Invertible entanglement matching method in probabilitic teleportation

Invertible entanglement matching method in probabilitic teleportation

The Journal of China Universities of Posts and Telecommunications December 2013, 20(Suppl. 2): 113–116 www.sciencedirect.com/science/journal/10058885 ...

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The Journal of China Universities of Posts and Telecommunications December 2013, 20(Suppl. 2): 113–116 www.sciencedirect.com/science/journal/10058885

http://jcupt.xsw.bupt.cn

Invertible entanglement matching method in probabilitic teleportation ZHAO Mei-xia () School of Science, Xi’an University of Posts and Telecommunications, Xi’an 710121, China

Abstract This paper proposes an invertible entanglement matching method to realize an optimal probabilitic teleportation by introducing an auxiliary quantum channel. This requires the channel parameter matrix (CPM) of the auxiliary quantum channel to be equal to the invertible matrix of CPM of the transmission channel. By using this method, the probability of successful teleportation is more than 1/2. Keywords

probabilitic teleportation, entanglement matching, CPM, tansformation matrix

1 Introduction Quantum entanglement plays a key role in quantum information processing, and entanglement state can be prefabricated, evolved, controlled, exchanged, and matched by means of some suitable operation [1–3]. Since the seminal teleportation protocol of Bennett et al. [4], the investigation of teleportation has been a hot topic and there has been a great development in theory and experiments [5–11]. Until now the research of teleportation has been developed to many branches, such as directly and network controlled teleportation [12–16], prefect and probabilistic teleportation [17–19] and so on. In fact, one of the main tasks of teleportation is how to construct usable quantum channels, different channel will yield different result, some channels can realize perfect teleportation, and some channels can only realize probabilitic teleportation. Because of the unavoidable environment influence, the quantum channels are not always maximally entangled. Therefore, the probabilitic teleportation [20–22] has been widely discussed nowadays. Recently, a tensor expression method of teleportation has been proposed [23–25]. Using this tensor expression, Received date: 28-10-2013 Corresponding author: ZHAO Mei-xia, E-mail: [email protected] DOI: 10.1016/S1005-8885(13)60210-1

one has been proposed a necessary and sufficient condition for realizing perfect teleportation or successful teleportation. Based on the Bell Basis measurement, one can conclude that if the CPM X is unitary, one can always realize a perfect teleportation (i.e, the whole transmissive probability p = 1 ). If the CPM X is invertible but not unitary, one can only realize a probabilitic teleportation (i.e, p < 1 ). For the probabilitic teleportaton, Li et al. [17] firstly presented a protocol of probabilitic teleportation by introducing an auxiliary qubit state | 0〉 A and operating an unitary transformation in the Bob’s state to realize a successful teleportation. The whole probability of successful teleportation is 2b 2 < 1/ 2 in probabilitic teleportation. In this paper, we introduce an auxiliary quantum channels instead of an auxiliary qubit state | 0〉 A in Bob’s hand, demanding ACPM Y to be equal to the invertible matrix of CPM X, i.e, Y = X −1 .That is called invertible entanglement matching. By using this method, the whole probability of successful teleportation is more than 1/2 in probabilitic teleportation. Moreover, compared with the method of introducing an auxiliary qubit [17], one essential advantage of our method need not make unitary evolution at Bob’s site and only needs Bob’s complete Bob’s Bell Basis measurement to realize an optimal probabilitic teleportation.

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2 Invertible entanglement matching method in teleportation

transformation matrix σ 1 = X σ 0 , σ 2 = X σ z , σ 3 = X σ x ,

Suppose Alice wants to send an unknown one-qubit state | ϕ 〉1 to Bob via a general two-qubit entangled state

