Local Operations can Generate Quantum Entanglement: The

Correlation Conversion Property of Quantum Channels

Laszlo Gyongyosi*1,2

1Quantum Technologies Laboratory, Department of Telecommunications

Budapest University of Technology and Economics

2 Magyar tudosok krt, H-1111, Budapest, Hungary 2Information Systems Research Group, Mathematics and Natural Sciences

Hungarian Academy of Sciences

H-1518, Budapest, Hungary

*gyongyosi@hit.bme.hu

Transmission of quantum entanglement will play a crucial role in future networks and long-distance quantum communications. Quantum Key Distribution, the working mechanism of quantum repeaters and the various quantum communication protocols are all based on quantum entanglement. On the other hand, quantum entanglement is extremely fragile and sensitive to the noise of the communication channel over which it has been transmitted. To share entanglement between distant points, high fidelity quantum channels are needed. In practice, these communication links are noisy, which makes it impossible or extremely difficult and expensive to distribute entanglement. In this work we first show that quantum entanglement can be generated by a new idea, exploiting the most natural effect of the communication channels: the noise itself of the link. We prove that the noise transformation of quantum channels that are not able to transmit quantum entanglement can be used to generate entanglement from classically correlated input. We call this new phenomenon the Correlation Conversion property (CC-property) of quantum channels. The proposed solution does not require any local operation or local measurement by the parties, only the use of standard quantum channels. Our results have implications and consequences for the future of quantum communications, and for global-scale quantum communication networks. The discovery also revealed that entanglement generation by local operations is possible.

One of the most important goals of current research in quantum computation and

communications is the development of global-scale quantum communication networks. The

success of worldwide Quantum Key Distribution and quantum repeater networks is based on

quantum entanglement [1-7]. On the other hand, the process of entanglement sharing and

distribution is an expensive task. The practical quantum channels are noisy, which makes it

very hard or even impossible to send entangled particles over these links. The main reason is

that quantum information is very fragile and extremely sensitive to the noise of the

communication links. The current solutions under development for entanglement transmission

are based on various entanglement purification methods, which could make it possible to

share entanglement between distant points, but only if the noise of the communication links

is low enough to allow the realization of the post-purification processes in the nodes.

However, these purification methods are very expensive and inefficient, since many entangled

pairs have to be shared between the parties with relatively high fidelity. One of the most

fundamental questions in the development of future communication networks is the process of

entanglement transmission. If it were possible to find quantum channels that could generate

entanglement between two distant points (let us refer to them as Alice and Bob) without

sending the entanglement itself, then we could dramatically reduce the cost of development of

future quantum communication networks. It would also have very serious consequences for

current knowledge about the nature of the information itself.

Over a quantum channel many types of information can be transmitted. Sending

entanglement would be possible only if the noise of channel is low (i.e., it is a high fidelity

channel which can transmit quantum information). On the other hand, if the noise of is

high (assuming it is a practical communication channel) then entanglement might be

transmitted with much difficulty, or it could be completely impossible.

,

Let us assume that there is a quantum channel between Alice and Bob, which is so noisy

that it cannot function as a transmission venue for any quantum information, but it can be

used to send classical information over it (i.e., it has quantum capacity , but has

positive classical capacity ).

( ) 0Q =

( ) 0C >

For simplicity, we will refer to this quantum channel as a classical-quantum channel * (or

low fidelity channel), since it can transmit classical correlations only. If Alice would like to

send entanglement to Bob over channel , she will find that it is not possible, since the

noise of makes it impossible to preserve the quantum entanglement. Alice must choose a

different solution.

Since the channel between Alice and Bob is so noisy and the transmission of quantum

entanglement is a difficult task, she might think the following: “Since the channel is noisy

and quantum entanglement is very fragile, would it be possible to feed only classical

correlations to classical-quantum channel , to get quantum entanglement between my

system, A, and system B on Bob’s side?”

In that case the problem of entanglement sharing would be reduced to the following process:

Alice prepares a classically correlated system, AB, in which she keeps A and feeds B to

channel . Bob receives B, and the result is quantum entanglement between Alice and Bob,

generated simply by the noise of the channel.

If it were possible, Alice could use the classical-quantum channel to send entanglement to

Bob, except for the fact that she prepared a classically correlated input and the channel can

transmit classical correlation only. This idea might seem to be unimaginable and completely

impossible at first sight, and our intuitions also strictly dictate that it cannot be true.

Up to this point, the possibility of entanglement generation between Alice and Bob has been

based on the transmission of quantum entanglement, which requires high fidelity quantum

channels between the parties.

As we have found, this is not the case. There exist low fidelity channels which can transmit

only classical correlation, but the noise transformation of the channel can re-transform the

input density matrix in such a way that it will result in quantum entanglement between

Alice and Bob. From this point onward, Alice has a much better choice than to send the

entanglement directly over . Alice can feed only a classically correlated input system to

* The term „classical-quantum channel” has several different interpretations in the literature. It is used mainly in the HSW (Holevo-Schumacher-Westmoreland) setting to describe the transmission of classical information over quantum channels. However, from the „classical-quantum” term it does not follow unambiguously that the quantum channel could not transmit quantum correlations. In our setting, under the „classical-quantum channel” we mean only those quantum channels, that can transmit classical correlations only.

, and the process of entanglement transmission will be made by the most natural property

of these communication channels: by the noise transformation of the channel, itself. We called

this new phenomenon the “Correlation Conversion property” (CC-property) of quantum

channels.

B

A

B

A

Quantum entanglement

Legends

(a)

B

A

B

A

(b)

Bob Alice Bob

Classical correlation

Alice

Legends

Low fidelity channelHigh fidelity channel

Quantum entanglement

Fig. 1(a): Alice’s standard solution for sending entanglement to Bob. If she would like to send part B

of her entangled system AB to Bob, then the channel has to be a high fidelity link, since quantum

entanglement is extremely fragile and sensitive to noise. If the channel is noisy, the transmission of

entanglement is a difficult task or completely impossible. (b): In our solution, Alice feeds only a

classically correlated input to the classical (low fidelity) channel , which will result in quantum

entanglement between her system A and Bob’s system B. The process does not require high fidelity

channels, since the entanglement is generated by the noise transformation of the channel. This

property is called the Correlation Conversion property of the communication link.

Producing entanglement from classical correlation by the noise of quantum channels seemed

to be impossible before our results. However, it has already been shown that separable states

can be used to distribute entanglement [8-12], but these protocols require ideal or nearly

noiseless channels between Alice and Bob, which is completely unattainable in a practical

communication system. These schemes also have another drawback: the requirement of local

operations and local measurement. Our solution does not require ideal channels nor any local

operation or local measurement on the encoder or decoder side to generate quantum

entanglement, only the use of standard quantum channels, i.e., local operations [15].

The Correlation Conversion property of quantum channels is summarized as follows. There

exist channels and which can produce quantum entanglement from classical

correlated inputs and , between systems and channel output ,

1

r

2

ACrAB Ar ( )1B Bs r=

where neither channel nor can transmit any quantum entanglement,

. The noise transformation of the channel can re-transform the

density matrix in such a way that it results in entanglement between systems A and B.

1

Ä

Ar

2

Ar

Br

1Bs =

( ) ( )1 2 0Q Q=

ACr

2

=

2

B )

The channel construction is summarized in Fig. 2. Neither channel nor

can transmit any quantum information or entanglement. The two channels can transmit

classical correlation only. In the initial phase, Alice prepares the fully separable systems

and . The input density matrices and are classically correlated and contain no

quantum entanglement. Alice then feeds to the input of , and a flag system , to the

input of . The quantum entanglement will be generated by the output of ,

between density matrices and . The output s of the second

channel will be also received by Bob, and will be simply traced out in the calculations. The

final system state will be referred to as s r , in which the

system state will contain quantum entanglement between and .

1 1

Cr

Cr

2

ABr

( )Br

Br

)r

Tr

1

B

A

1

(

AB

(2C =

( ) ( )( )1 2 Cr rÄ

Bs

C A=

r

1B

A

B

2

Classical correlation

C

A

C

Quantum entanglement

Legends

Fig. 2. The CC-property of quantum channels. Neither channel 1 nor 2 can transmit any

quantum information or entanglement (i.e., these channels are referred as classical-quantum channels);

however, the noise transformation of the channels can generate quantum entanglement between

and from classically correlated, unentangled inputs, and .

