Monogamy of quantum correlations - a review

Himadri Shekhar Dhar, Amit Kumar Pal, Debraj Rakshit, Aditi Sen(De), and Ujjwal SenHarish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad 211 019, India and

Homi Bhabha National Institute, Training School Complex, Anushaktinagar, Mumbai 400 085, India

Monogamy is an intrinsic feature of quantum correlations that gives rise to several interesting quantum char-acteristics which are not amenable to classical explanations. The monogamy property imposes physical restric-tions on unconditional sharability of quantum correlations between the different parts of a multipartite quantumsystem, and thus has a direct bearing on the cooperative properties of states of multiparty systems, includinglarge many-body systems. On the contrary, a certain party can be maximally classical correlated with an arbi-trary number of parties in a multiparty system. In recent years, the monogamy property of quantum correlationshas been applied to understand several key aspects of quantum physics, including distribution of quantum re-sources, security in quantum communication, critical phenomena, and quantum biology. In this chapter, we lookat some of the salient developments and applications in quantum physics that have been closely associated withthe monogamy of quantum discord, and “discord-like” quantum correlation measures.

I. INTRODUCTION

Quantum correlations, shared between two or more par-ties [1, 2], boast of novel features, which are exclusive tothe quantum world and are central to quantum informa-tion science. On one hand, they play a significant role inefficient quantum communication [3–5] and computational[6, 7] tasks, while on the other hand, they help us to under-stand cooperative phenomena in quantum many-body sys-tems [8–10]. A key difference between classical and quan-tum correlations is the way they can be shared among variousparts of a multiparty quantum system. Unlike classical cor-relations, which can be freely shared, the sharability of quan-tum correlations is restricted by the non-classical propertiesof the quantum system. For example, in a tripartite quantumstate, ρABC , if two parties A and B are maximally quantumcorrelated, then none of A and B can share any quantumcorrelation with the third party, C [4, 11–14] (For a socialrepresentation, see Fig. 1). However, there exists no suchconstraints for the classical correlations, and the pairs of par-ties, say AB and AC, can simultaneously share maximumclassical correlations. Similar situation arises in multipartyquantum systems of more than three parties. This exclusivetrade-off between quantum correlations of different combi-nations of parties in a multiparty quantum system is knownas monogamy of quantum correlations [4, 11–14]. The no-gotheorems, like the no-cloning theorem [15–18], put restric-tions on the available options in quantum cryptography [3].Similarly, the monogamy of quantum correlations is a re-striction on the sharability of quantum correlations, and yethelp in obtaining advantages in a quantum system over theirclassical counterparts.

In the seminal work by Coffman, Kundu, and Wootters(CKW) [13], a monogamy relation for three-qubit pure stateswas established by using an entanglement measure, namely,the squared concurrence [19, 20]. This scenario was latergeneralized by Osborne and Verstraete [21] for pure as wellas mixed states of an arbitrary number of qubits. Since quan-tum correlations do not have a unique quantification evenin the bipartite domain, the immediate question that followsis whether the monogamy inequality proposed by CKW isnecessarily obeyed by all kinds of quantum correlation mea-

sures. In general, quantum correlations can be broadly clas-sified into two categories – entanglement measures [1], andthe information-theoretic measures of quantum correlations[2]. While entanglement of formation [12, 22, 23], concur-rence [19, 20], distillable entanglement [24, 25], negativity[26–31], logarithmic negativity [28–31], relative entropy ofentanglement [32–34], etc. belong to the first category, quan-tum discord [35, 36], and quantum work deficit [37–40] areexamples of the second kind. Although quantum correlationsare qualitatively monogamous, not all of them are limited byonly the form of monogamy constraint proposed by CKW.In particular, while the squared concurrence and negativitysatisfy the CKW monogamy inequality for all three-qubitpure states [13, 21, 41], many others, such as logarithmicnegativity and information-theoretic measures, do not satisfythe same [41–45]. Deliberations on the monogamy of quan-tum correlations have led to important insights including a“conservation law” between entanglement and information-theoretic quantum correlations in multiparty quantum states[42]. In Ref. [46], the authors formulate the requirements fora bipartite entanglement measure to be monogamous for allquantum states, and show that additive and suitably normal-ized entanglement measure, which can faithfully describe thegeometric structure of the fully antisymmetric state, are non-monogamous. However, it is also understood that all kinds ofquantum correlations obey the CKW monogamy constraintfor a given state when raised to a suitable power, providedthey follow certain conditions [47].

Interestingly, the limitation imposed by the quantum me-chanical principles, in form of monogamy constraints, is notexclusive to quantum correlations. There exists no-go the-orems which place parallel restrictions such as monogamyof Bell inequality violation [48, 49] and exclusion principleof classical information transmission over quantum channels[50] (cf. [51–54]). More precisely, within the considerationof a multiparty set-up, for example, of an editor with sev-eral reporters, if the shared quantum state between the editorand a single reporter violates a Bell inequality [55, 56] or isquantum dense codeable [57], then the rest of the channelsshared between the editor and the other reporters are prohib-ited from possessing the same quantum advantage. Analo-gous monogamy constraints have also been addressed in the

arX

iv:1

610.

0106

9v1

[qu

ant-

ph]

4 O

ct 2

016

2

FIG. 1. (Color online) Monogamy: Two persons sharing an um-brella are unmindful to the presence of the third person.

context of quantum steering [58, 59], quantum teleportationfidelity [60], and contextual inequalities [61, 62].

The monogamy properties of quantum correlations findpotential applications in quantum information based proto-cols like quantum cryptography [3], entanglement distilla-tion [22], quantum state and channel discrimination [44, 45,63], and in characterizing quantum many-body systems [64–72] as well as in biological processes [73, 74]. The keyconcept of entanglement-based quantum cryptography es-sentially exploits the trade off in monogamy of quantum cor-relations, which limits the amount of information that aneavesdropper can extract about the secret key, shared be-tween a sender and a receiver, obtained via measurement onboth sides of an entangled state between the sender and thereceiver [4, 11, 75]. The constraints on shareability of entan-glement finds further application in enhancing quantum pri-vacy via entanglement purification [76]. Another importanceof monogamy relations, arising due to the constraints thatthey put on the distribution of quantum correlation amongmany parties, is their ability to capture multipartite quantumcorrelations present in the system, the latter being, in general,a challenging task [13, 77, 78]. Moreover, they play a deci-sive role in designing the structure of eigenstates of quantumspin models, which are expected to obey the no-go principlesarising from the monogamy constraints [67].

In this chapter, our main aim is to review the results onmonogamy of quantum discord, and other “discord-like”measures of quantum correlations. We survey the proper-ties of a proposed multiparty quantum correlation measure,called the “monogamy score”, and its relations with othermeasures of quantum correlations. Finally, we also takea look at the usefulness of the monogamy score for quan-tum discord in quantum information science, and in quan-tum many-body physics. The structure of this chapter is asfollows. In Section II, we present a short review on the stud-ies that have been carried out on the monogamy propertiesof entanglement. Section III consists of the definitions of

several quantum correlation measures, such as quantum dis-cord, quantum work deficit, and geometric quantum discord,that have been defined independent of the entanglement-separability paradigm, The monogamy properties of thesemeasures are discussed in the following sections. In SectionIV, we focus on the monogamy of quantum discord. Themonogamy property of other information-theoretic quantumcorrelation measures are discussed in Section V. Section VIprovides a report on the relation between the monogamyof quantum correlations with other multiparty measures. InSection VII, we review some noteworthy applications of themonogamy property. Section VIII contains some concludingremarks.

II. MONOGAMY RELATIONS OF ENTANGLEMENT

In this section, we provide a brief discussion on themonogamy properties of different entanglement measures,such as concurrence, negativity, etc. Starting from tripar-tite quantum systems, we expand the discussion to the morecomplex cases in multiparty systems. For a review, see [14].

We also formally define the “monogamy score” corre-sponding to an arbitrary quantum correlation measure.

A. Tripartite system: CKW inequality and beyond

In this subsection, we review monogamy properties of en-tanglement in the tripartite scenario, using concurrence asthe entanglement measure. For an arbitrary two-qubit quan-tum state ρAB , concurrence is defined as [19, 20] CAB ≡C(ρAB) = max{0, λ1 − λ2 − λ3 − λ4}, where {λi}, , i =1, . . . , 4, are square roots of the eigenvalues of the positiveoperator ρρ in descending order, and the spin-flipped den-sity matrix ρ is given by ρ = (σy ⊗ σy)ρ∗(σy ⊗ σy). Hereand henceforth, σx, σy and σz denote the standard Paulispin matrices. Note that C vanishes for the separable statesand attains the value 1 for the maximally entangled statesin C2 ⊗ C2 systems. The physical significance of concur-rence stems from the fact that the entanglement of formationis a monotonic function of concurrence, and vice versa, inC2 ⊗ C2 [19, 20]. A formal definition of entanglement offormation is given later in this section.

Consider a three-qubit pure state, ρABC = (|φ〉〈φ|)ABC .The concurrences corresponding to the the reduced densitymatrices ρAB = TrC(ρABC) and ρAC = TrB(ρABC) sat-isfy the inequalities, C2

AB ≤ Tr(ρAB ρAB) and C2AC ≤

Tr(ρAC ρAC), respectively. It can further be shown thatTr(ρAB ρAB) + Tr(ρAC ρAC) = 4 detρA. This leads to fol-lowing inequality [13]:

C2AB + C2

AC ≤ 4 det ρA, (1)

where ρA = TrBC(ρABC). Even though BC is a four-dimensional system, the support of ρBC = TrA(ρABC) isspanned by the eigenstates corresponding to at most twonon-zero eigenvalues of the reduced density matrix ρBC ,and hence is effectively defined on a two-dimensional space.

3

This allows to treat the bipartite split of A and BC as an ef-fective two-qubit system whose concurrence, CA:BC , is sim-ply 2

√det ρA. As a result, the inequality in (1) becomes

C2AB + C2

AC ≤ C2A:BC , (2)

and is referred to as the monogamy inequality for squaredconcurrence in the case of three-qubit pure state. For a three-qubit mixed state, the state ρBC may, in principle, have allfour non-zero eigenvalues. However, a generalization of theabove inequality for the mixed state is prescribed by replac-ing the right hand side of (2) by the minimum average con-currence squared over all possible pure state decompositions{pi, |ψi〉} of ρBC , and hence (2) also holds for arbitrarymixed three-qubit states.

In this context, let us define the tangle for a three-qubitpure state, ρABC , expressed in terms of squared concur-rences, as [13]

τABC = C2A:BC − C2

AB − C2AC . (3)

It turns out that τABC , which is also known as the three-tangle, or the residual entanglement, is independent of thechoice of the “node”, which is the siteA here. The tangle hasbeen argued to characterize three-qubit entanglement. Thegeneralization of the tangle to mixed states can be obtainedby the convex roof extension, the computation of which isdifficult.

Next, let us present the definition of the entanglementof formation (EoF) [20], a measure of bipartite entangle-ment, and its relation with concurrence for two-qubit states.Consider a bipartite quantum state ρAB , and the ensemble{pi, |ψi〉} denoting a possible pure state decomposition ofρAB , satisfying ρAB =

∑i pi|ψi〉〈ψi|. The EoF is defined

as

Ef (ρAB) = min{pi,|ψi〉}

∑i

piS(TrB [|ψi〉〈ψi|]), (4)

where S(TrB [|ψi〉〈ψi|]) is the von Neumann entropy of thereduced density matrix corresponding to the A part of ρAB .A compact formula for the EoF is known for two-qubit sys-tems in terms of the concurrence, CAB . For a two-qubitmixed state ρAB , Ef (ρAB) = h

((1 +

√1− C2

AB)/2)

[20], where h(x) = −x log2 x − (1 − x) log2(1 − x). TheEoF, being a concave function of squared concurrence, doesnot obey the CKW inequality. However, the EoF can alsonot be freely shared amongst the constituents of a multipartysystem. In fact, the square of the EoF does obey the same re-lation as the squared concurrence for tripartite systems [79].

Several studies have addressed the monogamy property ofconcurrence within the tripartite scenario. We briefly men-tion some important findings in this direction. It is natural toask whether the monogamy inequality is satisfied in tripartitesystems consisting of higher-dimensional parties. The analy-sis understandably becomes complex with increasing dimen-sion of the Hilbert space, as much less is known about thequantification of entanglement in higher dimensional cases.For example, the exact formula for EoF and concurrence aremissing in higher dimensions. However, there have beendefinitive efforts to understand the monogamy constraints in

higher dimensional systems. The results reported in Ref. [80]indicate that the CKW inequality cannot be directly extendedto higher-dimensional states. However, it has been demon-strated that the squared concurrence satisfies the monogamyrelation for arbitrary pure states in C2 ⊗ C2 ⊗ C4 [81] (cf.[82, 83]). The monogamy property of squared concurrencein higher-dimensional systems of more than three parties hasalso been addressed [84]. Other approaches to constructmore monogamy inequalities for entanglement in tripartitestates have made use of the generalized concurrence [85],which is a multipartite measure of entanglement.

B. Monogamy score

Just like for squared concurrence, one can formulate theproblem of monogamy for arbitrary quantum correlationmeasures. For a given bipartite quantum correlation mea-sure, Q, and a three-party quantum state, ρABC , we call thestate to be monogamous for the measure Q if

Q(ρA:BC) ≥ Q(ρAB) +Q(ρAC). (5)

Otherwise, the state is non-monogamous for that measure.The measure is called monogamous for a given tripartitequantum system if it is monogamous for all tripartite states.Similar to the spirit of tangle defined in Eq. (3), one candefine a quantity called monogamy score [77], of a bipartitequantum correlation measure, Q, based on the monogamyrelation, given in (5). It is given by

δQ = Q(ρABC)−Q(ρAB)−Q(ρAC), (6)

where we call the party “A” as the nodal observer. With thisnotion, the tangle [13] can be called monogamy score forsquared concurrence, and can also alternatively be denotedby δC2 . This definition can be extended to an N -party quan-tum state, ρ12...N . An arbitrary N -party state is said to bemonogamous with respect to Q, if

Q(ρj:rest) ≥∑k 6=j

Q(ρjk), (7)

and the corresponding monogamy score, for any quantumcorrelation measure, Q, is defined as

δjQ(ρ1:23...N ) = Q(ρj:rest)−∑k 6=j

Q(ρjk), (8)

with j as the node. Here, ρjk represents the two-party den-sity matrix, which can be obtained from ρ12...N by tracingout all the other parties except j and k (j, k = 1, 2, . . . , N ).It has been argued [13, 77] that the monogamy score quan-tifies a multiparty quantum correlation measure, and henceconstitutes a method of conceptualizing a multiparty mea-sure using bipartite ones. A monogamous quantum corre-lation measure, for a given node j, have δjQ ≥ 0 for allmultipartite states. Unless otherwise stated, we always usefirst party as the nodal observer and in that case, we denotemonogamy score as δQ, discarding the superscript. Hence-forth, we shall always describe an N -party quantum state,

4

pure or mixed, by ρ12...N , while for ease of notations, wedenote a three-party quantum state by ρABC .