After Alice’s measurement, the total state will collapse to Bob, which is of the following form 1 〈φ Aλ | ψ 〉 B = R iσ i( λ ) k | k 〉 (6) 2 Obviously, if the transformation matrix σ α is unitary for all λ , Bob can always perfectly retrieve the original state by acting (σ λ ) −1 on | ψ 〉 B . Regardless of the

| ϕ 〉 2,3 as the quantum channel: | ϕ 〉1 = R i | i〉 = R 0 | 0〉 + R1 | 1〉 = α | 0〉 + β | 1〉

| ϕ 〉 2,3 =

1

1

X jk | jk 〉 =

( X 00 | 00〉 + X 01 | 01〉 +

2 2 X 10 | 10〉 + X 11 | 11〉)

(2) jk

where R R = α + β = 1 , X X i

* i

2

2

(1)

* jk

= 2 and the indexes

( i, j , k ) are 0 or 1. In the method of tensor analysis, a repeated index summation. If Alice adopts Bell Basis measurement (BM) φijλ , here

λ = {1, 2,3, 4} and φij2 = (1 2 ) (| 00〉− | 11〉), φij1 = (1 2 ) ⋅

and σ 4 = −iX σ y [23–25].

unknown state | ϕ 〉1 , the four Bell basis measurement outcomes are equally, each occurs with probability 1/4, this is just the square of the coefficient in Eqs. (4) and (6). After Alice finishing all the measurement and Bob making the corresponding operation, Bob can perfectly retrieve the original state with probability p = 1 . If the matrix σ λ is

φij4 = (1 2 ) ⋅

invertible but not unitary, Bob can only realize a successful teleportation with nonzero probability p < 1 .

(| 01〉+ | 10〉) . One can express the system state with Bell

When considering the probabilitic teleportaton, after Alice’s Bell Basis measurement, Bob introduces an auxiliary quantum state | ϕ 〉 4,5 for the state 〈φ Aλ | ψ 〉 B

(| 00〉+ | 11〉) ,

φij3 = (1

2 ) (| 01〉+ | 10〉) ,

Basis. The transformation relation L between the standard Bell Basis φijλ and the two-qubit entangled state Basis

(| 00〉,| 01〉,| 10〉,| 11〉) is shown as follows

⎛ φij1 ⎞ ⎛ L100 ⎜ 2⎟ ⎜ 2 ⎜ φij ⎟ = 1 ⎜ L00 3 ⎜φ 3 ⎟ 2 ⎜ L00 ⎜ ij4 ⎟ ⎜ ⎜ L4 ⎜φ ⎟ ⎝ 00 ⎝ ij ⎠

L101

L110

2 01 3 01 4 01

2 10 3 10 4 10

L L L

L L L

L111 ⎞ ⎛ 2 ⎟⎜ L11 ⎟⎜ 3 ⎟⎜ L11 ⎟⎜ 4 ⎟⎜ L11 ⎠⎝

00 01 10 11

⎞ ⎟ ⎟= ⎟ ⎟⎟ ⎠

0 0 1 ⎞ ⎛ 00 ⎞ ⎟ ⎟⎜ 0 0 −1⎟ ⎜ 01 ⎟ (3) 1 1 0 ⎟ ⎜ 10 ⎟ ⎟ ⎟⎜ 1 −1 0 ⎠ ⎜⎝ 11 ⎟⎠ Under the Bell Basis φijλ , the total state of the system ⎛1 ⎜ 1 ⎜1 2 ⎜0 ⎜ ⎝0

can be rewritten as: 1 i jk λ 1 1 i jk | ψ 〉 tot = 2 Lλij | λ k 〉 = R X Lij | λ k 〉 = RX 2 2 2 1 i jk λ 1 i (λ )k R X Tij | λ k 〉 = R σ i | λ k 〉 (4) 2 2 where σ i( λ ) k = X jk 2 Lλij = X jk Tijα is the element of

⎛ σ 0λ 0 σ 1λ 0 ⎞ ⎛ X 00 X 10 ⎞ ⎛ T00λ T10λ ⎞ = ⎜ 01 (5) ⎟ ⎟⎜ λ1 λ1 ⎟ X 11 ⎠ ⎝ T01λ T11λ ⎠ ⎝ σ 0 σ1 ⎠ ⎝ X Under the condition of BM, one can easily obtain T 1 = σ 0 , T 2 = σ z , T 3 = σ x and T 4 = −iσ y = σ xσ z ,

