Locally Inaccessible Information as a Fundamental Ingredient to QuantumInformation

F. F. Fanchini,1, ∗ L. K. Castelano,2 M. F. Cornelio,3 and M. C. de Oliveira3

1Departamento de Fısica, Universidade Federal de Ouro Preto, CEP 35400-000, Ouro Preto, MG, Brazil2Departamento de Fısica, Universidade Federal de Sao Carlos, CEP 13565-905, Sao Carlos, SP, Brazil

3Instituto de Fısica Gleb Wataghin, Universidade Estadual de Campinas,P.O. Box 6165, CEP 13083-970, Campinas, SP, Brazil

(Dated: August 9, 2011)

We restate the quantum discord (QD) as the fraction of the mutual information that is locallyinaccessible. We show that important measures, as the entanglement of formation (EOF) and theconditional entropy can be written exclusively as a function of the QD. For mixed states, the EOFbetween the subsystems A and B is given by the average locally inaccessible information (LII) ofthe pair minus the balance of LII that the pair shares with a environment E that purifies A andB. We also show that the conditional entropy can be obtained through the discord between thepair minus the discord that one of the subsystems has with the environment. These new relationselucidate important and yet not well understood phenomenon as the entanglement sudden deathand the difference between entanglement and discord. Moreover, we discuss the meaning of thenegativity of the conditional entropy, which is intimately related to entanglement irreversibility andstate merging protocol.

PACS numbers:

I. INTRODUCTION

Different ways to measure quantum correlations havebeen widely studied in the last years [1–3]. Among thesequantum correlations, quantum discord [1] has playedan important role. Based on the difference of two dis-tinct definitions of the mutual information, Ollivier andZurek developed the first measure of quantum correla-tion. This new feature was explored in its various aspects[4, 5], intriguing the community by its peculiar proper-ties - for instance, asymmetry and sudden changes [6]. Itwas recently shown that the Entanglement of Formation(EOF) and Quantum Discord (QD) obey a very specialmonogamic relationship [7]. This important result givesrise to new operational aspects to quantum discord, suchas the net amount of entanglement processed in a quan-tum computer [7], as the difference between the entan-glement cost and entanglement distillation [8], and as theamount of entanglement consumed in the state mergingprotocol [9].

Differently from classical systems, a fraction of thequantum mutual information can not be accessed locally.Based on this idea, other interesting operational interpre-tation of QD emerges - as a measure of the mutual infor-mation fraction that is not accessible locally or, shortly,the locally inaccessible information (LII) [10]. In this pa-per, we show that EOF can be understood by using thissame point of view, thus leading to improved monogamicrelations between both quantities. In that sense, the EOFbetween any two subsystems A and B can always be writ-ten exclusively as a function of the LII. Moreover it is

∗Electronic address: fanchini@iceb.ufop.br

possible to write the EOF between two subsystems Aand B as average LII of the pair minus the balance ofLII of the pair with a purifying environment E. As seenbellow, EOF between two subsystems can be describedas a LII balance function, giving to EOF a new opera-tional meaning. We derive several relations between EOFand symmetrized and antisymmetrized versions of the LIIthat essentially quantify the average of the LII and thedirectional balance of LII, when measurements are madeat A and B, respectively. Furthermore, we relate the QDwith the conditional entropy in a simple manner for anarbitrary bipartite system. Such a relation gives a newway to understand the negative signal of the conditionalentropy.

II. LOCALLY INACCESSIBLE INFORMATION

The QD between two subsystems is defined by the dif-ference of two distinct, but equivalent in the classicaldomain, quantum versions of mutual information (MI).In classical information theory MI is defined as

IAB = HA +HB −HAB , (1)

where HA is the Shannon entropy of the subsystem Aand analogous to HB and HAB , and

IAB = HA −HA|B . (2)

that is a classically equivalent definition of the MI (givenby the Bayes rule [11]) and results from the fact that theMI also can be written as the reduction in the uncertaintyof A due to the knowledge of B. Here HA|B and HB|Aare the conditional entropy (HA|B = HAB −HB). In theclassical information theory, the MI essentially measures

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FIG. 1: (Color Online) The Venn diagram showing the rela-tions between the entropies of a bipartite classical system.

the amount of information that one subsystem has aboutanother subsystem (or stochastic variable). The relationbetween HA, HB , HAB , HA|B , HB|A, and IAB can beexpressed in a Venn diagram, Fig. (1).

