# Emergence of Monogamy under Static and Dynamic Scenarios

EEmergence of Monogamy under Static and Dynamic Scenarios

Rivu Gupta , Saptarshi Roy , Shiladitya Mal , Aditi Sen (De) Harish-Chandra Research Institute and HBNI, Chhatnag Road, Jhunsi, Allahabad - , India Department of Physics and Center for Quantum Frontiers of Research and Technology (QFort),National Cheng Kung University, Tainan , Taiwan (Dated: February , )Characterizing multipartite quantum correlations beyond two parties is of utmost importancefor building cutting edge quantum technologies, although the comprehensive picture is still miss-ing. Here we investigate quantum correlations (QCs) present in a multipartite system by exploringconnections between monogamy score (MS), localizable quantum correlations (LQC), and genuinemultipartite entanglement (GME) content of the state. We ﬁnd that the frequency distribution ofGME for Dicke states with higher excitations resembles that of random states. We show that there isa critical value of GME beyond which all states become monogamous and it is investigated by con-sidering different powers of MS which provide various layers of monogamy relations. Interestingly,such a relation between LQC and MS as well as GME does not hold. States having a very low GME(low monogamy score, both positive and negative) can localize a high amount of QCs in two parties.We also provide an upper bound to the sum of bipartite QC measures including LQC for randomstates and establish a gap between the actual upper bound and the algebraic maximum. I. INTRODUCTION

Correlations play one of the fundamental roles inproviding insight about the laws describing nature atvarious scales. The features possessed by these corre-lations depend on the theory under which they havebeen analyzed – some characteristics are common to allthe theories while others are exclusive to a particularone. Correlations in the quantum domain, commonlyreferred to as quantum correlations (QCs) possess manysuch unique characteristics that are qualitatively differ-ent from classical correlations (CCs). From entangle-ment to nonlocality [ , ], these special properties con-stitute and in turn help to understand the intricaciesof quantum mechanics. Importantly, these specialitiesof QCs are responsible for fueling tasks like quantumteleportation [ ], quantum dense coding [ ], genuinerandomness certiﬁcation [ ], quantum computation [ ],etc. which are impossible via the sole use of CCs.Rapid developments in realizing quantum technolo-gies demand complete characterization of multisite en-tangled states. A concrete way to assess quantum-ness and to study the distribution of QCs in a mul-tipartite state is typically hard. In this direction, twoeffective methods have been developed – one dealswith the constraints on the shareability of correlationsamong the various parties of a multiparty state knownas monogamy of QCs [ , ] which is absent for CCs (cf.[ ]) and other one is measurement-based, referred to asentanglement of assistance or localizable entanglement(LE) [ – ]. In the usual setting of monogamy rela-tion, different bipartite QCs in a multipartite state areconsidered where for ﬁxed two-party reduced states,the other (spectator) parties play a passive role and aremerely forgotten (traced out). On the other hand, thelatter concept of LE emerges from endorsing an activestatus to these spectator parties where they performlocal projective measurements to enhance the bipartite QCs. Both the quantities can capture different perspec-tives of multipartite states, depending on the choiceof bipartite QC measures which are well-studied andquantiﬁed [ ]. Therefore, apart from their fundamen-tal importance, they also possess some utilitarian ap-plications like distinguishing classes of quantum states[ ], in quantum cryptography [ – ] and characteriz-ing phases in many-body systems [ , , , – ] (seealso [ , – ]).Nevertheless, once a correlation is uniquely ascribedas quantum, it becomes natural to seek the relationshipbetween other exclusive QCs. Associating nonlocalitywith entanglement is perhaps one of the best examplesof such intersectional investigation [ , – ]. In thiswork, we establish a connection between monogamy,localizable QCs and genuine multipartite entanglementwith varying numbers of parties (for schematic rep-resentation, see Fig. ). The genuine entanglementcontent of a multipartite pure state can be deﬁned asthe minimum geometric distance of a genuinely entan-gled state from non-genuine ones, known as general-ized geometric measure (GGM) [ , ] (cf. [ ]). In thecase of monogamy and localizable QCs, different QCmeasures with a deﬁnite power [ , , , , , ]are investigated to obtain different landscapes of QCs.Note that QCs with some powers in monogamy- andmeasurement-based measures are also valid QC mea-sures.In this respect, we introduce two quantities, the crit-ical GGM, and the critical exponent, beyond whichall states satisfy the monogamy inequality. We pro-ceed to show that the critical GGM shows a univer-sal diminishing character with the number of partiesfor all quantum correlation measures, thereby provid-ing a sufﬁcient criteria in terms of GGM in states sat-isfying monogamy inequality. The different QC mea-sures that we employ for our investigation involve mea-sures from both the entanglement-separability [ , ] a r X i v : . [ qu a n t - ph ] F e b GME Measurement-basedMonogamy-based

