Lower bounds on violation of monogamy inequality for quantum correlation measures

Asutosh Kumar∗ and Himadri Shekhar Dhar†

Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad 211 019, India(Dated: June 10, 2016)

In multiparty quantum systems, the monogamy inequality proposes an upper bound on the distri-bution of bipartite quantum correlation between a single party and each of the remaining parties inthe system, in terms of the amount of quantum correlation shared by that party with the rest of thesystem taken as a whole. However, it is well-known that not all quantum correlation measures uni-versally satisfy the monogamy inequality. In this work, we aim at determining the non-trivial valueby which the monogamy inequality can be violated by a quantum correlation measure. Using aninformation-theoretic complementarity relation between the normalized purity and quantum corre-lation in any given multiparty state, we obtain a non-trivial lower bound on the negative monogamyscore for the quantum correlation measure. In particular, for the three-qubit states the lower boundis equal to the negative von Neumann entropy of the single qubit reduced density matrix. Weanalytically examine the tightness of the derived lower bound for certain n-qubit quantum states.Further, we report numerical results of the same for monogamy violating correlation measures usingHaar uniformly generated three-qubit states.

I. INTRODUCTION

The concept of monogamy [1–3] is an intrinsic aspect inthe study of quantum correlations in multiparty systemsand plays a precursory role in quantum security [3–5] andmultiport communication protocols such as super dense-coding [6]. In its most primitive avatar, monogamy ofquantum correlation states: if two parties A and B ina multiparty quantum state (say, %ABC) are maximallyquantum correlated (Q(%AB) = 1), then neither A nor Bcan possess any quantum correlation with a third party,i.e., Q(%AC) = Q(%BC) = 0. The above statement is sat-isfied by all quantum correlation measures, and is a cleardeparture from classical correlations which are not boundto such constraints. Interestingly, monogamy property isnot limited to quantum correlations but is manifested inother quantum properties such as Bell inequality viola-tions [7], quantum steering [8], channel capacities [9], andcontextual inequalities [10].

However, in general, two parties in a quantum stateneed not necessarily share maximal quantum correla-tion, and are thus able to share some correlations withother parties, albeit in a constrained way. The con-cept of monogamy thus restricts the distribution of bi-partite quantum correlations in multiparty quantum sys-tems but its quantification is not always achievable. Forthree-qubit systems, an important figure of merit, themonogamy inequality, was obtained for an entanglementmeasure called the tangle [11]. It was shown that, fora general three-qubit state, %ABC , the amount of tangle(τ) shared between the party A, individually with par-ties B and C, is bounded above by the tangle sharedby A with BC taken as a whole. Mathematically, thisimplies: τ(%A:BC) ≥ τ(%A:B) + τ(%A:C). It is easy toverify, that the inequality satisfies the aforementioned

∗ asukumar@hri.res.in† himadrisdhar@hri.res.in

statement for monogamy. If τ(%A:B) = 1, then τ(%A:C)is necessarily zero, as τ(%A:BC) ≤ 1. Moreover, themonogamy inequality for tangle was shown to exist evenin n-qubit quantum states [12]. In literature, states andquantum correlations that satisfy the above monogamyinequality are termed as monogamous. However, it iswell known that the monogamy inequality is not sat-isfied universally for all quantum correlation measures,even for three-qubit states [13–18]. Entanglement mea-sures such as entanglement of formation [19], apart frominformation-theoretic measures such as quantum discord[20] and known to be, in general, non-monogamous. Re-cent results on monogamy have shown that quantum cor-relation measures may satisfy the monogamy inequalityby minute extensions of the upper bound [21], increasingfractional and integer exponents of quantum correlationmeasures [22], or by considering large number of parties[23] (cf. [24]).