The first quantum generalization of the classical MI isanalogous, and it is generally accepted as the measureof the total amount of correlations (quantum and classi-cal) of a quantum state. For a bipartite state ρAB , thequantum MI is written as:

IAB = SA + SB − SAB , (3)

where SAB ≡ S(ρAB), SA ≡ S(TrB{ρAB}), and SB ≡S(TrA{ρAB}), where S(·) denotes the von Neumann en-tropy. The second definition of the quantum MI is writ-ten as a function of the measurement-based density op-erator. Classically the MI can be written as IAB =HA − HA|B , but in the quantum regime the amount ofMI obtained from a measurement-based density operatordiverges from Eq. (3). In this case, an amount of the MIcan not be locally accessed. In fact, a local measurementover the subsystem B depends on the basis of the meter,and even with a good choice, generally the total mutualinformation can not be accessed. Therefore a fraction ofthis mutual information is non-local, and consequently itis a LII.

Given this fundamental difference between a quantumand a classical system Henderson and Vedral [2] and, in-dependently, Zurek [1] defined a quantity that measuresthe maximum amount of locally accessible information[2],

J←AB = max{Πk}

[SA −

∑k

pkSA|k

], (4)

where SA|k is the conditional entropy after a measure-ment in B. Explicitly, SA|k ≡ S(ρA|k) where ρA|k =TrB(ΠkρABΠk)/TrAB(ΠkρABΠk) is the reduced state ofA after obtaining the outcome k in B and {Πk} is a com-plete set of positive operator valued measurement thatresults in the outcome k with probability pk. In thiscase, since a measurement might give different results de-pending on the basis choice, a maximization is required.

Thus J←AB is the locally accessible mutual informationand gives the maximum amount of AB mutual informa-tion that one can extract by measuring at B only [10].An illustration of this arrangement is shown in Fig. (2),where the arrows represent the maximization involved inthe calculation of the locally accessible mutual informa-tion. Note that a slice of the MI is not locally accessiblebecause it can be divided in two terms: one given by theJ←AB and another given by the LII. The LII is then givenby the MI minus J←AB , which is exactly the definition ofthe QD,

δ←AB = IAB − J←AB . (5)

In other words, the QD above gives the amount of in-formation that is not accessible locally by measurementson B. The QD δ←AB vanishes if and only if the density

FIG. 2: (Color Online) An extended Venn diagram where thequantum entropies are exposed. Here a part of the mutualinformation is not locally accessible and it is divided in twoparts: the classical correlation and the quantum discord.

matrix of the composed system ρAB remains unaffectedby a measurement in B. In this case, all the MI betweenthe pair is locally accessible. Based on this fact, we canrephrase the definition of δ←AB as the fraction of the ABMutual Information locally Inaccessible by B. This stressout that the smaller δ←AB , the smaller LII, and conse-quently, the larger the mutual information of AB thatcan be accessed by B through a measurement in his ownsubsystem. While in δ←AB the measurements over the ba-sis that minimizes the inaccessible information are madeover B (meaning that it is the mutual information of ABthat is inaccessible by B, which is being minimized), inδ←BA those measurements are made over A (meaning thatthe mutual information of AB is inaccessible by A). In-deed, there are states such that δ←BA 6= 0 though δ←AB = 0and vice versa.

By using the asymmetry of δ←AB and δ←BA, we can definetwo important quantities: The first one is the average ofthe LII when measurements are made on A and B,

$+A|B =

1

2(δ←AB + δ←BA) , (6)

and the second one is the balance of LII when measure-

3

ments are made on A and B,

$−A|B =1

2(δ←AB − δ←BA) . (7)

The average LII (Eq. (6)) is a symmetric functionsince $+

A|B = $+B|A. Due to the symmetry of the mu-

tual information (IAB = IBA), and by the definitionof quantum discord, Eq. (5), we can relate the averageLII to the average of the classical correlations through$+