FIG. . Schematic representation of interplay between dif-ferent multipartite quantum correlations, thereby providingclassiﬁcations among multipartite quantum correlation mea-sures – genuine multipartite entanglement, monogamy- andmeasurement-based quantum correlations are related. Anal-ysis shows that features of multipartite QCs are more promi-nently present in monogamy-based measures compared tothe measurement-based ones, considered in this paper. and information-theoretic [ ] paradigms – negativity[ – ] and concurrence [ , ] being measures of theformer category, while quantum discord [ – ] is ourchoice of the measure of QC from the latter category.Moreover, for states with less number of qubits anda ﬁxed amount of GGM, the critical exponent can as-sume very high values but as the number of qubitsgrows, it saturates to its lower limits, thereby show-ing the increase of quantumness in randomly simulatedstates with number of parties. We prove that for a ﬁxedamount of genuine multipartite entanglement (GME)content of an arbitrary three-qubit state and the gener-alized Greenberger-Horne-Zeilinger (gGHZ) state [ ],the monogamy score of entanglement for the former isalways lower than that of the gGHZ state. Such an up-per bound obtained from the gGHZ state does not holdfor states having more than three-qubits. We also showthat Dicke states with higher excitations behave morelike the random states while the Dicke states with a sin-gle or low excitations are not. A usual way of analyzingmonogamy is to look at the sum of all possible bipartitecorrelations with respect to a particular party of a multi-partite state. The monogamous feature is reﬂected as anupper bound to this sum which turns out to be muchsmaller than the algebraic maximum of the sum. Weprovide an estimate of the sum for different bipartitequantum correlation measures of random multipartitestates as well as Dicke states, thereby revealing the gapbetween the algebraic maximum and the actual valuewhich leads to the violation of monogamy inequality.In the case of measurement-based QC measures, weshow that even if the original state has low GME aswell as low (both positive and negative) monogamyscore, substantial quantum correlation can be localizedusing projective measurements, which becomes morepronounced in the case of states having more num- ber of qubits. We support such observation both qual-itatively and quantitatively by considering the mini-mum localizable QC produced from states with a ﬁxedamount of GME or monogamy score. A slight contrast-ing behavior is observed for randomly generated Dickestates with a single excitation for which states possess-ing high GME (monogamy score) can always producemoderate amount of localizable entanglement althoughlow GME can also achieve high localizable entangle-ment. This is due to the fact that the sample spaceof Dicke states having high GGM is low in numberwith the increase of number of parties. Both the re-sults illustrate that the monogamy score can capture thefeatures of multipartite QCs more prominently com-pared to the measurement-based QCs, thereby showingthe interplay between measurement - and monogamy-based measures with GME. We also report that unlikemonogamy scores, the sum of the localizable QCs ofmultipartite random states can reach close to their alge-braic maximum, especially for states with a low numberof qubits.The paper is organized in the following way. Sec. IIestablishes a relation between monogamy of QCs andgenuine multipartite entanglement for random multi-partite states by varying parties from three to six, Dickestates with different excitations [ ] and three-qubit W-class states [ ]. We characterize the set of states whichare non-monogamous with respect to certain bipartiteQC measures, in terms of genuine multipartite entan-glement content in SubSec. II B. In Sec. III, we ﬁnallyrelate the three quantities, monogamy score, localizableentanglement, and GGM as well as report an upperbound on the distribution of localizable entanglement.The summary of results and their implications are pre-sented in Sec. IV. II. MONOGAMY VS. GENUINE MULTIPARTITEENTANGLEMENT

Before providing the relation, let us ﬁrst present theprerequisites to carry out the investigation. We ﬁrstgive deﬁnitions of monogamy score of an arbitrary QCmeasure, classes of multiqubit states under study andgenuine multipartite entanglement measure.

Monogamy of QC.

The restrictions on the distribu-tion of bipartite quantum correlations, Q , in a multi-party state, ρ N , is referred to as the monogamy ofQCs. Quantitatively, it constrains the sum of all bipar-tite QCs of a quantum state with a given nodal party,say, 1, i.e., it provides an upper bound, Q ( ρ ) , on ∑ Ni = Q ( ρ i ) where without loss of generality, we as-sume the nodal party to be the ﬁrst party. Hence, astate is said to be monogamous with respect to Q if itsatisﬁes Q ( ρ ) ≥ ∑ Ni = Q ( ρ i ) . This is evaluated viathe monogamy score, which for any power, α , of a given Q , is deﬁned as [ ] δ Q α = Q α rest − N ∑ i = Q α i , ( )where Q α rest ≡ Q α ( ρ ) and Q α i ≡ Q α ( ρ i ) . In thiswork, the QC measures are considered to be negativity( N ), concurrence ( C ) and quantum discord ( D ). Simulation of quantum states.

