# An error-free protocol for quantum entanglement distribution in long-distance quantum communication

**ABSTRACT** Quantum entanglement distribution is an essential part of quantum communication and computation protocols. Here, linear optic

elements are employed for the distribution of quantum entanglement over a long distance. Polarization beam splitters and wave

plates are used to realize an error-free protocol for broadcasting quantum entanglement in optical quantum communication.

This protocol can determine the maximum distance of quantum communication without decoherence. Error detection and error correction

are performed in the proposed scheme. In other words, if there is a bit flip along the quantum channel, the end stations (Alice

and Bob) can detect this state change and obtain the correct state (entangled photon) at another port. Existing general error

detection protocols are based on the quantum controlled-NOT (CNOT) or similar quantum logic operations, which are very difficult

to implement experimentally. Here we present a feasible scheme for the implementation of entanglement distribution based on

a linear optics element that does not need a quantum CNOT gate.

Keywordsquantum entanglement–quantum communication–quantum error detection–decoherence–error correction

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Page 1

An Error-Free Protocol for Quantum

Entanglement Distribution in Long Distance

Quantum Communication

Shamsolah Salemian Shahram Mohammadnejad

Nanoptronics Research Center

Electronics and Electrical Engineering Department

Iran University of Science and Technology

salemian@iust.ac.ir shahramm@iust.ac.ir

Abstract- Quantum entanglement distribution is an essential

part of quantum communication and computation

protocols. Here, linear optic elements are employed for

distribution of quantum entanglement over long distance.

Polarization beam splitters and wave plates are used for

realization of an error-free protocol for broadcasting

quantum entanglement in optical quantum communication.

This protocol can determine maximum distance of

quantum communication to transmit quantum information

without decoherence. Error detection and error correction

are performed in the proposed scheme. In other words, if

bit-flip occurs along the quantum channel, the end stations

(Alice and Bob) can detect this state changing and obtain

the correct state (entangled photon) on the another port.

Existing general error detection protocols are based on the

quantum controlled-NOT (CNOT) or similar quantum

logic operations, which are very difficult to implement

experimentally. Here we present a feasible scheme for the

implementation of entanglement distribution based on

linear optics element that does not need quantum CNOT

gate.

Keywords: quantum entanglement, quantum communication,

quantum error detection, decoherence, error correction

I. INTRODUCTION

Quantum entanglement is an important concept of

quantum physics and is the basis of most quantum

communication and computation protocols [1-11]. Each

of these protocols allows efficient communication and

computation beyond the capabilities of classical

communication, which makes it attractive as a new

emerging quantum information technology. This might

result in to construct

communication network, that the distribution of quantum

entanglement on a global scale is a central task of this

network. Up to now, only photon is suitable system for

long distance quantum communication. Other systems

such as atoms or ions are studied, but their applicability

for quantum communication schemes is presently not

feasible within the near future. Therefore photons are the

only choice for long-distance quantum communication.

One of the problems of photon-based schemes is the loss

of photons in the quantum channel. This limits the

maximum distance of single photon transmission to

about 10 km in present silica fibers [12-13]. This

a worldwide quantum

problem can be solved by subdividing the larger distance

into smaller segment over which entanglement can be

teleported. Eventually, entanglement swapping [14] is

used for transporting of entanglement over long

distances.

In this paper, an error-free protocol will be proposed to

extend the bridgeable distance of single photons and

subsequently reduce the number of quantum repeater

along the quantum communication network. Error

detection and error correction are performed in the

proposed scheme.

(

10 →

and

01 →

) along the quantum channel, the

state evolution can be detected by Alice and Bob and

corrected states are obtained on another port. Here the

linear polarization states of photons, H for ‘horizontal’

and

V for ‘vertical’, will serve as the physical

representation of logical

0

≡

H

and

1

≡

V

.

If qubit changes

bit values, with

II. ERROR MODEL FOR QUANTUM INFORMATION

Understanding the nature of the errors is first step in

protecting information against errors. In the error models

for classical communication and computation, errors

may not influence bits independently, and so the error

models would have to consider any correlation between

errors on different bits. The same is true for errors on

quantum bits, but we must consider that the quantum

alteration is a continuous process as opposed to the

classical discrete case; the encoding operation cannot

make multiple copies of arbitrary quantum states, and

the corruption of encoded quantum state cannot be

detected through the complete measurement of all the

qubits.