For all quantum correlation measures, one can de-fine a monogamy score [25], which captures the differ-ence between the non-negative quantities, Q(%A:rest), and∑kQ(%A:Bk

), in a quantum state, %AB1B2···Bn. The

monogamy score (δQ) is postive for all measures thatsatisfy the monogamy inequality, and is bounded aboveby the term Q(%A:rest). In recent works, for n-qubit purequantum states, the upper bound for all quantum mea-sures, regardless of the fact that they satisfy the inequal-ity, has been expressed as functions of the genuine mul-tipartite entanglement [26, 27]. The lower bound for δQis trivially zero for situations where the inequality is sat-isfied (as δQ ≥ 0). However, for quantum correlationmeasures that do not satisfy the monogamy inequality,there does not exist a non-trivial lower bound. It is thennatural to ask whether there exists a bound on the valueby which the monogamy inequality can be violated. Theexistence of a non-trivial lower bound on δQ is an impor-tant aspect in the study of various quantum informationprotocols. Monogamy poses a fundamental restrictionon the sharability and distribution of quantum resources,

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and while this provides a significant advantage in obtain-ing bounds on secret key rates in quantum cryptography[4], it also restricts the availability of resources in multi-partite protocols such as dense coding [9]. Moreover, themonogamy score has been used as an important figure ofmerit in studies on multiparty quantum states [14, 25],identification of quantum channels [28], and distinguish-ing phases of many-body systems [29]. However, themonogamy score is a difficult quantity to compute andestimate for generic quantum states and generic quantumcorrelations. It is therefore interesting to derive both up-per and lower bounds of the monogamy score. While theupper bounds of monogamy score are known in the liter-ature [26, 27], our study aims at obtaining a non-triviallower bound of the quantity.

In this work, we determine a non-trivial lower boundon the negative monogamy score of quantum correlationmeasures that violate the monogamy inequality, usingan information-theoretic complementarity between thenormalized purity of the reduced state (%A) and quan-tum correlation shared between two parties (%AB). Weobserve that, under certain conditions, for three-qubitquantum states the lower bound reduces to the nega-tive of the von Neumann entropy [30] of the single-qubitreduced density matrix. Using a set of well-known quan-tum states, we analytically derive the lower bound forlarger number of parties and examine its variation andtightness. Moreover, we numerically evaluate the boundfor a set of entanglement and information-theoretic quan-tum correlation measures by randomly generating Haar-uniform three-qubit states, to show that the bound issatisfied. This paper is organized as follows. In Sec. II,we briefly review the monogamy of quantum correlationmeasures, and the complementarity relation between thenormalized purity of a subsystem with respect to the bi-partite quantum correlation Q in the quantum system.Lower bounds on monogamy scores, in terms of the com-plementarity relation, are determined in Sec. III. We thenanalyze, in Sec. IV, the characteristics of the lower boundusing a few well-known quantum states and quantum cor-relation measures. We conclude in Sec. V.

II. MONOGAMY AND COMPLEMENTARITY

We begin with a brief discussion on the monogamy in-equality, as introduced by Coffman et al. for the tangleof three-qubit states [11], and later expanded for highernumber of qubits [12] and other quantum correlationmeasures. Consider that Q is a bipartite quantum cor-relation measure. For a multipartite quantum systemdescribed by state %AB1B2...Bn

≡ %AB , the monogamyproperty is captured by the inequality

n∑j=1

Q(%ABj) ≤ Q(%AB), (1)

where the party A is the nodal observer. Quantum cor-relation measures that universally satisfy the above in-

equality is said to satisfy monogamy or is monogamous.Similarly, the quantum state, %AB1B2...Bn

, is said to bemonogamous under the quantum correlation measure Q,if it satisfies Eq. (1). We note that the state need notbe monogamous for all quantum correlations measures,and in general, can be non-monogamous. An importantfigure of merit to investigate the monogamy inequalityis the monogamy score [25]. It encapsulates the deficitbetween the quantities in the monogamy inequality, andis defined as

δQ = Q(%AB)−n∑j=1

Q(%ABj). (2)

An obvious extension of the monogamy inequality is thatthe δQ is non-negative (δQ ≥ 0) for all monogamous Q.One can interpret a positive δQ as “residual” quantumcorrelation of an n-party quantum state that is not cap-tured by the distributed bipartite quantum correlations,which in essence captures some form of multipartite cor-relations in the system [26]. Several quantum correla-tion measures, including entanglement monotones suchas tangle [11], squashed entanglement [31] and squarednegativity [32], are monogamous, as opposed to mea-sures such as entanglement of formation [19], quantumdiscord [20], negativity [33], logarithmic negativity [34],and quantum work-deficit [35, 36].