An N-qubit random purestate chosen Haar uniformly reads as [ ] | ψ R (cid:105) = N ∑ i = α i | i i ... i N (cid:105) ( )where α j = a j + i b j with a j and b j ∈ R being sam-pled from a Gaussian distribution of mean 0 and unitstandard deviation ( G (

0, 1 ) ) and {| i k (cid:105)} s constituting thecomputational basis. For N =

3, the state space splitsinto two inequivalent classes of states under stochasticlocal operations and classical communication, the GHZ-and the W-class states [ ]. The GHZ class states takethe same form as in Eq. ( ), while the W-class states,constituting a set of measure zero are given by | ψ W (cid:105) = a | (cid:105) + b | (cid:105) + c | (cid:105) + d | (cid:105) , ( )where a , b , c , d are complex numbers whose real partsare taken from G (

0, 1 ) . For states with higher numberof qubits, i.e., for ( N ≥ ) , we consider another classof states, the Dicke states [ ], which reduces to thegeneralized W state (obtained from Eq. ( ) by putting a =

0) for three-qubit case. A Dicke state of N qubitshaving r excitations is deﬁned as | ψ rD (cid:105) = ∑ c P P ( | (cid:105) ⊗ ( n − r ) ⊗ | (cid:105) ⊗ r ) , ( )where P denotes the permutation of all states with n − r excitations, | (cid:105) and r ground states, | (cid:105) . The coefﬁcients c P = c P + ic P are again chosen from G (

0, 1 ) duringtheir simulation, so that random Haar uniformly cho-sen Dicke states are numerically generated. For four-and ﬁve-party states, the excitations are taken to be asingle or two while we have upto three excitations forsix-qubit Dicke states. Genuine multipartite entanglement: generalized geometricmeasure.

The genuine multiparty entanglement (GME)content of these random pure states can be computedusing the generalized geometric measure (GGM). It isa distance-based measure of GME and is deﬁned asthe minimum distance of a given state from the setof all non-genuinely entangled states in the state space[ , ]. For general mixed states, carrying out the mini-mization is very hard [ ]. However, for pure states, theSchmidt decomposition makes the optimization proce-dure tractable and the GGM can be expressed in termsof Schmidt coefﬁcients in different bipartitions of themultipartite pure state, | ψ N (cid:105) , as G ( | ψ N (cid:105) ) = − max (cid:8) λ A : B |A ∪ B = {

1, 2, . . . , N } , A ∩ B = ∅ (cid:9) ,( ) where λ A : B is the maximum Schmidt coefﬁcient in the A : B bipartition of | ψ N (cid:105) , and maximization is per-formed over all such possible bipartitions. Before ex-ploring the monogamy features, let us discuss some ofthe GGM characteristics of random states which willmake it a convenient reference point when comparisonswith the monogamy scores will be made. GGM

GHZ-classW-classDicke

GGM

RandomDicke-1Dicke-2

GGM

RandomDicke-1Dicke-2

GGM

RandomDicke-1Dicke-2Dicke-3

FIG. . (Normalized) Frequency distribution of GGM (ordi-nate) vs. GGM (abscissa). Haar uniformly random (red), ran-dom Dicke class with single (blue), two (green) and three ex-citations (black) for three- (bottom left), four- (bottom right),ﬁve- (top left) and six- qubit states (top right) are generated.Number of states (all kinds) simulated is 5 × . Althoughthe ordinate is dimensionless, the abscissa is in ebits. Frequency distribution of GGM.