σ λ = XT λ = ⎜

σ 0 = I is 2 × 2 identity matrix and σ x , y , z are the Pauli matrices. From these equations, one can get the

which is expressed as follows 1 lm 1 | ϕ 〉 4,5 = (Y 00 | 00〉 + Y 01 | 01〉 + Y | lm〉 = 2 2 Y 10 | 10〉 + Y 11 | 11〉)

(7)

Now Bob’s state can be expressed as 1 〈φ Aλ | ψ μ 〉 BY = Riσ i( λ ) k Y lm | lmk 〉 (8) 2 2 Under Bob’s Bell Basis φklμ , where μ = {1, 2,3, 4} , the state 〈φ Aλ | ψ μ 〉 BY can be rewritten as

〈φ Aλ | ψ μ 〉 BY =

1

Riσ i( λ ) k Y lm Lμkl | μ m〉 =

2 2 1 i ( λ ) k lm μ R σ i Y Tkl | μ m〉 = 4 1 i (λ ) k (μ ) m 1 R σ i σ k | μ m〉 = R iσ i( λμ ) m | μ m〉 (9) 4 4

where σ k( μ )m = Y lmT σ klμ

⎛ σ 0μ 0 σ 1μ 0 ⎞ ⎛ Y 00 Y 10 ⎞ ⎛ T00μ = ⎜ 01 ⎟⎜ μ1 μ1 ⎟ Y 11 ⎠ ⎝ T01μ ⎝ σ 0 σ1 ⎠ ⎝ Y = σ λ ⋅ σ μ = ( XT λ )(YT μ )

σ μ = YT μ = ⎜ σ λμ

T10μ ⎞ ⎟ T11μ ⎠

(10) (11)

After Bob’s Bell Basis measurement, Bob’s state is changed into 1 1 〈φBμ 〈φ Aλ | ψ μ 〉 BY = R iσ i( λ ) k σ k( μ ) m | m〉 = R iσ i( λμ ) m | m〉 (12) 4 4

Supplement 2

ZHAO Mei-xia / Invertible entanglement matching method in probabilitic teleportation

In this case, if the matrix σ λμ is unitary for all λ and μ , Bob can always perfectly retrieve the original state by acting (σ λμ ) −1 on 〈φBμ 〈φ Aλ | ψ μ 〉 BY . If the matrix σ λμ is

σ 1,1 = XT 1YT 1 = σ 0 σ

1,3

= XT YT = σ x

σ

2,1

= XT 2YT 1 = σ z

1

3

invertible but not unitary, Bob can realize a successful teleportation with probability p < 1 . If the matrix σ λμ is

σ 2,3 = XT 2YT 3 = iσ y

irreversible, one can never realize this teleportation. Now we try to analyze invertibility and unitarity of the transformation matrix σ λμ for successful teleportation, the invertible matrix (σ λμ ) −1 of the transformation matrix

σ 3,1 = XT 3YT 1 =

is (σ λμ )−1 = [( XT λ )(YT μ )]−1 = (T μ )−1Y −1 (T λ ) −1 X −1

(13)

Fortunately, T = σ 0 , then 1

(σ 1μ ) −1 = (T μ )−1Y −1 (σ 0 )−1 X −1 = (T μ ) −1Y −1 X −1 If we let Y = X

−1

, then (σ

λμ −1

)

(14)

have four unitary

transformation matrices,which are (σ 1μ ) −1 = (T μ )−1Y −1 X −1 = (T μ ) −1

(15)

so one can always realize a successful teleportation by acting (σ λμ ) −1 on 〈φBμ 〈φ Aλ | ψ μ 〉 BY , this is the reason that we introduce the invertible entanglement matching to the channel of probabilitic teleportaton. In order to state our method easily, we consider two simple states as the transmission channel | ϕ 〉 2,3 and the auxiliary channel | ϕ 〉 4,5 , as is demonstrated in Fig. 1.