We now discuss an information-theoretic complemen-tarity relation, for multipaty quantum states, betweenthe normalized purity of a subsystem of the state andthe bipartite quantum correlation the subsystem shareswith rest of the system, as derived in Ref. [37]. Letus consider a bipartite quantum correlation measure Q,for quantum state %XY ∈ CdX ⊗ CdY , where X and Yare two subsystems of the state. Let us say, Q satis-fies the conditions (i) Q(%XY ) ≤ S(%X), which is truefor a host of quantum correlation measures [38], and (ii)0 ≤ Q(%XY ) ≤ log2 dY . For such measures, it has beenshown that Q obeys a non-trivial complementarity rela-tion between purity of the reduced state, %X , and theshared bipartite quantum correlation in the state, %XY ,as given below [37]:

P(%X) + Q(%XY ) ≤ b |

{b = 1, if dX ≤ dY ,b = 2− log2 dY

log2 dX, if dX > dY ,

(3)where P is defined as,

P(%X) =log2 dX − S(%X)

log2 dX, (4)

and quantifies the normalized purity of the system [39]in the X-part, and Q(%XY ), given by

Q(%XY ) =Q(%XY )

min{log2 dX , log2 dY }, (5)

represents the normalized quantum correlation of the sys-tem in the X:Y bipartition. For example, an impor-tant quantity that satisfies the complementarity relation

3

is the quantum mutual information between two par-ties, defined as, I(%XY ) = S(%X) + S(%Y ) - S(%XY )[40]. I(%XY ), which captures the total correlationsin a system, %XY , satisfies the condition I(%XY ) ≤2 min{S(%X), S(%Y )} ≤ 2 min{log2 dX , log2 dY }. Henceif one considers the correlation, Q(%XY ) = I(%XY )/2,the complementarity is naturally satisfied. Moreover, fortripartite quantum states, %ABC ∈ Cd ⊗ Cd ⊗ Cd, thecomplementarity relation reduces to

P(%AB) + Q(%AB:C) ≤ 3

2, (6)

and is independent of the dimension of the system, andcan be shown to be saturated by the Greenberger-Horne-Zeilinger (GHZ) state [41].

This complementarity relation has potential applica-tion in quantum information protocol. In particular, ithas been used to obtain a security proof of quantum cryp-tography for individual attacks via a variation [42] of theEkert key distribution [1] protocol. We note that thecomplementarity relation in Eq. (3), is satisfied by a hostof quantum correlation measures, irrespective of the con-ditions (i) and (ii), as shown in Appendix A (cf. [37]).

III. LOWER BOUNDS ON MONOGAMYSCORES

An important aspect of the monogamy score, inEq. (2), is that the quantity has a distinct upperbound given by the term Q(%AB), where we use %AB= %A(B1···Bn). This implies that for monogamous Q, theamount of distributed bipartite entanglement in a multi-partite state is bounded by the amount of block entangle-ment shared by a single party with the rest of the system.Interestingly, it has been shown that the upper bound ofδQ, for a large number of pure n-qubit states, is deter-mined by a set of entropic or quadratic functions of thegenuine multipartite entanglement [43, 44] of the state%AB [26, 27]. Moreover, for monogamous Q, there existsa definite lower bound on δQ, given by δQ > 0. How-ever, for Q that does not satisfy monogamy inequality, ingeneral, the situation is not straightforward. Though theupper bound remains the same, a non-trivial lower boundis not obvious. For cases where %AB is not monogamouswith respect to Q, δQ can be negative. One of the pri-mary motivation of our work is to find out the non-trivialdegree or limit to which this violation of monogamy in-equality occurs. In other words, we question, what is theworst negative value of the quantity δQ?