Ar

Bs Ar Br

We can easily find such kinds of channels; for example, any channel with error

probability , could handle quantum entanglement transmission, but only if

1

x yp p p p= + + z

( ) ( )1 1 2 0x y z x y x z y zQ p p p p p p p p p= - + + + + + > (1)

holds true [13]. The error probability p of is so high that it makes it impossible to

transmit quantum entanglement, thus . The noise parameters and

affect the eigenvalues of the input density matrix of as will be proven in the

Supplementary Material. For the second channel, , the condition Q also has to

hold. For an entanglement-breaking channel this condition is trivially satisfied since it

destroys every quantum entanglement on its output.

1

( )1 =

2

0Q

2

,x yp p

0=

zp

,v v+ - ABr

( )2

To measure the amount of noise-generated entanglement we consider the use of the

relative entropy of entanglement, from the set of other entanglement measures [9-10], such as

the negativity, concurrence or entanglement of formation [8, 12]. By definition, the

relative entropy of entanglement function of the joint state of subsystems A and B is

defined by the

( )E ⋅

( )E r

r

(D ⋅ ⋅) quantum relative entropy function, as

( ) ( ) ( ) ((min min log logAB AB

AB ABE D Tr Trr r

r r r r r r r= = - ))

,r

)v

, (2)

where is the set of separable states . As we have found, the

following connection holds for the amount of noise-generated entanglement. The achievable

entanglement between and is , where v ,v are the

eigenvalues of channel output density matrix . We characterized an input system, for

which the amount of entanglement between the separable input system and the

entangled channel output density matrix is

ABr ,1

n

AB i A i B ii

pr r=

= Äå

( ) (maxAB v vE vs

+ -+-

=

ABs

Ar Bs --

ABs

+

r

-

AB

( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( )

( )

( ) ( )

00 00 10 10

min

1 1 1 32 2 2 21 1 1 32 2 2 2

max

1 ,

ABAB AB AB

v v

in

E D

v v E v v E

v v v v E

v v

p v v

rs s r

b b b b

+ -

+ - + -

+ - + -

+ --

+ -

=

æ ö æ ö÷ ÷ç ç= - - - - -÷ ÷ç ç÷ ÷÷ ÷ç çè ø è øéæ ö æ öù÷ ÷ç çê ú= - - - - - Y Y÷ ÷ç ç÷ ÷÷ ÷ç çê úè ø è øë û

= -

= - ⋅ -

(3)

where ( ) ( ) ( )00 00 10 10 1E E Eb b b bY Y = = = , ( is the difference of the

eigenvalues in input system , p is the noise of channel , while

)inv v+ --

ABr 1

( )001 00 112

b = + , ( )10 00 11b -

)

12

= are the maximally entangled states.

For the relative entropy of entanglement of channel output system the

inequality

( ABE s ABs

( ) ( ) ( ) 20 19AB inE p v vs + -< £ - ⋅ - = (4)

holds, since ( ) 213

p- £ and ( ) 103inv v+ -< - £ . In Fig. 3, the amount of the noise-

generated entanglement is summarized in the function of the difference of

eigenvalues v and v of .

( ABE s

ABs

)

+ -

Fig. 3. The amount of noise-generated entanglement in the function of the difference of the

eigenvalues of the channel output density matrix.

Our results confirmed that the CC-property works for the most natural and simplest channel

models—for example, the Pauli channels. We found a combination of two very simple

channels, the so called phase flip channel and the entanglement-breaking channel ,

that can transmit classical correlation only [14]; however, they can be used to generate

entanglement. The

1 2

13

p ³ error probability of the channel results in , the

entanglement-breaking channel has also since it measures the input system and

outputs a classically correlated density matrix. However, over these channels the maximum of

the noise-generated entanglement is

1 ( )1 0Q =

( )2 0Q =

( ) ( ) ( ) 2minr

m9AB

AB AB vs s r-

= = -axv v

v+ -

+ -ABE D .

For details and further derivation of the various correlation measures, see the Supplementary

Information.

=

Conclusion

In this work we first proved that quantum entanglement can be produced by the noise

transformation of classical (low fidelity) channels, the result of which has dramatic

consequences for future quantum communications. Our results make it possible to generate

entanglement between distant points from classically correlated inputs over quantum

channels that have no capability of transmitting quantum information. We developed a new

idea which exploits the most natural property of the communication channels and which

opens new dimensions in the fields of quantum communications. The solution does not

require any local operation or local measurement by the parties, only the use of standard

quantum channels. It also constrains us to revise our current knowledge about quantum

channels and the nature of information itself. We have to reveal those deeply involved,

currently hidden and uncharacterized possibilities that quantum information still holds.

Acknowledgements

The author would like to thank Kamil Bradler and Tomasz Paterek for useful discussions and

comments. The results discussed above are supported by the grant TAMOP-4.2.1/B-

09/1/KMR-2010-0002, 4.2.2.B-10/1--2010-0009 and COST Action MP1006.

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Local Operations can Generate Quantum Entanglement: The

Correlation Conversion Property of Quantum Channels

Laszlo Gyongyosi*1,2

1Quantum Technologies Laboratory, Department of Telecommunications

Budapest University of Technology and Economics

2 Magyar tudosok krt, H-1111, Budapest, Hungary 2Information Systems Research Group, Mathematics and Natural Sciences

Hungarian Academy of Sciences

H-1518, Budapest, Hungary

*gyongyosi@hit.bme.hu

Supplementary Information

1 Theorems and Proofs In the Supplementary Information we provide the theorems and proofs. First, we discuss

properties of the channel structure, then we characterize the input system. Finally, we show

the results on the channel output system.

1.1 Channel System Description

First, we show that channels and can transmit classical correlation only. 1 2

Proposition 1. The channels and in the joint structure can transmit

only classical information.

1 2 1 Ä 2

z

First channel: the phase flip channel

Channel transmits Alice’s input system and generates output system . 1 Br ( )1 B Br s=

In the current work we demonstrate the results for a phase flip channel with error

probability where

1

x yp p p p= + +

1 1, ,6 6x y zp p p³ ³ = 0, and 1 ,

3p ³ (A.1)

characterize the noise transformation of the channel. For this parameterization, we get a

channel that can transmit classical correlation only, since the channel has no quantum

capacity [14]:

( ) ( )1 1 2 0x y z x y x z y zQ p p p p p p p p p= - + + + + + = . (A.2)

We use this channel as the first channel in the joint construction . The noise

parameters and affect the eigenvalues of the input density matrix as will

be shown in Section 1.3.

1 Ä 2

2

,x yp p 0zp = ABr

Second channel: the entanglement-breaking channel

The second channel in is the entanglement-breaking channel . Giving an

entangled system to input of an entanglement-breaking channel , it will destroy

every entanglement on its output . Formally, a noisy quantum channel is

entanglement-breaking if for a half of a maximally entangled input

2 1 Ä

A¢

EB

EB

B EB

AA¢Y , the output of the

channel is a separable state [37]. Let us assume that the maximally entangled input system of

an entanglement-breaking channel is EB1

0

1 d

AA A Ai

i id

-

¢ ¢=

Y = å . The output of

can be expressed as follows:

EB

( ) ( ) AEB X x xAA

xp x r r¢Y Y = Äå B , (A.3)

where represents an arbitrary probability distribution, while and are the

separable density matrices of the output system. The noise-transformation of an

entanglement-breaking channel can be described as follows: it performs a complete von

Neumann measurement on its input system , and outputs a classically correlated (or

completely uncorrelated, depending on the measurement outcome) density matrix

. It can be formalized as follows:

( )Xp x

( )EB

Axr

Bxr

EB

r

s= r

( ) { } ,EB x xx

Trr r= På s (A.4)

where { }xP

x

represents a POVM (Positive Operator Valued Measure) measurement on

and is the output density matrix of the channel [37]. Any entanglement-breaking channel

can be decomposed into three parts: channel that acts as a noisy transformation

on , a measurement operator

r

s

r

EB 1EB

{ }xP , and a second channel , that outputs the density

matrix :