A|B = IAB − (J←BA + J←AB) /2. So the average LII is

equal to the mutual information minus the average clas-sical (locally accessible) information when measurementsare performed in A and B. On the other hand, the LIIbalance [10] is asymmetric and gives the difference in theefficiency that each subsystem has to determine the mu-tual information by local measurements. The LII balanceis directly related to the balance of the classical correla-tions $−A|B = J←BA − J←AB . As expected, such a result

demonstrates that the imbalance on the classical correla-tions has the same value in modulus as the imbalance ofthe quantum information described by the QD. Suppose,for example, that $−A|B > 0. In this case, a well chosen

measurement in A is more efficient for inferring mutualinformation of AB than a well chosen measurement in B.Thus, A has less LII than B and this imbalance increasesas $−A|B increases. On the other hand, if $−A|B < 0, then

measurements in A are less efficient for inferring the stateof B than vice versa. As seen below, these quantities arevery useful to uniquely relate EOF to LII.

To present the relation between EOF and the LII, webegin by considering a pure joint state |ψAB〉. In thiscase, QD is symmetric (δ←AB = δ←AB) and is equal to EOF.Thus we can write

EAB = $+A|B , (8)

where $+A|B is given by Eq. (6). Eq. (8) tells that the

EOF of a pure state |ψAB〉 is exactly the average LII.We can also write

EAB = IAB − (J←BA + J←AB) /2, (9)

and the EOF of a pure state |ψAB〉 is equal to the mutualinformation minus the average locally accessible informa-tion by A and B.

Now we extend our consideration for an arbitrarymixed state ρAB shared by A and B. In such a case,a new subsystem E that purifies the pair A and B mustbe considered. In this new situation, a cost must be paidto include an additional subsystem - the exceeded knowl-edge that the environment E has over the pair needs tobe considered. As seen bellow, the EOF for the resultingmixed state ρAB cannot be simply written as in Eqs. (8)and (9). Instead, it is given by the average LII of the pair(A,B) minus the LII balance of each of the subsystems Aand B with E. To prove this relationship, let us supposea pure state described by ρABE = |φABE〉〈φABE | where

ρAB = TrE{ρABE}. We begin with a conservation re-lation for the distributed EOF and QD derived earlier[7].

EAB + EAE = δ←AB + δ←AE . (10)

This equation states that the sum of bipartite entangle-ment with A, as given by the EOF, can not be changedwithout changing by the same rate the sum of QD be-tween A and B and between A and E, when measure-ments are made in the subsystems B and E. Similarly,we can write

EAB + EBE = δ←BA + δ←BE , (11)

and

EAE + EBE = δ←EA + δ←EB . (12)

Rearranging Eqs.(10-12) and writing them out in func-tion of the average LII, given by Eq. (6), and the LIIbalance (Eq. (7)), we can rewrite EAB as follows

EAB = $+A|B −$

−E|A −$

−E|B , (13)

which is a curious expression for the EOF that, for anarbitrary quantum state, can be written exclusively asa function of the LII. Eq. (13) allow us to give an op-erational meaning, independently on number of systemcopies [12], to the entanglement of formation for the pairA,B: The entanglement of formation EAB, for an ar-bitrarily mixed quantum state ρAB is the average locallyinaccessible information of A,B minus the balance of thelocally inaccessible information between each subsystemA and B with a purifying ancilla E. We can readily seethat the EOF lowers compared to the pure state version,Eq. (8) if the ancilla E locally has less access to the mu-tual information with A and B than the subsystems Aand B together.

In other words, when the efficiency of a well chosenmeasurement in E in order to infer the MI either withA or with B is worse than measurements in either A orB to infer its own MI with E, the EOF between A andB will be smaller than its pure state version given byEq. (8), because $−E|A + $−E|B > 0 in Eq. (13). When

the opposite is true, then EOF will be larger, because$−E|A+$−E|B < 0. It is important to note that the entan-

glement is not given by the shared non-local information,but contrarily it is given by its balance.

Moreover, from Eq. (13) we can yet extend Eq. (10)to include the other bipartition EBE since, given a puretripartite quantum system, the sum of all entanglementbipartition is equal to the sum of all average LII,

EAB + EAE + EBE = $+A|B +$+

A|E +$+B|E . (14)

Thus, the sum of all bipartite entanglement for an ar-bitrary pure tripartite state is equal to the sum of allpossible average LII between pairs of A, B and E.