In an N -party scenario, Coffman, Kundu, and Woottersconjectured a generalization of the CKW inequality to N -qubit pure states, ρ12...N , i.e., the inequality in (7), by replac-ing Q with C2. Some time later, the conjecture was provenfor arbitrary N -qubit states by Osborne and Verstraete [21],by using an inductive strategy. Also, there have been severalattempts to construct generalized monogamy inequalities forentanglement in qubit systems [85–90].

C. Monogamy of negativity and other entanglementmeasures

The negativity [26–31], N (ρAB), corresponding to thequantum state ρAB defined on the Hilbert space CA ⊗ CBfor two parties A and B, is defined by NAB ≡ N (ρAB) =

(‖ρTA

AB‖1 − 1)/2, where ρTA

AB is obtained by performing thepartial transposition on the state ρAB with respect to the sub-system A [26, 27], i.e., (ρTA)ij,kl = (ρ)kj,il, and where‖ρ‖1 = Tr

√ρρ† denotes the trace norm of the matrix ρ.

For systems in C2 ⊗ C2 and C2 ⊗ C3 systems, N > 0 isthe necessary and sufficient for non-separability. In order toachieve a maximum value of unity in C2⊗C2, the negativitycan be redefined as NAB = ‖ρTA

AB‖1 − 1.For any pure three-qubit state ρABC , the monogamy in-

equality for squared negativity [41],

N 2AB +N 2

AC ≤ N 2A:BC , (9)

holds, where A has been chosen as the nodal party, and in(5), Q is replaced by N 2. For any three-qubit pure state,it turns out that NA:BC = CA:BC . Moreover, as shown inRef. [91] for arbitrary two-qubit mixed states, we have

NAB = ‖ρTA

AB‖1 − 1 ≤ CAB . (10)

Thus for any three-qubit pure state, we obtain NAB ≤ CABand NAC ≤ CAC , and consequently, the proof of (9) au-tomatically follows from the corresponding one for squaredconcurrence. For an N -qubit pure state, the generalizedmonogamy inequality for squared negativity, given in Eq. (7)can be proven by using N1:23...N = C1:23...N for pure states,and the relation in Eq. (10).

As mentioned in the previous subsection, there have beenefforts to propose stronger versions of the monogamy rela-tion for entanglement. Recently, a stronger monogamy in-equality for negativity has been proposed [92], and a de-tailed study has been performed for four-qubit states. An-other interesting proposal pitches the square of convex-roofextended negativity as an alternative candidate to character-ize strong monogamy of multiparty quantum entanglement[93].

Monogamy has also been studied for other entanglementmeasures, including entanglement of assistance (EoA) [94],and squashed entanglement [95, 96]. The definition of EoA,which was originally introduced in terms of entropy of en-tanglement, can, in principle, be generalized for other mea-sures of entanglement. For example, concurrence of as-

sistance (CoA) is an entanglement monotone for pure tri-partite states [97], and similar to the squared concurrence,monogamy properties of the squared CoA have been studiedextensively [79, 97–99]. Ref. [100] found lower and upperbounds of EoA, among which the upper bound was shown toobey monogamy constraints for arbitrary N -qubit states. InRef. [96], Koashi and Winter showed that for arbitrary tripar-tite states, the one-way distillable entanglement, the one-waydistillable secret key [101], and the squashed entanglement[95] satisfy the monogamy relation, given in (5). For tripar-tite pure states, the entanglement of purification [102] wasshown to be non-monogamous in general [103].

The monogamy inequality of entanglement sharing hasbeen investigated also for continuous variable systems. InRefs. [104, 105], monogamy inequality for the “continu-ous variable tangle”, or the “contangle”, has been providedfor arbitrary three-mode Gaussian states and for symmetricarbitrary-mode Gaussian states, where contangle is definedas the convex roof of the square of the logarithmic negativ-ity. Further generalization of the results have been achievedin [106–108].

III. INFORMATION-THEORETIC MEASURES OFQUANTUM CORRELATION

In this section, we define a few information-theoretic mea-sures of quantum correlations, whose monogamy propertiesare discussed in the subsequent sections.

A. Quantum discord

Let us consider a bipartite quantum state ρAB , for whichthe uninterrogated, or unmeasured quantum conditional en-tropy is defined as

S(ρA|B) = S(ρAB)− S(ρB), (11)

where ρB = TrA(ρAB) is the reduced density matrix of thesubsystemB, obtained by tracing over the subsystemA. Onecan also define an interrogated conditional entropy, given by

S(ρA|B) = min{ΠB

i }

∑i

piS(ρA|i), (12)

where the minimization is performed over all complete setsof projective measurements, {ΠB

i }, performed on subsystemB. The corresponding post-measurement state for subsystemA is given by ρA|i = TrB [(IA ⊗ ΠB

i )ρAB(IA ⊗ ΠBi )]/pi,

where IA is the identity operator on the Hilbert space of thesubsystem A, and pi = TrAB [(IA ⊗ ΠB

i )ρAB(IA ⊗ ΠBi )] is

the probability of obtaining the outcome i. These two defi-nitions of quantum conditional entropy are two different ex-tensions of the two equivalent expressions of the classicalmutual information. The former is used to define the unin-terrogated quantum mutual information, given by

I(ρAB) = S(ρA)− S(ρA|B), (13)

5

which is interpreted as the “total correlation” of ρAB[35, 36,109–113]. On the other hand, the latter provides the defini-tion of the interrogated quantum mutual information,

I←(ρAB) = S(ρA)− S(ρA|B), (14)

also interpreted as the “classical correlation” present in thequantum state ρAB [35, 36]. The arrow in the superscript be-gins from the subsystem on which the measurement is per-formed. The quantum discord of the state ρAB is the differ-ence between the uninterrogated and interrogated quantummutual informations [35, 36], and is given by

D←(ρAB) = I(ρAB)− I←(ρAB)

= S(ρA|B)− S(ρA|B). (15)

Note here that one can also define quantum discord,D→(ρAB), by performing local measurement over thesubsystem A instead of the subsystem B. In general,D←(ρAB) 6= D→(ρAB). Unless otherwise stated, here andthroughout in this chapter, we consider D←(ρAB) as themeasure of quantum discord. Note also that the definition ofquantum discord is provided by using local projective mea-surement. However, quantum discord can also be defined interms of positive operator valued measurements (POVMs).

B. Quantum work deficit

The amount of extractable pure states from a bipartite stateρAB , under a set of global operations, called the “closed op-erations”(CO), is given by [37–40]

ICO = log2 dim (H)− S(ρAB), (16)

where the set of closed operations consists of (i) unitary op-erations, and (ii) dephasing the bipartite state by a set of pro-jectors, {Πk}, defined on the Hilbert space H of ρAB . Onthe other hand, considering the set of “ closed local opera-tions and classical communication” (CLOCC), the amountof extractable pure states from ρAB is given by [37–40]

ICLOCC = log2 dim (H)−minS (ρ′AB) . (17)

Here, CLOCC consists of (i) local unitary operations, (ii)dephasing by local measurement on the subsystem, say, B,and (iii) communicating the dephased subsystem to the otherparty,A, via a noiseless quantum channel. The average quan-tum state, after the local projective measurement {ΠB

k } isperformed on B, can be written as ρ′AB =

∑k pkρ

kAB with

ρkAB and pk being defined in a similar fashion as in the caseof quantum discord. The minimization in ICLOCC is achievedover all complete sets {ΠB

k }. The “one-way” quantum workdeficit is then defined as [37–40]

W←(ρAB) = ICO − ICLOCC

= min{ΠB

k }[S (ρ′AB)− S(ρAB)] , (18)

where similar to quantum discord, the the arrow in the super-script starts from the subsystem over which the measurementis performed.

C. Geometric measure of quantum discord

Geometric quantum discord [114, 115] for a bipartitequantum state ρAB can be defined as the minimum squaredHilbert-Schmidt distance of ρAB from the set, SQC , of all“quantum-classical” states, given by σAB =

∑i piσ

iA ⊗

|i〉 〈i|, where {|i〉} forms a mutually orthonormal set of theHilbert space of the subsystem B. Mathematically, geomet-ric quantum discord is given by

DG(ρAB) = minσAB∈SQC

||ρAB − σAB ||22, (19)

where ||ρ − σ||2 = Tr(ρ − σ)2, for two arbitrary densitymatrices ρ and σ (however, see [116]). Although geometricmeasures can be defined by using general Schatten p-norms[117] (see also [118, 119]), it has been shown in Ref. [120]that the geometric quantum discord can be consistently de-fined by using the one-norm, i.e., the trace-distance only.Note here that the asymmetry in the definition of quantumdiscord due to a local measurement over one of the partiesalso remains here in the choice of σAB . One can also con-sider “classical-quantum” states, σAB =

∑j pj |j〉 〈j|⊗σ

jB ,

to define the geometric quantum discord.

IV. MONOGAMY OF QUANTUM DISCORD

The monogamy score for quantum discord, defined after(6), is denoted by δ←D or δ→D , depending on the subsystemover which the measurement is performed while computingthe quantum discord. In Refs. [44, 45], it was found thatquantum discord violates the monogamy relation already forcertain three-qubit pure states.

Before discussing the monogamy properties of quan-tum discord in detail, let us first ask the question as towhether a measure of quantum correlation, chosen from theinformation-theoretic domain, can be monogamous. Al-though this is a difficult question to answer in its full gen-erality, some insight can be obtained by noting that there is amarked difference between such a measure and the ones be-longing to the entanglement-separability category. The for-mer may have a non-zero value in the case of a separablestate, while by definition, entanglement measures vanish forall unentangled states. Let us consider a general bipartitequantum correlation measure, Q, which, for an arbitrary bi-partite quantum state ρAB , obeys a set of basic properties, asenumerated below [43].

P1. Positivity: Q(ρAB) ≥ 0.

P2. Invariance under local unitary transforma-tion: Q(ρ′AB) = Q(ρAB), with ρ′AB =

(UA ⊗ UB)ρAB(U†A ⊗ U†B). Here, UA and UBare unitary operators defined on the Hilbert spaces ofthe subsystems A and B.

P3. Non-increasing upon the introduction of a local pureancilla: Q(ρAB) ≥ Q(ρA:BC), with ρA:BC = ρAB ⊗(|0〉 〈0|)C .

6

Note here that the first two properties are standard require-ments for any measure of quantum correlations, i.e., bothentanglement and information-theoretic measures. The thirdproperty is satisfied, for example, by quantum discord, ir-respective of whether the ancilla is attached to the mea-sured, or the unmeasured side. The direction of the inequal-ity in P3 is interesting. It may seem that we should haveQ(ρAB) ≤ QA:BC(ρAB ⊗ (|0〉 〈0|)C), as throwing out theC-part, of a state in A : BC, may only “harm” (i.e., reduceQ), if at all. However, if we look at the definition of quantumdiscord, we find that having the extra C-part may do harm,as it is only the I← term in D← that can get affected due tothe extra C-part. The extra C-part leads to a larger class ofpossible measurements that can be performed for the maxi-mization in I←A:BC , than in I←AB .

A generic separable state of the AC system is given by

ρAC =∑i

piPA(|ψi〉)⊗ PC(|φi〉), (20)

where P (|α〉) = |α〉 〈α|. Let us now consider a special formof the tripartite separable state ρABC =

∑i piPA(|ψi〉) ⊗

PB(|i〉) ⊗ PC(|φi〉) with {|i〉} being a set of mutually or-thonormal states. The quantum correlation,Q, in theA : BCbipartition, has the same value as that in the unitarily con-nected state σABC =

∑i piPA(|ψi〉) ⊗ PB(|i〉) ⊗ PC(|0〉).

Also, the amount of quantum correlation present in thestate σABC in the A : BC bipartition can not be higherthan that present in the A : B bipartition in the stateσAB = TrC(σABC), i.e., Q(σAB) ≥ Q(ρA:BC). If wenow assume that Q satisfy the monogamy relation, thenQ(σAB) ≥ Q(ρAB) +Q(ρAC). Since ρAB ≡ σAB , we ob-tain Q(ρAC) ≤ 0, which, due to the positivity of Q, impliesQ(ρAC) = 0. Hence, a quantum correlation measure, Q,which is monogamous, and which obeys the properties P1–P3, must be zero for a generic separable state, ρAC . Con-trapositively, a general bipartite quantum correlation mea-sure, Q, which has a non-zero value for at least one separa-ble state, and which obeys a set of basic properties must benon-monogamous [43].

The above discussion indicates that in the three-qubit sce-nario, monogamy for a general measure of quantum correla-tion, which belongs to the information-theoretic domain andwhich can be non-zero for a two-qubit separable state of rank2, can be violated for pure three-qubit states. This includesquantum discord, and other “discord-like” measures [2]. Asan example, let us consider the three-qubit generalized Wstates [121, 122], parametrized using two real parametersand given by

|gW〉 = sin θ cosφ |100〉+ sin θ sinφ |010〉+ cos θ |001〉 ,(21)

with θ, φ (0 ≤ θ ≤ π, 0 ≤ φ < 2π) being the real pa-rameters. The plot of the negative of the quantity δ←D =D←(ρA:BC) − D←(ρAB) − D←(ρAC) as a function of θand φ is presented in Fig. 2. Note that over the entire planeof (θ, φ), −δ←D has positive values, indicating that general-ized W states always violate monogamy of quantum discord[44, 45].

0 π/4 π/2 3π/4 π

θ

0

π/2

π

3π/2

2π

φ

0

0.05

0.1

0.15

0.2

0.25

FIG. 2. (Color online) Plot of −δ←D as a function of θ and φ forthree-qubit generalized W states given in Eq. (21). In agreementwith the results reported in [44, 45], −δ←D is positive over the en-tire plane of (θ, φ). Reproduced figure with permission from theAuthors and the Publisher of Ref. [44]. Copyright (2012) of theAmerican Physical Society.