2EB

xs

1 22 EB EB= P . (A.5)

In our setting , and the input of the channel is the flag , from the classically

correlated density matrix . After the channel has got the flag , measures it and

outputs a density matrix ,

2 EB=

r

Cr

AC

=

2 Cr

( )2C Cs r

( ) { }2 ,C Cx

Trr r= Lå C Cs (A.6)

where { }CL defines a projective measurement in the standard basis { }0 , 1 , while the

output flag system is an arbitrary density matrix. Cs

The decomposition of the entanglement-breaking channel is depicted in Fig. A.1. It

contains two I ideal channels as and , and a projective measurement, as

follows its noisy evolution can be rewritten as

2

CL1EB 2

EB

2 CI I= L . (A.7)

The channel measures the input flag system , then outputs the density matrix . As

the result of measurement flag system C, system AB collapses into a well specified state. The

output density matrix contains the result of the measurement

2 Cr Cs

Cs { }CL , which will be

referred as a one-bit classical message ‘0’ or ‘1’ that will inform Bob about the measurement

result. Using the classical information from , Bob will be able to determine whether he

received an entangled or a classically correlated system B. The measurement

2

{ }CL of

and the identification processes together called post-selection. It is immediately follows that

the classical information from encoded in , is a required information to Bob to

determine whether system AB has become entangled, or not. If the post-selection process is

successful then Bob localized the entanglement to AB, and we will refer it as entanglement-

localization.

2

2 Cs

I ICLC C

2 CI I

Fig. A.1. The decomposition of the entanglement-breaking channel 2 . It measures the flag system

C and outputs the density matrix to Bob, which encodes a classical bit (conditional state

preparation). From the one-bit message, Bob will be able to identify the result of the projective

measurement of the channel for the post-selection process.

Cs

EB

CL

The quantum capacity of any entanglement-breaking channels is trivially zero, since

due to the { }CL

2 =

measurement operator of the channel every entanglement vanishes. As

follows, for , after EB { }CL has been applied on we will have Cr

( )2Q 0= , (A.8)

which makes no possible to transmit quantum entanglement over channel . 2

Kraus Representation

The map of the quantum channel can also be expressed with a special representation called

the Kraus representation. For a given input system and the quantum channel , this

representation can be expressed as [4], [32-35]

Ar

( ) †A i

iN Nr = å A ir

, (A.9)

where are the Kraus operators, and . The isometric extension of by

means of the Kraus representation can be expressed as

iN †i i

iN N I=å

( ) ( )†A i A i A BE A i E

i iN N U N ir r r= =å Äå . (A.10)

The action of the quantum channel on an operator k l , where { k } is an

orthonormal basis, also can be given in operator form using the Kraus operator

(klN k= )l . By exploiting the property , for the input quantum system †BEUU P= Ar

( ) † †

,.A BE A A i A j i A jE E

i j i jU U U N i N j N N ir r r r

æ öæ ö ÷ç÷ç ÷ç÷= = Ä Ä = Äç ÷÷ çç ÷÷ç ÷çè ø è øå å å †

Ej

i

(A.11)

Tracing out the environment, we get

( )( ) †E A BE A i A

iTr U N Nr r = å . (A.12)

Kraus Representation of the Phase Flip Channel

The effect of the phase flip channel on the subsystem of can be expressed in

Kraus representation as follows [15-17]:

1 Br ABr

( ) ( ) ( ) ( ) ( )†1 1 ,B B

AB A B i AB ii

I Nr r r r= Ä = å N

)

(A.13)

where denotes the identity transformation on subsystem A and ( AI r

( ) ( )( ) ( )0

1

1 2, 1 2

2, 2 ,

BA

BA

N I diag p p

N I diag p p

= Ä - -

= Ä -

, (A.14)

while is the error probability of the channel . x yp p p p= + + z 1

Kraus Representation of the Entanglement-breaking Channel

The entanglement-breaking channel on the subsystem of can be expressed as 2 Cr ACr

( ) ( ) ( ) ( ) ( )†2 2 ,C C

AC A C i AC ii

I Nr r r r= Ä = å N (A.15)

where

( ) ,Ci A i C C

N I x V¢= Ä (A.16)

where and C denote the input and output systems, and the Kraus-operators C ¢ ( )CiN are

unit rank. The sets { }i Cx ¢ and { }C

V each do not necessarily form an orthonormal set

[37].

1.2 Characterization of Channel Input System

Theorem 1. There exists fully separable, classically correlated input system , that can

be characterized by the ( difference of the eigenvalues v , of the separable

subsystem .

ABCr

)

2

)r

inv v+ -- + v-

ABr

Proof.

Note: The results will be demonstrated for qubit (d=2 dimensional) inputs and qubit channels.

Before the sending phase, Alice prepares the separable system , which contain no

quantum entanglement between and . Alice holds , while she feeds the systems

and , which will be the inputs of the joint channel structure , where is the

valuable system, and is a flag state. The quantum entanglement will be prepared between

systems and s , after Bob has received the flag system . The

process of decoherence on two qubit states has been studied in the literature [15-31]. In our

case, the noise of the channel will affect only one system state, which requires further

investigation in the mathematical description.

ABCr

1 Ä

Ar

)Br

Br Ar Br

Cr Br

(C C

Cr

B = Ar (1 2s =

The channel input system with the separable systems A, B, and flag state C, is

prepared by Alice as follows

ABCr

†:

( ) ( )

( ) ( )

( ) ( )

( ) ( )

( )

1 1 000 000 000 110 110 000 110 1104 41 3 000 000 000 110 110 000 110 1104 41 001 001 011 011 101 101 111 1112

1 000 000 110 11021 2

ABC

in

in

in

in

v v

v v

v v

v v

v v

r

+ -

+ -

+ -

+ -

+ -

=æ ö÷ç - - + + +÷ç ÷÷çè øæ ö÷ç - - - - +÷ç ÷÷çè øæ ö÷ç - + + +÷ç ÷÷çè ø

=æ ö÷ç - - + +÷ç ÷÷çè ø

-

+

+

000 110 110 000 001 001,

011 011 101 101 111 111in

æ ö+ + +æ ö ÷ç÷ç ÷ç÷ç ÷ç÷÷ç ÷÷çè ø + +è ø

(A.17)

where is a separable Bell diagonal state [15-16], which can be expressed as ABr

( ) ( )

( ) ( )

( ) ( )

1 1 00 00 11 112 21 00 11 11 0021 01 01 10 10 .2

AB in

in

in

v v

v v

v v

r + -

+ -

+ -

æ ö÷ç= - - +÷ç ÷÷çè øæ ö÷ç - +÷ç ÷÷çè øæ ö÷ç - +÷ç ÷÷çè ø

+

+ (A.18)

where , are the eigenvalues of density matrix (will be defined in v+ v-

)

ABr (A.24)) and

( 13inv v+ -- = (the eigenvalues of the input system are ABr

12

v+ = and 16

v- = ), while

the separable (from ) mixed system : ABr Cr

C i ii

pr y= å iy (A.19)

in the probabilistic mixture of the pure systems 0 0y = and 1 1y = , is called the flag.

The noise of channel will transform the eigenvalues into the range 1

(( ) ( ) ) 10 13inv v p v+ v+ - -< - = - ⋅ - £

.

† Note: The separable initial system in (A.17) contains no quantum entanglement between systems A and B, and will be referred as classically correlated. For the complete correctness, it is not pure classical correlation, since it has some positive quantum discord, see (A.70). We note that we are not interested in the further partitions [35-36] of the initial state. To generate entanglement between A and B, local operations will be applied on B and C. These local operations make no possible to preserve entanglement in B and C, or in any partitions of ABC that contain these subsystems.

To see that AB is a separable Bell diagonal state and the flag C together is a fully separable

system, we also give here the density matrix of (A.17).