4

FIG. 3: (Color Online) Depiction of Clockwise (red arrows)and Counterclockwise (blue arrows) flow of Locally Inacces-sible Information. The sum of the two possible directions ofLII Flow results in the sum of all possible EOF between pairsA, B and E.

Now let us define

L� ≡ δ←AB + δ←EA + δ←BE , (15)

and

L ≡ δ←AE + δ←BA + δ←EB . (16)

In Eq. (15) the QD between pairs is computed sequen-tially as E → B → A→ E, i.e., for the tripartite systemABE we start computing the bipartite QD, by doingmeasurements on E to infer the mutual information be-tween E and B, then we measure B to infer the mutualinformation between B and A, and finally we measure Ato infer the mutual information between A and B, rep-resenting, as shown in Fig. (3), a clockwise, L� flow ofLII. Reversely, the computation of the QDs in Eq. (16)for the sequence E → A→ B → E represents a counter-clockwise (see Fig. (3)), L flow of LII. An interestingpicture can be formed by rewriting Eq. (14) as follows

EAB + EAE + EBE =1

2(L� + L) . (17)

In this sense, the sum of all possible EOF between pairsA, B and E is the sum of the clockwise and counterclock-wise flow of LII. It is also interesting to check what is thedifference between both LII flows, L� and L. It turnsout that

L� − L

2= (EAB − δ←AE)+(EAE − δ←EB)+(EBE − δ←BA) .

(18)Interestingly, the right hand side of Eq. (18) is equal tothe sum of the conditional entropies SA|E+SE|B+SB|A =

0, when the joint system ABE is pure [7]. Thus, we findthat for pure states the amount of LII flow in one direc-tion is equal to the amount of LII flow in the oppositedirection in a closed cycle, i.e.,

L� = L

and Eq. (17) results in

EAB + EAE + EBE = L�. (19)

Therefore, for a given tripartite pure state ρABE the sumof the bipartite Entanglement of Formation between allpossible pairs from A, B, and E is simply given by theflow of Locally Inaccessible Information in a closed cycle.Also if L� = L, the sum of the LII balance in a cyclicsequence (described as the flow given by E → B → A→E or E → A→ B → E) is equals to zero,

ω−A|B + ω−B|E + ω−E|A = 0. (20)

As shown bellow, this result is the basis to obtain themost fundamental expressions relating the entanglementof formation and discord. Furthermore, it gives a verysimple relation between the conditional entropy and QD.

III. DIFFERENCE BETWEENENTANGLEMENT OF FORMATION AND

QUANTUM DISCORD

Another intriguing aspect of quantum information isthe difference between EOF and QD. Because QD canbe different from zero for separable states, it is usuallyassumed that QD could include extra quantum correla-tions when compared to entanglement. However, thereare some mixed states where the QD is smaller than theEOF. Thus, a fundamental question emerges: what infact measures the difference between them? By usingEq. (13), we can write the difference between the EOFand the QD as:

EAB − δ←AB = $−B|A +$−A|E +$−B|E . (21)

Notice that EAB − δ←AB can be either larger or smallerthan zero, because it depends on the efficiency of de-termining the locally mutual information by performingmeasurements on each subsystem. It is natural that de-pending on the quantum state ρABE , the efficiency thatmeasurements performed in E in order to determine themutual information of the pairs AE and BE is differentfrom the efficiency that measurements performed in Bto determine the mutual information of BE and AB, aswell as from the one where measurements on A in or-der to determine the mutual information of AB and AE.Thus, the difference between the EOF and the QD givesthe difference of such efficiency. Furthermore, it is in-teresting to note that if EAB − δ←AB is positive certainlyEAE − δ←AE is negative and vice-versa, which comes fromthe conservation law given in Eq. (10).

5

We can also interpret Eq. (21) from the flow of LIIaspect. As depicted in Fig. (4), in Eq. (21) all LIIdepart from measurements in E and concentrate in B, ordepart from B and concentrate in E after measurementsin A. We can split this in the flow of locally inaccessibleinformation departing from measurements in E as

LE→A→B ≡ δ←BE + δ←AE + δ←BA, (22)

and the flow of LII departing from B and concentratingin E as

LB→A→E ≡ δ←EB + δ←AB + δ←EA, (23)

Note that the definitions in Eq. (22) and Eq. (23) aredistinct from the LII flux given in Eq. (15) and Eq. (16).So it is possible to write Eq. (21) as