In this context, one must note that the monogamy in-equality of quantum discord, as given by D→(ρA:BC) ≥D→(ρAB) + D→(ρAC), when the measurement is alwaysperformed over the nodal observer, can be shown to be equiv-alent to an inequality between the uninterrogated mutual in-formation of the subsystem BC, and the EoF of the same, asgiven by [123]

Ef (ρBC) ≤ I(ρBC)

2, (22)

for three-qubit pure states. Li and Luo [124] have shownthat the inequality given in (22) does not hold for all tripar-tite pure states, thereby indicating that quantum discord canbe both monogamous as well as non-monogamous, comple-menting the results obtained in [44, 45], where the measure-ments involved in the calculation of quantum discord are per-formed over the non-nodal observers.

The question that naturally arises next is whether a quan-tum correlation measure, Q, which violates one or more ofthe properties P1-P3, can be monogamous. In [43], it hasbeen shown that a quantum correlation measure, Q, which ismonogamous, and remains finite when the dimension of oneof the subsystems, say, A, is fixed, i.e.,

Q(ρAB) ≤ f(dA) <∞, (23)

must be zero for all separable states. Here, dA representsthe dimension of A, and f is some function. To prove this,for a generic separable state ρAB , let us consider a symmetricextension of the form ρAB1B2...BN

, where ρAB = ρABi∀i =1, 2, . . . , N , N being an arbitrary positive integer, such thatQ(ρAB) = Q(ρABi), 1 ≤ i ≤ N [125–128]. This implies∑Ni=1Q(ρABi

) = NQ(ρAB). Now, the monogamy of Qimplies

Q(ρA:B1B2...BN) ≥ NQ(ρAB). (24)

Using (23), we have that Q(ρA|B1...BN) is finite ∀N , includ-

ing N → ∞. Therefore, (24) can be violated with a large

7

enough choice of N if Q(ρAB) 6= 0 for the separable stateρAB .

A. Relation of entanglement of formation to quantum discord

Let us consider a tripartite pure state ρABC , which is a pu-rification of the bipartite density matrices ρAB and ρAC , i.e.,TrB(C)[ρABC ] = ρAC(AB). Let us now consider I←(ρAC),where the only difference with the quantity defined in Eq.(14) is that here we assume an optimization over POVMs[35, 129]. Let {pi, |ψi〉} be a pure state decomposition ofρAB that achieves the minimum in the definition of the EoFof ρAB . Now, there must exist a particular measurement set-ting {Mi} corresponding to the subsystem C of the stateρABC , for which the joint state of the rest of the system,AB, turns out to be |ψi〉 with probability pi correspondingto the ith outcome [130]. This leaves the subsystem A in thestate TrB(|ψi〉〈ψi|). Following the definition of I←(ρAB),this implies [96]

I←(ρAC) ≥ S(ρA)−∑i

piS [TrB(|ψi〉〈ψi|)] (25)

= S(ρA)− Ef (ρAB). (26)

One may also consider an alternative approach, where aparticular measurement {Mi}, when performed over the sub-system C, attains the maximum in I←(ρAC) = S(ρA) −∑i piS(ρi). In general, {Mi} may have a rank more than

1. It is now suggestive to decompose {Mi} into rank-1 non-negative operators, Mij , satisfying Mi =

∑jMij . Let us

assume that the action of Mij on the subsystem C leaves thesubsystem A in ρij with probability pij , where the follow-ing relations hold: pi =

∑j pij and ρi =

∑j ρij . Con-

cavity of the von Neumann entropy implies that S(ρA) −∑ij pijS(ρij) ≥ S(ρA) −

∑i piS(ρi) = I←(ρAC). How-

ever, this conflicts with the definition of I←(ρAC), unlessS(ρA)−

∑ij pijS(ρij) = I←(ρAC). Now consider the ac-

tion of {Mij} on the subsystem C. The state, |φij〉, of thesubsystem AB, corresponding to the outcome ij, is a purestate, and the measurement, {Mij}, therefore, leads to anensemble {pij , |φij〉}, satisfying

∑ij pij |φij〉 〈φij | = ρAB .

This also signifies that ρij = TrB [|φij〉〈φij |]. Consequently[96],

Ef (ρAB) ≤∑ij

pijS(ρij) (27)

= S(ρA)− I←(ρAC). (28)

Combining (26) and (28), we have [96]

Ef (ρAB) + I←(ρAC) = S(ρA). (29)

This directly leads to a relation between entanglement of for-mation and quantum discord. The above relation turns out tobe extremely important to prove several monogamy relationsfor different quantum correlation measures as can be seen insubsequent sections. However, note that the use of the rela-tion implies that quantum discord is computed by performingPOVMs.

Note here that ρ⊗nABC is a purification of the states, ρ⊗nABand ρ⊗nAC . Moreover, the additivity of von Neumann entropyimplies that S(ρ⊗nA ) = nS(ρA). As a result, one obtainsEf (ρ⊗nAB) + I←(ρ⊗nAC) = nS(ρA). Dividing both sides by nand taking the limit n→∞, Eq. (29) reduces to

EC(ρAB) + C←D (ρAC) = S(ρA), (30)

where EC(ρAB) = limn→∞

1nEf (ρ⊗nAB) is the entanglement

cost for creating the ρAB by local operations and classicalcommunication (LOCC) from a resource of shared singlets[131], and C←D (ρAC) = lim

n→∞1nI←(ρAC) is the one-way

distillable common randomness of ρAC [129].

B. Conservation law: Entanglement vs. quantum discord

The above discussion directly leads to a conservation rela-tion between EoF and quantum discord of an arbitrary three-qubit pure state, ρABC . Note here that since ρABC is pure,S(ρA) is a good measure of quantum correlation of ρABC inthe A : BC partition. Eq. (29) implies that the amountof quantum correlation between the subsystem A and therest of the system is the sum of the amount of quantumcorrelation present between A and B, and the amount ofclassical correlation present between A and C, thereby im-posing a constraint over the distribution of the correlationsbetween A and the rest of the system. Adding the unin-terrogated mutual information between A and C, given byI(ρAC) = S(ρA) + S(ρC) − S(ρAC), to both sides of Eq.(29), one obtains

Ef (ρAB) = D←(ρAC) + S(ρA|C). (31)

Proceeding in a similar fashion, one can write Ef (ρAB) andEf (ρAC) as

Ef (ρAB) = D←(ρBC) + S(ρB|C),

Ef (ρAC) = D←(ρAB) + S(ρA|B). (32)

Since the tripartite state is pure, Ef (ρC:AB) = S(ρC) andEf (ρB:AC) = S(ρB), which, from Eq. (32), implies [42]

D←(ρAB) = Ef (ρAC)− Ef (ρC:AB) + Ef (ρB:AC).

(33)

Also, noticing that S(ρA|B) = −S(ρA|C) since the stateρABC is pure, and using Eqs. (31) and (32), one obtains[42]

Ef (ρAB) + Ef (ρAC) = D←(ρAB) +D←(ρAC). (34)

Note here that in the above discussion, we have consideredthe party A to be the nodal observer, and while computingquantum discord, the measurement is performed over thenon-nodal observer. The above equation suggests that foran arbitrary tripartite pure state ρABC , the sum of all pos-sible bipartite entanglements shared by the nodal observerwith the rest of the individual subsystems, as measured bythe EoFs, can not be increased without increasing the sum ofcorresponding quantum discords by the same amount. This

8

seems to indicate a “conservation relation” between EoF andquantum discord with respect to a fixed nodal observer in thecase of a given tripartite pure state.

In this context, we point out that multipartite measures ofquantum correlations have been defined as the sum of quan-tum correlations for all possible bipartitions in a multipartyquantum state, by using EoF, quantum discord, and geomet-ric quantum discord as measures of bipartite quantum corre-lations. For tripartite pure states, the conservation law im-plies that certain such multipartite measures correspondingto EoF and quantum discord are equivalent [132, 133]. Inthe same vein, Ref. [134] investigates the above multipar-tite quantum correlations, in terms of EoFs and quantum dis-cords for even and odd spin coherent states.

Since D←(ρA:BC) = Ef (ρA:BC) = S(ρA) for an arbi-trary tripartite pure state ρABC , Eq. (34) implies an equiva-lence between the monogamy relation of EoF and quantumdiscord [45]. In the case of mixed states, the conservationlaw changes into [42]

D←(ρAB) +D←(ρAC) ≥ Ef (ρAC) + Ef (ρAB) + ∆,

(35)

where ∆ = S(ρB)− S(ρAB) + S(ρC)− S(ρAC).

C. Relation with interrogated information

We now derive a relation which gives a physical insightinto the monogamy property of quantum correlation mea-sures. For an arbitrary tripartite state ρABC , an uninterro-gated conditional mutual information is defined as

I(ρA:B|C) = S(ρA|C)− S(ρA|BC), (36)

while the corresponding interrogated version can be ex-pressed as

I(ρA:B|C) = S(ρA|C)− S(ρA|BC), (37)

involving measurement over one or more of the subsystems.Here, I(ρA:B|C) and I(ρA:B|C) are non-negative, which isa direct consequence of the non-increasing nature of con-ditional entropy with an increase in the number of partiesover which it is conditioned. The definitions of S and Sare as given in Sec. III A. Given a tripartite quantum stateρABC , the interaction information [135], I(ρABC), is de-fined as the difference between the information shared bythe subsystem AB when C is present, and when C is tracedout. Since S(ρA|C) = S(ρAC) − S(ρC) and S(ρA|BC) =S(ρABC)− S(ρBC), one can write an uninterrogated inter-action information as [44]

I(ρABC) = I(ρA:B|C)− I(ρAB)

= S(ρAB) + S(ρBC) + S(ρAC)− S(ρABC)

−(S(ρA) + S(ρB) + S(ρC)). (38)

One can also define an interrogated interaction informa-tion, where the conditional entropies are defined so that acomplete measurement has to be performed on one of the

subsystems. In the case of the tripartite state ρABC , an inter-rogated interaction information is given by

I(ρABC){ΠBk ,Π

Ci ,Π

BCj } = I(ρA:B|C){ΠC

i ,ΠBCj } − I(ρAB){ΠB

k },

(39)

where the suffix on I(ρABC){ΠBk ,Π

Ci ,Π

BCj } indicates that

the measurements are performed over B, C, and BC. Asimilar notation is used to define I(ρA:B|C){ΠC

i ,ΠBCj } =

S(ρA|C){ΠCi } − S(ρA|BC){ΠBC

j } and I(ρAB){ΠBk }

=

S(ρA) − S(ρA|B){ΠBk }≡ S(ρA) −

∑k pkS(ρA|k). For

an arbitrary tripartite state ρABC , the value of the interro-gated interaction information is obtained by performing anoptimization over the measurements. One can show that thequantum interaction information (i) can be either positive ornegative, (ii) is invariant under local unitaries, and (iii) obeysthe inequality I(ρABC) ≥ I(ρABC) under unilocal measure-ments, which can be seen directly from the fact that quantumdiscord is non-negative [44].

If the optimization over the complete set of measurementsis performed, the monogamy relation of quantum discord,i.e.,

D←(ρAB) +D←(ρAC) ≤ D←(ρA:BC) (40)

directly leads to

I(ρA:B|C)− I(ρAB) ≤ I(ρA:B|C)− I(ρAB). (41)

On the other hand, assuming the relation (41) im-plies that minΠBC

iS(ρA|BC){ΠBC

i } − S(ρA|BC) ≥[minΠB

iS(ρA|B){ΠB

i } − S(ρA|B)] + [minΠCiS(ρA|C){ΠC

i }

− S(ρA|C)], which, in turn, implies the monogamy ofquantum discord. Therefore, an arbitrary tripartite quantumstate ρABC is monogamous with respect to quantum discordif and only if I(ρABC){ΠB

k ,ΠCi ,Π

BCj } ≤ I(ρABC) [44]. For

a tripartite pure state, since I(ρABC) = 0, the interrogatedinteraction information is non-positive.

Consider the space of arbitrary three-qubit pure states,formed by the union of the GHZ and W classes, which areinequivalent under stochastic local operations and classicalcommunication (SLOCC) [136]. The monogamy of quan-tum discord has been tested numerically for the GHZ and theW classes [44]. Evidence for both satisfaction and violationof monogamy relation of quantum discord in the former classhas been found, while for the latter, monogamy of quantumdiscord is always found to be violated [44], if the measure-ment is performed over non-nodal observers. The violationof monogamy of quantum discord in the case of three-qubitpure states belonging to the W class can be proved analyt-ically using the equivalence of monogamy relations of EoFand quantum discord in the case of tripartite pure states (seeSec. IV B) [45]. Up to local operations, an arbitrary three-qubit pure state can be parametrized as [137, 138]

|ψABC〉 = λ0 |000〉+ λ1eiγ |100〉+ λ2 |101〉+ λ3 |110〉

+λ4 |111〉 , (42)

where {λi : i = 1, . . . , 4} and γ are real parameters. Forλ4 = 0, Eq. (42) represents an arbitrary state from the W

9

0 2 4 6−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

φ 0 0.5 1−0.2

0

0.2

0.4

0.6

0.8

p

FIG. 3. (Color online) Variation of the monogamy score for squaredquantum discord (blue solid line) in comparison to that of quan-tum discord (red dash-dotted line) [139]. Left: Variations of themonogamy score for squared quantum discord and quantum dis-cord for the generalized W state, |gW〉, as functions of the parame-ter φ, where the parameter θ is set to π/4. Right: Variations of themonogamy score for squared quantum discord and quantum dis-cord in the case of the two-parameter state |ψ(p, ε)〉, as a functionof the state parameter p, where the other parameter is chosen to beε = 0.5. Reprinted figure with permission from the Authors and thePublisher of Ref. [139]. Copyright (2012) of the American PhysicalSociety.

class, for which the tangle vanishes [13], i.e.,

C2AB + C2

AC = C2A:BC , (43)

where C represents the concurrence [13, 20]. Since Ef(0 ≤ Ef ≤ 1) is a concave function of C2(0 ≤ C2 ≤ 1),for two-qubit states, Ef (ρAB) + Ef (ρAC) ≥ Ef (ρA:BC).Hence by using Eq. (34), we obtain the proof of violationof monogamy for quantum discord for the states from the Wclass.

