Quantum Teleportation Channel Capacity Coding

2017 2nd _{International Conference on Computer Science and Technology (CST 2017)} ISBN: 978-1-60595-461-5

Quantum Teleportation Channel Capacity Coding

Dong-fen LI, Rui-jin WANG

a*

_{,Hao LUO, Edward Baagyere,}

Feng-li ZHANG, Xin-yun WANG and Xue-qin CHENG

School of Information and Software Engineering, University of Electronic Science and Technology of China, Chengdu, Sichuan 610054, China

a_{[email protected]}

*Corresponding author

Keywords: Quantum teleportation, Multiple degrees of freedom, Quantum capacity, Graph-state basis, Concatenated code.

Abstract. In this paper, we use the idea of quantum Graphic Cascade Coding, information security and quantum random coding to construct an immune noise channel, by using the normal state of pattern tree chart and forest chart, to make large diagonal matrices or blocks, in order to perform quantum coding operations under multiple degrees of freedom. Through calculation and analysis of different concatenated code channel capacity, we get the formula of different noise channels in multi-channel coherence information under the coder multiple degrees of freedom. By this, we can quickly calculate the coherence information of various concatenated codes in the channel, the approximation of the channel capacity and the noise margin of the channel transmission quantum, and analyze the different anti-noise performance of different cascaded codes in different parameter channels. And, thus, we can obtain the security regions where different noise channels can transmit quantum information.

Introduction

In recent years, we witnessed the eavesdropping scandal and user privacy disclosure event that exacerbated the information security concerns within the community of users of network services [1]. The "Prism door" and "Jindong Mall" incidents leaked a large number of sensitive information, events that point to the fact that more need to be done at China's national level in order to manage some of these information security crises [2]. However, the existing classical communication encryption method is not safe. These classical cryptographic algorithms are mostly constructed using large prime numbers couple with the use of huge amount of computing power in order to transmit secret information in a secure manner by protecting the privacy and integrity of sensitive information. Thus, this kind of security is depended on the computational complexity of the cryptographic algorithm used [3]. For example, the RSA algorithm, whose security is reduced to large integer factorization, is an NP problem. However, according to Moore's Law, the computing power of the processor is exponentially increasing, and thus, under the attack of malware and Trojan, it makes the cracking of such classical encryption algorithms easier, and which often lead to the leakage of sensitive information.

carry and efficient parallel processing capabilities, is therefore hard to crack the password using these technology, which is as strong as "spear".

Quantum teleportation (QT) is a technique of quantum entanglement distribution and quantum measurement. It transmits an unknown quantum state to a distant place, by realizing the spatial transfer of the quantum state without the transport of the physical carrier, it has the advantages of high reliability, low communication complexity and resource saving. Therefore, the QT solves traditional communication encryption algorithm problems of been easily cracked and that of monitoring of the information transmission process. These properties of the QT technique are the "shield" that guarantee the security of network information.

QT is an essential element of quantum information processing and also an important element of practical quantum technology. It plays an important role in large-scale quantum computing, long-range quantum communication and practical quantum computing networks. The core resource of QT is quantum entanglement, that is, when the quantum state superposition principle is reflected in the multi-particle or multi-degree of freedom system, there will be a unique phenomenon of quantum mechanics—quantum entanglement [5][6]. When the physical system and the noise environment are coupled to each other, it will accelerate the increase of quantum decoherence. At the same time, quantum entanglement will also occur leading to entanglement attenuation, and even entanglement sudden death problem [7] [8]. The channel capacity problem seriously affects the quality of quantum communication under noise environment and it is quantum teleportation under single degree of freedom. The reason is that in many applications, people often do not need to do more than the degree of freedom of quantum positioning, but only in a state to operate. How to improve low capacity communication under single degree of freedom is what these works seeks to address.

Related Work

In 2008, Winter et al. proposed the quantum capacity of symmetric quantum channels and private capacity, and gives the corresponding theorem and expression [9]. In the literature [10] the advantage of the asymmetric channel is used, but did not take into account the practical problems in the quantum channel which are the difficulties involve in maintaining a maximum entanglement state and the consuming of large entangled resources and other issues. In 2013, Kesting calculated the noise margins of different quantum codes under the Pauli environment, in the depolarized channel; they get the optimal encoding of different quantum concatenated codes [11]. In the same year, Chen et al. used information theory security and quantum random coding theory, secure capacity coding model based on noise channel to prove the message authentication capacity under strong security conditions [12]. From 2014 to 2015, Li et al. use the properties of hyper-entanglement state which is easy to prepare, easy to measure and easy to implement, to construct entangled exchange of quantum cryptography channel, and proposed a highly efficient ultra-dense coding method based on hyper-entanglement, effectively improving the channel utilization and capacity in quantum communication [13] [14] [15]. Afterwards, scholars spread and form the CSS construction theorem of quantum coding, it becomes an effective way of classical coding to construct quantum coding, and analyzes the quantum channel capacity in entanglement assistance, and which been widely used [16-17].

construct the unitary transformation matrix, purify the quantum channel with a certain probability to achieve dense coding and quantum error correction code. And thus, the receiver cannot completely distinguish the transmitted quantum state, the channel capacity also does not reach the theoretical value, and it cannot contain the needs of long-distance quantum teleportation.