To calculate the fre-quency distribution, f ( G ) , we count the number ofstates having GGM between, say, a and b which is thendivided by the total number of states simulated. In restof the paper, wherever we calculate frequency distribu-tion, we use this normalized version.For random pure states of three- to six-qubits, the dis-tribution takes a bell shape whose mean increases with N while the standard deviation (SD) decreases with theincrease of number of parties as shown in Table II andFig. . The maximum value of GGM for random statesthat can be simulated also increases when N increasesfrom three to six and it is close to its algebraic maxi-mum, i.e., 0.5 for random six-qubits which can also beobtained for the N -party GHZ state [ ].On the other hand, as one expects, the trends in fre-quency distribution for GGM are drastically differentfor the Dicke states with low excitations. In particular, ifwe consider Haar uniformly generated three-qubit W-states, f ( G ) is steadily decreasing with a peak around0 − . with the standard deviationbeing . . Hence, most of the states in this class pos-sess a low genuine multipartite entanglement, whichreaches its maximum at 0.326. The maxima as well asthe average value of GGM for the Dicke states with asingle excitation sharply decreases with the increase of TABLE I. Mean and SD of GGM, G for random states withdifferent number of parties.Mean SD .

162 0 . .

231 0 . .

295 0 . .

347 0 . TABLE II. Mean and SD of G for random Dicke states havingdifferent number of excitations. | ψ D (cid:105) | ψ D (cid:105) | ψ D (cid:105) Mean SD Mean SD Mean SD .

11 0 . .

062 0 .

048 0 .

21 0 . .

039 0 .

033 0 .

22 0 . .

028 0 .

023 0 .

183 0 .

049 0 .

313 0 . the number of parties ( see Tables II and III). For ex-ample, the fraction of states residing in the GGM bin of0 − N – 50% for three-qubits, 70% forﬁve-qubits and almost all the simulated states for six-qubits. Interestingly, with the increase of excitations inDicke states, the distribution follows the same patternas in the random states as we will show in the followingproposition. Proposition . The average GGM of an N-qubit Dicke statewith N /2 excitations for even N and N /2 + excitationsfor odd N is almost the same as that of the random states ofN-qubits.Proof. Let us ﬁrst consider the situation when N is even.The logic behind the statement remains similar for odd N . For an N -qubit Dicke state comprising N /2 exci-tations having equal coefﬁcients, the maximum eigen-value comes from the :N- bipartition and is given by N /2 ( N − ) as shown in Ref. [ ]. Thus the GGM ofsuch a state is G eq = ( N − ) /2 ( N − ) , where super-script "eq" indicates that the coefﬁcients are all equal.To obtain the GGM of an N -qubit random state, we ob-serve from our numerical calculations that the largesteigenvalue actually comes from a single party reduceddensity matrix. To that end, we try to estimate it by ap-proximating the average value of the von-Neuman en- TABLE III. Actual maximum of G by varying number ofqubits for randomly generated and Dicke states.N Random | ψ D (cid:105) | ψ D (cid:105) | ψ D (cid:105) .

429 0 .

334 0 .

435 0 .

246 0 .

455 0 .

449 0 .

194 0 . .

453 0 .

154 0 .

325 0 . tropy (given by − tr ρ log ρ ) of the reduced state, usingthe formula [ ] (cid:104) S (cid:105) = log ( M ) − M K , ( )where M is the dimension of the density matrix of thereduced state and MK represents the total dimensionof the pure state from which the reduced system is ob-tained upon tracing out. Since (cid:104) S (cid:105) is the entropy of asingle qubit state, we can ﬁnd the largest eigenvalue,say, x by solving − x log ( x ) − ( − x ) log ( − x ) = (cid:104) S (cid:105) . ( )Then, the average GGM of the random state is, (cid:104)G(cid:105) = − x . For example, by solving Eq. ( ), for N = (cid:104) S (cid:105) = M = K =

8, giving x = (cid:104)G(cid:105) = qubits, M = K =

32 gives (cid:104) S (cid:105) = x = (cid:104)G(cid:105) = (cid:104)G uneqD (cid:105) is less than theDicke states with equal coefﬁcients i.e., (cid:104)G uneqD (cid:105) ≤ G eq .Interestingly, we note from Table II, the gap between theaverage GGM value for Dicke states with unequal coef-ﬁcients and the upper bound for the Dicke state withequal coefﬁcients becomes small with the increase of N and at the same time, (cid:104)G(cid:105) for random states also ap-proaches to G eq ≈ G uneq for large N . Hence the proof.We will discuss the distribution of localizable QCs insubsequent sections, but before that, we shall be inves-tigating the connection between monogamy score andGGM of a multiparty entangled state. A. Relationship between monogamy score and GGM