D. Monogamy of quantum discord raised to an integer power

Let us consider a bipartite quantum correlation measure,Q, which is monotonically decreasing under discarding sys-tems, and remains unchanged under discarding systems onlyfor quantum states satisfying monogamy. Suppose thatρABC is a state that violates the monogamy relation for Q.This implies

Q(ρA:BC) < Q(ρAB) +Q(ρAC),

Q(ρA:BC) > Q(ρAB) > 0,

Q(ρA:BC) > Q(ρAC) > 0. (44)

We have additionally assumed that Q(ρAB) > 0 andQ(ρAC) > 0. However, the vanishing Q cases can be han-dled separately. These directly lead to

limn→∞

[Q(ρAB)

Q(ρA:BC)

]n= 0,

limn→∞

[Q(ρAC)

Q(ρA:BC)

]n= 0. (45)

Therefore, for all values of ε > 0, there exists two positive in-tegers, n1(ε) and n2(ε), such that [Q(ρAB)/Q(ρA:BC)]m <ε for all positive integers m ≥ n1(ε), and similarlyfor [Q(ρAB)/Q(ρA:BC)]m with respect to n2(ε). With achoice of ε < 1/2, one obtains [Q(ρAB)/Q(ρA:BC)]m,[Q(ρAC)/Q(ρA:BC)]m < ε for all positive integers m ≥n(ε), where n(ε) = max[n1(ε), n2(ε)], leading to[

Q(ρAB)

Q(ρA:BC)

]m+

[Q(ρAC)

Q(ρA:BC)

]m< 2ε < 1, (46)

for all positive integers m ≥ n(ε). Therefore, consideringthe bipartite quantum correlation measure Qm, we find thatmonogamy is obeyed for the state ρABC [47].

In the case of quantum discord, monogamy property isobeyed for m ≥ 2 [47, 139], while in the subsequent sec-tions, we shall be discussing similar results regarding othermeasures of quantum correlations. Note also that if a quan-tum correlation measure Q is monogamous in the case of athree-party quantum state, any positive integer power of themeasure is also monogamous for the same state, which canbe shown directly by expanding (Q(ρAB)+Q(ρAC))m, andconsidering the non-negativity of Q [47].

We illustrate this by considering the monogamy of squaredquantum discord for two specific cases: (1) The three-qubit generalized W state, discussed in Sec. IV, and (2) atwo-parameter three-qubit pure state, given by |ψ(p, ε)〉 =√pε |000〉+

√p(1− ε) |111〉+

√(1− p)/2(|101〉+ |110〉),

where p and ε are real. The variations of the monogamyscore of squared quantum discord and quantum discord, asfunctions of the state parameters, are plotted in Fig. 3 [139],which clearly indicates that over the entire range of the stateparameters, the squared quantum discord is monogamous,but the quantum discord itself is not.

The case of non-integer powers was considered in [140].It was shown that a monogamous measure remains monoga-mous on raising its power, i.e., if Q(ρA:BC) ≥ Q(ρA:B) +Q(ρAC), then Q(ρA:BC)m ≥ Q(ρA:B)m + Q(ρAC)m,where m ≥ 1. Similarly, one can also prove that a non-monogamous measure of quantum correlation remains non-monogamous with a lowering of the power [140]. Moreover,it has been pointed out in Ref. [140] that if a convex bipar-tite quantum correlation measure Q, when raised to a powerr = 1, 2, is monogamous for pure tripartite states, then Qr isalso monogamous for the mixed states on the given Hilbertspace. Note also that all the results mentioned in this subsec-tion, except the monogamy of squared quantum discord, hasbeen generalized to the N -partite case [47, 140].

E. N-partite quantum states

Research on the monogamy properties of quantum corre-lations belonging to either entanglement-separability or theinformation-theoretic paradigm predominantly deals withtripartite quantum states [13, 21, 41, 44, 45]. It is observedthat “good” entanglement measures [141], which are knownto be non-monogamous, in general, for tripartite quantumstates, tend to obey monogamy, when considered for quan-tum states of a moderately large number of parties [142].

10

Irrespective of the genre of the measure used, it is possi-ble to determine certain other independent sufficient condi-tions for an arbitrary bipartite quantum correlation measureto satisfy monogamy for arbitrary multiparty states. Let usconsider an N -partite pure state ρ12...N , where each of theparties has a dimension d. As the first condition, we assumeour chosen quantum correlation measure Q to be monoga-mous for all tripartite quantum states in dimension d⊗d⊗dmwith m ≤ N − 2. The monogamy of such an N -partite statecan be expressed as

Q(ρ1:23...N ) ≥ Q(ρ12) +Q(ρ1:34...N ), (47)

where a partitioning of ρ12...N , given by 1 : 2 : 34 . . . N , isassumed, and qubit 1 is chosen to be the nodal observer. Onemay, in turn, partition the state ρ134...N = Tr2[ρ12...N ] as1 : 3 : 4 . . . N , and can continue to do so till the last coupleof parties, labeled by N − 1, and N . Recursively applyingthe tripartite monogamy relation, (47) can be reduced to

Q(ρ1:23...N ) ≥ Q(ρ12) +Q(ρ1:34...N ),

≥ Q(ρ12) +Q(ρ13) +Q(ρ1:45...N ),

. . .

≥N∑k=1

Q(ρ1k), (48)

the intended monogamy of Q for the state ρ12...N [21, 142].As the second condition, let us assume the convexity of

Q, and a convex roof definition of Q in the case of a mixedstate ρ12...N . For a tripartition 1 : 2 : 34 . . . N of an N -partymixed state ρ12...N , one obtains

Q(ρ12...N ) = Q

(∑k

pk(|ψ〉 〈ψ|)k12...N

),

=∑k

pkQ(|ψ〉i12...N

), (49)

where the optimal convex roof decomposition providingQ(ρ12...N ) is {pk, |ψi12...N 〉}. If one additionally assumesthat Q is monogamous for all tripartite pure states in dimen-sion d⊗d⊗dm, withm ≤ N−2, then by using the convexityof Q, from Eq. (49), one can write

Q(ρ12...N ) ≥ Q(ρ12) +Q(ρ1:34...N ). (50)

Continuing as before, the monogamy of the mixed stateρ12...N with respect to Q, i.e., the relation (48), can beproven.

The above result has been numerically tested in Ref.[142], by Haar-uniformly generating three-, four-, and five-qubit pure states. For several quantum correlation mea-sures, it is found that the percentages of multiqubit purestates increase with increasing the number of parties. Thesemeasures include quantum discord and qantum work deficit.For example, the percentages of three-, four-, and five-qubitstates, which are monogamous for quantum discord withmeasurement performed on the nodal observer, are respec-tively 90.5%, 99.997%, and 100%. This result indicates thatin the case of a moderately large number of parties in thesystem, quantum correlation measures, which are known to

be non-monogamous for tripartite quantum states, tend toobey monogamy for almost all states [142]. The adjective“almost” is necessary, since for a fixed number of parties,the set of Haar-uniformly generated states may exclude thesets of measure zero in the state space. Therefore, theremay exist measure-zero non-monogamous multipartite stateswhich can not be made monogamous for a specified quan-tum correlation measure by increasing the number of par-ties. Indeed, it is found that the family of N -qubit Dickestates [143] can not be made monogamous with respect toquantum discord by increasing the number of parties. Morespecifically, it has been shown that an N -partite pure statewith vanishing tangle, i.e., C2(ρ12...N ) =

∑Nj=2 C2(ρ1j),

violates the monogamy relation for quantum discord if thesum of the uninterrogated conditional entropy conditionedon all the non-nodal observers is a negative quantity [142].In other words, C2(ρ12...N ) =

∑Nj=2 C2(ρ1j) implies δ←D ≤∑N

j=2 S(ρ1|j),and hence, for pure states having vanishing

tangle and∑Nj=2 S(ρ1|j) < 0, the monogamy score for

quantum discord is negative, and this is the case for the N -qubit Dicke states.

F. Monogamy of quantum discord in open quantum systems

Although being of extreme importance, studies of themonogamy property of quantum correlations under noisy en-vironments is limited, possibly due to the mathematical diffi-culties. Recently, there have been experimental evidence, inphotonic systems, of a flow of quantum correlations whichoccurs between a two-qubit system, AB, and its environ-ment, E [144]. In this scenario, initially present bipartiteentanglement in the system AB decays with time, whilemultipartite entanglement and multipartite quantum discordemerges in the multipartite system constituted by the bipar-tite system and its environment. In another work, the dy-namics of the monogamy property of quantum discord, inthe case of global noise1, and dissipative and non-dissipativesingle-qubit quantum channels is discussed in [63], by us-ing generalized GHZ [145] and generalized W [121, 122]states as input states to the noise. As a representative of thedissipative noise, amplitude-damping (AD) channel is used,while phase-damping (PD) and depolarizing (DP) channelsare chosen as examples of non-dissipative channels2. In

1 The global noisy channel considered corresponds to the completely pos-itive trace preserving (CPTP) map, ρ → ρ′ = ζ(ρ), given by ρ′ =γ I

d+ (1− γ)ρ, where I is the identity matrix, d is the dimension of the

Hilbert space on which ρ is defined, and γ ∈ [0, 1] is the mixing factor.2 The CPTP maps, ρ → ρ′ = ζ(ρ), corresponding to these local noisy

channels, can be given in the form of their respective Kraus operators,{Ek}, such that ρ′ =

∑k EkρE

†k , where

∑k E†kEk = I. For the

single-qubit AD, PD, and DP channels, the Kraus operators are given by{Ead

k }, {Epdk }, and {Edp

k }, respectively. The Kraus operators for theAD channel are given by

Ead0 =

(1 00√1− γ

), Ead

1 =

(0√γ

0 0

),

11

case of the three-qubit generalized GHZ state, given by|gGHZ〉 = a0 |000〉+a1 |111〉, the monogamy score of quan-tum discord has been shown to be decaying monotonicallywith increasing strength of the noise parameter. On the otherhand, in the case of three-qubit generalized W states given by|gW〉 = a0 |001〉+a1 |010〉+a2 |100〉3, monogamy score forquantum discord exhibits non-monotonic behaviour whenthe value of the noise parameter is increased. Here, eachqubit of the three-qubit states is used as input to the quantumchannel under study. For example, when the DP channel isbeing studied, the three qubits of the three-qubit state are fedinto these independent DP channels. A characteristic valueof the noise parameter, called the dynamics terminal [63],is introduced to quantify the robustness of the monogamyscore against a particular type of noise applied to the inputstate, and depolarized channel is identified as the one thatdestroys monogamy score faster than the other channels con-sidered. A related statistics of three-qubit states belonging tothe sets of generalized GHZ states and generalized W statesis obtained numerically, which leads to a conclusive two-stepdistinguishing protocol to identify the type of noise appliedto the three-qubit system. We discuss the details of the two-step channel discrimination protocol in Sec. VII B. In [146],dynamics of quantum dissension [147], and the monogamyscore of quantum discord in the case of amplitude-damping,dephasing, and depolarizing channels are discussed, whenthe input is from a set of three-qubit states including mixedGHZ states, mixed W states, and a certain mixture of sep-arable and bi-separable states. It was also found there thatcertain non-monogamous states become monogamous withthe increase of noise.

V. MONOGAMY OF OTHER QUANTUMCORRELATIONS

In this section, we discuss the monogamy properties ofinformation-theoretic quantum correlation measures otherthan quantum discord. We will present results that are spe-cific to the measures, while the results for generic measuresof quantum correlations, as discussed in the previous sec-tions, remain valid. We start the discussion with quantumwork deficit, which, apart from obeying the properties re-lated to monogamy for general quantum correlation mea-sures, has a direct relation with the monogamy of quantum

while for the PD and the DP channels, they are

Epd0 =

√1− γI, Epd

1 =

√γ

2(I+ σ3), E

pd2 =

√γ

2, (I− σ3),

and

Edp0 =

√1− γI, Edp

i =

√γ

3σi; i = 1, 2, 3,

respectively. Here, γ is the local noise parameter, with γ ∈ [0, 1].3 Note that |gW〉, given in Eq. (21), has been parametrized in the spher-

ical polar co-ordinates, and during numerical simulation, θ and φ aregenerated continuously, while in this case, a0 and a1 are chosen Haar-uniformly to simulate the |gW〉 state.

discord in the case of tripartite pure states [47]. Assumingthat the optimizations for both quantum discord and quan-tum work deficit of the bipartite state ρAB takes place for thesame ensemble {pk, ρkAB}, from the definition of quantumdiscord and quantum work deficit, one can show that

W←(ρAB) = D←(ρAB)− S(ρB) +H({pk}), (51)

where H({pk}) is the Shannon entropy originating fromthe local measurement on the party B. Since H({pk}) ≥S(ρB), W←(ρAB) ≥ D←(ρAB). This implies that ifquantum work deficit is monogamous, i.e., W←(ρA:BC) ≥W←(ρAB) + W←(ρAC), and since W←(ρA:BC) =D←(ρA:BC) = S(ρA) [35–39] for pure states, we have

D←(ρA:BC) = W←(ρA:BC) ≥W←(ρAB) +W←(ρAC)

≥ D←(ρAB) +D←(ρAC),

(52)

which implies monogamy of quantum discord. Note that thereverse is not true. Note also that although one can showthat W←(ρAB) + W←(ρAC) ≥ D←(ρAB) + D←(ρAC)for a three party mixed state under the same assumption ofoptimization.

In the case of arbitrary three-qubit pure states, quan-tum work deficit can be both monogamous and non-monogamous. As discussed in Sec. IV D, similar to quan-tum discord, a state that is non-monogamous with respectto quantum work deficit can be made monogamous by rais-ing quantum work deficit to an appropriate integer power,m. For quantum discord, one requires m ≥ 2 to obtainmonogamy, while for quantum work deficit, the percentageof non-monogamous three-qubit pure states with m ≥ 4 isapproximately 0.22, implying that a higher integer power ofquantum work deficit is necessary to achieve monogamy foralmost all three-qubit pure states.

Similar to the findings for quantum discord and quantumwork deficit, and from the discussion in Sec. IV, geomet-ric quantum discord is not monogamous in general, sinceit is non-zero in the case of some separable states. How-ever, in contrast to quantum discord and quantum workdeficit, geometric quantum discord is always monogamousfor an arbitrary three-qubit pure state ρABC [43]. Notethat the geometric quantum discord, in the present case, iscomputed by performing the minimization over the set ofall “classical-quantum states” instead of “quantum-classicalstates” as defined in Sec. III C. This is proved by show-ing the existence of a classical-quantum state, σABC , forwhich DG(ρA:BC) ≥ ||ρAB − σAB ||22 + ||ρAC − σAC ||22,where σAB(AC) = TrC(B)(σABC), whenever ρABC is pure.Since the right hand side of the inequality is always big-ger than DG(ρAB) +DG(ρAC), due to the minimization in-volved in geometric quantum discord, we obtain the claimedmonogamy of DG. Monogamy of geometric quantum dis-cord is also considered in Ref. [148], while in the multi-qubit scenario, the monogamy of DG is addressed in Refs.[149, 150]. Ref. [151] investigates the monogamy of geo-metric quantum discord in photon added coherent states.