EAB − δ←AB =1

2(LE→A→B − LB→A→E) , (24)

i.e., the difference on the entanglement of formation of Aand B and the QD, when measurements are made in B,is the difference of flow of LII from and to the purifyingancilla E. Similarly, we can write

EAB − δ←BA =1

2(LE→B→A − LA→B→E) , (25)

where the order of A andB has been changed to explicitlydiffer it from Eq. (24) due to the distinct sequence ofmeasurements, this process of LII flow is depicted in Fig.(5). Combining these last two equations, it is possible torewrite Eq. (13) as

EAB −$+A|B =

1

2

(LE→(A

B) − L(AB)→E

), (26)

where

LE→(AB) ≡ δ

←AE + δ←BE , (27)

and

L(AB)→E ≡ δ

←EA + δ←EB , (28)

allowing us to understand that the difference between theEOF and the average LII of the pair A,B is equal to thenet flow of LII in and out of the ancilla E. Again, we seethat EAB = ω+

A|B when the systems are fully symmetric.

But the net flow of LII in and out the ancilla E canvanish, also when δ←AE = δ←EB , and δ←BE = δ←EA. In such acase, even though EAB 6= δ←BA 6= δ←AB , the entanglementof Formation EAB is equal to the average LII of the pairAB.

IV. LII AND ENTANGLEMENT SUDDENDEATH

By using the relations here presented, we can investi-gate another important aspect about the distribution of

FIG. 4: (Color Online) Depiction of flow of Locally Inacces-sible Information departing from measurements in E (bluearrows) and concentrating in E (red arrows). The net resultof these two flows is the difference between the EOF and theQD for AB when measurements are made on B.

FIG. 5: (Color Online) Depiction of flow of Locally Inacces-sible Information departing from measurements in E (bluearrows) and concentrating in E (red arrows). The net resultof these two flows is the difference between the EOF and theQD for AB when measurements are made on A.

the entanglement and the quantum discord in a multipar-tite system. We consider a four qubit system, where twoinitially pure entangled qubits A and B interact individ-ually with their own reservoir RA and RB , respectively(for details see [13]). We suppose an amplitude dampingchannel at temperature T = 0 K and we write a map toeach qubit as

Σ (|0〉A|0〉RA) → |0〉A|0〉RA

Σ (|1〉A|0〉RA) →

√1− p|1〉A|0〉RA

+√p|0〉A|1〉RA

, (29)

6

where p = 1 − e−Γt and identically for B interactingwith RB . We choose as the initial condition |Ψ(0)〉 =2√3|0〉A|0〉B + 1√

3|1〉A|1〉B , which is an example where the

phenomenon known as entanglement sudden death [14]occurs. As one can observe by Eq. (13), the entanglementbetween AB suddenly vanishes when the average LII be-tween AB is equal to the balance of the LII between theenvironment and each subsystem (A and B). Actually,as soon as measurements over the environment allowsmore inference about the mutual information with thepair A,B, their entanglement decreases. As illustratedin Fig. (6), when the entanglement between A and Bvanishes (entanglement sudden death), the excess of theknowledge that the environment E has about the subsys-tem A and B, as measured by

$−RARB |A +$−RARB |B = LRARB→(AB) − L(A

B)→RARB,

becomes equal to the average LII in a finite time.To obtain the results plotted in Fig. (6), we analytically

solve the dynamics of EAB and, more importantly, theQD between each subsystem A and B with the wholeenvironment E ≡ RA⊗RB . These results extend furtherthe investigation performed in Ref. [5] and provides anew way to calculate the QD and the EOF for higherdimensional systems. Indeed, the monogamic relationcan be used to calculate the QD and the EOF betweentwo subsystems with dimension 2 × N and rank 2 (seealso [15]). For example, in the exposed case above, wehave [7]

EA(RARB) = δ←AB + SA|B , (30)

where SA|B is the conditional entropy, and

δ←A(RARB) = EAB + SA|B . (31)

V. FUNDAMENTAL RELATIONS

Based upon the previous results, we are able to ob-tain fundamental expressions relating the entanglementof formation and quantum discord. First of all, we com-bine Eq. (13) and Eq. (20) to show that

EAB = δ←AB + δ←BE − δ←EB , (32)

EAB = δ←BA + δ←AE − δ←EA. (33)

