The Correlation Conversion Property of Quantum Channels

The Correlation Conversion Property of Quantum Channels

Laszlo Gyongyosi

1Quantum Technologies Laboratory, Department of Telecommunications

Budapest University of Technology and Economics

2 Magyar tudosok krt, Budapest, H-1117, Hungary

2Information Systems Research Group, Mathematics and Natural Sciences

Hungarian Academy of Sciences

Budapest, H-1518, Hungary

Supplementary Information

S.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.

S.1.1 Channel System

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

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

First channel: the phase flip channel

Channel transmits Alice’s input system and generates output system .

In the current work we demonstrate the results for a phaseflip channel with error probability where

and (S.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]:

. (S.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.

Second channel: the entanglement-breaking channel

The second channel in isthe 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, 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 . The output of can be expressed as follows:

, (S.3)

where represents an arbitrary probability distribution, while and are the separable density matrices of the output system.The noise-transformationof 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 an uncorrelated, depending on the measurement) density matrix . It can be formalized as follows:

(S.4)

where 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 , and a second channel , that outputs the density matrix :

. (S.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 ,

(S.6)

where defines aprojective measurement in the standard basis , whilethe output flag system is an arbitrary density matrix.

The decomposition of the entanglement-breaking channel is depicted in Fig. S.1. It contains two Iideal channels as and , and a projective measurement, as follows its noisy evolution can be rewritten as

. (S.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, 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 (distillable) or a classically correlated system B. The measurement 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 entanglement to AB, and we will refer it as entanglement-localization.

Fig. S.1. The decomposition of the entanglement-breaking channel . It measures the flag system C and outputs the density matrixto 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.

The quantum capacity of any entanglement-breaking channels is trivially zero, since due to the measurement operator of the channel every entanglement vanishes. As follows, for , after has been applied on we will have

, (S.8)

which makes no possible to transmit quantum entanglement over channel.

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]

, (S.9)

where are the Kraus operators, and . The isometric extension of by means of the Kraus representation can be expressed as

. (S.10)

The action of the quantum channel on an operator , where is an orthonormal basis, also can be given in operator form using the Kraus operator . By exploiting the property , for the input quantum system

(S.11)

Tracing out the environment, we get

. (S.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], [41]:

(S.13)

where denotes the identity transformation on subsystem A and

(S.14)

while is the error probability of the channel .

Kraus Representation of the Entanglement-breaking Channel

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

(S.15)

where

(S.16)

where and denote the input and output systems, and the Kraus-operators are unit rank. The sets and each do not necessarily form an orthonormal set [37].

S.1.2 Characterization of Input System

Theorem1. There exists input system , that can be characterized by the difference of the eigenvalues , of the separable, classically correlated subsystem .

Proof.

Note: The results will be demonstrated for qubit channels. Before the sending phase, Alice prepares theseparablesystem, 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 distillable entanglement will be prepared between systems and , after Bob has received the flag system . The process of decoherence on two qubit states has been exhaustively studied in the literature [15-31], [39], [41]. However, in our case the noise of the channel will affect only one system state, which still requires further investigation in the mathematical description.

The channel input system with the separable systems A, B, and flag state C, is prepared by Alice as follows[*]:

(S.17)

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

(S.18)

where , are the eigenvalues of density matrix (will be defined in (S.24)) and (the eigenvalues of the input system are and ), while the separable (from ) mixed system :

(S.19)

in the probabilistic mixture of the pure systems and ,is called the flag.

The noise of channel will transform the eigenvalues into the range .To see that AB and the flag C together is a separable system, we also give here the density matrix of (S.17).

(S.20)

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

(S.21)

while is the difference of the eigenvalues in input system . Systemisclearlyseparable and contains no distillable 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 andcontains no quantum entanglement since the partial transpose of with respect to C is positive, i.e., , see later in (S.34). The eigenvalues , of matrix can be expressed as follows. First, we rewrite system in the following representation [15-22], [41]:

, (S.22)

where and are the Bloch vectors, with the Pauli matrices , while are real parameters . For a Bell diagonal state . Choosing and , the input state in(S.22) can be given in matrix representation as follows:

. (S.23)

Then, the eigenvalues , of are defined as

(S.24)

The other two eigenvalues , can be defined as follows:

(S.25)

System can be expressed in the same way, as

, (S.26)

and the eigenvalues of this matrix will be denoted by

(S.27)

and

(S.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

, (S.29)

and

, (S.30)

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

(S.31)

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

■

Corollary 1.The separability of input system for any is satisfied, since .