The monogamy of quantum correlations in the case ofthree-qubit pure symmetric states, in the Majorana represen-

12

tation [152], was addressed in [153], where the Rajagopal-Rendell quantum deficit (RRQD) [154, 155] is used as thequantum correlation measure. For a bipartite quantum stateρAB , RRQD is defined as the relative entropy distance[34, 156] of the state ρAB from ρdAB =

∑ij pij |i〉 〈i| ⊗

|j〉 〈j|, which is diagonal in the eigenbasis of the marginals,ρA, and ρB , given by {|i〉} and {|j〉}, respectively. Here,pij = 〈j|〈i|ρAB |i〉|j〉, with

∑i,j pij = 1. Mathematically,

DRR = S(ρAB ||ρdAB), (53)

where for two arbitrary density matrices ρ and σ, S(ρ||σ) =Tr(ρ log2 ρ− ρ log2 σ). In the case of three-qubit pure sym-metric states, RRQD was shown to be non-monogamous ingeneral [153]. In particular, it was shown that although gen-eralized W states can satisfy as well as violate the monogamyinequality for RRQD, the generalized GHZ states always sat-isfy the relation.

We conclude this section by mentioning the monogamyproperty of measurement induced non-locality, introducedby Luo and Fu [157], and defined as

N(ρAB) = max{ΠA}||ρAB − ρ′AB ||22, (54)

where ρ′AB =∑i Πi

A ⊗ IBρABΠiA ⊗ IB , IB is the iden-

tity matrix in the HIlbert space of B, and {ΠiA} is the

set of elements of a projective measurement for which∑i Πi

AρAΠiA = ρA. For this measure, three-qubit pure

states belonging to the GHZ and the W class can be non-monogamous in general [158], although unlike quantum dis-cord, both the generalized GHZ and generalized W statessatisfy the monogamy relation.

VI. RELATION WITH OTHER MULTIPARTYMEASURES

An important perspective of the monogamy inequality ofquantum correlations can be obtained by harvesting its inher-ent multipartite nature, and establish relations between themonogamy scores and the other quantum correlation mea-sures including bipartite and multipartite entanglement. Inseveral works, monogamy scores of a given quantum corre-lation measure has been used as important markers of multi-partite quantum correlations. In the seminal paper by Coff-man, Kundu, and Wootters [13] introducing the monogamyinequality for tripartite states, tangle has been described as“essential three-qubit entanglement”. Over the years, quan-tities such as monogamy scores for different quantum cor-relation measures, including quantum discord and quantumwork deficit, have been used as bona-fide measures of multi-partite quantum correlations [44, 77].

Recently, the relations of monogamy score for quantumdiscord with different multipartite quantum correlation mea-sures, such as tangle [13], genuine multiparty entangle-ment measures quantified by generalized geometric measure(GGM) [159, 160], and global quantum discord [161], havebeen established [44, 162, 163]. Significant approaches toquantify multipartite entanglement and quantum correlationsusing the monogamy principle have also been undertaken

[79, 139, 164, 165]. Monogamy scores of quantum discordand violation of Bell inequalities have also been connected[166, 167]. Tripartite dense coding capacities are also shownto have relations with the monogamy score of quantum dis-cord [168, 169].

A. Monogamy score versus other multi-site quantumnessmeasures

In Ref. [170], the monogamy score of a quantum cor-relation measure for an N -qubit pure multipartite state,ρ12...N = |Ψ〉 〈Ψ|, is shown to be intrinsically related tothe genuine multipartite entanglement, as quantified by theGGM [159, 160]. An N -party pure quantum state |Ψ〉 isgenuinely multipartite entangled if there exist no bipartitionacross which the state is product. The GGM, G, of |Ψ〉 isdefined as

G(|Ψ〉) = 1− max{|Φ〉}

|〈Φ|Ψ〉|2, (55)

where the maximization is over the set of states {|Φ〉},which are not genuinely multiparty entangled. The GGM isknown to be an entanglement monotone [159]. The aboveexpression for GGM can be simplified to the form, G =1−maxk∈[1,N/2]

[{ξm(ρ(k))

}], where

{ξm(ρ(k))

}is the set

of highest eigenvalues of all possible reduced k-qubit states,where k ranges from 1 to N/2.

For three-qubit pure states, the maximum eigenvalue hasto arise from single-qubit density matrices. Hence, an im-mediate relation between the monogamy score δQ and theGGM can be established in this case [44, 78]. Let a =max{ξm(ρ(1))} be the maximum eigenvalue correspond-ing to the single-qubit reduced state, ρ(1) of |Ψ〉, so thatG = 1 − a. From Eq. (8), δjQ ≤ Q(ρj:rest), where j isthe nodal qubit. Now, the quantity Q(ρj:rest) is a functionof a, say FQ(a). Moreover, we have G = 1 − a, whichgives us FQ(a) = FQ(1 − G), which gives us the bound,δjQ ≤ FQ(a) = FQ(1 − G). Now the monogamy score,δQ, is defined as the minimum score over all possible nodes,implying δQ ≤ δjQ. Hence, we obtain an upper-bound on themonogamy score in terms of a function of GGM, given by

δQ(|Ψ〉) ≤ FQ(1− G(|Ψ〉)). (56)

For quantum correlation measures that reduce to the vonNeumann entropy for pure states, such as distillable entan-glement [12], entanglement cost [171], entanglement of for-mation [22], squashed entanglement [95, 96], relative en-tropy of entanglement [32, 34], quantum discord [35, 36],and quantum work-deficit [37–40], the function FQ(1−G) =h(G), where h(x) is the Shannon entropy. For the entangle-ment monotones, squared concurrence [13] and squared neg-ativity [30, 41], the quantity FQ(1−G) is equal to zG(1−G),where z = 4 and 1 for C2 and N 2, respectively. We presenta plot in Fig. 4 of the upper bound for the case of three-qubit pure states, with the quantum correlation being chosenas quantum discord. The above relation can be generalized toN -qubit pure states, where the upper bound on δQ in terms

13

of the entropic or quadratic functions of GGM, can be shownto exist for all states that satisfy a set of necessary conditions.For instance, for allN -qubit states |Ψ〉with G = 1−a, the up-per bound is universally valid. In other words, for N -qubitstate |Ψ〉, if the maximum eigenvalue, among eigenvaluesof all local density matrices, is obtained from a single-qubitdensity matrix, it has been shown that the upper-bound re-mains valid [170].

An important implication of the above bound is that it iseven for those quantum correlation measures for which thecorresponding δQ can not be explicitly computed for arbi-trary states. Examples of such measures include distillableentanglement, entanglement cost, and relative entropy of en-tanglement. The theorem implies that any possible value forthese measures will always result in a δQ that lies on or abovethe boundary.

Apart from entanglement measures, violations of Bell in-equalities are important indicators of quantumness presentin compound systems [56]. The two-point correlation func-tion Bell inequality violations, and their monogamy proper-ties [48, 49] have been connected with the monogamy scoresof entanglement and quantum discord in three-qubit quan-tum systems [166, 167]. It has been shown that for three-qubit pure states, the monogamy scores for quantum corre-lations including quantum discord can be upper-bounded bya function of the monogamy score for Bell inequality vio-lation [166]. Moreover, it was shown in Ref. [167] that ifthe monogamy score for quantum discord in the case of anarbitrary three-qubit pure state, |Ψ〉, is the same as that ofthe three-qubit generalized GHZ state, then the monogamyscore corresponding to the Bell inequality violation of |Ψ〉is bounded below by the same as that of the generalizedGHZ state, when the measurements in quantum discord areperformed on the non-nodal observer. In case the measure-ments are performed on the nodal observer, the role of thegeneralized GHZ state is replaced by the “special” GHZstate of N qubits [167], given by |sGHZ〉 = |00 . . . 0〉N +

|11〉 ⊗ (β |00 . . . 0〉 +√

1− β2e−iθ |11 . . . 1〉)N−2, whereβ ∈ [0, 1], and θ is a phase.

B. Information complementarity: Lower bound onmonogamy violation

For a multipartite system in a pure quantum state, ρ12...N ,let us first divide the whole system into two parts, x and y,such that x ∪ y = {1, 2, . . . , N}. It is possible to derive aninformation-theoretic complementarity relation between thepurity of the subsystem ρx, where ρx = Tryρ12...N+1, and thebipartite quantum correlation shared between the subsystemxwith the rest of the system, i.e., with y, and is given by [75]

P (ρx) +Q(ρxy) ≤ b

{= 1, if dx ≤ dy,= 2− log2 dy

log2 dx, if dx > dy,

. (57)

Here, dx(y) is the Hilbert-space dimension of x(y), P =log2 dx−S(ρx)

log2 dxis the normalized purity of the subsystem x,

S(ρx) is the von Neumann entropy of ρx, and Q(ρxy) =Q(ρxy)

min{log2 dx,log2 dy}is the normalized quantum correlation

(with Q(ρxy) being the corresponding quantum correlation)shared between the subsystems x and y. The proof of therelation (57) requires that Q(ρxy) satisfies the conditionsQ(ρxy) ≤ S(ρx). However, it is independently satisfied byseveral important quantum correlation measures [75, 172].The above complementarity relation has useful applicationin quantum key distribution [75].

If we now consider a non-monogamous normalized bipar-tite quantum corelation measure, Q, which could, for ex-ample, be normalized quantum discord or normalized quan-tum work deficit, we can obtain a useful lower bound on themonogamy score in terms of purity. For an N -qudit state,using (8) and (57), one obtains the relation [172],

δQ ≥ −(N − 2)

(1− P (ρn0

) +1− x0

N − 2

), (58)

where n0 is the nodal qubit and x0 = P (ρn0) + Q(ρn0:rest).

For x0 ≥ 1 and large N , we obtain δQ ≥ −(N − 2)(1 −P (ρn0

)), which provides a nontrivial lower bound for themonogamy score of Q. For three-qubit states, the lowerbound of δQ reduces to δQ ≥ −S(ρn0

). See Fig. 4.

C. Relation with multiport dense coding capacity

Another application of the monogamy inequality and themonogamy score is their role in estimating optimal clas-sical information transfer in multiport dense coding pro-tocols [57, 173–177]. A complementarity relation of themonogamy score for quantum correlation measures, such assquared concurrence and quantum discord, with the maximaldense coding capacity for pure tripartite quantum states wasestablished in Ref. [168]. The dense coding capacity of abipartite quantum state ρ12 is given by [173–177]

C(ρ12) = max [log2 d1, log2 d1 + S(ρ2)− S(ρ12)] . (59)

Without a shared entangled resource, the capacity wouldbe log2 d1, and hence the quantum advantage is Cadv =max{0, S(ρ2)−S(ρ12)}. One can consider a multiport com-munication with ρ12...N as the resource state, where 1 is thesender, and the rests are the receivers. The quantum advan-tage in multiport dense coding for the transfer of classicalinformation from 1 to N − 1 individuals can be defined asCadv = max [{S(ρi)− S(ρ1i)|∀ i = 2, . . . , N}, 0]. Now, ifone considers the set of pure tripartite quantum states, themonogamy score for squared concurrence and quantum dis-cord are intrinsically related to the quantum advantage indense coding. Specifically, it was shown that for any fixedmonogamy score, the maximum quantum advantage is ob-tained from a single parameter family of three-qubit purestates, given by |ψα〉 = |111〉 + |000〉 + α(|101〉 + |010〉)[168]. It is possible to derive a complementarity relationbetween the monogamy score of quantum discord and thequantum advantage. A similar complementarity relation ex-ists between the monogamy score of squared concurrenceand Cadv.

14

Monogamy scores of entanglement and quantum discordhave also been related to the multiparty dense coding ca-pacity between several senders and a single receiver [169].In particular, it was shown that in the noiseless scenario,among all multiqubit pure states with an arbitrary but fixedmultiparty dense coding capacity, the generalized GHZ statehas the maximum monogamy score for quantum discord,i.e., if C(|ψ〉) = C(|gGHZ〉), it implies δD(|ψ〉) ≤δD(|gGHZ〉), where C(ρ12...N ) = log2 d1...N−1+S(ρN )−S(ρ1...N ), with ρ1...N being a state shared between N −1 senders, 1, 2, . . . , N − 1, and the receiver, N . Here,d1...N−1 = d1d2 . . . dN−1. We have suppressed the arrowin the superscript of δD here, as the result is true indepen-dent of the direction of the arrow. The above result is alsotrue if δD is replaced by the tangle. Note also that Ref. [169]also considers the noisy channel case.

VII. PHYSICAL APPLICATIONS

In recent years, several works have been undertaken toelucidate the role of monogamy of quantum correlations instudying quantum systems and their dynamics. In particu-lar, the concept of monogamy has been used to characterizequantum states [44, 45, 153, 158] and channels [63], and alsoto provide deeper understanding of many physical propertiessuch as critical phenomena in many-body systems [68–72]involving complex quantum models such as frustrated spinlattices [72], and biological compounds [73, 74]. Moreover,monogamy also provides an important conceptual basis toquantify quantum correlations in multiparty mixed states, byusing the concept of the monogamy score in situations wherethe usual measures of quantum correlations are neither easilyaccessible nor computable. More precisely, given a bipartitequantum correlation measure, the monogamy score for thatmeasure defined for a given multiparty system, leads us toa measure of multipartite quantum correlation, without anincrease in the complexity on both experimental and theoret-ical fronts as compared to those at the level of the bipartitemeasure.

A. State discrimination

An important aspect in the study of the monogamy proper-ties of quantum correlation measures, such as quantum dis-cord and quantum work-deficit, is that these measures are notuniversally monogamous. In other words, for these quan-tum correlation measures, the monogamy inequality is notuniversally satisfied for all quantum states. This dichotomyallows the monogamy of quantum correlations to be an im-portant figure of merit in state discrimination. In particu-lar, for three-qubit pure states, it was shown that while thegeneralized GHZ states are always monogamous with re-spect to quantum discord, all generalized W states violatethe monogamy inequality [44, 45]. The above results wereextended to the more general sets, viz. the GHZ and theW class states, where it was shown that more than 80%of the Haar-uniformly generated GHZ class states satisfy

0

0.1

0.2

0.3

0.4

0.5

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

G

δ←

D

GHZ Class

W Class

FIG. 4. (Color online) Upper and lower bounds on the monogamyscore of quantum discord for pure three-qubit GHZ- and W-classstates. For three-qubit pure states, the upper bound using the GGM,G, and the lower bound from the information complementarity, aregiven by the quantity h(G) and −h(G), respectively, where h isthe binary entropy function. The scatter points in the figure cor-respond to 106 three-qubit pure states generated Haar uniformly.An equal number of W-class states are also Haar uniformly gen-erated. The figure shows that all W-class states have a negativemonogamy score that is weakly bounded below by−S(G), whereasGHZ-class states can have both positive and negative monogamyscores. Reprinted figure with permission from the Authors and thePublisher of Ref. [78]. Copyright (2012) of the American PhysicalSociety.

monogamy, in contrast to W class states which are alwaysnon-monogamous (see Fig. 4). The monogamy score, there-fore, plays a role that is akin to entanglement witnesses[27, 178–182]. Indeed a given linear entanglement witnessallots values (real numbers) with a certain sign (say, nega-tive), to all separable states while for entangled states, thesame witness can have values of both signs. So, a positivevalue of the witness for a certain state immediately impliesthat the state is entangled. Similarly, a positive value of themonogamy score for quantum discord for a three-qubit purestate implies that the state is from the GHZ class. Subse-quently, the comparative studies of the SLOCC inequivalentclasses was discussed using the monogamy score of anothermeasure of quantum correlation [153], namely, the quantumdeficit [154, 155]. It was shown that while generalized Wstates may violate monogamy, generalized GHZ states al-ways satisfy the monogamy inequality of quantum deficit.We therefore find that the state discrimination protocol us-ing monogamy inequalities is dependent on the choice ofthe quantum correlation. For instance, using the monogamyproperties of measurement induced non-locality [157], it wasshown that both tripartite generalized GHZ and generalizedW states are monogamous[158].