To establish a connection between monogamy scoresin Eq. ( ) with respect to negativity, concurrence andquantum discord for various values of the exponent, α and GGM, we address the following questions:• Is there a pattern in the distribution of non-monogamous states in terms of their GGM con-tent? How does that depend on the exponent, α ?• Is it possible to ﬁnd a critical value of GGMbeyond which no non-monogamous states arepresent (Eq. ( )) and is it independent of thechoice of QC measure for a ﬁxed exponent? Ananswer to this question can shed light on theproperties of the non-monogamous nature of QCmeasures, thereby giving a sufﬁcient condition onstates satisfying monogamy relation in terms ofGME. As we know, qualitatively and in an ex-treme situation, bipartite quantum states havingmaximal QCs follow the monogamy relation. Apossible reason can be that the violation obtained is due to the stringent bound that we put on ∑ Ni = Q i . Hence, it will also be interesting to ﬁndthe actual upper bound on the sum for randomstates.• If the distribution of GGM with respect to monog-amous and non-monogamous states is consid-ered, depending on the set of states, how doessuch distribution change?To examine the relational properties of randomlygenerated states, we deﬁne the following quantities.Firstly, we segregate the random states into bins pos-sessing deﬁnite ranges of GGM values and compute thefraction of non-monogamous states in each bin, which,in turn, is computed as f NM Q α = Number of non-monogamous statesTotal number of states within GGM range ,( )for a ﬁxed QC measure. Such a quantity is useful to ad-dress the ﬁrst and the last questions while we computethe content of GGM above which all randomly gener-ated states turn out to be monogamous which we referto as critical value of GGM, given by G c = maximum GGM beyond which δ α Q ≥

0, ( )to obtain the answer to the second one. Our aim is toﬁnd the change occurred in the critical GGM dependingon the choice of the QC measure, Q and power α inmonogamy score. GGM

GGM

GGM

GGM δ D < 0 δ D > 0 δ N < 0 δ N > 0 FIG. . Frequency distribution of both non-monogamous andmonogamous states with GGM. Both discord and negativitymonogamy scores are studied with α =

1. All other speciﬁca-tions are same as in Fig. . . Random states Let us ﬁrst resolve the questions for random Haaruniformly generated states. As we will show, com-pletely different picture emerges for a speciﬁc class of -0.1 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5

GGM

GGM

GGM

GGM

FIG. . δ N (red) (vertical axis) against GGM (horizontal axis).Again negativity monogamy scores for random three- (bot-tom left), four- (bottom right), ﬁve- (top left) and six-qubit(top right) states for a given GGM are plotted. Blue solid linerepresents the generalized GHZ state. Both the axes are inebits. α = 0.2GGM α = 0.3GGM α = 0.4GGM α = 0.6 GGM δ D α (random) < 0 δ N α (random) < 0 δ D α (Dicke) < 0 δ N α (Dicke) < 0 FIG. . f NM Q α (vertical axis) vs. GGM (horizontal). δ α D and δ α N are plotted for different α values, for ﬁve-qubit random states(red and blue) and ﬁve-qubit Dicke states with one excitation(green and black in the bottom row). The y -axis is dimension-less while the x -axis is in ebits. states. If we ﬁrst focus on the non-monogamous statesas they vary with their respective GGM content, weobserve that with increase in the number of parties,non-monogamous states cease to exist, especially forentanglement. Also, at higher values of GGM, suchstates decrease in number, especially in case of entan-glement but for discord, δ D stays negative for a largerGGM range (see Fig ). With an increase in the num-ber of parties, the minimum monogamy score goesfrom being negative to positive and the correspondingnon-monogamous states possess low amount of gen-uine multipartite entanglement. This is possibly ex-pected, since with more number of parties, the inher- α = 0.1 α = 0.5 α = 1.0 α = 1.5 α = 2.0 0 0.1 0.2 0.3 0.4 0.5 3 4 5 6NConcurrence FIG. . The critical value of GGM above which all the states are monogamous for ﬁxed QC measures, i.e., G c ( y -axis) with respectto N ( x -axis). As a QC measure, we consider negativity (left), concurrence (middle) and discord (right). Different exponents, α ,involved in monogamy are considered. The horizontal axis is dimensionless while the vertical one is in ebits. ent quantum correlations present in the system increaseand non-monogamous states exist only at small val-ues of multipartite entanglement as depicted in Fig. .Moreover, we ﬁnd that the gGHZ state provides an up-per bound for three-qubit pure states which we willprove analytically both for negativity- and concurrence-monogamy score. Interestingly, as shown in Fig. ,the upper bound does not hold with an increase in thenumber of parties. Theorem . For random three-qubit pure states, | ψ (cid:105) , whichhave the same GGM as the generalized GHZ state | ψ GG (cid:105) ,the entanglement monogamy score is bounded above by thatof the gGHZ state.Proof. The reduced density matrices of the gGHZ state, | ψ GG (cid:105) = α | (cid:105) + β | (cid:105) , are separable and hence themonogamy score for negativity reduces to δ GG N = N GG = (cid:113) α ( − α ) , ( )assuming α ≥ ≥ β , while its GGM is alwaysgiven by G ( | ψ GG (cid:105) ) = − α , since it is symmetric withrespect to the permutation of parties. On the otherhand, suppose the tripartite state has Schmidt coefﬁ-cient, λ ≥ rest -bipartition and GGM comesfrom that bipartition. If the GGM of gGHZ and arbi-trary tripartite state coincide, we have λ = α . More-over, δ N ≤ N = (cid:113) λ ( − λ ) = δ GG N ( )and hence the proof. In a similar fashion, one can getthe proof for concurrence as for pure states, negativityand concurrence are different by a factor of 2.Let us assume that the largest eigenvalue contribut-ing to the GGM of the random state comes from a party,other than the nodal party, i.e., G ( | ψ R (cid:105) ) = − λ with λ < λ and λ ≥ λ = α by equating GGM for the gGHZ and arbitrary state. Thus, from Eq. ( ), we have δ N ≤ (cid:113) λ ( − λ ) ≤ (cid:113) λ ( − λ ) = (cid:112) α ( − α ) = δ GG N and the second in-equality is due to the fact that λ ≥ ]. Speciﬁcally,when α ≤ δ Q α ≤ < . For 0.5 < α < α = α > ) [ ]. . W class Among the states from the three-qubit W-class, therange of multipartite entanglement for which non-monogamous states exist is larger for a given QC mea-sure, than the random states. The fraction of such statesis also larger for a particular GGM interval. Thus, thecritical GGM is also higher in this case compared torandom states.Considering negativity and concurrence, we see thatwhen α <