( ) ( )

( )

( )

( )

( ) ( )

( )

1 10 0 0 0 0 02 2

10 0 0 0 0 02

0 0 0 0 0 0 0 010 0 0 0 0 0 02

0 0 0 0 0 0 0 010 0 0 0 0 0 02

1 10 0 0 0 0 02 2

10 0 0 0 0 0 02

ABC

in in

in

in

in

in in

in

v v v v

v v

v v

v v

v v v v

v v

r

+ - + -

+ -

+ -

+ -

+ - + -

+ -

=æ ö÷ç - - - ÷ç ÷ç ÷ç ÷ç ÷ç ÷÷ç ÷ç - ÷ç ÷ç ÷ç ÷ç ÷ç ÷ç ÷÷ç ÷ççç -çççççççççç -ççççççç - -ççççççç -çè ø

,

÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷

0

-

(A.20)

where was given in ABr (A.18), and can be expressed in matrix form as:

( ) ( )

( )

( )

( ) ( )

1 1 10 02 2 2

10 02 ,

10 0 02

1 10 02 2

AB

in in

in

in

in in

v v v v

v v

v v

v v v v

r

+ - + -

+ -

+ -

+ - + -

=æ ö÷ç - - - ÷ç ÷ç ÷ç ÷ç ÷ç ÷÷ç ÷ç - ÷ç ÷ç ÷ç ÷ç ÷ç ÷ç ÷- ÷ç ÷ç ÷ç ÷ç ÷ç ÷ç ÷ç - - ÷÷çè ø

0

12

-

)

(A.21)

while is the difference of the eigenvalues in input system . System is

clearly separable and contains no quantum entanglement, which can also be easily checked by

the Peres-Horodecki criterion [31-32]: the partial transposes will be positive, i.e., ( )

and , which trivially follows since is a separable Bell diagonal state. The

flag system is also separable and contains no quantum entanglement since the partial

transpose of with respect to C is positive, i.e., ( ) , see

( inv v+ --

( ) 0BTABr ³

Cr

ABCr

ABr ABr

AT³ 0ABr

ABr

0CTABCr ³ (A.33).

The eigenvalues v , of matrix can be expressed as follows. First, we rewrite system

in the following representation [15-22]:

+ v- ABr

ABr

3

1

14AB i i i

iI I I I cr s s

=

æ ö÷ç ÷ç= Ä + ⋅ Ä + Ä ⋅ + Ä ÷ç ÷ç ÷çè øår s

s s , (A.22)

where and s are the Bloch vectors, r , ,x y zs s s sé ù= ë û with the Pauli matrices , while

are real parameters . For a Bell diagonal state . Choosing

and s , the input state in

is

r

ic

( )r1 ic- £ £ 1

)

0= =r s 0,0,=

(0,0,s= (A.22) can be given in matrix representation as follows:

3 1

3 1 2

1 2 3

1 2 3

1 0 00 1 01

4 0 1 00 0 1

AB

r s c c cr s c c cc c r s c

c c r s c

r

æ ö+ + + - ÷ç ÷ç ÷ç ÷+ - - +ç ÷ç ÷= ç ÷÷ç + - + - ÷ç ÷ç ÷ç ÷ç - -è ø

2

- +

. (A.23)

Then, the eigenvalues v , v of are defined as + - ABr

( ) ( )

( ) ( )

2 23 1 2

2 23 1 2

1 1 041 1 04

v c r s c c

v c r s c c

+

-

æ ö÷ç= - + - + + ³÷ç ÷çè øæ ö÷ç= - - - + + ³÷ç ÷çè ø

,

. (A.24)

The other two eigenvalues u , u can be defined as follows: + -

( ) ( )

( ) ( )

2 23 1 2

2 23 1 2

1 1 041 1 04

u c r s c c

u c r s c c

+

-

æ ö÷ç= + + + + - ³÷ç ÷çè øæ ö÷ç= + - + + - ³÷ç ÷çè ø

,

. (A.25)

System can be expressed in the same way, as ACr

3 1

3 1 2

1 2 3

1 2 3

1 0 00 1 01

4 0 10 0 1

AC

r s c c cr s c c cc c r s c

c c r s c

r

æ ö+ + + - ÷ç ÷ç ÷ç ÷+ - - +ç ÷ç ÷= ç ÷÷ç + - + - ÷ç ÷ç ÷ç ÷ç - -è ø

2

0- +

, (A.26)

and the eigenvalues of this matrix will be denoted by

( ) ( )

( ) ( )

2 23 1 2

2 23 1 2

1 1 041 1 04

c r s c c

c r s c c

k

k

+

-

æ ö÷ç= - + - + + ³÷ç ÷çè øæ ö÷ç= - - - + + ³÷ç ÷çè ø

,

, (A.27)

and

( ) ( )

( ) ( )

2 23 1 2

2 23 1 2

1 1 041 1 04

c r s c c

c r s c c

t

t

+

-

æ ö÷ç= + + + + - ³÷ç ÷çè øæ ö÷ç= + - + + - ³÷ç ÷çè ø

,

, (A.28)

respectively.

Using this representation form, the required conditions for the separability of the input

system can be given as follows. For separable systems AB and AC, the conditions

{ } 1max , , ,2

v v u u+ - + - £ , (A.29)

and

{ } 1max , , ,2

k k t t+ - + - £ , (A.30)

have to be satisfied. Furthermore, assuming a Bell diagonal state ( ), the

condition

0, 0r s= =

1 2 3 1c c c+ + £ (A.31)

also trivially follows for the separability for each systems, AB and AC.

These results conclude the proof of Theorem 1.

■

Corollary 1. The separability of input system for any ABCr ( ) 103

v v+ -< - £ is satisfied,

since { } 1max , , ,2

v v u u+ - + - £ .

Remark 1. (On the role of classical communication). The proposed scheme uses only

quantum channels between Alice and Bob and no classical channels applied in the process.

The entanglement generation requires only the use of quantum channels and does not contain

any further local operation or classical communication between the parties. The post-selection

process is also realized by itself the noise of quantum channel . The one-bit classical

message is produced by the local measurement

2

{ }CL of , and the result will be

communicated to Bob by , itself. Alice does not send any classical information to Bob,

nor Bob to Alice.

2

2

Note: The proposed scheme could be reduced to classical communication between Alice and

Bob, if and only if in the input system the quantum discord would be , ABCr ( ) 0ABCr =

however this not the case: , see ( ) 0ABCr >

1

(A.61), (A.65) and (A.70), which makes no

possible to interpret the transmission of C as classical communication [9].

Remark 2. (On the impossibility of entanglement generation by local operations). We are

interested in the entanglement between A and B. The theorem on the impossibility of

entanglement generation by local operations [38] is not violated, because the local operations

will be applied to B and C, instead of A and B. Channels and are CPTP

(Completely Positive Trace Preserving) maps, which can be interpreted as local operations on

systems B and C. The first channel acts as a local operation on B, the entanglement-

breaking channel performs a local measurement

1 2

2 { }CL on C, then conditionally prepares

a density matrix depending on the measurement outcome (conditional state preparation) and

sends it to Bob. Since channel sends the output density matrix only to Bob, channel

also represents a local operation.

2

1 2

ABCr

2

As the results have confirmed, quantum entanglement can be generated only by the use of

standard quantum channels and , from which Corollary 2 follows.

Corollary 2. Local operations on B and C can result in quantum entanglement between A

and B. These local operations are two CPTP maps, which makes no possible to preserve

entanglement in B and C.

Required Conditions on Separability of Input System

Lemma 1. The input system is fully separable system and AB is classically correlated,

which stands the following requirement on . The partial transposes of with respect

to the subsystems have to be positive.

ABCr ABr

The input density matrix has to be classically correlated and system has to be

separable, which also can be given by different conditions. Using the Peres-Horodecki

criterion [31-32] it is summarized as:

ABr ABCr

( )

( )

( )

( )

0,

0,

0,

0,

A

B

B

C

TAB

TAB

TABC

TABC

r

r

r

r

³

³

³

³

(A.32)

hold true, and ( ) 103

v v+ -< - £ by the initial assumption on the input system.