The monogamy inequality and the related monogamyscores have also been used to characterize pure tripartite

15

FIG. 5. (Color online) Schematic representation of the two-stepchannel discrimination protocol proposed in [63]. Reproduced fig-ure with permission from the Authors and the Publisher of Ref.[63]. Copyright (2016) of Physics Letter A (Elsevier).

quantum states [77], by finding the relation of the monogamyscores for those states with the corresponding values for mea-sures of genuine multipartite entanglement, viz. the GGM[159, 160] and the multipartite Mermin-Klyshko Bell in-equalities [55, 183–185]. In particular, tripartite states thathave a vanishing monogamy score for quantum discord havebeen explored in this way [77]. Some of these aspects havebeen discussed in Sec. VI.

B. Channel discrimination

Another application of the monogamy considerations ofquantum correlations comes from the study of their behav-ior under the action of global and local noisy channels.It has been observed that an analysis of the dynamics ofthe monogamy scores of quantum discord and entangle-ment, quantified by negativity, for initial tripartite general-ized GHZ and generalized W states can conclusively iden-tify the noisy channel acting on the system [63]. By ana-lyzing the monogamy scores for quantum discord (δD) andnegativity (δN ), with generalized GHZ and generalized Wstates as inputs, a two-step discrimination protocol to iden-tify the above channels has been developed. To describe theprotocol, let us consider an apriori unknown noisy channel,chosen from a set containing a global noise channel, and theAD, PD, and DP channels. See Sec. IV F for descriptions ofthese channels. In step 1 of the channel-discrimination pro-tocol, one feeds generalized W state to the unknown channelwith moderate noise, i.e., γ ∈ [0.4, 0.6]. The monogamyscore of discord is the primary indicator in this step. For theglobal noise and DP channel, δD ≥ 0, while it is strictly neg-ative for the AD and PD channels. In step 2 of the protocol,the same unknown channel, with moderate noise, is appliedto an generalized GHZ state, and in this instance, both δNand δD of the output state are estimated. It is observed thatδN > 0 for the global noisy channel, but vanishes for theDP channel. Note that step 2 distinguishes between the in-stances which exhibit δD ≥ 0 in step 1. On the other hand,in step 2, δD ≥ 0.13 for the AD channel and δD ≤ 0.09 for

0.0 0.5 1.0 1.5 2.0 2.5

0.00

0.05

0.10

0.15

D

Q3HΡ123L

0

0

0

0

0

0

0

1

1

1

1

1

1

1

33

3

3

3

3

3

3

3

5

5

5

5

5

55

5555 77777

7

7

7

7

7

77

3

0

1

1

FIG. 6. (Color online) Variation of the monogamy score of squaredquantum discord as a function of ∆ with increasing system-size inthe case of the XXZ model. Reprinted figure with permission fromthe Authors and the Publisher of Ref. [70]. Copyright (2014) of theEurophysics Letters (Institute of Physics).

the PD channel. Therefore, the values of δD and δN togethercan discriminate between the global noise and the three localnoisy channels [63]. A schematic representation of the two-step channel discrimination protocol can be found in Fig. 5.

Identification of quantum channels using the dynamics ofmonogamy scores of quantum correlations is potentially animportant addition to the literature on channel identification[186] and estimation [187], which is a significant yet less dis-cussed part of the vast literature on quantum state estimation[188].

C. Characterization of quantum many-body systems

Although several studies have attempted to characterizemany-body quantum systems using quantum correlations be-yond entanglement (for a review, see Ref. [2]), those engag-ing monogamy of quantum correlations to understand co-operative properties in strongly-correlated many-body sys-tems are relatively scarce. In Ref. [68], monogamy prop-erty of quantum discord is used to characterize the groundstate of the one dimensional bond-charge Hubbard model. Aparadigmatic quantum spin model in one dimension is theXYZ model, represented by the Hamiltonian

H =J

4

N∑i=1

{(1 + g)σxi σ

xi+1 + (1− g)σyi σ

yi+1 + ∆σzi σ

zi+1

}+hf2

N∑i=1

σzi , (60)

where J is the strength of the nearest-neighbour exchangeinteraction, g is the x − y anisotropy parameter, ∆ and hfare the anisotropy, and the strength of the external mag-netic field, respectively, in the z direction. Several importantone-dimensional quantum spin Hamiltonians emerge fromEq. (60). For example, the Hamiltonian in Eq. (60), withg = 0 and hf = 0, represents the one-dimensional XXZmodel, while for ∆ = 0 and g = 1, the model corresponds

16

to the one-dimensional quantum Ising model in an externaltransverse magnetic field. In Ref. [70], the monogamy scoreof squared quantum discord is used to investigate the crit-ical points of the one-dimensional XXZ model. See Fig.6 for the variation of δ←D2 against the anisotropy parame-ter, ∆. Monogamy properties of other quantum correla-tion measures such as geometric discord [69] and measure-ment induced disturbance [71] have also been investigated inthe XXZ model. In an experimental study investigating theground state of the one-dimensional quantum Ising modelin a transverse magnetic field, by using a nuclear magneticresonance (NMR) setup, monogamy scores of negativity andquantum discord are shown to distinguish between the casesof positive (frustrated phase) and negative (non-frustratedphase) values of J in the ground state of the system [72].It is interesting to note that monogamy of entanglement hasbeen used to constrain the bipartite entanglement of resonat-ing valence bond states [64–66].

D. Quantum biological processes

An interesting recent development in physics has been theinvestigation of the possibility of quantum effects in certaincomplex biological processes. In particular, light-harvestingprotein complexes have been modeled to investigate pho-tosynthetic processes in certain bacteria, with specific in-terest in the role of quantum coherence and quantum cor-relations [189–192]. Several studies have investigated thewell-known Fenna-Mathews-Olson (FMO) complex, whichmediates energy transfer from the receiving chromophoresto the central reaction center, and attempted to character-ize the efficiency of the energy transfer in terms of quan-tum correlations [190, 191]. The role of monogamy of

quantum correlations in the dynamics was recently investi-gated in Refs. [73, 74]. In Ref. [74], it is shown how themonogamy of quantum correlations, as quantified by nega-tivity and quantum discord, is able to detect the arrangementof the different chromophore nodes in the FMO complex andsupport the predicted pathways for the transfer of excitationenergy. The results also reiterate the predominance of mul-tiparty quantum correlation measures over bipartite correla-tions between the nodes of the FMO complex.

VIII. CONCLUSIONS

Like other no-go theorems [15–18, 193–196] in quantuminformation science, in a multipartite domain, restrictions onsharability of quantum correlations, named as monogamy ofquantum correlations, play a crucial role in achieving suc-cesses in, and in understanding of several quantum infor-mation processing tasks. In this chapter, we have discussedthe monogamy properties of information-theoretic quantumcorrelations, specifically quantum discord, and highlightedtheir significant features. Computable multipartite quantumcorrelation measures are rare, although there are a handfulof bipartite quantum correlation measures, including quan-tum discord and several “discord-like” measures, which arepossible to calculate, at least numerically. The concept ofmonogamy opens up a new avenue where multipartite prop-erties of a system can be studied via bipartite quantum cor-relations and becomes extremely useful to study differentphysical systems. This leads to another interesting aspectof monogamy, namely, its application in several key phe-nomena in quantum physics, ranging from quantum commu-nication to the emerging research on quantum spin models,quantum biology, and open quantum systems, which is alsoreviewed.

[1] R. Horodecki, P. Horodecki, M. Horodecki, and K.Horodecki, Quantum entanglement, Rev. Mod. Phys. 81, 865(2009).

[2] K. Modi, A. Brodutch, H. Cable, T. Patrek, and V. Vedral,The classical-quantum boundary for correlations: Discordand related measures, Rev. Mod. Phys. 84, 1655 (2012).

[3] N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, Quantumcryptography, Rev. Mod. Phys. 74, 145 (2002).

[4] B. M. Terhal, Is entanglement monogamous?, IBM J. Res.Dev. 48, 71 (2004).

[5] A. Sen(De) and U. Sen, Quantum advantage in communica-tion networks, Phys. News, 40, 17 (2010).

[6] R. Raussendorf and H. J. Briegel, A one-way quantum com-puter, Phys. Rev. Lett. 86, 5188 (2001).

[7] H. J. Briegel, D. E. Browne, W. Dur, R. Raussendorf, andM. Van den Nest, Measurement-based quantum computa-tion, Nat. Phys. 5, 19 (2009).

[8] M. Lewenstein, A. Sanpera, V. Ahufinger, B. Damski, A.Sen(De), and U. Sen, Ultracold atomic gases in optical lat-tices: mimicking condensed matter physics and beyond, Adv.Phys. 56, 243 (2007).

[9] L. Amico, R. Fazio, A. Osterloh, and V. Vedral, Entangle-ment in many-body systems, Rev. Mod. Phys. 80, 517 (2008).

[10] R. Augusiak, F. M. Cucchietti, M. Lewenstein, Many bodyphysics from a quantum information perspective, LectureNotes in Physics 843, 245 (2012).

[11] A. K. Ekert, Quantum cryptography based on Bell’s theorem,Phys. Rev. Lett. 67, 661 (1991).

[12] C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Schu-macher, Concentrating partial entanglement by local oper-ations, Phys. Rev. A 53, 2046 (1996).

[13] V. Coffman, J. Kundu, and W. K. Wootters, Distributed en-tanglement, Phys. Rev. A 61, 052306 (2000).

[14] J. S. Kim, G. Gour, and B. C. Sanders, Limitations to SharingEntanglement, Contemp. Phys. 53, 417 (2012).

[15] W. K. Wootters and W. H. Zurek, A single quantum cannotbe cloned, Nature 299, 802 (1982).

[16] D. Dieks, Communication by EPR devices, Phys. Lett. A 92,271 (1982).

[17] H. P. Yuen, Amplification of quantum states and noiselessphoton amplifiers, Phys. Lett. A 113, 405 (1986).

[18] H. Barnum, C. M. Caves, C. A. Fuchs, R. Jozsa, and B. Schu-macher, Noncommuting mixed states cannot be broadcast,

17

Phys. Rev. Lett. 76, 2818 (1996).[19] S. Hill and W. K. Wootters, Entanglement of a Pair of Quan-

tum Bits, Phys. Rev. Lett. 78, 5022 (1997).[20] W. K. Wootters, Entanglement of Formation of an Arbitrary

State of Two Qubits, Phys. Rev. Lett. 80, 2245 (1998).[21] T. J. Osborne and F. Verstraete, General Monogamy Inequal-

ity for Bipartite Qubit Entanglement, Phys. Rev. Lett. 96,220503 (2006).

[22] C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K.Wootters, Mixed-state entanglement and quantum error cor-rection, Phys. Rev. A, 54, 3824 (1996).

[23] P. M. Hayden, M. Horodecki and B. M. Terhal, The Asymp-totic Entanglement Cost of Preparing a Quantum State, J.Phys. A: Math. Gen. 34, 6891 (2001).

[24] E. M. Rains, Rigorous treatment of distillable entanglement,Phys. Rev. A 60, 173 (1999).

[25] E. M. Rains, Bound on distillable entanglement, Phys. Rev.A 60, 179 (1999).

[26] A. Peres, Separability criterion for density matrices, Phys.Rev. Lett. 8, 77 (1996).

[27] M. Horodecki, P. Horodecki, and R. Horodecki, Separabilityof mixed states: necessary and sufficient conditions, Phys.Lett. A 223, 1 (1996).

[28] K. Zyczkowski, P. Horodecki, A. Sanpera, and M. Lewen-stein, Volume of the set of separable states, Phys. Rev. A 58,883 (1998).

[29] J. Lee, M. S. Kim, Y. J. Park, and S. Lee, Partial teleporta-tion of entanglement in a noisy environment, J. Mod. Opt. 47,2151 (2000).

[30] G. Vidal and R. F. Werner, Computable measure of entangle-ment, Phys. Rev. A 65, 032314 (2002).

[31] M. B. Plenio, Logarithmic Negativity: A Full EntanglementMonotone That is not Convex, Phys. Rev. Lett. 95, 090503(2005).

[32] V. Vedral, M. B. Plenio, M. A. Rippin, and P. L. Knight,Quantifying Entanglement, Phys. Rev. Lett. 78, 2275 (1997).

[33] V. Vedral and M. B. Plenio, Entanglement measures and pu-rification procedures, Phys. Rev. A 57, 1619 (1998).

[34] V. Vedral, The role of relative entropy in quantum informa-tion theory, Rev. Mod. Phys. 74, 197 (2002).

[35] L. Henderson and V. Vedral, Classical, quantum and totalcorrelations, J. Phys. A 34, 6899 (2001).

[36] H. Ollivier and W. H. Zurek, Quantum Discord: A Mea-sure of the Quantumness of Correlations, Phys. Rev. Lett.88, 017901 (2002).

[37] J. Oppenheim, M. Horodecki, P. Horodecki, and R.Horodecki, Phys. Rev. Lett. 89, 180402 (2002).

[38] M. Horodecki, K. Horodecki, P. Horodecki, R. Horodecki, J.Oppenheim, A. Sen(De), and U. Sen, Local Information as aResource in Distributed Quantum Systems, Phys. Rev. Lett.90, 100402 (2003).

[39] M. Horodecki, P. Horodecki, R. Horodecki, J. Oppenheim,A. Sen(De), U. Sen, and B. Synak-Radtke, Local versus non-local information in quantum-information theory: Formal-ism and phenomena, Phys. Rev. A 71, 062307 (2005).