1, a signiﬁcant percentage of states re-mains non-monogamous while with α ≥

1, the numberof such states decreases with G but non-monogamousstates exist for substantially high values of the expo-nent, upto 1.9. In case of discord, however, states vio-lating the monogamy inequality exist for all values ofexponent upto α =

3, although the number is decreas-ing with GGM for α > . Dicke states As the number of excitations and parties increase, thesituation is similar to the random states as already ar-gued for GGM. The non-monogamous states fall in frac-tion more and more sharply and with an increase in thenumber of parties, the GGM range for the existence ofsuch states also decreases. For ﬁve- and six-qubits, allstates become monogamous for two or more excitationswhen α ≥

1. It indicates that multipartite quantum cor-relations get enhanced with an increase in excitationsand they behave in a similar fashion to random states.Similarly, Dicke states having high excitations and mul-tipartite entanglement content can violate monogamyscore with low α which does not remain true when α isincreased. On the other hand, Dicke states with a singleor low excitations show a large fraction of states to benon-monogamous even for a moderate α values. (seeFig. ) B. Criticalities in GGM and Monogamy power

An interesting feature in the relationship betweenGGM and monogamy score is the existence of a criti-cal value of GGM, G c as deﬁned in Eq. ( ). It meansthat if a random state possess a GGM value above G c , itis guaranteed to be monogamous. We track the changesin the values of G c with the number of parties, N andthe monogamy power α .When the monogamy power is set to unity, i.e., α =

1, we ﬁnd that all QC measures show similar features,where the G c decreases with N , hitting zero for N = . For α values different from unity, we getvarying responses of G c , as seen in Fig. .To associate G c with α in monogamy score, for a ﬁxedmultipartite entanglement content of a state, we ﬁnd acritical exponent beyond which, the monogamy score isalways non-negative. We denote it by α C . To enunci-ate its variation with GGM and its dependence on thenumber of parties, we consider negativity and discordas the correlation measures. Based on the observationsfrom Fig. , we note the following points: . States which require a high value of α to sat-isfy the monogamy inequality are present for lownumber of parties and the number of such statesdecreases signiﬁcantly for N ≥ . α C ≥ G . This is because,states possessing signiﬁcant genuine multipartitecorrelations are monogamous over a large rangeof the exponent. . Near the tail of the GGM spectrum, where statesare strongly quantum correlated, α C is low, evenfor low number of qubits which is nicely depictedin Fig. for three-qubits. . Non-monogamous states are mostly observed for α ≤ GGM N D GGM N D GGM N D GGM N D FIG. . Critical exponent, α C (see text for deﬁnition) againstGGM both for discord (solid circles) and negativity (pluses)monogamy scores. All other speciﬁcations are same as in Fig. . C. Maximum of the sum of bipartite QC measures