Proposition 2. These conditions on systems and are satisfied in the initial state. ABCr ABr

These conditions will be checked by the Peres-Horodecki criterion [31-32], by taking the

partial transposes ( ) , , and ( of the input system of ATABr

ABCs

( ) BTABr

) ATAB

( ) BTABCr

) BTABr

2

) CTABCr ABCr

ABr

ABC

(A.20). The positivity of ( and ( trivially follows from r (A.18), since is a

separable Bell diagonal state. For simplicity we will show the partial transpose of with

respect to C, where (before channel has applied

r

{ }CL to the flag ) is: Cr

( ) ( )

( )

( )

( )

( ) ( )

( )

1 10 0 0 0 0 02 2

10 0 0 0 0 0 02

0 0 0 0 0 0 0 010 0 0 0 0 0 02

0 0 0 0 0 0 0 010 0 0 0 0 0 02

1 10 0 0 0 0 02 2

10 0 0 0 0 0 02

ABC

v v v v

v v

v v

v v

v v v v

v v

s

+ - + -

+ -

+ -

+ -

+ - + -

+ -

=æ ö÷ç - - - ÷ç ÷ç ÷ç ÷ç ÷ç ÷÷ç ÷ç - ÷ç ÷ç ÷ç ÷ç ÷ç ÷ç ÷÷ç ÷ç ÷ç ÷ç ÷-ç ÷ç ÷ç ÷ç ÷÷ç ÷ç ÷ç ÷ç ÷ç ÷ç ÷-ç ÷÷çççççç - -ççççççç -çè ø

.

÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷

-

(A.33)

System ( can be expressed as follows: ) CTABCr

( )

( )

( ) ( )

( )

( )

( ) ( )

( )

1 0 0 0 0 0 0 02

1 10 0 0 0 02 2

0 0 0 0 0 0 0 010 0 0 0 0 0 02

0 0 0 0 0 0 0 010 0 0 0 0 0 02

1 10 0 0 0 02 2

10 0 0 0 0 0 02

CTABC

in

in in

in

in

in in

in

v v

v v v v

v v

v v

v v v v

v v

r

+ -

+ - + -

+ -

+ -

+ - + -

+ -

=æ ö÷ç - - ÷ç ÷ç ÷ç ÷ç ÷ç ÷÷ç ÷ç - - ÷ç ÷ç ÷ç ÷ç ÷ç ÷ç ÷çççç -çççççççççç -ççççççç - - -ççççççç -çè ø

,

÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷÷

0

0

)

)

(A.34)

where v , v are the eigenvalues of the input density matrix . One readily can check by

the Peres-Horodecki criterion [31-32] that the partial transpose is positive, hence

+ - ABr

( ) 0CTABCr ³ , (A.35)

and

( ) . (A.36) 0BTABCr ³

Tracing out flag system C from , one can check easily that the partial transpose of the

resulting matrix with respect to A and B is positive, since ( ) and

. Since these conditions on are all satisfied, it also proves that in the

separable input system ABC, system AB contains no quantum entanglement.

ABCr

(C ABCTr r 0ATABr ³

( ) 0BTABr ³ ABCr

Proposition 3. The noise of affects the eigenvalues of . The noise of

can transform the initial eigenvalues of in the output system , as such

1 ,v v+ - ABCr

ABs r=

1

ABr (1A Br

( ) 209

v v+ -< - £ will hold. In this domain positive quantum entanglement can be generated

between and channel output . Ar ( )r s=1 B B

1.3 The Correlation Conversion Property

Required Conditions on the Entangled Output System

Lemma 2. In the output system of two conditions

have to be satisfied. First, the flag system C has to be separable from systems A and B.

Second, for positive quantum entanglement in the difference between the eigenvalues

of output matrix , the condition has to hold.

( ) (1 2ABC A B Cs r r r= Ä

ABs

( )0 v v+ -< -

) 2

)Br

=

2

2

)

1 Ä

,v v+ - ABs

The Correlation Conversion property of quantum channels is summarized in Theorem 2.

Theorem 2. (On the Correlation Conversion property of quantum channels). There exists

channels and which can generate quantum entanglement from fully separable,

classically correlated inputs and , between systems and channel output

, where neither channel , nor can transmit any quantum

entanglement, . The noise transformation of the channel can

retransform the density matrix in such a way that it results in entanglement between systems

A and B.

1

(

2

( )

ABr

( )

ACr

Ar

1Bs = 1 2

1 2 0Q Q=

Proof.

Here we prove that the output system of contains quantum entanglement between

Alice’s density matrix and the channel output . According to the Theorem 2, the noise

of channel system generates quantum entanglement between Alice’s density matrix

and channel output from the classically correlated input systems and .

After Bob has received systems and , the resulting system state will be referred as

follows:

1 Ä

s

Cs

Ar

1 Ä

B

BsAr ABr ACr

Bs

( ) (1 2ABC A B Cs r r r= Ä , (A.37)

in which system the flag C remains separable, since the partial transposes of are non-

negative, see

ABCs

(A.34), and the v , eigenvalues of density matrix affected by the noise

of , and with relations

+

- =

v-

(

ABs

1 Ä 2 )( ) ) (1 inv v p v v+ - + -- ⋅ -

( ) 209

v v+ -< - £ , (A.38)

and

( ) ( )1 2 2 1v v v v+ - + -- + + - = . (A.39)

After the flag system C has been removed (since it was fed to the entanglement-breaking

channel ), the system state reduces to 2

( )C ABC ABTr s s= . (A.40)

The density matrix between Alice’s system and channel output can be expressed as

follows (before channel has applied

Ar Bs

2 { }CL to the flag ): Cr

( ) ( )

( )

( )

( ) ( )

1 1 10 02 2 2

10 02 ,

10 0 02

1 10 02 2

AB

v v v v

v v

v v

v v v v

s

+ - + -

+ -

+ -

+ - + -

=æ ö÷ç - - - ÷ç ÷ç ÷ç ÷ç ÷ç ÷÷ç ÷ç - ÷ç ÷ç ÷ç ÷ç ÷ç ÷ç ÷- ÷ç ÷ç ÷ç ÷ç ÷ç ÷ç ÷ç - - ÷÷çè ø

0

12

-

(A.41)

where , are the eigenvalues of the channel output density matrix , and v+

(

v- ABs

) 209

v v< -

A

2

+ - £ , according to the characterization of the input system . One can

further readily check by the Peres-Horodecki criterion [31-32], that matrix in

ABr

ABs (A.41) has

no negative partial transpose, which shows that and still have not become entangled:

, To achieve the entanglement in AB, the matrix

Ar Bs

( )TABr

( )AB 0.BTr ³

2

(A.41) has to be

decomposable into two different matrices, and its decomposition in determined by the flag

system C. This post-selection process [8-12], [36-37] will be made by the entanglement-

braking channel . It will be possible if and only if the flag system C has been transmitted

over , and after B has been received by Bob, i.e., there is a causality in the post-selection

process: the flag C cannot be measured by before Bob would have not received B from

. On the other hand, without any information from , Bob will not be able to

determine whether he received an entangled system B, or he owns just a classically correlated

system.

2

2

1

Q

2

Lemma 3. The entanglement-breaking channel determines whether Bob received an

entangled system B, or not. The output of is a one-bit classical message that informs

Bob about the result.

2

The flag system will be fed to the input of the entanglement-breaking channel , with

and . The input flag is assumed to be in the probabilistic

mixture of the pure systems

Cr

C

2

( )2 0= ( )2 0> Cr

0 0y = and 1 1y = , hence the output of will C=0 or

C=1, after the channel has applied the measurement operator 2 { }CL to , using the

standard basis

Cr

{ }0 , 1 . The probabilities of the measurement outcomes will be quantified

in Theorem 3.

According to Fig. A.1, the channel can be decomposed as and , and

a

21EB I= 2

EB I=

{ }CL projective measurement . After the flag system has been

transmitted over , it will simply be traced out by the partial transpose operator

and the final system state will reduce to . The flag state has no impact on

the amount of the generated entanglement over in . On the other hand, the

measurement

2 CI= L

( )C ABCTr s =

I Cr

Tr2 ( )C ⋅

ABs

1 ABs

{ }CL of is a probabilistic process, which causes a decrease in the amount

of generable entanglement, as will be quantified in Theorem 3.