[40] I. Devetak, Distillation of local purity from quantum states,Phys. Rev. A 71, 062303 (2005).

[41] Y.-C. Ou and H. Fan, Monogamy inequality in terms of nega-tivity for three-qubit states, Phys. Rev. A 75, 062308 (2007).

[42] F. F. Fanchini, M. F. Cornelia, M. C. de Oliveira, and A. O.Caldeira, Conservation law for distributed entanglement offormation and quantum discord, Phys. Rev. A 84, 012313(2011).

[43] A. Streltsov, G. Adesso, M. Piani, D. Bruß, Are GeneralQuantum Correlations Monogamous? Phys. Rev. Lett. 109,050503 (2012).

[44] R. Prabhu, A. K. Pati, A. Sen(De), and U. Sen, Conditions formonogamy of quantum correlations: Greenberger-Horne-Zeilinger versus W states, Phys. Rev. A 85, 040102(R)(2012).

[45] G. L. Giorgi, Monogamy properties of quantum and classicalcorrelations, Phys. Rev. A 84, 054301 (2011).

[46] C. Lancien, S. Di Martino, M. Huber, M. Piani, G. Adesso,and A. Winter, Should Entanglement Measures be Monoga-mous or Faithful?, Phys. Rev. Lett. 117, 060501 (2016).

[47] K. Salini, R. Prabhu, A. Sen(De), and U. Sen, Monotonicallyincreasing functions of any quantum correlation can make allmultiparty states monogamous, Ann. Phys. 348, 297 (2014).

[48] B. Toner, Monogamy of non-local quantum correlations,Proc. R. Soc. A 465, 59 (2009).

[49] P. Kurzynski, T. Paterek, R. Ramanathan, W. Laskowski, andD. Kaszlikowski, Correlation Complementarity Yields BellMonogamy Relations, Phys. Rev. Lett. 106, 180402 (2011).

[50] R. Prabhu, A. K. Pati, A. Sen(De), and U. Sen, Exclusionprinciple for quantum dense coding, Phys. Rev. A 87, 052319(2013).

[51] T. R. de Oliveira, A. Saguia and M. S. Sarandy, Nonviolationof Bell’s inequality in translation invariant systems, Euro-phys. Lett. 104, 29901 (2013).

[52] Z.-Y. Sun, Y.-Y. Wu, J. Xu, H.-L. Huang, B.-J. Chen, andB. Wang, Violation of Bell inequality in perfect translation-invariant systems, Phys. Rev. A 88, 054101 (2013).

[53] R. Prabhu, A. Sen(De), and U. Sen, Genuine multipartyquantum entanglement suppresses multiport classical infor-mation transmission, Phys. Rev. A 88, 042329 (2013).

[54] D. Sadhukhan, S. Singha Roy, D. Rakshit, A. Sen(De), andU. Sen, Beating no-go theorems by engineering defects inquantum spin models, New J. Phys. 17, 043013 (2015).

[55] J. S. Bell, On the Einstein Podolsky Rosen paradox, Physics1, 195 (1964).

[56] J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, Pro-posed Experiment to Test Local Hidden-Variable Theories,Phys. Rev. Lett. 23, 880 (1969).

[57] C. H. Bennett and S. J. Wiesner, Communication via one- andtwo-particle operators on Einstein-Podolsky-Rosen states,Phys. Rev. Lett. 69, 2881 (1992).

[58] M. D. Reid, Monogamy inequalities for the Einstein-Podolsky-Rosen paradox and quantum steering, Phys. Rev.A 88, 062108 (2013).

[59] A. Milne, S. Jevtic, D. Jennings, H. Wiseman, and T.Rudolph, Quantum steering ellipsoids, extremal physicalstates and monogamy, New J. Phys. 16, 083017 (2014).

[60] S. Lee and J. Park, Monogamy of entanglement and telepor-tation capability, Phys. Rev. A 79, 054309 (2009).

[61] R. Ramanathan, A. Soeda, P. Kurzynski, and D. Kasz-likowski, Generalized Monogamy of Contextual Inequalitiesfrom the No-Disturbance Principle, Phys. Rev. Lett. 109,050404 (2012).

[62] P. Kurzynski, A. Cabello, and D. Kaszlikowski, Fundamen-tal Monogamy Relation between Contextuality and Nonlo-cality, Phys. Rev. Lett. 112, 100401 (2014).

[63] A. Kumar, S. Singha Roy, A. K. Pal, R. Prabhu, A. Sen(De),and U. Sen, Conclusive Identification of Quantum Channelsvia Monogamy of Quantum Correlations, Phys. Lett. A 380,3588 (2016).

[64] A. Chandran, D. Kaszlikowski, A. Sen(De), U. Sen, and V.Vedral, Regional Versus Global Entanglement in Resonating-

18

Valence-Bond States, Phys. Rev. Lett. 99, 170502 (2007).[65] H. S. Dhar and A. Sen(De), Entanglement in resonating va-

lence bond states: ladder versus isotropic lattices, J. Phys.A: Math. Theor. 44, 465302 (2011).

[66] S. Singha Roy, H. S. Dhar, D. Rakshit, A. Sen(De), and U.Sen, Response to defects in multi- and bipartite entangle-ment of isotropic quantum spin networks, arXiv: 1607.05195(2016).

[67] D. Sadhukhan, S. Singha Roy, D. Rakshit, R. Prabhu, A.Sen(De), and U. Sen Quantum discord length is enhancedwhile entanglement length is not by introducing disorder ina spin chain, Phys. Rev. E 93, 012131 (2016).

[68] M. Allegra, P. Giorda, and A. Montorsi, Quantum dis-cord and classical correlations in the bond-charge Hubbardmodel: Quantum phase transitions, off-diagonal long-rangeorder, and violation of the monogamy property for discord,Phys. Rev. B 84, 245133 (2011).

[69] X.-K. Song, T. Wu, and L. Ye, The monogamy relation andquantum phase transition in one-dimensional anisotropicXXZ model, Quantum Inf. Process. 12, 3305 (2013).

[70] L. Qiu, G. Tang, X.-Q. Yang, and A.-M. Wang, Relatingtripartite quantum discord with multisite entanglement andtheir performance in the one-dimensional anisotropic XXZmodel, Europhys. Lett. 105, 30005 (2014).

[71] M. Qin, Z.-Z. Ren, and X. Zhang, Renormalization of theglobal quantum correlation and monogamy relation in theanisotropic Heisenberg XXZ model, Quantum Inf. Process.15, 255 (2016).

[72] K. R. K. Rao, H. Katiyar, T. S. Mahesh, A. Sen(De), U.Sen, and A. Kumar, Multipartite quantum correlations re-veal frustration in a quantum Ising spin system, Phys. Rev. A88, 022312 (2013).

[73] J. Zhu, S. Kais, A. Aspuru-Guzik, S. Rodriques, B. Brock,P. J. Love, Multipartite Quantum Entanglement Evolutionin Photosynthetic Complexes, J. Chem. Phys. 137, 074112(2012).

[74] T. Chanda, U. Mishra, A. Sen(De), and U. Sen, Time dynam-ics of multiparty quantum correlations indicate energy trans-fer route in light-harvesting complexes, arXiv:1412.6519(2014).

[75] A. Bera, A. Kumar, D. Rakshit, R. Prabhu, A. Sen(De), andU. Sen, Information complementarity in multipartite quan-tum states and security in cryptography, Phys. Rev. A 93,032338 (2016).

[76] D. Deutsch, A. Ekert, R. Jozsa, C. Macchiavello, S. Popescu,and A. Sanpera, Quantum privacy amplification and the se-curity of quantum cryptography over noisy channels, Phys.Rev. Lett. 77, 2818 (1996).

[77] M. N. Bera, R. Prabhu, A. Sen(De), and U. Sen, Characteri-zation of tripartite quantum states with vanishing monogamyscore, Phys. Rev. A 86, 012319 (2012).

[78] R. Prabhu, A. K. Pati, A. Sen(De), and U. Sen, Relatingmonogamy of quantum correlations and multisite entangle-ment, Phys. Rev. A 86, 052337 (2012).

[79] Y.-K. Bai, Y.-F. Xu, and Z. D. Wang, General Monogamy Re-lation for the Entanglement of Formation in Multiqubit Sys-tems, Phys. Rev. Lett. 113, 100503 (2014).

[80] Y.-C. Ou, Violation of monogamy inequality for higher-dimensional objects, Phys. Rev. A 75, 034305 (2007).

[81] X.-J. Ren and W. Jiang, Entanglement monogamy inequalityin a 2⊗ 2⊗ 4 system, Phys. Rev. A 81, 024305 (2010).

[82] C.-S. Yu and H.-S. Song, Entanglement monogamy of tripar-tite quantum states, Phys. Rev. A 77, 032329 (2008).

[83] J.-H. Huang and S.-Y. Zhu, Entanglement monogamy in athree-qubit state, Phys. Rev. A 78, 012325 (2008).

[84] Y.-C. Ou, H. Fan, and S.-M. Fei, Proper monogamy in-equality for arbitrary pure quantum states, Phys. Rev. A 78,012311 (2008).

[85] M. F. Cornelio, Multipartite monogamy of the concurrence,Phys. Rev. A 87, 032330 (2013).

[86] B. Regula, S. D. Martino, S. Lee, and G. Adesso, StrongMonogamy Conjecture for Multiqubit Entanglement: TheFour-Qubit Case, Phys. Rev. Lett. 113, 110501 (2014).

[87] C. Eltschka, A. Osterloh, and J. Siewert, Possibility of gen-eralized monogamy relations for multipartite entanglementbeyond three qubits, Phys. Rev. A 80, 032313 (2009).

[88] C. Eltschka and J. Siewert, Quantifying entanglement re-sources, J. Phys. A: Math. Theor. 47, 424005 (2014).

[89] G. Gour and N. R. Wallach, All maximally entangled fourqubits states, J. Math. Phys. 51, 112201 (2010).

[90] X.-N. Zhu and S.-M. Fei, Entanglement monogamy relationsof qubit systems, Phys. Rev. A 90, 024304 (2014).

[91] K. Chen, S. Albeverio, and S. M. Fei, Concurrence of arbi-trary dimensional bipartite quantum states, Phys. Rev. Lett.95, 040504 (2005).

[92] S. Karmakar, A. Sen, A. Bhar, and D. Sarkar, Strongmonogamy conjecture in a four-qubit system, Phys. Rev. A93, 012327 (2016).

[93] J. H. Choi and J. S. Kim, Negativity and strong monogamy ofmultiparty quantum entanglement beyond qubits, Phys. Rev.A 92, 042307 (2015).

[94] D. P. DiVincenzo, C. A. Fuchs, H. Mabuchi, J. A. Smolin, A.Thapliyal, A. Uhlmann, Entanglement of Assistance, LectureNotes in Computer Science 1509, 247 (1999).

[95] M. Christandl and A. Winter, “Squashed entanglement” - Anadditive entanglement measure, J. Math. Phys. (N.Y.) 45, 829(2004).

[96] M. Koashi and A. Winter, Monogamy of quantum entan-glement and other correlations, Phys. Rev. A 69, 022309(2004).

[97] G. Gour, D. A. Meyer, and B. C. Sanders, Deterministic en-tanglement of assistance and monogamy constraints, Phys.Rev. A 72, 042329 (2005).

[98] X.-N. Zhu and S.-M. Fei, Generalized monogamy rela-tions of concurrence for N -qubit systems, Phys. Rev. A 92,062345 (2015).

[99] T. R. de Oliveira, M. F. Cornelio, and F. F. Fanchini,Monogamy of entanglement of formation, Phys. Rev. A 89,034303 (2014).

[100] Z.-G. Li, S.-M. Fei, S. Albeverio, and W. M. Liu, Bound ofentanglement of assistance and monogamy constraints, Phys.Rev. A 80, 034301 (2009).

[101] I. Devetak and A. Winter, Distillation of secret key and en-tanglement from quantum states, Proc. R. Soc. Lond. A, 461,207 (2005).

[102] B. M. Terhal, M. Horodecki, D. W. Leung, and D. P. DiVin-cenzo, The entanglement of purification, J. Math. Phys. 43,4286 (2002).

[103] S. Bagchi and A. K. Pati, Monogamy, polygamy, and otherproperties of entanglement of purification, Phys. Rev. A 91,042323 (2015).

[104] G. Adesso and F. Illuminati, Continuous variable tangle,monogamy inequality, and entanglement sharing in Gaus-sian states of continuous variable systems, New J. Phys. 8,15 (2006).

[105] G. Adesso, A. Serafini, and F. Illuminati, Multipartite en-tanglement in three-mode Gaussian states of continuous-

19

variable systems: Quantification, sharing structure, and de-coherence, Phys. Rev. A 73, 032345 (2006).

[106] G. Adesso and F. Illuminati, Strong Monogamy of Bipar-tite and Genuine Multipartite Entanglement: The GaussianCase, Phys. Rev. Lett. 99, 150501 (2007).

[107] T. Hiroshima, G. Adesso and F. Illuminati, Monogamy In-equality for Distributed Gaussian Entanglement, Phys. Rev.Lett. 98, 050503 (2007).

[108] G. Adesso and F. Illuminati, Genuine multipartite entangle-ment of symmetric Gaussian states: Strong monogamy, uni-tary localization, scaling behavior, and molecular sharingstructure, Phys. Rev. A 78, 042310 (2008).

[109] W. H. Zurek, in Quantum Optics, Experimental Gravitationand Measurement Theory, edited by P. Meystre and M. O.Scully (Plenum, New York, 1983).

[110] S. M. Barnett and S. J. D. Phoenix, Entropy as a measure ofquantum optical correlation, Phys. Rev. A 40, 2404 (1989).

[111] B. Schumacher and M. A. Nielsen, Quantum data processingand error correction, Phys. Rev. A 54, 2629 (1996).

[112] N. J. Cerf and C. Adami, Negative Entropy and Informationin Quantum Mechanics, Phys. Rev. Lett. 79, 5194 (1997).

[113] B. Groisman, S. Popescu, and A. Winter, Quantum, classical,and total amount of correlations in a quantum state, Phys.Rev. A 72, 032317 (2005).

[114] S. Luo and S. Fu, Geometric measure of quantum discord,Phys. Rev. A 82, 034302 (2010).

[115] B. Dakic, V. Vedral, and C. Brukner, Necessary and SufficientCondition for Nonzero Quantum Discord, Phys. Rev. Lett.105, 190502 (2010).

[116] M. Piani, Problem with geometric discord, Phys. Rev. A 86,034101 (2012).

[117] T. Debarba, T. O. Maciel, and R. O. Vianna, Witnessed en-tanglement and the geometric measure of quantum discord,Phys. Rev. A 86, 024302 (2012).