To understand the criticalities deeply, let us considerthe actual maximum of ∑ Ni = Q i and its difference withthe algebraic maximum. It is clear from previous inves-tigation that the monogamy-based bound is too strin-gent when α is small. However, the sum of bipartiteQCs is still lower than the sum of the individual max-ima, i.e. ∑ Ni = Q i < ( N − ) for any QC measure in aqubit-scenario. Since the monogamy score is negativefor low values of α , the above quantity has a substan-tial strength at those values and decreases sharply witha rise in the exponent. With higher number of partiesin the parent multipartite state, it decreases for moder-ate to high α and also drops down to zero much morerapidly. For negativity, it is lower than that of concur-rence. Our observations are illustrated in Fig. forrandom states, and Dicke states. III. INTERPLAY BETWEEN MEASUREMENT-,GEOMETRY- AND MONOGAMY-BASED QUANTUMCORRELATIONS

Let us now move to relate measure-based QCs withboth the monogamy-based QC measures and geomet-ric measure of entanglement. The measurement-basedmeasures as well as geometric measures quantify QCsin an active way while monogamy-based measures dothe job in a passive way as explained in the introduc-tion. This is due to the fact that instead of tracing out α α α FIG. . (Left) Plot of ∑ i Q i (ordinate) for three- (solid points) and four-qubit (hollow points) Haar uniformly generatedstates against α (abscissa) in case of negativity (red), discord (blue) and concurrence (black). Similar study has been carriedout for random Dicke states with a single excitation (middle) and two excitations (right). The vertical axis is in ebits in case ofentanglement, in bits in case of discord while the horizontal axis is dimensionless. N − N − N -qubit state. These local measurements concen-trate the global correlations of the state into a particularbipartite pair and are known as localizable correlations[ – ]. Therefore, the localized bipartite correlationshave potential to capture quantumness distributed inmultipartite states [ , ].Since we want to relate measurement-based QC mea-sures with the monogamy-based one, we introduce alocalized version of QC measure, Q , with a power α ,denoted by LQ α , when the local measurements are per-formed in the all the parties except ﬁrst two parties 1and 2, and for a multipartite pure state, | ψ N (cid:105) and givenQC measure, Q , it can mathematically be representedas LQ α ( | ψ N (cid:105) ) = max { Π } N − ∑ k = p k Q α ( | φ k (cid:105) ) , ( )where { Π } denotes the set of local rank- projectivemeasurements on the N − k is a particular outcome combination of the N − | φ k (cid:105) is the normalized postmeasurement state for the k th outcome with p k beingthe corresponding probability. We report the connec-tion of LQ α with G and δ Q α , as well as the variationof LQ α with the power, α . For concurrence, negativityand discord as QC measures, the respective localizedversions are denoted by LC α ( | ψ N (cid:105) ) , LN α ( | ψ N (cid:105) ) , and LD α ( | ψ N (cid:105) ) . LQ negconcdisc LQ negconcdisc LQ negconcdisc LQ negconcdisc FIG. . f ( LQ α = ) (ordinate) vs. localizable QC mea-sures. The QCs localized here are negativity (red), concur-rence (blue) and discord (black) for three- (bottom left), four-(bottom right), ﬁve- (top left) and six-qubit (top right) randomstates. All other speciﬁcations are same as in Fig. . A. Relation of LQ α with δ Q α and GGM Frequency distribution of localizable QCs.

Before per-forming this relational analysis in a systematic way,let us study the frequency distribution of LQ α = (see[ ] for entanglement of formation). We ﬁnd that likemonogamy score and GGM, the shape of the distribu-tion for random states is bell-like and it shifts towardsits algebraic maximum with the increase of N and be-comes sharper with N since the average value of LQ α = increases and SD decreases with the increase of num-ber of parties as shown in Fig. . The observation isindependent of the choice of QC measures and for dif-ferent values of α for Haar uniformly generated ran- dom states. The opposite picture emerges for the Dickestates with low excitations — (cid:104)LQ α = (cid:105) decreases with N , i.e., the distribution shifts towards the low value ofthe respective measure for high N although the widthof the distribution decreases with the increase of num-ber of parties. However, the increase of mean with N ismuch slower than the one observed for GGM. For ex-ample, the average obtained for negativity and discordfor Haar uniformly generated states are respectively0.337, 0.378, 0.397 and 0.58, 0.714, 0.727 with N =

3, 4, 5(compare them with Table I). With the increase of α ,mean decreases and SD increases both for random andDicke states. Relation of measurement-based QCs with generalized ge-ometric measure as well as monogamy score.

