operation on the state of atom 3, 4 or not so that he obtains the wanted state.
For example, in step 2, supposing that Alice and Bob obtain the measurement result 0, 0 > A 0, 0 > B , the state of the whole system collapses into

>=
1 {(agg>12 + bge>12 + ceg>12 + dee>12 )gg>34 4
+ (aeg>12 + bee>12 + cgg>12 + dge>12 )eg>34 + (age>12 + bgg>12 + cee>12 + deg>12 )ge>34 + (aee>12 + beg>12 + cge>12 + dgg>12 )ee>34 }
(22)
Table 2 In teleportation process a variety of measurement results and the transformations made by Bob (measurement results for light ﬁelds is in set of (R1A R2B , R4A R2B , R1A R3B , R4A R3B )). Measurement results on light ﬁelds made by Alice andMeasurement Bob results on atoms 1, 2 made by Alice States of atoms 3 and 4
Transformation operated by Bob
00 > A + − > B 00 > A − + > B + + > A + − > B + + > A − + > B − − > A + − > B − − > A − + > B + 0 > A + − > B + 0 > A − + > B 0+ > A + − > B 0+ > A − + > B
gg > 12
4x bgg > 34 + age > 34 + deg > 34 + cee > 34
ge > 12
I agg > 34 + bge > 34 + ceg > 34 + dee > 34
eg > 12
3x 4x dgg > 34 + cge > 34 + beg > 34 + aee > 34
ee > 12
3x cgg > 34 + dge > 34 + aeg > 34 + bee > 34
00 > A − 0 > B + + > A − 0 > B − − > A − 0 > B 00 > A 0− > B + + > A 0− > B − − > A 0− > B
gg > 12 ge > 12 eg > 12 ee > 12
x z bgg > 34 + age > 34 − deg > 34 − cee >34 4 3 z agg > 34 + bge > 34 − ceg > 34 − dee >34 3 x x z dgg > 34 + cge > 34 − beg > 34 − aee >34 4 3 3 x z cgg > 34 + dge > 34 − aeg > 34 − bee >34 3 3
+ 0 > A − 0 > B 0+ > A − 0 > B
gg > 12 ge > 12 eg > 12 ee > 12
z x z bgg > 34 − age > 34 − deg > 34 + cee >34 3 4 4 z z agg > 34 − bge > 34 − ceg > 34 + dee >34 3 4 dgg > 34 − cge > 34 −beg > 34 + aee > 34 3z 3x 4x 4z cgg > 34 − dge > 34 −aeg > 34 + bee > 34 3x 3z 4z
+ 0 > A 0− > B 0+ > A 0− > B
gg > 12 ge > 12 eg > 12 ee > 12
bgg > 34 + age > 34 +deg > 34 + cee > 344x agg > 34 + bge > 34 +ceg > 34 + dee > 34I dgg > 34 + cge > 34 +beg > 34 + aee > 343x 4x cgg > 34 + dge > 34 +aeg > 34 + bee > 343x
Please cite this article in press as: Z. Wang, X. Fang, Quantum teleportation of arbitrary twoqubit state via entangled cavity ﬁelds, Optik  Int. J. Light Electron Opt. (2014), http://dx.doi.org/10.1016/j.ijleo.2014.05.038
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Table 3 In teleportation process a variety of measurement results and the transformations made by Bob (measurement results for light ﬁelds is in set of (R2A R2B , R3A R3B , R2A R3B , R3A R2B )). Measurement results on light ﬁelds made by Alice and Bob
Measurement results on atom 1, 2 made by Alice
States of atom 3 and 4
Transformation operated by Bob
+ − > A + − > B + − > A − + > B − + > A + − > B − + > A − + > B + 0 > A 0− > B − + > A 0− > B − 0 > A 0− > B 0− > A − 0 > B + − > A − 0 > B
gg > 12
dgg > 34 + cge > 34 + beg > 34 + aee > 34
3x 4x
ge > 12
cgg > 34 + dge > 34 + aeg > 34 + bee > 34
3x
eg > 12
bgg > 34 + age > 34 + deg > 34 + cee > 34
4x
ee > 12
agg > 34 + bge > 34 + ceg > 34 + dee > 34
I
− 0 > A − 0 > B − + > A − 0 > B − 0 > A − 0 > B 0− > A 0− > B
gg > 12 ge > 12 eg > 12 ee > 12
dgg > 34 − cge > 34 + beg > 34 − aee > 34 cgg > 34 − dge > 34 + aeg > 34 − bee > 34 bgg > 34 − age > 34 + deg > 34 − cee > 34 agg > 34 − bge > 34 + ceg > 34 − dee > 34
3x 4x 4z 3x 4z 4x 4z 4z
+ 0 > A + − > B − 0 > A − + > B 0− > A + − > B 0− > A − + > B
gg > 12 ge > 12 eg > 12 ee > 12
dgg > 34 + cge > 34 − beg > 34 − aee > 34 cgg > 34 + dge > 34 − aeg > 34 − bee > 34 bgg > 34 + age > 34 − deg > 34 − cee > 34 agg > 34 + bge > 34 − ceg > 34 − dee > 34
3x 4x 3z 3x 3z 3z 4x 3z
Table 4 In teleportation process a variety of measurement results and the transformations made by Bob (measurement results for light ﬁelds is in set of (R2A R1B , R3A R1B , R2A R4B , R3A R4B )). Measurement results on light ﬁelds made by Alice and Bob
Measurement results on atoms 1, 2 made by Alice
States of atoms 3 and 4