[118] S. Rana and P. Parashar, Comment on “Witnessed entangle-ment and the geometric measure of quantum discord”, Phys.Rev. A 87, 016301 (2013).

[119] T. Debarba, T. O. Maciel, and R. O. Vianna, Reply to “Com-ment on ‘Witnessed entanglement and the geometric measureof quantum discord’ ”, Phys. Rev. A 87, 046301 (2013).

[120] F. M. Paula, T. R. de Oliveira, and M. S. Sarandy, Geometricquantum discord through the Schatten 1-norm, Phys. Rev. A87, 064101 (2013).

[121] A. Zeilinger, M. A. Horne, and D. M. Greenberger, inProceedings of Squeezed States and Quantum Uncertainty,edited by D. Han, Y. S. Kim, and W. W. Zachary, NASAConf. Publ. 3135, 73 (1992).

[122] A. Sen(De), U. Sen, M. Wiesniak, D. Kaszlikowski, and M.Zukowski, Multiqubit W states lead to stronger nonclassi-cality than Greenberger-Horne-Zeilinger states, Phys. Rev.A 68, 062306 (2003).

[123] X.-J. Ren and H. Fan, Non-monogamy of Quantum Discordand Upper Bounds for Quantum Correlation, Quant. Inf.Comp. 13, 469 (2013).

[124] N. Li and S. Luo, Total versus quantum correlations in quan-tum states, Phys. Rev. A 76, 032327 (2007).

[125] R. F. Werner, Quantum states with Einstein-Podolsky-Rosencorrelations admitting a hidden-variable model, Phys. Rev.A 40, 4277 (1989).

[126] R. F. Werner, An application of Bell’s inequalities to aquantum state extension problem, Lett. Math. Phys. 17, 359(1989).

[127] A. C. Doherty, P. A. Parrilo, and F. M. Spedalieri, Com-plete family of separability criteria, Phys. Rev. A 69, 022308

(2004).[128] D. Yang, A simple proof of monogamy of entanglement, Phys.

Lett. A 360, 249 (2006).[129] I. Devetak and A. Winter, Distilling common randomness

from bipartite quantum states, IEEE Trans. Inf. Theory 50,3183 (2004).

[130] L. P. Hughston, R. Jozsa, and W. K. Wootters, A completeclassification of quantum ensembles having a given densitymatrix, Phys. Lett. A 183, 14 (1993).

[131] P. M. Hayden, M. Horodecki, and B. M. Terhal, The asymp-totic entanglement cost of preparing a quantum state, J.Phys. A 34, 6891 (2001).

[132] Z. H. Ma, Z. H. Chen, and F. F. Fanchini, Multipartite quan-tum correlations in open quantum systems, New J. Phys. 15,043023 (2013).

[133] F. F. Fanchini, L. K. Castelano, M. F. Cornelio, and M. C. deOliveira, Locally inaccessible information as a fundamentalingredient to quantum information, New J. Phys. 14, 013027(2012).

[134] M. Daoud, R. A. Laamara, and W. Kaydi, Multipartite quan-tum correlations in even and odd spin coherent states, J.Phys. A: Math. Theor. 46, 395302 (2013).

[135] T. M. Cover and J. A. Thomas, Elements of Information The-ory (Wiley, New Jersey, 2006).

[136] W. Dur, G. Vidal, and J. I. Cirac, Three qubits can be en-tangled in two inequivalent ways, Phys. Rev. A 62, 062314(2000).

[137] A. Acın, A. Andrianov, L. Costa, E. Jane, J. I. Latorre, andR. Tarrach, Generalized Schmidt Decomposition and Clas-sification of Three-Quantum-Bit States, Phys. Rev. Lett. 85,1560 (2000).

[138] A. Acın, D. Bruss, M. Lewenstein, and A. Sanpera, Clas-sification of Mixed Three-Qubit States, Phys. Rev. Lett. 87,040401 (2001).

[139] Y.-K. Bai, N. Zhang, M.-Y. Ye, and Z. D. Wang, Exploringmultipartite quantum correlations with the square of quan-tum discord, Phys. Rev. A 88, 012123 (2013).

[140] A. Kumar, Conditions for monogamy of quantum correla-tions in multipartite systems, Phys. Lett. A 380, 3044 (2016).

[141] M. Horodecki, P. Horodecki, and R. Horodecki, Limits forEntanglement Measures, Phys. Rev. Lett. 84, 2014 (2000).

[142] A. Kumar, R. Prabhu, A. Sen(de), and U. Sen, Effect of alarge number of parties on the monogamy of quantum corre-lations, Phys. Rev. A 91, 012341 (2015).

[143] R. Dicke, Coherence in Spontaneous Radiation Processes,Phys. Rev. 93, 99 (1954).

[144] G. H. Aguilar, J. Farıas, A. Valdes-Hernandez, P. H. SoutoRibeiro, L. Davidovich, and S. P. Walborn, Flow of quan-tum correlations from a two-qubit system to its environment,Phys. Rev. A 89, 022339 (2014).

[145] D. M. Greenberger, M. A. Horne, and A. Zeilinger, in BellsTheorem, Quantum Theory, and Conceptions of the Uni-verse, edited by M. Kafatos (Kluwer Academic, Dordrecht,1989).

[146] P. K. Sarangi and I. Chakrabarty, A study of Quantum Cor-relation for Three Qubit States under the effect of QuantumNoisy Channels, arXiv:1411.7579 (2014).

[147] I. Chakrabarty, P. Agrawal, and A. K. Pati, Quantum Dissen-sion: Generalizing Quantum Discord for Three-Qubit States,Eur. Phys. J. D 65, 605 (2011).

[148] X.-K. Song, T. Wu, and L. Ye, Monogamy properties ofquantum discord for a three-qubit entangled state, Mod.Phys. Lett. B 27, 1350049 (2013).

20

[149] S. Rana and P. Parashar, Tight lower bound on geometric dis-cord of bipartite states, Phys. Rev. A 85, 024102 (2012).

[150] Y.-K. Bai, T.-T. Zhang, L.-T. Wang, and Z. D. Wang, Cor-relation evolution and monogamy of two geometric quan-tum discords in multipartite systems, Eur. Phys. J. D 68, 274(2014).

[151] M. Daoud, W. Kaydi, and H. El Hadfi, Distribution of ge-ometric quantum discord in photon-added coherent states,Mod. Phys. Lett. B 29, 1550239 (2015).

[152] E. Majorana, Atomi orientati in campo magnetico variabile,Nuovo Cimento 9, 43 (1932).

[153] Sudha, A. R. Usha Devi, and A. K. Rajagopal,Monogamy ofquantum correlations in three-qubit pure states, Phys. Rev.A 85, 012103 (2012).

[154] A. K. Rajagopal and R. W. Rendell, Separability and corre-lations in composite states based on entropy methods, Phys.Rev. A 66, 022104 (2002).

[155] A. R. Usha Devi and A. K. Rajagopal, Generalized Infor-mation Theoretic Measure to Discern the Quantumness ofCorrelations, Phys. Rev. Lett. 100, 140502 (2008).

[156] A. Wehrl, General properties of entropy, Rev. Mod. Phys. 50,221 (1978).

[157] S. Luo and S. Fu, Measurement-Induced Nonlocality, Phys.Rev. Lett. 106, 120401 (2011).

[158] A. Sen, D. Sarkar, and A. Bhar, Monogamy of MeasurementInduced Non-Locality, J. Phys. A: Math. Theor. 45, 405306(2012).

[159] A. Sen (De) and U. Sen, Channel capacities versus entan-glement measures in multiparty quantum states, Phys. Rev.A 81, 012308 (2010).

[160] A. Biswas, R. Prabhu, A. Sen (De), and U. Sen, Genuine-multipartite-entanglement trends in gapless-to-gapped tran-sitions of quantum spin systems, Phys. Rev. A 90, 032301(2014).

[161] C. C. Rulli and M. S. Sarandy, Global quantum discord inmultipartite systems, Phys. Rev. A 84, 042109 (2011).

[162] H. C. Braga, C. C. Rulli, T. R. de Oliveira, and M. S. Sarandy,Monogamy of quantum discord by multipartite correlations,Phys. Rev. A 86, 062106 (2012).

[163] S.-Y. Liu, Y.-R. Zhang, L.-M. Zhao, W.-L. Yang, and H. Fan,General monogamy property of global quantum discord andthe application, Ann. Phys. 348, 256 (2014).

[164] Y.-K. Bai, Y.-F. Xu, and Z. D. Wang, Hierarchicalmonogamy relations for the squared entanglement of forma-tion in multipartite systems, Phys. Rev. A 90, 062343 (2014).

[165] F. Liu, F. Gao, S.-J. Qin, S.-C. Xie, and Q.-Y. Wen, Multipar-tite entanglement indicators based on monogamy relations ofn-qubit symmetric states, Sci. Rep. 6, 20302 (2016).

[166] P. Pandya, A. Misra, and I. Chakrabarty, Complementar-ity between Tripartite Quantum Correlation and Bipar-tite Bell Inequality Violation in Three Qubit Pure States,arXiv:1512.01770 (2015).

[167] K. Sharma, T. Das, A. Sen(De), and U. Sen, Distribu-tion of Bell-inequality violation versus multiparty-quantum-correlation measures, Phys. Rev. A 93, 062344 (2016).

[168] R. Nepal, R. Prabhu, A. Sen(De), and U. Sen, Maximally-dense-coding-capable quantum states, Phys. Rev. A 87,032336 (2013).

[169] T. Das, R. Prabhu, A. Sen(De), and U. Sen, Multipartitedense coding versus quantum correlation: Noise inverts rel-ative capability of information transfer, Phys. Rev. A 90,022319 (2014).

[170] A. Kumar, H. S. Dhar, R. Prabhu, A. Sen(De), andU. Sen, Forbidden Regimes in Distribution of Bipartite

Quantum Correlations due to Multiparty Entanglement,arXiv:1505.01748 (2015).

[171] C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A.Smolin, and W. K. Wootters, Purification of Noisy Entangle-ment and Faithful Teleportation via Noisy Channels, Phys.Rev. Lett. 76, 722 (1996).

[172] A. Kumar and H. S. Dhar, Lower bounds on violation ofmonogamy inequality for quantum correlation measures,Phys. Rev. A 93, 062337 (2016).

[173] S. Bose, M. B. Plenio, and V. Vedral, Mixed state dense cod-ing and its relation to entanglement measures, J. Mod. Opt.47, 291 (2000).

[174] T. Hiroshima, Optimal dense coding with mixed state entan-glement, J. Phys. A: Math. Gen. 34, 6907 (2001).

[175] M. Ziman and V. Buzek, Correlation-assisted quantum com-munication, Phys. Rev. A 67, 042321 (2003).

[176] D. Bruß, G. M. DAriano, M. Lewenstein, C. Macchiavello,A. Sen(De), and U. Sen, Distributed quantum dense coding,Phys. Rev. Lett. 93, 210501 (2004).

[177] D. Bruß, M. Lewenstein, A. Sen(De), U. Sen, G.M. DAriano,and C. Macchiavello, Dense coding with multipartite quan-tum states, Int. J. Quant. Inf. 4, 415 (2006).

[178] B. M. Terhal, A family of indecomposable positive linearmaps based on entangled quantum states, Lin. Alg. Appl.323, 61 (2001).

[179] O. Guhne, P. Hyllus, D. Bruß, A. Ekert, M. Lewenstein,C. Macchiavello, and A. Sanpera, Detection of entangle-ment with few local measurements, Phys. Rev. A 66, 062305(2002).

[180] M. Lewenstein, B. Kraus, J. I. Cirac, and P. Horodecki, Opti-mization of entanglement witnesses, Phys. Rev. A 62, 052310(2000)

[181] D. Bruß, J. I. Cirac, P. Horodecki, F. Hulpke, B. Kraus, M.Lewenstein, and A. Sanpera, Reflections upon separabilityand distillability, J. Mod. Opt. 49, 1399 (2002).

[182] O. Guhne, P. Hyllus, D. Bruß, A. Ekert, M. Lewenstein, C.Macchiavello, and A. Sanpera, Experimental detection of en-tanglement via witness operators and local measurements, J.Mod. Opt. 50, 1079 (2003).

[183] N. D. Mermin, Extreme quantum entanglement in a superpo-sition of macroscopically distinct states, Phys. Rev. Lett. 65,1838 (1990).

[184] M. Ardehali, Bell inequalities with a magnitude of violationthat grows exponentially with the number of particles, Phys.Rev. A 46, 5375 (1992).

[185] A. V. Belinskii and D. N. Klyshko, Interference of light andBell’s theorem, Phys. Usp. 36, 653 (1993).

[186] A. Fujiwara, Quantum channel identification problem, Phys.Rev. A 63, 042304 (2001).

[187] M. Sarovar and G. J. Milburn, Optimal estimation of one-parameter quantum channels, J. Phys. A: Math. Gen. 39,8487 (2006).

[188] M. Paris and J. Rehacek J (eds.), Quantum State Estimation:Lecture Notes in Physics, Vol. 649 (Springer, Berlin, 2004).

[189] G. S. Engel, T. R. Calhoun, E. L. Read, T.-K. Ahn, T. C. M.Caronal, Y.-C. Cheng, R. E. Blankenship, and G. R. Fleming,Evidence for wavelike energy transfer through quantum co-herence in photosynthetic systems, Nature 446, 782 (2007).

[190] F. Caruso, A. W. Chin, A. Datta, S. F. Huelga, and M.B. Plenio, Highly efficient energy excitation transfer inlight-harvesting complexes: The fundamental role of noise-assisted transport, J. Chem. Phys. 131, 105106 (2009).

[191] M. Sarovar, A. Ishizaki, G. R. Fleming, and K. B. Whaley,Quantum entanglement in photosynthetic light-harvesting

21

complexes, Nat. Phys. 6, 462 (2010).[192] N. Lambert, Y.-N. Chen, Y.-C. Cheng, C.-M. Li, G.-Y. Chen,

and F. Nori, Quantum biology, Nat. Phys. 9, 10 (2013).[193] J. S. Bell, On the problem of hidden variables in quantum

mechanics, Rev. Mod. Phys. 38, 447 (1966).

[194] S. Kochen and E.P. Specker, The problem of hidden variablesin quantum mechanics, J. Math. Mech. 17, 59 (1967).

[195] A. K. Pati and S. L. Braunstein, Impossibility of deleting anunknown quantum state, Nature 404, 164 (2000).

[196] A. Kalev and I. Hen, No-broadcasting theorem and its clas-sical counterpart, Phys. Rev. Lett. 100, 210502 (2008).