Remark 1.(On the role ofclassical communication). The proposed scheme uses only quantum channels between Alice and Bob andno classical channels applied in the process. The entanglement generation requires only the use of quantum channels and does not contain any non-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 of , and the resultwill be communicated to Bob by . Alice does not send any classical information to Bob, nor Bob to Alice.

Note: The proposed scheme couldbe reduced to classical communication between Alice and Bob, if and only if in the input system the quantum discord would be , however this not the case: , see later (S.61), (S.65) and (S.70)which makes no possible to interpret the transmission of Cas classical communication [9].

Remark 2. (On the impossibility of entanglement generation by LOCC).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 operationson systemsB and C. The first channel acts as a local operation on B, the entanglement-breakingchannel performs a local measurement on C, then conditionally prepares a density matrix depending on the measurement outcome (conditional state preparation). Since channel sends the output density matrix only to Bob, channel also represents a local operation.

As the results have confirmed, distillable 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 distillable quantum entanglement between A and B. These local operations are two CPTP maps, which makes no possible to preserve entanglement in subsystemsB and C.

Required Conditions

In input system the subsystem AB is classically correlated. The partial transposes of with respect to the subsystems have to be positive.

The input density matrixhas 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:

(S.32)

hold true, and by the initial assumption on the input system.

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

These conditions will be checked by the Peres-Horodecki criterion [31-32], by taking the partial transposes , , and of the input system of (S.20).The positivity of and trivially follows from(S.18), since is a separable Belldiagonal state. For simplicity we will show the partial transpose of with respect to C, where (before has applied to the flag) is:

(S.33)

System can be expressed as follows:

(S.34)

where , 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

, (S.35)

and

. (S.36)

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 systemABC,system AB contains no quantum entanglement.

Proposition 3.The noise of affects the eigenvalues of . The noise of can transform the initial eigenvalues of in the output system , as such will hold. In this domain positive quantum entanglement can be generated between and channel output .

S.1.3 The Correlation Conversion Property

In theoutput 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.

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

Theorem 2. (On the CorrelationConversion property of quantum channels).There exists channels and which can generatedistillable entanglement from classically correlated input , 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 distillable entanglement between systems A and B.

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:

, (S.37)

in which system the flag C remains separable, since the partial transposesof are non-negative, see(S.34), and the, eigenvalues of density matrix affected by the noise of , and with relations

, (S.38)

and

. (S.39)

After the flag system C has been removed (since it was fed to the entanglement-breaking channel ), the system state reduces to

. (S.40)

The density matrix between Alice’s system and channel output can be expressed as follows (before channel has applied to the flag ):

(S.41)

where , are the eigenvalues of the channel output density matrix , and , according to the characterization of the input system . One can further readily checkthat matrix in (S.41)has no negative partial transpose, which shows that and still have not become entangled: , To achieve the entanglement in AB, the matrix(S.41)has to be decomposable into two different matrices, and its decomposition in determined by the flag system C. This post-selectionprocess [8-12], [36-37] will be made by the entanglement-breaking channel .It will be possible if and only if the flag system Chas been transmitted over , andafter 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.Theentanglement-breaking channelwill give the answer to Bob. The output of is a one-bit classical message that informs Bob about the result [42].

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 and ,hence the output of will C=0 or C=1, after the channel has applied the measurement operator to,using the standard basis . Channel can be decomposed as and , and a 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 of is a probabilistic process, which causes a decrease in the amount of generable entanglement, as will be quantified in Theorem 3.

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 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 .

After the channel has applied to the flag system C, the output system in(S.33)can be rewritten as follows:

, (S.42)

and can be decomposed as:

(S.43)

From it follows that system in(S.41)can be decomposed into

, (S.44)

where

(S.45)

and

. (S.46)

Due to the measurement on C of the channel , systemin(S.42)collapses into (S.45) or (S.46)17H. 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. 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 has been applied on C by , the partial transposesof will be negative:, , which makes possible to achieve entanglement between A and B. The systems or cannot be post-selectedwithoutthe output of the entanglement-breaking channel . The selection ofsystem in , i.e., the localization of entanglement into ABcould not be made until the output of the entanglement-breaking channel has not received by Bob, only their probabilistic mixture existsfor Bob.After the channel has applied on the flag C, the entangled system can be post-selected by Bob, pending the classical information from.

Note: In the input system the density matrix could be selected by Alice if and only if she would have applied a measurement operator on C. However at that initial stage the flag Ccannot be measured, she can send only the classically correlated system to Bob. Assuming the case that Alice would apply a measurement 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 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.