These equations are the simplest expressions relatingEOF exclusively to QD. They show that the differencebetween the entanglement and the quantum discord isproportional (twice) to the LII balance of one of the sub-system with the environment. In Eq. (32), we see thatfor an arbitrarily mixed system AB, when the subsystemB is measured in order to know about the mutual infor-mation of the pair AB, some additional information isacquired about the pair BE, and so it needs to be takenin account (2ω−B|E needs to be summed). We also can

FIG. 6: (Color Online) the red curve (solid) shows the entan-glement between the pair AB, while the blue curve (dotted)shows the average LII $+

A|B . The cyan curve (traced) rep-

resents the sum of the balance LII between the environmentand the pair AB. When p ≈ 0.65 the average becomes equalto the sum of the balance and the entanglement sudden deathoccurs.

derive an important result: if the systems B and E aresymmetric, we have that EAB = δ←AB , which is a directconsequence of the fact that for this case δ←BE = δ←EB .Then, we have that the quantum discord δ←AB is equal tothe entanglement of formation EAB , not only when thesystem AB is pure, but also when the systems E andB are symmetric. The same is valid for Eq. (33), δ←ABis equal to EAB , not only when the system AB is pure,but also when the systems E and A are symmetric. Ofcourse, if we have full symmetry between A, B, and Ethen EAB = ω+

A|B , extending this equivalence for pure

AB states only.By using our results, we are able to find a very use-

ful relation between the quantum discord and the condi-tional entropy. The conditional entropy is an importantquantity in information theory that is intimately relatedto the entanglement irreversibility and to the state merg-ing protocol. As well exposed by Horodecki, Oppenheim,and Winter [16], given two subsystems A and B, the con-ditional entropy measures the amount of quantum com-munication that is needed to transfer the full state of onesubsystem (A) to the other (B) given the knowledge ofB. Intriguingly, the conditional entropy can be negativeand it means that B can obtain the full state AB usingonly classical communication. Additionally A and B willbe able to transfer quantum information in the future atno further cost [16]. First of all, let us pay attention tothe case of pure states. For a pure state, we can writethe conditional entropy as

− SA|B = δ←BA. (34)

As exposed above, δ←BA measures the amount of mutualinformation of AB inaccessible by measurements on A.

7

Clearly, A can not transfer this information to the sub-system B and consequently it is preserved for a futurecommunication. In this sense, what could we say aboutmixed states? To calculate the conditional entropy formixed states, in terms of the LII, we use the relationpresented in Ref.[7] EAB = δ←BE + SB|E and Eq. (32).Based on this equation, it is simple to show that

− SA|B = δ←BA − δ←EA. (35)

It is clear now, observing equation Eq. (35), what infact happens for mixed states. As one can see, there isan amount of LII that A shares with E once it can not besent to B. Furthermore, since this information mutuallybelongs to A and E, it can not be used jointly with Bfor further tasks. In fact, it has to be subtracted fromδ←BA. Moreover, by using Eq. 35, it is easy to analyze thenegativity of the conditional entropy, which depends onthe balance of LII. The sign of the conditional entropy hasan important meaning in important tasks like quantumstate merging and entanglement distillation. Again, asfor the EOF relations, more important than the amountof LII it is the balance of LII between the subsystems Aand B and the purifying ancilla (environment). By usingthe balance of LII, we are able to identify the sing ofthe conditional entropy. For instance, if the subsystem

A shares the same amount of LII (independently of theamount) with B and E, the conditional entropy is null,SA|B = 0. If A LII with B is larger that the LII withE, the conditional entropy is negative. Obviously, SAB

is positive when A LII with E is larger than the LII withB.

VI. SUMMARY

To conclude, we present a new operational meaning tothe Entanglement of Formation in terms of the LocallyInaccessible Information flow. Our relations based on av-erage LII and LII balance demonstrates that the EOF canbe understood for a general quantum system exclusivelyas a function of the QD. These new relations elucidate im-portant and yet not well understood phenomenon as theentanglement sudden death and the difference betweenthe entanglement and the quantum discord. Moreover,we relate the QD to the conditional entropy and we finda simple relation between these two quantities. This newrelation presents a different way to look at the negativesignal of the conditional entropy that is ruled by the QDbalance between the environment and the system.

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