Efficient methods for choosing quantum channel in teleportation and bidirectional teleportation

The Journal of China Universities of Posts and Telecommunications December 2013, 20(Suppl. 2): 101–104 www.sciencedirect.com/science/journal/10058885

http://jcupt.xsw.bupt.cn

Efficient methods for choosing quantum channel in teleportation and bidirectional teleportation HU Yang (), FU Hong-zi, TIAN Xiu-lao School of Science, Xi'an University of Posts and Telecommunications, Xi'an 710121, China

Abstract The tensor representation of quantum teleportation shows the internal connection between Bell basis measurement matrix T α and channel parameter matrix (CPM) R. An explicit CPM criterion based on the tensor representation is derived. This criterion can be used to judge whether a quantum state can be employed as the quantum channel or not in teleportation and bidirectional teleportation, and it provides us more flexible choices to construct the quantum channel. Keywords quantum bidirectional teleportation, tensor representation, CPM, CPM criterion

1 Introduction Since the pioneer work of Bennett et al. [1] presented in 1993, quantum teleportation turns out to be a hot topic in the past two decades. There has been a great progress of quantum teleportation both in the theory [2–9] and experiments [10–15]. Up to now, many classical schemes of teleportation has been proposed, such as controlled teleportation [16–17], probabilistic teleportation [18–19] and bidirectional teleportation [20–21]. Karlsson and Bourennane proposed the controlled teleportation protocol by using the GHz state as the quantum channel in 1998 [16–17], Li et al. gave the protocol of probabilistic teleportation and entanglement matching method by introducing an auxiliary qubit [18–19]. Recently, a scheme of bidirectional quantum teleportation is proposed by employing a special quantum state as the quantum channel [20–21]. Selecting different quantum channels will get different transmission schemes, so choosing a proper quantum channel is an essential question in teleportation. Recently, a tensor representation method in quantum teleportation is proposed [22–24], it can help us to avoid tedious equation and complex computations and make the Received date: 08-11-2013 Corresponding author: HU Yang, E-mail: [email protected] DOI: 10.1016/S1005-8885(13)60212-5

teleportation processing succincter. Importantly, it shows the tight connection between Bell basis measurement matrix T α ∈ C 2 and the CPM R. This paper derives an explicit CPM criterion based on the tensor representation method. This criterion can be used to judge whether a quantum state can be employed as the quantum channel or not in teleportation and bidirectional teleportation. According to this criterion, one can select a variety of apposite quantum channels, which can provide more flexible choices for the experimenters.

2 Quantum channel satisfy conditions in teleportation Suppose the sender Alice has a general unknown one-qubit state ϕ 1 , and she wants to sent it to the receiver Bob by adopting the two-qubit state ϕ quantum channel. The forms of ϕ

1

and ϕ

23 23

expressed as fllows ϕ 1 = X i i = X 0 0 + X1 1

ϕ

23

=R

jk

jk = R

00

00 + R

as the can be (1)

01

10

11

01 + R 10 + R 11 ( 2)

The total state of the system before the Bell basis measurement is ϕ tot = ϕ 1 ⊗ ϕ 23 = X i R jk i 1 jk 23 (3)

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where each index take value 0 or 1 and it implicit summation when an index is repeated. Unknown coefficients satisfy X i X i∗ = 1 , R jk R∗jk = 1 and R 00 R11 ≠ 01

10

R R . Alice wants to perform Bell basis measurement ψ ijα = α on ϕ tot , where α = 1, 2,3 or 4 and the Bell basis

1

ψ ijα

1

ψ ij1,2 =

are

2

( 00

± 11 ) , ψ ij3,4 =

( 01 ± 10 ) . As the Bell basis is orthogonal with each

2 other, we can express the total state ϕ

tot

by using the

α

Bell basis ψ ij , let T denote the transformation matrix between the computation basis ij

and Bell basis ψ ijα ,

then T is as follows: ⎛ T001 T002 T003 T004 ⎞ ⎛1 1 ⎜ 1 ⎜ 2 3 4 ⎟ T T01 T01 T01 ⎟ 1 ⎜ 0 0 T = ⎜ 011 = ⎜T T102 T103 T104 ⎟ 2 ⎜0 0 ⎜⎜ 101 ⎟ ⎜ 2 3 4 ⎟ ⎝ 1 −1 ⎝ T11 T11 T11 T11 ⎠ The tensor representation of the total the Bell state is ϕ tot = X i R jk i