In this section, we aim at obtaining the lower boundon monogamy scores, for non-monogamous measures Q,by making use of the complementarity relation, given inEq. (3). In deriving the lower bounds on δQ below, wewill use the notation QXY ≡ Q(%XY ), QXY ≡ Q(%XY ),PX ≡ P(%X), among others, and where %X = TrY %XY .Consider an arbitrary multipartite quantum state %AB ≡%A(B1B2...Bn) of n+ 1 parties. For a normalized bipartite

quantum correlation measure Q, a trivial lower bound onthe monogamy score is provided in terms of the numberof parties in the system B (= B1B2 . . . Bn), as given by

δQ = QAB −n∑k=1

QABk≥ −(n− 1). (7)

To obtain a non-trivial lower bound on the δQ for non-monogamous measures, we use the complementarity re-lation as shown below. For ρAB , using the relation inEq. (3), we can write

PA + QAB = x0 (say) ≤ b0or , QAB = x0 − PA. (8)

Similarly, for the reduced density matrices, %ABk,

where all parties except A and Bk have been tracedout, i.e., %ABk

= TrABk(%AB), we obtain PA + QABk

= xk ≤ bk. Summing over all k, we get the relation

nPA +

n∑k=1

QABk=

n∑k=1

xk ≤n∑k=1

bk. (9)

When dA ≤ dBk(say, dA = dBk

= d), using the com-plementarity relation (3), one has bk = 1, ∀ bk. Hence,Eq. (9) can be written as

nPA +

n∑k=1

QABk=

n∑k=1

xk ≤ n

or,

n∑k=1

QABk≤ n(1− PA). (10)

Subtracting Eq. (10) from Eq. (8), we obtain an expres-sion for the monogamy score, δQ, which upon arranging,gives us

δQ = QAB−n∑k=1

QABk≥ −(n−1)(1−PA)−(1−x0). (11)

By rearranging Eq. (11), one can show that the obtainedlower bound on monogamy score can be improved, incomparison to Eq. (7), as is evident from

δQ ≥ −(n− 1)

(1− PA +

1− x0

n− 1

). (12)

For large n or for x0 ≥ 1 [45], the lower bound onmonogamy score reduces to

δQ ≥ −(n− 1)(1− PA), (13)

where PA is the normalized purity defined in Eq. (4),and ranges from 0 ≤ PA ≤ 1. For the important three-qubit states, which have received a lot of attention in thestudy of monogamy, we observe that the lower bound onthe monogamy score is given by, δQ ≥ −S(%A), whereS(%A) is the von Neumann entropy of the subsystem A.This shows that the lower bound satisfies the limits formonogamy (δQ & 0) for states that are weakly entangledalong the A : rest bipartition, such that S(%A) ≈ 0.

4

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FIG. 1. Histograms depicting the frequency (f) of states for the quantity δQ + S(%A), for different quantum correlationmeasures, Q. The graph corresponds to a set (∼ 106) of random Haar uniform generated rank-1 and rank-2 three-qubit states.The monogamy score is calculated for the quantum correlation measures given by (a) negativity (N ), (b) logarithmic negativity(L), (c) quantum discord (D), (d) quantum work-deficit (∆), (e) quantum mutual information (I), and (f) conditional mutualinformation (J ). The histogram shows that δQ + S(%A) ≥ 0, and thus satisfies δQ ≥ −S(%A).

IV. ANALYZING THE LOWER BOUND

Let us now examine the lower bound of the monogamyscore, δQ, for certain quantum states. For example, weconsider (n + 1)-party quantum states %AB1···Bn

∈ C2 ⊗CdB1 ⊗ · · · ⊗ CdBn . For these states, the lower bound inrelation (13) becomes δQ ≥ −(n−1)S(%A). For dBk

= 2,∀ k, the above expression for the lower bound of δQ holdsfor all multiqubit quantum states.