Errors occur on a qubit when its alteration differs from

the desired one. This difference can happen because of

inexact control over the qubits or by interaction of the

qubits with an environment. A ‘quantum channel’ is a

formal description of how qubits in a given setting are

affected by their environment. The general change of a

qubit in the state 0 interacting with an environment in

the state E will yield a superposition state of the form:

Page 2

2211

100

E

β the qubit remains in the basis

EE

ββ+→

(1)

That is, with amplitude

state 0 and the environment evolves to some state

E . With amplitude

2

state 1 and the environment evolves to some state

Similarly, when the qubit is initially in state 1 with the

environment in state E , we have

011

EEE

ββ

+→

More generally, when a qubit in a general pure state

interacts with the environment in state E , we will have

(

0

E

α β+

We can rewrite the state after the interaction as

01

1

0

2

1

0

2

1

0

2

1

0

2

Let

10

10

ααψ+=

. Then we have

(

(

4422

2

(5)

This represents the most general evolution that can

happen on a single qubit, whether or not it interacts with

an environment.

The interesting point is that a generic continuous

evolution has been rewritten in terms of a finite number

discrete transformations; with various amplitudes the

state is either unaffected, or undergoes a phase flip Z, a

bit flip X or a combination of both XZ. This is possible

because these operators form a basis for the linear

operators on a Hilbert space of a single qubit.

Specific errors can be described as special cases of the

right side of equation (5). For example, suppose we

know that the error is a ‘bit flip’, which has the effect of

the not gate X with some amplitude and leaves the qubit

unaffected (applies the identity) with possibly some

other amplitude. This would correspond to states of the

environment such that

EE

ββ=

. Equation (5) for the general evolution

thus simplifies to

EXEE

ψβψβψ+→

Single qubit errors resulting from an uncontrolled

situation leading to an inexact rotation of the qubit about

the x-axis of the Bloch sphere and will have

EcE

ββ=

for some constant c, so that the

1

1

β the qubit evolves to the basis

2

E

.

4433

(2)

)

01011022

133144

0101

1

EEE

E

αα α β α β

α β

+→+

+

(3)

( )()

(

(

(

)(

)(

)(

)

)

)

011022133144

011133

011133

012244

012244

01

1

1

1

1

EEEE

EE

EE

EE

EE

α βα β α βα β

ααββ

ααββ

ααββ

ααββ

+++

=++

+−−

+++

+−−

(4)

)()(

)(

)

))()(

4422

33113311

2

11

2

1

2

1

EE XZEEX

EEZEEE

ββψββψ

ββψββψψ

−+++

−++→

3311

EE

ββ=

and

4422

2211

(6)

2211

environment’s state factors from the qubit’s state and the

operator

XIc

22

ββ

+

is unitary. In other words,

(

22

EXIcE

⊗+→

ψββψ

Therefore the error is called coherent and will be

incoherent if the environment state does not factor out.

When

1

β

is orthogonal to

flip error model, the operator X (bit flip) is applied with

probability

p

=

2

β

and remains

probability

p

−=1

. The generic evolution of this

latter case is non-unitary.

The case of the generic evolution of a qubit can be

generalized to the situation of a larger quantum system

(e.g. a register of qubits in a quantum computer) in some

logical state ψ , interacting through some error process

with an environment initially in state E . Suppose this

process is described by a unitary operator

the joint state of the system and the environment. Then

the state of the joint system after the interaction

is

EUerrψ

. Its density matrix is

UEEU

ψψρ =

(8)

By applying trace operator on both sides of equation (8),

the following relation is obtained,

(

)(

errEE

EEUTR TR

=ψψρ

)

()

2

(7)

1E

22E

β

, in the quantum bit-

2

unaffected with

2

1 β

err

U

acting on

†

errerr

)

†

†

erri

i

i

AAU

?

=ψψ

(9)

Where,

(not including the environment). The error model is

completely described by the

For instance, the bit-flip error explained above can be

described as the interaction between a qubit and the

environment that applies the identity operator with

probability

p

−

1

and the X operator with probability p .

If the qubit is in the initial state ψ , then the state after

the error process is described by the density matrix

(

pXp

flip

ψψψψρ

+−= 1

So the

IpA

−=

0

1

(11)

iA are operators acting on the system of interest

iA .