Transformation operated by Bob
+ − > A 00 > B − + > A 00 > B + − > A + + > B − + > A + + > B + − > A − − > B − + > A − − > B
gg > 12
cgg > 34 + dge > 34 + aeg > 34 + bee > 34
3x
ge > 12
dgg > 34 + cge > 34 + beg > 34 + aee > 34
3x 4x
eg > 12 ee > 12
agg > 34 + bge > 34 + ceg > 34 + dee > 34 bgg > 34 + age > 34 + deg > 34 + cee > 34
I 4x
+ − > A + 0 > B − + > A + 0 > B + − > A 0+ > B − + > A 0+ > B
gg > 12 ge > 12 eg > 12 ee > 12
cgg > 34 − dge > 34 + aeg > 34 − bee > 34 dgg > 34 − cge > 34 + beg > 34 − aee > 34 agg > 34 − bge > 34 + ceg > 34 − dee > 34 bgg > 34 − age > 34 + deg > 34 − cee > 34
3x 3z 3x 4z 4z 4x 4z
− 0 > A 00 > B − 0 > A + + > B − 0 > A − − > B 0− > A 00 > B 0− > A + + > B 0− > A − − > B
gg > 12 ge > 12 eg > 12 ee > 12
cgg > 34 + dge > 34 − aeg > 34 − bee > 34 dgg > 34 + cge > 34 − beg > 34 − aee > 34 agg > 34 + bge > 34 − ceg > 34 − dee > 34 bgg > 34 + age > 34 − deg > 34 − cee > 34
3x 3z 3x 3z 4x 3z 3z 4x
− 0 > A + 0 > B − 0 > A 0+ > B 0− > A + 0 > B 0− > A 0+ > B
gg > 12 ge > 12 eg > 12 ee > 12
cgg > 34 − dge > 34 − aeg > 34 + bee > 34 dgg > 34 − cge > 34 − beg > 34 + aee > 34 agg > 34 − bge > 34 − ceg > 34 + dee > 34 bgg > 34 − age > 34 − deg > 34 + cee > 34
3x 3z 4z 3x 4x 3z 4z 3z 4z 3z 4x 4z
In step 3, supposing that the measurement result for atoms 1 and 2 is gg > 12 , the state of atoms 3 and 4 then becomes into 
>= agg>34 + bge>34 + ceg>34 + dee>34
(23)
This is exactly teleported state what we want. Thus the whole teleportation process ends. The other cases are represented in Tables 1–4. The measurement results on light ﬁelds made by Alice and Bob can be divided into four groups according to whether they appear in sets of (R1A R1B , R1A R4B , R4A R1B , R4A R4B ), (R2A R2B , R3A R3B , R2A R3B , R3A R2B ) and (R1A R2B , R4A R2B , R1A R3B , R4A R3B ), A B A B A B A B (R2 R1 , R3 R1 , R2 R4 , R3 R4 ) or not. Each group of measurement results are labeled in Tables 1–4, respectively. From Tables 1–4, we can see that the success probability for the teleportation scheme presented here is 100%.
as quantum entanglement channel in our teleportation protocol which is considered to be mesoscopic features; (ii) it operates without the help of Bell state measurements for atom pairs, but needs the Belllike measurement for single cavity ﬁeld; (iii) in our teleportation protocol, the state for a pair of atom is transmitted directly to another pair of atom without the help of the other atoms; (iv) the cavity ﬁeld entanglement state to be used as quantum entanglement channel has been prepared theoretically [16]. This scheme can be realized in the experiment. In conclusion, we have proposed a novel deterministical teleportation protocol for arbitrary twoqubit state. In this protocol, two pairs of the entanglement cavity ﬁelds are used as quantum entanglement channel, and the state for a pair of atoms is transmitted directly to another pair of atoms without the help of Bell state measurements for atomic states.
4. Discussion and conclusion Finally, we compare this scheme with the other schemes. The scheme proposed in this paper, has some prominent features: (i) unlike the usual schemes proposed in Refs. [13,14], the entanglement cavity ﬁelds instead of the entanglement qubits is taken
Acknowledgements This work was supported by the Natural Science Foundation of Anhui Province of China (no. 090412060)
Please cite this article in press as: Z. Wang, X. Fang, Quantum teleportation of arbitrary twoqubit state via entangled cavity ﬁelds, Optik  Int. J. Light Electron Opt. (2014), http://dx.doi.org/10.1016/j.ijleo.2014.05.038
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Please cite this article in press as: Z. Wang, X. Fang, Quantum teleportation of arbitrary twoqubit state via entangled cavity ﬁelds, Optik  Int. J. Light Electron Opt. (2014), http://dx.doi.org/10.1016/j.ijleo.2014.05.038