Let us consider the superposition of the generalizedGHZ [41] and the W state [46, 47], which are permuta-tionally invariant multiqubit states given by

|Ψα,γ〉 = α|0〉⊗n + β|1〉⊗n + γ|Wn〉, (14)

where |Wn〉 is the normalized n-qubit W state, givenby |Wn〉 = 1√

n

∑P(|0〉⊗(n−1) ⊗ |1〉

), where the sum in

the expression is over all P permutations of the prod-uct state containing (n − 1) |0〉’s, and a single |1〉. α,β, and γ are complex numbers that satisfy the normal-ization constraint |α|2 + |β|2 + |γ|2 = 1. The n-qubitstate, |Ψα,γ〉, satisfies the lower bound for the monogamyscore, given by δQ ≥ −(n − 2)S(%α,γ), where %α,γ =Trn−1(|Ψα,γ〉〈Ψα,γ |) is obtained by tracing out n − 1qubits. %α,γ can be written as

%α,γ =

|α|2 + n−1n |γ|

2 αγ∗√

1n

α∗γ√

1n

1n |γ|

2 + |β|2

. (15)

The largest eigenvalue of the single qubit state %α,γ isgiven by

e =1

2

(1 +

√1− 4|α|2|β|2 − 4(n− 1)

n|γ|2(|β|2 +

|γ|2n

)

).

Hence, S(%α,γ) is equal to h(e), where h(x) is the Shan-non (binary) entropy [48] of the variable x. The lowerbound of the monogamy score is given by, δQ ≥ −(n −2)h(e). Hence, a tight lower bound on the monogamyscore is obtained for low values of h(e).

If we set, γ = 0, then we obtain the generalizedGHZ state, |ΨGHZ〉 = α|0〉⊗n + β|1〉⊗n, and thelargest eigenvalue of the single qubit reduced state is12

(1 +

√1− 4|α|2|β|2

), which is independent of n. For

states with |α|2 = |β|2 = 1/2, e = 1/2, and h(e) = 1,which gives us a weak bound. However, for states with|α|2|β|2 ≈ 0, e ≈ 1, which implies h(e) ≈ 0. For thesestates, the lower bound of the monogamy score is givenby, δQ ≥ −ε, where ε → 0. Alternately, if one sets, α= β = 0, such that γ = 1, one obtains the n-qubit Wstate, given by |ΨW 〉 = |Wn〉. The maximum eigenvalueof the reduced state is then given by, e = n−1

n , and h(e)= h(1/n). Hence, as n increases, the quantity h(e)→ 0,and hence we obtain, δQ & 0.

To expand the study on W states, we consider the n-qubit Dicke state [49] containing all P permutations ofn − r qubits in |0〉 and r qubits in |1〉 state, as given in

5

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δ

-1

-0.8

-0.6

-0.4

-0.2

0

-S(ρA)

JNLD∆I

FIG. 2. (Color online.) Tightness of the bound δQ ≥ −S(%A).The score, δ ≡ δQ, of monogamy inequality violating quantumcorrelation measures for a set of randomly generated Haaruniform three-qubit states, plotted along the x-axis, with re-spect to the negative von Neumann entropy (−S(%A)) of partyA, plotted along the y-axis. The measures are negativity(black-circle), logarithmic negativity (red-square), quantumdiscord (blue-diamond), quantum work-deficit (green-cross),quantum mutual information (indigo-up-triangle), and condi-tional mutual information (violet-down-triangle). The figureshows that δQ ≥ −S(%A). It is evident that the lower boundis tight for states with low reduced entropy. For the easeof viewing and without affecting the results, the plot is pro-vided for a set of 2×104 rank-1 and rank-2 three-qubit states,drawn from a larger sample set (∼ 106) of randomly generatedHaar uniform states. Along with Fig. 1, the figure numeri-cally reasserts the lower bound on monogamy score, given byδQ ≥ −S(%A).

the equation

|Ψr,n〉 =

(n

r

)−1/2∑P(|0〉⊗(n−r) ⊗ |1〉⊗r

). (16)

For r = 1, the Dicke state is the same as an n-qubit Wstate, |ΨW 〉, mentioned earlier. The single qubit reducedstate eigenvalue, e, is equal to r

n (or n−rn ), and hence

h(e) = h(r/n). Hence, as the rn ratio decreases, h(e)

decreases, and we obtain, δQ & 0. However, for r = n/2,h(e) = 1, and the lower bound on the monogamy scoreis weak.