The Design of Cascade Coding Schemes for Quantum State Under Multiple Degrees of Freedom

In this paper, we use the idea of quantum Graphic Cascade Coding, information theory security and quantum random coding theory to construct an immune noise channel by using the normal state of the pattern of tree chart and forest chart to make large diagonalzed matrices or blocks in order to performs quantum coding operations under multiple degrees of freedom. Through calculation and analysis of different concatenated code channel capacity, we can get the formula of different noise channel in multi-channel coherence information under the coder multiple degrees of freedom. We can then quickly calculate the coherence information of various concatenated codes in the channel, the approximation of the channel capacity and the noise margin of the channel transmission quantum, and analyzes the different anti-noise performance of different cascaded codes in different parameter channels, and obtains the security regions where different noise channels can transmit quantum information.

The coding properties are as follows:

Suppose N is the channel capacity, ρis input state, channel coherence information

is ( , )I ρ N =S N[ ( )]ρ −S I[ A⊗N(ψAP)], ψAP is the purified state ofρ , IA is the

identity operator that is acting on the auxiliary systemA, quantum single shot capacity

is Q N₁( ) max ( , )I N ρ

ρ

= . Suppose _ρn is the total input state of the input channel,

n

N⊗ is the function of n parallel channels N , the quantum coding capacity

is ( ) lim1 ₁( n)

n

Q N Q N

n ⊗

→∞

= . For quantum bit channels, suppose ρc_is

n bit code quantum

state, the output state of quantum channel isσc _{= N}⊗n(ρc)_{. Suppose}σc _=N⊗n₍ρc_{) is} the purified state ofρc_{, then} c n₍ AC AC₎

A

I N

ρ _{= ⊗} ⊗ ψ ψ _{, the average information for}

each channel is CN 1[ ( C) ( AC)]

I S S

n σ σ

= − .

Then cascade coding schemes for quantum state under multiple degrees of freedom can be designed as follows:

Step 1: If C

k k k k

G G

ρ =



π is Figure diagonal state of figure state basis, π_k is

the probability of G_k , 0≤π_k ≤1and _k 1

k

π =



, k is a bit string of length n,

then

1, ,...,2 n

k k k k

G = G . In the kraus operator summation representation,

, , , [0,1], 1

x y z x y z

p p p f∈ f = −p −p −p is channel fidelity, Single-channel pairs of

state is C C †

aEa Ea

σ

=



η

ρ

. If the channel combined output state isσAC_{, then it can be} expressed as AC_,

sm tl

σ

, and according to orthogonal basis of chart state, we can

get ( ), ( ) ( 1)a

k a

P AC

i i j a

a E Z K

i k j j k

σ π π η

∈

⊕ ⊕ =



− .

Step 2: Suppose n₁ is Internal Code of Tree diagram, n₂ is outside code, the

concatenation code is n1×n2 , then the total input quantum state

is 1( _{00...01 00...01})

2

C _{G G G G}

ρ = + . The joint output state can be divided and the matrix can be constructed if divided, where _{e a}

a

η =



η satisfy k e a

E ∈Z K , and

0 a

a

η =



η satisfy k 0

a

E ∈Z K . η_eand η₀is eigenvalue of σAC, When the error operator

is Z Z1 2...Zi, then 

1

11...100...0 i n i

k

−

= , and ηe =ai, η0=bi, if bit string x is unchanged, The

eigenvaluesη_eand η₀are invariant, when the error operator is

1 1 2... i n

Z Z Z Z , then degree

of simplicity is

1 1

i n

C ₋ . Related information of tree diagram

is 1 1

1 1 1 0 ( , ) ( ) ( ) ( ) n n

C C AC i

n i

I

ρ

N S

σ

S

σ

C A B

− ⊗

− =

= − =



− .

Step 3: If

2

1 2 ... n

K =K ⊗K ⊗ ⊗K is stable subgroups of n₁×n₂Forest diagram, each

subgroup

2

j

K corresponds to j₂ tree, accompany set isE=Z Kk , consists of n₂ parts

of k

Z . In the tree diagram, it have l_i and s_i belongs to Z Z_{1 2}...Z_i type and

1 1 2... i n

Z Z Z Z type, it satisfy 2 1 2 0 ( ) n i i i

l s n

−

=

+ =



. The contribution value of the eigenvalue of

the joint output state is 1

1

0

o o o o i i i i i i n

l l l s s s

i i i i

i

a b c d

−

− −

=

∏

. Then sum all the elementsK, and analyze

the forest diagram, output the eigenvalues of σc_{, and then get the corresponding degree}

of degeneracy which is 2

1 1

!

2 ! ! 1

0

1 _{( )}i i n

l s i

n i i i

n C l s − + − =

Δ =

∏

and channel average coherence

information of _{2 2 2}

, 1 2

1 _{{ [ 'log ( ') log ( ) log ( )]}} CN

e e o o

l s

I

n n η η η η η η

=



− + +

   .

For different concatenated codes, when tan

θ

=P_y /P_x , take the two

[image:5.612.115.497.71.223.2]

Figure 1. tan

θ

=P_y/P_x,

θ

=0. Figure 2. tan

θ

=P_y /P_x,

π

/ 4.

Analysis and Summary

The two figures show that quantum teleportation channel capacity coding is a very effective method to analyze using the pattern base, and it can quickly calculate the coherent information of the various concatenated codes in the general bubble channel, by calculation and analysis of the noise margin of different concatenated codes in order to select the optimal encoding. Compared with the conventional method, it has been proven that the method of the scheme can significantly improve the calculation speed; the coding method of this paper is helpful to the research of quantum cryptography in the future.

Acknowledgement

This work is supported by Fundamental Research Funds for the Central Universities (ZYGX2014J051, ZYGX2014J066), Science and Technology projects in Sichuan Province (2015JY0178, 2016ZC2575, 2014GZ0109, 2015KZ002, 2015JY0030), National Natural Science Foundation of China (61472064).

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