2

The one-bit classical information encoded by s is a required condition for Bob

on the entanglement localization, as stated in Remark 3.

(2C = )Cr

Remark 3. The output of the is a necessary condition to achieve entanglement in .

Before the output of the entanglement-breaking channel the localization of entanglement

2 ABs

2

is not possible since the matrix is in the in the probabilistic mixture of the two possible

systems , where 0 and 1 is the one-bit classical output of channel

.

ABs

(0 +( )AB AB ABs s s=

2

)1

2

After the channel has applied { }CL to the flag system C, the output system in ABCs

(A.33) can be rewritten as follows:

( ) ( )0 10 0 1 1ABC AB ABC Cs s s= Ä + Ä , (A.42)

and after the measurement { }CL of it can be decomposed as: 2

( ) ( )

( ) ( )

( )

v v

v v

v v

+ -

+ -

+ --

1 10 0 0 0 0 02 2

0 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 0

1 10 0 0 0 0 02 2

0 0 0 0 0 0 0 00 0 0 0 0 0 0 0

10 0 0 0 0 0 02

0 0 0 0 0 0 0 0

0 0

ABC

v v

v v

s

+ -

+ -

=æ ö÷ç - - - ÷ç ÷ç ÷ç ÷ç ÷ç ÷÷ç ÷ç ÷ç ÷ç ÷ç ÷ç ÷ç ÷ç ÷÷ç ÷ +ç ÷ç ÷ç ÷ç ÷ç ÷ç ÷ç ÷÷ç ÷ç ÷ç ÷ç ÷- - -ç ÷ç ÷ç ÷÷ç ÷ç ÷çè ø

( )

( )

( )

10 0 0 0 02 .

0 0 0 0 0 0 0 010 0 0 0 0 0 02

0 0 0 0 0 0 0 010 0 0 0 0 0 02

v v

v v

v v

+ -

+ -

+ -

æ ö÷ç ÷ç ÷ç ÷ç ÷ç ÷ç ÷÷ç ÷ç ÷ç ÷ç ÷ç ÷ç ÷ç ÷ç ÷÷ç - ÷ç ÷ç ÷ç ÷ç ÷ç ÷ç ÷ç ÷÷ç ÷ç ÷ç ÷ç - ÷ç ÷ç ÷ç ÷÷ç ÷ç ÷ç ÷ç ÷ç ÷ç ÷ç ÷-ç ÷çè ø

(A.43)

From it follows that system in AB (A.41) can be decomposed into s

( ) ( )

( ) ( ) ( )

( )

( )

( ) ( ) ( )

0 1

1 1 10 02 2 2

10 02

10 0 02

1 10 02 2

AB AB AB

v v v v v v

v v

v v

v v v v v v

s s s

+ - + - + -

+ -

+ -

+ - + - + -

= + =

æ ö÷ç - - + - - ÷ç ÷ç ÷ç ÷ç ÷ç ÷÷ç ÷ç - ÷ç ÷ç ÷ç ÷ç ÷ç ÷ç ÷- ÷ç ÷ç ÷ç ÷ç ÷ç ÷ç ÷ç - - ÷÷çè ø

0

12

- + -

,

(A.44)

where

( )

( ) ( )

( ) ( )

0

1 10 02 2

0 0 0 00 0 0 0

1 10 02 2

AB

v v v v

v v v v

s

+ - + -

+ - + -

æ ö÷ç - - - ÷ç ÷ç ÷ç ÷ç ÷ç ÷÷ç ÷ç= ÷ç ÷ç ÷ç ÷ç ÷ç ÷ç ÷÷ç - - ÷ç ÷çè ø-

(A.45)

and

( )

( )

( )

( )

( )

1

1 0 0 02

10 02

10 0 02

10 0 02

AB

v v

v v

v v

v v

s

+ -

+ -

+ -

+ -

æ ö÷ç - ÷ç ÷ç ÷ç ÷ç ÷ç ÷÷ç ÷ç - ÷ç ÷ç ÷ç= ÷ç ÷ç ÷ç ÷- ÷ç ÷ç ÷ç ÷ç ÷ç ÷ç ÷ç - ÷÷çè ø

0. (A.46)

Due to the measurement { }CL

on C of the channel , system in 2 ABs (A.42) collapses into

(A.45) or (A.46). If measured C=0, then the entanglement-localization was successful,

and Bob in the post-selection process will be able to use the entangled system B, after he

received the output from .

2

Cs 2

During the process the flag system C is trivially separable in from the remaining parts,

. Moreover, the partial transposes ( ) , ( ) , ( ) , ( are both still

non-negative. On the other hand, after

ABCs

BTABCABs AT

ABs BTABs s ) CT

ABCs

{ }CL

( s

has been applied on C by , the partial

transposes of ( will be negative: , , which makes

2

) 0BT<)0ABs ( ) )0

AT< 0 ( )AB ( ABs 0

possible to achieve entanglement between A and B. The systems ( or ( cannot

be post-selected without the output of the entanglement-breaking channel .

)0ABs

2

1

)

)

1ABs

2

The selection of system ( in , i.e., the localization of entanglement into AB could

not be made until the output of the entanglement-breaking channel has not received by

Bob, only their probabilistic mixture exists for Bob. After the

channel has applied

)0ABs ABs

( ) (0AB AB ABs s s= +

2 { }CL on the flag C, the entangled system ( can be post-

selected by Bob, pending the classical information from .

)0ABs

Cs

Note: In the input system the density matrix ( could be selected by Alice if and

only if she would have applied a measurement operator

ABr )0ABr

{ }CL on C. However at that initial

stage the flag C cannot be measured, she can send only the classically correlated system

to Bob. Assuming the case that Alice would apply a measurement ( ) ( )0AB AB ABr r r= + 1

{ }CL

)0ABr

1 ( )1 0Q =

1

ABr

on the flag C in the initial phase (before the transmission) to get the entangled

density matrix ( , she will find that she is not able to send the entangled B to Bob over

, since . It is also not possible over , because Q by the initial

assumptions on and . As follows, in the input system , only the partial transpose

of can be used to analyze the entanglement in AB, which is positive.

2 ( )2

) 0B <

0=

2 ABr

These results conclude the proof of Theorem 2.

■

Corollary 3. The partial transposes , are negative in the

channel output system .

( )( )0 0ATABs < ( )( 0

TABs

ABs

While for the input system ( ) and ( ) , see 0ATABr ³

( )( )0BT

AB s

0BTABr ³

)0

(A.21), in (A.42) the partial

transposes , of are negative, see (A.45)( )( )0AT

ABs s ( AB . The negative partial

transposes prove that A and B have become entangled in the channel output system.

Remark 4. Entanglement generation over is possible if and only if the output of the

entanglement-breaking channel has been received by Bob. With no output from , the

channel output AB would be

1

2 2

(A.41) which system state would not result in entanglement

between A and B. If Bob receives 0 from , then he will know that he received the

entangled system ( , however the measurement of the entanglement-breaking channel

is a probabilistic process; will decrease the amount of maximally generated

entanglement over , as will be exactly quantified by the relative entropy of entanglement

function in Theorem 3.

2

)

2

0ABs

1

2 2

The proposed channel output system satisfies the separability requirements and the

condition for the entanglement of and . As follows, the noise of channel structure

can transform the input density matrices and in such a way that results in

quantum entanglement between Alice’s system and channel output .

ABCs

Ar Bs

r

1 Ä Br Cr

A ( )1B Bs r=

Corollary 4. The noise of channel can transform the eigenvalues of in such

a way that

1 ,v v+ - ABr

( ) 209

v v+ -< - £ in the channel output system is satisfied, and AB

becomes entangled.