Now we look at the monogamy score for a set of quan-tum correlation measures. We generate a large numberof random Haar uniform three-qubit states, and computeδQ for a set of monogamy inequality violating measuressuch as negativity (N ), logarithmic negativity (L), quan-tum discord (D), quantum work-deficit (∆), quantummutual information (I), and conditional mutual informa-tion (J ). For the generated pure and mixed three-qubitstates, and for the above measures, it is observed that themonogamy score satisfies the lower bound, δQ ≥ −S(%A),as shown in Fig. 1, which shows an histogram of the fre-quency of states against the quantity δQ + S(%A). Thefigure shows that δQ + S(%A) is always positive. To check

the tightness of the bound, we plot the monogamy score,δQ, against −S(%A) in Fig. 2. It is evident from the figurethat the lower bound is tight for states with low reducedvon-Neumann entropy.

V. CONCLUSION

Quantum correlations are an important aspect of mod-ern physics and a key enabler in quantum communi-cation and computation technologies. The concept ofmonogamy is a distinguishing feature of quantum cor-relations, which sets it apart from classical correlations,and has played a significant role in devising quantumsecurity in secret key generation and multiparty commu-nication protocols. In recent years, a significant amountof research has been devoted to understand the role ofmonogamy in various quantum phenomena, including vi-olation of Bell inequalities and contextuality, and also ininvestigating correlation properties beyond quantum me-chanics, such as in no-signalling theories. Lots of efforthave also been spent in devising stronger monogamy con-ditions [50–54] and to extend known features in discretequantum states to continuous variable systems [55].

An important tool in quantitatively capturing theconcept of monogamy of quantum correlations is themonogamy inequality, which bounds from above the dis-tribution of quantum correlations among different partiesin a multiparty state. The monogamy inequality can bestudied in terms of the monogamy score. All monogamyinequality satisfying measures have a positive monogamyscore, which is bounded above by a well defined valuethat can be derived as functions of the genuine multipar-tite entanglement of the quantum state. However, not allquantum correlation measures satisfy the monogamy in-equality, and in general, the monogamy score can be neg-ative. Interestingly, there exists no perceptible limitingvalue on the negative monogamy score. The main contri-bution of our work is the determination of a non-triviallower bound of the monogamy score for measures andstates where the monogamy inequality is violated. Thisis achieved using a complementarity relation between thenormalized purity of a subsystem and the bipartite quan-tum correlation in the system. Subsequently, we analyzethe strength and weakness of the lower bound for dif-ferent quantum states and measures of quantum corre-lation, and observe conditions that immediately lead tomonogamy. The results in the paper provide a unifyingframework to study monogamy relations in both entan-glement and information-theoretic quantum correlations,and are an important addition to the study of monogamythat opens possible directions for further investigation.

ACKNOWLEDGMENTS

The authors would like to thank Ujjwal Sen for fruitfulsuggestions and discussions on the results.

6

Appendix A: Quantum correlation measures and thecomplementarity relation

In our work, we investigate the lower bound on themonogamy score of quantum correlation measures thatviolate the monogamy inequality, using a complementar-ity relation between the normalized purity and normal-ized bipartite quantum correlation. In this Appendix,we discuss some entanglement, and information-theoreticquantum correlation measures that we have used in ourstudy (see Figs. 2 and 3), and look at the validity of thecomplementarity relation from the context of our results.

We begin with two measures of entanglement that are,in general, known to violate the monogamy inequality,namely, the negativity [33] and the logarithmic negativity[34]. Negativity (N ) is a computable measure of bipartiteentanglement, which is defined in terms of the eigenval-ues of the partially transposed matrix %ΓA

AB . N is thesum of the absolute value of the negative eigenvalues of

%ΓA

AB , or equivalently, N (%AB) =‖%ΓA

AB‖1−1

2 , where ‖ · ‖1is the trace-norm. For upto dimension 2 × 3, N = 0for separable states. An extension of the entanglementmeasure of negativity, is the non-convex entanglementmonotone, called the logarithmic negativity (L). Mathe-

matically, L = log2 ‖%ΓA

AB‖1, and in terms of N , it is equalto log2(2N +1). Both N and L are non-monogamous en-tanglement monotones.