)

X

(10)

iA describing this error model are

XpA

=

1

In the following sections, we focus on the practical

protocol for distributing entangled state and detecting the

bit flip error in single photons along the quantum

communication channel. Based on type of error, properly

selecting the linear optical element and adjustment of its

position and angle in the photon transmission path can

correct this error.

III. ERROR-FREE PROTOCOL FOR QUANTUM

ENTANGLEMENT DISTRIBUTION

In quantum communication, entangled photon pairs are

created and sent to Alice and Bob over the free space or

fiber link. These pairs need to be detected in the Alice

and Bob's stations. The detection method needs to select

a basis to measure in, measure the polarization, and

record enough information to match each photon Alice

detects with the corresponding photon Bob detects. This

Page 3

section describes the linear optics components used to

make the basis choice and polarization measurement, the

detectors used to detect error in single photons state, and

the wave plates used to correct single photons state.

Quantum entanglement distribution protocol, shown

schematically in Fig. 1, is proposed to distribute

entangled photons between quantum communication

stations and also to correct occurred error along the

quantum channel. Entangled photon pairs are produced

by EPR source using spontaneous parametric down

conversion (SPDC) process [15]. The photons pass first

through a polarizing beam splitter (PBS) which transmits

horizontally polarized photons while reflecting vertically

polarized photons. The outgoing photons lunch to the

separate quantum channel and transmit to Alice and Bob

stations. Note that the EPR source and Alice station can

be in one station. During the photon transmission in the

quantum channel, environmental can effect on the

photon and decoherence can be happened. Because of

this, preservation of quantum channel against the

environmental effects such as temperature and

electromagnetic field is very important. Anyway, if

quantum state changes during transmission line,

proposed protocol can correct it. Arrived photons pass

first through PBS in Alice and Bob stations. Because of

the effect of PBS1 and PBS4, there are four possible

output combinations: a single bit-flip error will result in

the output state emerging from either the (c1,d2) or

(c2,d1) mode pair; a double error from (c1, c2); no-error

from (d1,d2). The corresponding corrections to the state

are performed by HWP_1, or HWP_2, or both,

depending on the output mode-pair. Modules Mc1, Md1,

Mc2, Md2 represent the ‘applications packages’ and

consist of unitary operations and detectors. If the output

state is accepted only from mode pair (d1,d2), then the

scheme functions as a single-pair realization of error-

rejection. If the output state is accepted from all mode

pairs, then a limited form of error-correction is

performed.

The state of polarization-entangled photons pairs

produced by SPDC process may be written as

(

2121 12

2

where H ( V ) denotes the horizontal (vertical) linear

polarization state of a photon and the ket subscripts

denote the spatial propagation mode. Polarizing beam

splitters (PBS) transmit horizontally polarized photons

and reflect vertically polarized photons.

)

1

VVHH

+=φ

(12)

Fig.1: Entanglement distribution protocol

Page 4

So that the entangled state

four PBSs, transfers to

(

12

2

12

φ

, after passing through

)

()

()

212121

4_ , 3_

⎯

2121

2_ , 1_

⎯

2121

2

1

2

1

1

dddddd

PBS

⎯

PBS

⎯

bbaa

PBS

⎯

PBS

⎯

VVHH

VVHH

VVHH

φ

φ

=+⎯→⎯

+⎯→⎯

+=

(13)

As we see from equation (13), shared state between

Alice and Bob via output modes d1 and d2 is equivalent

to polarization-entangled state produced at the source

location. For these error-free transmissions no photons

are ever directed into modes c1and c2. But, we know

that quantum channels such as fiber is not ideal and

polarization-flip is occurred due to depolarizing optical

fiber. Therefore, for the case of a single bit-?ip, the

output state will be obtained from either the (c1,d2) or

(c2,d1) mode pair and will be of the following form

respectively,

(

12121

2

or

(

21212

2

In this case, the shared state between Alice and Bob is

different to initial entangled state and suffer an error.