ABs

The channel output system can also be expressed as follows: ABs

( ) ( )( )1 2 31 1 14AB z z x x y y z zI I I I p c p c cs s s s s s s= Ä + ⋅ Ä + Ä ⋅ + - Ä + - Ä + Är s ,s s

(A.47)

which can be expressed in matrix representation as [15]:

( ) ( )( ) ( )

( ) ( )( ) ( )

3 1 2

3 1 2

1 2 3

1 2 3

1 0 0 10 1 1 1 01 ,

4 0 1 1 1 01 1 0 0 1

AB

r s c p c p cr s c p c p c

p c p c r s cp c p c r s c

s

æ ö+ + + - - - ÷ç ÷ç ÷ç ÷ç + - - - + - ÷ç ÷ç ÷= ÷ç ÷ç - + - - + - ÷ç ÷ç ÷ç ÷÷ç - - - - - +è ø

1

z

(A.48)

where p is the error probability of channel . Due to the noise of ,

the eigenvalues v , v of are changed from the initial values to

x yp p p p= + + 1 1

+ - ABs

( ) ( ) ( )( )

( ) ( ) ( )( )

2 23 1

2 23 1

1 1 1 141 1 1 14

v c r s p c p c

v c r s p c p c

+

-

æ ö÷ç= - + - + - + - ÷ç ÷çè øæ ö÷ç= - - - + - + - ÷ç ÷çè ø

2

2

,

, (A.49)

satisfying the required condition

( ) 209

v v+ -< - £ . (A.50)

The other two eigenvalues u , of are irrelevant in the further calculations, since

they have no effect on the amount of noise-generated entanglement.

+ u+ ABs

Capacity Calculations

Next we discuss the amount of quantum entanglement in which can be produced by the

noise of , assuming the previously-shown input system characterization.

ABs

1 Ä 2

Theorem 3. (On the amount of noise-generated entanglement). The relative entropy of

entanglement between the fully separable, classically correlated input system and the

output system is

ABr

ABs

( ) ( ) ( ) ( ) (min max 1 ,AB

AB AB AB inv vE D v v p v

rs s r

+ -+ - + --

= = - = - ⋅ )v-

)

where is the relative entropy of entanglement, ( ABE s (D ⋅ ⋅)

)

is the relative entropy

function, is the difference of eigenvalues in , p is the noise of the channel

, while v ,v are the eigenvalues of channel output density matrix .

(v v+ -

+ -

in- ABr

1 ABs

Proof.

First we show that the entanglement generated by can be measured by the

quantum relative entropy function

1 Ä 2

(D ⋅ ⋅)

2

. Then we prove that the amount of achievable

quantum entanglement is determined by the noise characteristic of . To measure

the amount of entanglement we consider using the E relative entropy of entanglement

function [10-11], from the set of other entanglement measures, such as the negativity,

concurrence or entanglement of formation [9, 13]. By definition, the relative entropy of

entanglement function of the joint state of subsystems A and B is defined by the quantum

relative entropy function

1 Ä

( )E r

( )⋅

r

( ) ( ) (log ABTr r r )( )logABD Trr r r r= - , as

( ) (minAB

ABE Dr

r r= )r

,r

)v

, (A.51)

where the set of separable states . The amount of the noise-

generated entanglement between and is expressed by ,

where

ABr ,1

n

AB i A i B ii

pr r=

= Äå

BsAr ( ) (maxAB v vE vs

+ -+ --

= -

( ) 209ABE s< £ . The relative entropy of entanglement between the

separable channel input and the channel output density matrix is

( AB )E s

ABr ABs

( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( )( )

( ) ( )

00 00 01 01

min

1 1 1 32 2 2 21 1 1 32 2 2 2

max

1 ,

ABAB AB AB

v v

in

E D

v v E v v E

v v v v E

v v Ev v

p v v

rs s r

b b b b

+ -

+ - + -

+ - + -

+ -

+ --

+ -

=

æ ö æ ö÷ ÷ç ç= - - - - -÷ ÷ç ç÷ ÷÷ ÷ç çè ø è øéæ ö æ öù÷ ÷ç çê ú= - - - - - Y Y÷ ÷ç ç÷ ÷÷ ÷ç çê úè ø è øë û

= - ⋅ Y Y

= -

= - ⋅ -

(A.52)

where ( is the difference of the eigenvalues in input system , and )inv v+ -- ABr

( ) ( ) ( )00 00 01 01 1E E Eb b b bY Y = = = , (A.53)

while ( )001 00 112

b = + and (101 00 112

b = - )

)

are the maximally entangled

states. From the results on the relative entropy of entanglement in the output

system , the inequality

( ABE s

ABs

( ) ( ) ( ) 20 19AB inE p v vs + -< £ - ⋅ - = (A.54)

trivially follows, since ( ) 213

p- £ and ( ) 103inv v+ -< - £ .

Note: In the separable input system , there is no entanglement; the

relative entropy of entanglement of r is

,1

n

AB i A i B ii

pr r=

= Äå

AB

,r

( ) , ,1

min 0.AB

n

AB AB i A i B ii

E D pr

r r r r=

æ ö÷ç ÷ç= Ä ÷ç ÷ç ÷çè øå = (A.55)

According to the Theorem 3, (minAB

AB ABDr

s r )

)

)

taken between and is analogous to

the maximized difference of the eigenvalues v of output matrix . In

Fig. A.2, the amount of noise-generated entanglement is summarized in function of

the difference of eigenvalues v and v of .

ABr

,v+ -

ABs

(maxv v

v v+ -

+ ---

+ -

ABs

( ABE s

ABs

Fig. A.2. The amount of noise-generated entanglement in function of the difference of the eigenvalues

of channel output density matrix.

These results prove the statements of Theorem 3.

■

Illustration of CC-property

In Fig. A.3, the relative entropy of entanglement in in function of the noise

parameter p of the first channel is shown. To illustrate the effect of the noise of channel

on the amount of generated entanglement, we characterized the Bell diagonal input (see

( )E ⋅

AB

ABs

1

1

(A.18)) system r as:

( )11 ,3inc v v+ -= - = (A.56)

( )213inc v v+ -= - - = - (A.57)

and

( )3 21 2 1 2inc v v+ -= - ⋅ - = + ⋅ .c (A.58)

One can check readily that this input system is the same system given by formulas of (A.18)

and (A.21), assuming ( ) 13inv v+ -- = . This system is separable, since 1 2 3 1c c c+ + £ ,

and { } 1max , , ,2

v v u u+ - + - £ , where 12

v+ = and 1 .6

v- =

As shown in Fig. A.3, the 13

p ³ error probability of the phase flip channel results in the

decreasing amount of entanglement , for the increasing error probability p.

1

( ABE s )

Fig. A.3. The amount of noise-generated entanglement, assuming a phase flip channel 1 with

13

p ³ , an entanglement-breaking channel 2 , and 1 21 1,3 3

= = -c c and The

maximal entanglement

3 2c c⋅1 2 .= +

( ) ( ) )( )

(1 21

3AB in

pE p v vs + -

-= - ⋅ - = =

9 is obtained for 1 3p = .

For the given input system , the maximized amount of noise-generated entanglement

over the channels and is

ABr

21

( ) ( ) (

( ) ( )( )

)min max

1 21 ,3 9

ABAB AB AB v v

in

E D v

pp v v

rs s r

+ -+ --

+ -

= =

-= - ⋅ - = =

v-

(A.59)

since for 1 3p = ,

12

v+ = and ( ) ( ) ( )21 2 1 11 1 1 1 1

4 3 3 3v p p p-

æ ö÷æ ö æ öç ÷ç ÷ç ç ÷= - - ⋅ - - - - - =÷ç ç ç ÷÷ç ÷ç ç ÷è ø è øç ÷÷çè ø

5 .18

÷÷÷÷

)

)

2 v-

(A.60)

1.4 Correlation Measures and Quantum Capacity

In this section, we derive the various correlation measures [15-31] for the output system .

These correlation measures can help to analyze further the properties of the Correlation

Conversion property.