To analyze information-theoretic quantum correla-tions, we consider the measures of quantum discord [20]and quantum work-deficit [35]. Along the way, we alsoconsider the quantum mutual information [40], as a mea-sure of total correlations, and the measured mutual in-formation, which has been characterized as classical partof total correlations [20, 56]. Information-theoretic mea-sures of quantum correlations draw upon specific prop-erties of classical information theory [57], and puts themunder a quantum microscope. For example, consider twoexpressions of mutual information between two parties:I(%AB) = S(%A) + S(%B) − S(%AB), and J (%AB) =S(%B) − S(%B|A). From a classical context, if S(x) isthe Shannon entropy and %AB , along with %A and %B ,are classical variables, the quantities I and J are equal.However, from a quantum perspective, where S(x) is thevon Neumann entropy and %AB is a bipartite quantumdensity matrix, I is the quantum mutual information andJ is the measured mutual information, and are, in gen-eral, not equal. I, captures the total correlation betweenthe two parties in %AB , whereas J captures the classi-cal part of the mutual information. This is due to thefact the conditional entropy, S(%B|A), used to define Jinvolves local measurements on one of the subsytems,which washes away all quantum correlations. Quantumdiscord (D), in a bipartite quantum state, is defined asthe difference between the two quantities: D(%AB) =I(%AB)−J (%AB). Another measure defined using an in-formation perspective is the quantum work-deficit (∆),which is derived as the deficit between the amount of pure

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f

(d)

FIG. 3. Histograms depicting the frequency (f) of states fordifferent values of x0, in the complementarity relation, PA +QA:BC ≤ x0. The graph corresponds to a set (∼ 106) of ran-dom Haar uniform generated rank-1 and rank-2 three-qubitstates, and the quantum correlation measures, given by (a)negativity, (b) logarithmic negativity, (c) quantum discord,and (d) quantum work-deficit. For the measures N and L,in subfigures (a) and (b), the rank-1 states correspond to thetaller peaks at x0 & 1, while the rank-2 states are evenly dis-tributed in the region x0 . 1. We note that for rank-1 states,PA + QA:BC = x0 = 1, for the measures D and ∆. Hence,the graph for these measures, in subfigures (c) and (d), corre-spond only to rank-2 states. We observe that x0 is well belowthe trivial value, x0 < 2, for all states.

quantum states that are permitted for extraction underthe set of closed global operations and product statesthat are obtained under a class of closed local operationsand classical operations (closed LOCC). For a bipartitequantum state %AB , the number of pure qubits obtainedunder global operations, performed through sequentialapplication of unitary and dephasing operations, is givenby IG(%AB) = N − S(ρAB), where N = log2 d. Here,d is the dimension of the system. Under closed LOCCoperations, optimized via local unitary and dephasingoperations, and classical exchange of information, thenumber of pure classical states obtained is IL(%AB) =N−infΛ[S(%′AB)]. Λ belongs to the class of closed LOCCoperations. The quantum correlation measure ∆ is de-fined as: ∆ = IG − IL.

The complementarity relation, defined in Eq. (3), holdsfor quantum correlation measures that satisfy the condi-tions, (i) Q(%XY ) ≤ S(%X), which holds for a host ofquantum correlation measures, such as entanglement offormation, distillable entanglement, entanglement cost,quantum mutual information, relative entropy of en-tanglement, and classical correlation [37, 38], (ii) 0 ≤Q(%XY ) ≤ log2 dY , which is known to be definitely satis-fied by quantum mutual information. It should be noted

7

that there may exist quantum correlation measures Qthat may not satisfy one or both of the above conditions,or where the proof is not easily accessible, but may stillsatisfy the non-trivial complementarity relation given by,P(ρX) +Q(ρXY ) ≤ x0, where x0 < 2. An analysis of thequantum correlation measures of negativity, logarithmic

negativity, quantum discord, and quantum work-deficit,for random Haar uniform generated rank-1 and rank-2three-qubit states are shown in Fig. 3. A similar resultwas also observed in [37]. We note that for rank-1 three-qubit states, PA + QA:BC = 1, for D and ∆. This is notthe case for the entanglement measures, N and L.

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