Fig. 1 Teleportation channel of the entanglement matching between channel X and Y

The two states are as follows 1 1 ⎫ X jk | jk 〉 = | ϕ 〉 2,3 = (a | 00〉 + d | 11〉) ⎪ 2 2 ⎪ ⎬ 1 lm 1 Y | lm〉 = | ϕ 〉 4,5 = (Y 00 | 00〉 + Y 11 | 11〉) ⎪ ⎪⎭ 2 2

(16)

when the two channels satisfy our entanglement matching condition, i.e., Y = X −1 , then ⎛ Y 00 0 ⎞ 1 ⎛ d 0 ⎞ = (17) Y =⎜ ⎜ ⎟ 11 ⎟ ⎝ 0 Y ⎠ ad ⎝ 0 a ⎠ Accomplish such entanglement matching, the more explicitly expression of the transformation matrices σ λμ are as follows:

115

σ 1,2 = XT 1YT 2 = σ z

⎫ ⎪ σ = XT YT = −iσ y ⎪ ⎬ (18) σ 2,2 = XT 2YT 2 = σ 0 ⎪ σ 2,4 = XT 2YT 4 = −σ x ⎪⎭ 1,4

1

4

⎫ ⎪ ⎪ 2 ⎪ 1 ⎛ 0 −a ⎞ ⎪ σ 3,2 = XT 3YT 2 = ⎜ ⎟ ad ⎝ d 2 0 ⎠⎪ ⎪ 0 ⎞⎪ 1 ⎛ a2 3,3 3 3 σ = XT YT = ⎜ ⎟⎪ ad ⎝ 0 −d 2 ⎠ ⎪ ⎪ 1 ⎛ a2 0 ⎞ ⎪ σ 3,4 = XT 3YT 4 = ⎜ ⎟ ad ⎝ 0 d 2 ⎠ ⎪⎪ (19) ⎬ 1 ⎛ 0 a2 ⎞ ⎪ 4,1 4 1 σ = XT YT = ⎜ ⎟ ad ⎝ d 2 0 ⎠ ⎪ ⎪ 1 ⎛ 0 −a 2 ⎞ ⎪ 4,2 4 2 σ = XT YT = ⎜ ⎟⎪ ad ⎝ d 2 0 ⎠⎪ ⎪ 1 ⎛ −a 2 0 ⎞ ⎪ 4,3 4 3 σ = XT YT = ⎜ ⎟ ad ⎝ 0 d 2 ⎠ ⎪ ⎪ 2 ⎪ ⎛ ⎞ a 0 1 σ 4,4 = XT 4YT 4 = − ⎜ ⎪ 2⎟ ad ⎝ 0 d ⎠ ⎪⎭ After Alice’s Bell state measurement φ Aλ , there are 16 1 ⎛ 0 ⎜ ad ⎝ d 2

a2 ⎞ ⎟ 0⎠

kinds of collapsed state occurring at Bob’s site, they are expressed as follows 1 ⎫ 〈φ 1A | ψ 1 〉 BY = φB1 [+α | 0〉 + β | 1〉] ⎪ 4 ⎪ 1 2 1 2 〈φ A | ψ 〉 BY = φB [+α | 0〉 − β | 1〉] ⎪ ⎪ 4 ⎪ 1 〈φ 1A | ψ 3 〉 BY = φB3 [+ β | 0〉 + α | 1〉] ⎪ ⎪ 4 ⎪ 1 〈φ 1A | ψ 4 〉 BY = φB4 [− β | 0〉 + α | 1〉] ⎪ ⎪ 4 (20) ⎬ 1 2 1 1 ⎪ 〈φ A | ψ 〉 BY = φB [+α | 0〉 − β | 1〉] ⎪ 4 ⎪ 1 〈φ A2 | ψ 2 〉 BY = φB2 [+α | 0〉 + β | 1〉]⎪ 4 ⎪ ⎪ 1 3 2 3 〈φ A | ψ 〉 BY = φB [− β | 0〉 + α | 1〉] ⎪ 4 ⎪ ⎪ 1 4 2 4 〈φ A | ψ 〉 BY = φB [− β | 0〉 − α | 1〉] ⎪ ⎭ 4