1

jk

23

0 0⎞ ⎟ 1 1⎟ 1 −1⎟ ⎟ 0 0⎠ state ϕ

= X i R jk Tijα α k = X iσ i(

(4)

tot

α )k

under

α k

2013

invertible but not unitary, Bob can also realize a successful teleportation with p < 1 probability. If the matrix σ α is not invertible, Bob can never retrieve the state. Thus, from the above analysis, one can get the necessary and sufficient condition for realizing a non-zero probability of teleportation: the determinant of transformation matrix σ α is nonzero, i.e.

det ( R ) ≠ 0, ∀α , det (T α ) ≠ 0

(8)

Furthermore, one can write the necessary and sufficient condition for perfect teleportation: σ α is unitary, this condition demands that all of its sub-matrix must be unitary, i.e.

(T )

α −1

= (T α ) and R −1 = R +1 +

(9)

Considering standard Bell basis measurement, the measurement matrix 2T α is always unitary, so invertibility and unitarity of the CPM R determines whether a quantum state can be employed as the quantum channel or not in teleportation. This CPM criterion can tell us how to choose a quantum channel in teleportation, if the CPM R of the selected channel is unitary, one can realize a perfect teleportation, if CPM R of the selected channel is invertible, one can realize a probabilistic teleportation.

(5) where

⎛ σ α 0 σ 1α 0 ⎞ = σ i(α ) k = R iTijα , σ α = RT α = ⎜ 0α 1 α1 ⎟ ⎝ σ 0 σ1 ⎠ ⎛ R 00 R10 ⎞ ⎛ T00α T10α ⎞ (6) ⎟ ⎜ 01 ⎟⎜ R11 ⎠⎝ T10α T11α ⎠ ⎝R Obviously, T α ∈ C 2 is a 2 × 2 matrix ∈ C 2 , which is composed of the entries of the matrix T in Eq. (4). T α is called the measurement matrix, R is called the CPM, and σ α is the transformation matrix. α

After Alice performs the Bell basis measurement ψ ij on ϕ

Suppose Alice wants to teleport an unknown state ϕ

(7)

B

Obviously, as long as there exists the inverse matrix of by

σ α , Bob can obtain the teleported state Ri i

(σ )

α −1

on state

ϕα

B

. If the matrix σ α is

unitary for all α , Bob can always perfectly retrieve the state with probability p = 1 . If one of the matrices σ α is

A

to Bob, meanwhile, Bob wants to teleport an unknown state ϕ B to Alice. The forms of ϕ A and ϕ B are

ϕ

A

= xi i , ϕ i

B

= y j j ; i, j ∈ {0,1} j

∗ i

(10)

∗ j

where x x = 1 and y y = 1 . We choose a four-qubit state as the quantum channel and it can be written as ϕ A ,B ,B ,A = Rlmst lmst = R 0000 0000 + 1

1

2

2

R 0001 0001 +  + R1110 1110 +

, the total state will collapse to Bob and change tot

into the following form ϕ α = X iσ i(α ) k k

acting

3 Quantum channel satisfy conditions in bidirectional teleportation

R1111 1111 , l, m, s, t ∈ {0,1} where R

lmst

∗ lmst

R

(11)

= 1 , Alice owns particles (A1, A2) and

Bob owns particles (B1, B2). The four-qubit quantum channel has two kinds of forms: the single channel composed of a genuine quantum entangled state [8] as is illustrated in Fig. 1(a), and the double channel composed of two single channel as displayed in Figs. 1(b) and (c).

Supplement 2

HU Yang, et al. / Efficient methods for choosing quantum channel in teleportation and...

It is known that quantum state will collapse after it is measured. Seen in the Fig. 1(a), after Alice and Bob perform the Bell basis measurement on their own particles (A, A1) and (B, B1), the state of particle A and B can probably collapse to particle B2 and particle A2 respectively, so the single channel can achieve the bidirectional teleportation. For the bidirectional teleportation via the double channel, if one chooses the double channel as displayed in Fig. 1(b), it can not realize the bidirectional teleportation after Alice and Bob perform the Bell basis measurement on their own particles (A, A1) and (B, B1) individually; if one chooses the double channel as displayed in Fig. 1(c), after Alice and Bob perform the Bell basis measurement on their own particles (A, A1) and (B, B1) individually, although it can realize two state interchange, they are two disrelated and one-way teleportation. In order to realize bidirectional teleportation, we can only choose the single quantum channel which is composed of a genuine quantum entangled state [8]. Below, we will discuss the single channel satisfying conditions for bidirectional teleportation.