Error correction can be done on the output state very

easily by placing a properly positioned (at 45? with

respect to the

VH /

basis) half-wave plate (HWP) in

each of the two error channels c1 and c2. The HWP

rotates the linear polarization state of an incoming

photon into its orthogonal counterpart. As shown above,

when a single bit-?ip happens, it means that the flipped

qubit will appear in either mode c1or c2. Then the

corresponding HWP in these modes act on the

polarization state of that qubit and rotate it to correct

state. The ?nal two-photon output state will be

equivalent to the initial source state with regard to the

polarization entanglement and will be obtained in the

(c1,d2) or (c2,d1) mode pair. Suppose bit flip happens in

quantum channel (1), we have

(

2121 12

2

)

2

1

dcdcdc

VHHV

+=φ

(14)

)

1

1

dcdcdc

HVVH

+=φ

(15)

)

()

()

()

()

212121

1_

2121

4_ , 3_

⎯

2121

) 1 (

⎯

2121

2_ , 1_

⎯

2

1

2

1

2

1

2

1

1

dcdcdc

HWP

⎯ →⎯

dcdc

PBS

⎯

PBS

⎯

bbaa

channel

⎯⎯

quantum

⎯⎯

onflip

⎯

bit

⎯

bbaa

PBS

⎯

PBS

⎯

VVHH

VHHV

VHHV

VVHH

VVHH

φ

φ

=+⎯

+⎯→⎯

+→⎯

+⎯→⎯

+=

−

(16)

But if bit flip happens in quantum channel (2),

transformation is as following,

()

()

()

()

()

121212

2_

1212

4_ , 3_

⎯

2121

)2(

2121

2_ , 1_

⎯

2121 12

2

1

2

1

2

1

2

1

2

1

dcdcdc

HWP

⎯

dcdc

PBS

⎯

PBS

⎯

bbaa

channel

⎯⎯

quantum

⎯⎯

onflip

⎯

bit

⎯

bbaa

PBS

⎯

PBS

⎯

VVHH

HVVH

HVVH

VVHH

VVHH

φ

φ

=+⎯⎯→⎯

+⎯→⎯

+⎯→⎯

+⎯→⎯

+=

−

(17)

We can see from equation (16) and (17) that error

correction has been done and output states

φ

are equivalent to initial entangled state.

If bit-flip occurs in both transmitted photons in quantum

channel, then the output state will be obtained from

modes c1 and c2 and will be equivalent to the initial

state

12

(

212112

2

21dc

φ

and

12dc

φ

|,

)

()

()

()

()

21211211

2_

⎯

1_

21 211

4_, 3_

⎯

2

1

21

) 2

⎯

( , ) 1 (

⎯

2121

2_, 1_

⎯

2

1

2

1

2

1

2

1

1

cccccc

HWP

⎯

and

⎯

HWP

⎯

cccc

PBS

⎯

PBS

⎯

b

b

aa

channel

⎯⎯

quantum

⎯⎯

onflip

⎯

bit

⎯

bbaa

PBS

⎯

PBS

⎯

VVHH

HHVV

HHVV

VVHH

VVHH

φ

φ

=+⎯→⎯

+⎯→⎯

+→⎯

+⎯→⎯

+=

−

(18)

IV. CONCLUSION

Thus, it has been proved that it is possible to distribute

quantum entanglement on distant node by an error-free

protocol. It is clear that error rejection is performed if

Alice and Bob only received the two photon output state

from mode pair (d1,d2). Detection of both photons

within modules APD_1 and APD_3 means that no bit-

?ip occurred during the distribution of the entangled

state, ensuring that it is still of the initial entangled form.

If Alice and Bob accept the output state from all four

possible mode pairs, then an error correction can be

performed on the output state by using properly oriented

HWP.

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Shamsolah Salemian was born in 1976.

He received the B.Sc. degree from Shiraz

University and the M.Sc. degree from Iran

University of Science and Technology

(IUST). He is currently working toward

the Ph.D. degree at IUST where his

research interests

communication and computation and

optical integrated circuits.

include quantum

Shahram Mohammadnejad received the

B.Sc. degree in electrical engineering

from the University of Houston, Houston,

TX, in 1981 and the M.Sc. and Ph.D.

degrees in semiconductor material growth

and lasers from Shizuoka University,

Shizuoka, Japan, in 1990 and 1993,

respectively.

He invented the PdSrS laser for the first

time in 1992. He has published more than

80 scientific papers and books. His

research interests include semiconductor material growth, quantum

electronics, semiconductor devices, optoelectronics, and lasers. Dr.

Mohammadnejad is a scientific committee member of the Iranian

Conference of Electrical Engineering (ICEE), a member of Institute of

Engineering and Technology (IET), and a member of IET- CEng.