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1 ⎡ a d ⎤⎫ 〈φ A3 | ψ 1 〉 BY = φB1 ⎢ + β | 0〉 + α | 1〉 ⎥ ⎪ 4 ⎣ d a ⎦⎪ 1 ⎡ a d ⎤⎪ 〈φ A3 | ψ 2 〉 BY = φB2 ⎢ − β | 0〉 + α | 1〉 ⎥ ⎪ 4 ⎣ d a ⎦⎪ 1 3⎡ a d ⎤⎪ 3 3 〈φ A | ψ 〉 BY = φB ⎢ + α | 0〉 − β | 1〉 ⎥ ⎪ 4 ⎣ d a ⎦⎪ 1 a d ⎡ ⎤⎪ 〈φ A3 | ψ 4 〉 BY = φB4 ⎢ + α | 0〉 + β | 1〉 ⎥ ⎪ 4 ⎣ d a ⎦⎪ (21) ⎬ 1 a d ⎤⎪ 4 1 1 ⎡ 〈φ A | ψ 〉 BY = φB ⎢ − β | 0〉 + α | 1〉 ⎥ 4 ⎣ d a ⎦⎪ ⎪ 1 ⎡ a d ⎤ 〈φ A4 | ψ 2 〉 BY = φB2 ⎢ − β | 0〉 + α | 1〉 ⎥ ⎪ 4 ⎣ d a ⎦⎪ ⎪ 1 ⎡ a d ⎤ 〈φ A4 | ψ 3 〉 BY = φB3 ⎢ − α | 0〉 + β | 1〉 ⎥ ⎪ 4 ⎣ d a ⎦⎪ ⎪ 1 ⎡ a d ⎤ 〈φ A4 | ψ 4 〉 BY = φB4 ⎢ − α | 0〉 − β | 1〉 ⎥ ⎪ 4 ⎣ d a ⎦ ⎪⎭ From Eqs. (18)–(21), one can see that after Alice’s Bell basis measurement φ Aλ , Bob’s each collapsed state occurring with the probability PA1 = (1 4 ) = 1 16 , and 2

after Bob’s Bell Basis measurement φBμ , one can obtain the original state from 8 kinds of Bob’s collapsed states. So the whole probability of successful teleportation is P = 8 ×1 16 = 1 2 .

3 Conclusions In this paper, an invertible entanglement matching method is proposed by introducing an auxiliary quantum channel, using this method realize an optimal probabilitic teleportation. From Eqs. (18) and (20), Alice’s Bell state measurement φ 1A and φ A2 and Bob’s Bell Basis measurement φBμ , we obtain that the probability of successful teleportation is 1/2. From Eqs. (19) and (21), one can see that after Alice’s Bell state measurement φ A3 and φ A4 and Bob’s Bell Basis measurement φ Aμ , because

(σ λμ ) (λ = 3, 4, μ = 1, 2,3, 4) is invertible but not unitary, Bob can not directly obtain the original state by (σ λμ ) −1 acting on 〈φBμ 〈φ A3 | ψ μ 〉 BY and 〈φBμ 〈φ A4 | ψ μ 〉 BY . Therefore, one again can introduce an auxiliary quantum state | ϕ 〉6,7 for the states 〈φBμ 〈φ A3 | ψ μ 〉 BY

and 〈φBμ 〈φ A4 | ψ μ 〉 BY

on

Bob’s hand. Once again one obtaind that the probability of successful teleportation is 1/4, the whole probability of successful teleportation p = 1/ 2 + 1/ 4 > 1/ 2 .

2013

Acknowledgements This work was supported by the National Natural Science Foundation of China (10974247, 11175248), the Scientific Research Program of Education Department of Shaanxi Provincial Government (12JK0992).

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