⎛ T α T10α ⎞ ⎫ T α = ⎜ 00α α ⎟ ⎪ ⎝ T01 T11 ⎠ ⎪ ⎪ ⎛ T β T10β ⎞ ⎬ T β = ⎜ 00β ⎟ ⎪ β ⎝ T01 T11 ⎠ ⎪ αβ = R lmst Tilα T jmβ ⎪⎭ σ iljm

103

(15)

After Alice and Bob perform the Bell basis measurement on their own particles (A, A1) and (B, B1), the total state will collapse to Alice’s particle A2 and Bob’s particle B2 which has the following form: Ψ αβ = x i y jσ αβ s B t A (16) T

2

2

and the expression of the transformation matrix σ αβ is

σ αβ = R (T α ⊗ T β )

(17)

Here, the CPM R is a 4 × 4 complex matrix, and T , T β are 2 × 2 complex matrix. α

To achieve perfectly bidirectional teleportation between Alice and Bob, the transformation matrix σ αβ must be a unitary matrix first. And then, Alice and Bob should perform a unitary transformation on their own particles A2 and B2 respectively to ensure that their quantum state interchanged perfectly. So the matrix R must be expressed as a direct product form of two matrices R1 and R2, which are 2 × 2 unitary matrix. In this case, the transformation matrix σ αβ must be written as

σ αβ = R (T α ⊗ T β ) = ( R1 ⊗ R2 ) (T α ⊗ T β ) =

Fig. 1 The quantum channel of bidirectional teleportation: he single channel (a); the double channel (b) and (c). The black line represents quantum entanglement channel, the short black dot curve represents a Bell basis measurement

1

A

yj j

B

1

2

2

R lmst lmst

A1 ,B1 ,B2 ,A 2

(12)

Denoting T as the transformation matrix between the computational basis and the Bell basis, one can get Ψ T = x i y j R lmst Tilα T jmβ α β st B A = 2

αβ xi y jσ iljm α β st

B2 A 2

2

(α , β ∈ 1, 2,3, 4)

(13)

Here

R lmst

⎛ R 0000 ⎜ 0001 R = ⎜ 0010 ⎜R ⎜⎜ 0011 ⎝R

R 0100 R 0101

R1000 R1001

R 0110 R 0111

R1010 R1011

R1100 ⎞ ⎟ R1101 ⎟ R1110 ⎟ ⎟ R1111 ⎟⎠

1

α

2

β

α

σβ

So the collapsed state Ψ αβ

Ψ αβ

T

= x i y jσ α σ β s

B2

t

T

(18) can be rewritten as (19)

A2

Under the above circumstances, the matrix 2T α is unitary. If the two matrices R1 and R2 are also unitary, after Alice and Bob performing appropriate unitary

The total state of the system is Ψ T = ϕ A ⊗ ϕ B ⊗ ϕ A ,B ,B ,A =

xi i

(RT )⊗(R T ) = σ

(14)

transformations state Ψ αβ

T

(σ )

α −1

and

(σ ) β

−1

on the collapsed

respectively, the bidirectional teleportation

will be realized. To sum up, we find a criterion which can be used to determine whether a quantum state can be used as the quantum channel of bidirectional teleportation or not, in other words, the quantum channel for realizing bidirectional teleportation must satisfy the following necessary condition: (1) The quantum channel must be a single channel which is composed of a genuine quantum entangled state. (2) The CPM must be expressed as a direct product

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The Journal of China Universities of Posts and Telecommunications

form of two unitary matrices. If one wants to realize a probabilistic bidirectional teleportation, the CPM must can be expressed as a direct product form of two invertible matrices.

4 Conclusions How to choose the quantum channel is very important for teleportation. In this paper, we presented an explicit CPM criterion which can be used to judge whether a quantum state can be employed as the quantum channel or not in teleportation and bidirectional teleportation. Firstly, for unidirectional teleportation, the invertibility of the CPM R determines whether a quantum state can be employed as quantum channel or not in teleportation. Secondly, for bidirectional teleportation, if the CPM R of the quantum channel of a genuine quantum entangled state can be decomposed into the direct product forms of two unitary matrices U1 and U2, one can realize the bidirectional teleportation perfectly, if the CPM R of the quantum channel can be decomposed into the direct product forms of two invertible matrices R1 and R2, one can realize a probabilistic bidirectional teleportation. Moreover, one gets general methods of constructing quantum channel by means of the CPM criterion in teleportation and bidirectional teleportation. Acknowledgements This work was supported by the National Natural Science Foundation of China (10974247, 11175248), and the Scientific Research Program of Education Department of Shanxi Provincial Government (12JK0992).

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