ABs

Quantum Mutual Information

The quantum mutual information function measures the total (i.e., both classical

and quantum) correlation in the joint channel output state . The quantum mutual

information function of can be expressed as follows [15]:

( ABI s

ABs

ABs

( ) ( ) ( ) ( )AB A B ABI S S Ss r s s= + - . (A.61)

Using the eigenvalues of , can be rewritten as [15]: ABs ( ABI s

( ) ( ) ( ) 2 2 2log log log logAB A BI S S u u u u v v vs r s + + - - + + -= + + + + + , (A.62)

where are the eigenvalues of (defined in , , ,u u v v+ - + - ABs (A.24) and (A.25)), and

( ) ( ) ( ) ( ) (21 11 1 log 1 1 log 12 2AS r r rr = - - - - + + )2 r , (A.63)

( ) ( ) ( ) ( ) (21 11 1 log 1 1 log 12 2BS s s ss = - - - - + + )2 s . (A.64)

Classical Correlation

The classical correlation function measures the purely classical correlation in the

joint state . The amount of purely classical correlation in can be expressed

as follows [16-18]:

( ABs

ABs

)

( )ABs ABs

( ) ( ) ( )( ) ( )min ,

k

AB B

B BE k

S S B A

S pkS

s s

s s

= -

= - å

k (A.65)

where A BB k

A

k kk kr s

sr

= is the post-measurement state of , the probability of result k is Bs

kk q Ap d k kr= , while d is the dimension of system , makes up a normalized

probability distribution in the rank-one POVM elements

Ar kq

k kE q k k= of the POVM

measurement operator [15-16].

We can also use the following definition to compute the classical correlation:

( ) ( ) { }1 2 3min , , ,AB AS f fs r= - f (A.66)

where the functions and are defined as [15]: 1 2,f f 3f

( )( )

( )

( )( )

( )

( )( )

( )

( )( )

( )

1 3 2

3 2 3

3 2 3

3 2 3

1 11 log 14 2 1

1 11 log 14 2 11 11 log 14 2 11 11 log 14 2 1

f r s c r ss

r s c r s cs

r s c r s cs

r s c r s cs

= - + + + + + ++

- - + - - + -+

- + - - + - -+

- - - + - - ++

3

,

c

(A.67)

( ) ( ) ( ) ( )2 2 22 1 2 1 1 2

1 11 1 log 1 1 log 12 2

f r c r c r c= - - + - + - + + + + 21r c , (A.68)

and

( ) ( ) ( ) ( )2 2 23 2 2 2 2 2

1 11 1 log 1 1 log 12 2

f r c r c r c= - - + - + - + + + + 22r c

)

. (A.69)

Quantum Discord

The quantum discord function measures the purely quantum correlation in the joint

state . It is important to emphasize that this correlation measure does not identify the

( ABs

ABs

amount of entanglement in the joint system , hence it cannot be used to characterize the

entanglement that generated by the channel.

ABs

)

( ABs s

From the amount of quantum mutual information and the classical correlation

of output system , the quantum discord can be expressed as

( ABI s )

)( ABs ABs ( ABs

( ) ) ( )AB ABI s= - . (A.70)

Based on the previously-shown results, for the given channel output representation it can be

rewritten in the following form:

( ) ( ) ( )( ) ( )( ) { }( )( ) { }

2 2logu

u

-

-

+

+

ABs

)( )

B= +

u +

2 2

1 2 3

2 2 2 2 1 2

log log logmin , ,

log log log log min , , .

AB AB AB

A B

A

B

IS S u u u v v v vS f f fS u u u v v v v f f f

s s s

r s

r

s

+ + - + + - -

+ + - + + - -

= -

= + + + +

- -

= + + + +

3

) )

)

=

)2

)

(A.71)

Quantum Coherent Information

From the quantum discord and the classical correlation functions, the

quantum coherent information of can be expressed as follows:

( ABs ( ABs

(coh ABI s

( ) ( ( )( ) ( )( )

11

1.

coh AB A AB

AB AB AB

AB

III

s s s

s s s

s

-

= - + -

= -

(A.72)

Using the previously-derived results, it can also be rewritten as:

( )( )( ) ( ) 2 2 2 2

1log log log log 1.

coh AB

AB

A B

IIS S u u u v v v v

s

s

r s + + - - + + - -

= -

= + + + + -

(A.73)

Quantum Capacity

The of the joint structure can be given as the maximization of the quantum

coherent information of channel output system ,

( 1Q Ä

(coh ABI s ABs

( ) ( )

( ) ( )( )

( )( )

1 21lim max

1lim max 1

1lim max 1 .

A B

A B

A B

coh ABn

AB ABn

ABn

Q In

n

In

r r

r r

r r

s

s s

s

¥ "

¥ "

¥ "

Ä =

= +

= -

- (A.74)

From the previously-shown results it also can be expressed as follows:

( ) ( )

( ) ( )1 2

2 2

2 2

1lim max

log log1lim max .log log 1

A B

A B

coh ABn

A Bn

Q In

S S u u u un v v v v

r r

r r

s

r s¥ "

+ + - -

¥ "+ + - -

Ä =

æ ö+ + + ÷ç ÷ç= ÷ç ÷÷ç + -è ø

+ (A.75)

From the previously-shown consequences, the following connection can be derived:

( )( ) ( )

( )( ) ( )( )( )( ) ( )( )

2 2

1 2 2

2

log log1lim max log ,

log 1A B

A B

AB ABn

AB AB

S S u u u uQ E v E

nv E v E

r r

r s

s s

s s

+ + - -

-¥ "

+ +

æ ö+ + + ÷ç ÷ç ÷ç ÷çÄ = + + + ÷ç ÷ç ÷÷ç ÷ç+ - - -è ø

v- (A.76)

where ( ) 209ABE s< £ and are non-negative real numbers. Assuming a Bell

diagonal channel output state with , thus ,

reduces to

, , ,u u v v+ - + -

r s 0= = ( ) ( ) 1A BS Sr s= = ( )1 2Q Ä

( ) ( )( ) ( )( )( )( ) (( ))

( )( )

2 2

1 2 2

2

1 log log1lim max log

log1lim max 1 .

A B

A B

AB ABn

AB AB

ABn

u u u uQ E v E

nv E v E

Sn

r r

r r

s s

s

s

+ + - -

- -¥ "

+ +

¥ "

æ ö+ + ÷ç ÷ç ÷ç ÷çÄ = + + + ÷ç ÷ç ÷÷ç ÷ç+ - -è ø

= -

vs (A.77)

Correlation Measures for the Channel Output

Assuming the previously-characterized classically correlated input system with ABr

1 21 1,3 3

c c= - = - and , channel with error probability 3 21 2c = + ⋅ c 113

p ³ , and the

entanglement-breaking channel the previously introduced correlation measures ,

, , I and the amount of noise-generated entanglement E are

2

( )ABs

( )ABI s

( )ABs( )ABs ( )ABs coh

compared in Fig. A.4. The results are shown for the composite system AB, where system B is

affected by the noise of . 1

The coherent information , quantum discord and the quantum

entanglement are quantum correlations. The purely classical correlation is measured by

. The quantum mutual information measures both classical and quantum

correlations. From these correlations, the quantum entanglement can be achieved in if

only the measurement

(coh ABI s ) )

) )

( ABs

( ABs ( ABI s

ABs

{ }CL of the entanglement-breaking channel on C has been

resulted in 0. If measured 1, then only classical correlations will be available in s .

2

2 AB

Fig. A.4. The amount of total correlation , purely classical , purely quantum

correlations , quantum coherent information and the relative entropy of

entanglement in the channel output system , assuming

( ABI s ) )

) )

)

( ABs

( ABs

( ABE s

(coh ABI s

ABs 1 21 1,3 3

c c= = - and

with phase flip channel 3 1 2c = + 2,c⋅ 1 , 13

p ³ , and entanglement-breaking channel 2 .

Increasing p of , the total correlation and classical correlation start to

increase, while the discord and the coherent information I start to

decrease. At , reduces to , and to 0, while the

coherent information will be I , along with E , hence

the noise of the channel destroys every quantum correlations in the channel output system

.

1

1=

( ABI s

( ABs

( )1 AB

) )

)

) )

0

)

) )

( ABs

( )ABs

cohI

( )ABs =

( ABs

)

( )ABs

coh

p ( ABI s ( ABs

1- =

( )ABs

0s- =

ABs

Besides the fact that can be used to identify the amount of noise-generated

entanglement between subsystems A and B, these measures characterize completely the

amount of purely classical and purely quantum correlations in the channel output system

. On the other, neither nor can be used to identify the amount of

noise-generated entanglement in .

( ABE s

sABs ( AB

ABs

(coh ABI s

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