In stark con-trast to the relation of monogamy score and GGM,measurement-based QCs behave differently with GGMand monogamy score. Speciﬁcally, in ( G ( δ N ) , LQ α = ) -plane, random states are scattered, thereby showingthat states with low GGM (monogamy score) can re-sult with high amount of localizable entanglement andat the same time, states with high multipartite entangle-ment are able to localize small amount of entanglementas depicted in Figs. and , irrespective of number ofparties. Such a picture only changes when we considerthe Dicke state with a single excitation which only dis-plays a triangular structure, thereby showing a forbid-den region in that plane. It implies that although stateshaving low GGM can concentrate high localizable en-tanglement, a state with high GGM can always producemoderate amount of entanglement for Haar uniformlygenerated Dicke states.Next we will argue that the localizable entanglement(measured either by concurrence or negativity) canhave substantial value for sufﬁciently small GGM incase of random three-qubit states as well as three-qubit | ψ D (cid:105) . Proposition . For arbitrary three-qubit pure states, local-izable entanglement can have a moderately high value evenwhen the genuine multipartite entanglement content of thestate is small.Proof.

The Schmidt decomposition for a tripartite purestate is given by [ ] | ψ (cid:105) = a | (cid:105) + a exp i φ | (cid:105) + a | (cid:105) + a | (cid:105) + a | (cid:105) ( )where all parameters are real and positive semideﬁnitewith 0 ≤ φ ≤ π and ∑ i a i =

1. By performing projec-tive measurement on the third qubit of | ψ (cid:105) , the local-izable concurrence of the remaining two qubits is givenby 2 (cid:112) det ( ρ ) where ρ = Tr | ψ (cid:105) M (cid:104) ψ | , with | ψ (cid:105) M de-noting the post-measurement state. Suppose that LC achieves its optimum value due to measurements alongthe X , Y or Z direction, i.e. in the eigenvectors of σ i , i = x , y , z . Incidentally, for all three cases, the lo-calizable concurrence is given by LC σ = a a . ( )The actual LC can be higher than LC σ , i.e., LC σ ≤ LC . GGM

GGM

GGM

GGM

FIG. . Scattered plot of localizable negativity, LN , ( y -axis)against GGM ( x -axis) for random states (bottom left), randomDicke states with a single excitation (bottom right), two ex-citations (top left) and three excitations (top right) for three-(red), four- (blue), ﬁve- (green) and six-qubits (black). Boththe axes are in ebits. To obtain G ( | ψ (cid:105) ) , we note the eigenvalues of the sin-gle qubit reduced density matrices corresponding to thestate in Eq. ( ) to be λ ± = ( ± (cid:113) − LC σ − f ( a i )) , ( ) λ ± = ( ± (cid:113) − LC σ − f ( a i )) , ( ) λ ± = ( ± (cid:113) − LC σ − f ( a i )) , ( )where we have clubbed all the terms which cannot bewritten in terms of LC into f i ( a i ) . Hence, the GGM is G ( | ψ (cid:105) ) = − λ + i = λ − i . Depending on the values ofthe coefﬁcients, any one λ + i can give be maximum andthat contributes to the GGM. By ignoring f i ( a i ) whichare typically a very small numbers, the relationship be-tween the modiﬁed GGM and localizable concurrenceis found to be LC σ = (cid:113) − ( − G ( | ψ (cid:105) )) if λ + is maximum, LC σ = ( − ( − G ( | ψ (cid:105) )) ) λ + is maximum,where G ( | ψ (cid:105) ) ≥ G ( | ψ (cid:105) ) . Analysing the above rela-tions geometrically, we observe, that even for valuesof G ( | ψ (cid:105) ) ≤ LC with restricted set of mea-surement can be 0.6 or even higher. Since the original LC can be higher while the GGM can also take a lower value than the actual one, the above argument showsthat sizeable correlations can be localized even if theoriginal state possesses insigniﬁcant multipartite entan-glement. δ N δ N δ N δ N FIG. . LN , (vertical) vs. δ N (horizontal). All other speciﬁ-cations are similar to Fig. . Remark.

The similar argument can be given to havethe relation between localizable negativity and GGM.Since we have already established a relation betweenGGM and monogamy score, the above results alsoimply that it is possible to ﬁnd states having lowmonogamy score which can produce correspondinghigh localizable quantum correlations (see Fig. ).Like arbitrary three-qubit states, the three-qubitDicke state with a single excitation having low gen-uine multipartite entanglement can produce high lo-calizable entanglement as shown in Fig. . To showthat, let us consider the three-qubit Dicke state, | ψ D (cid:105) = a | (cid:105) + a | (cid:105) + a | (cid:105) with ∑ i a i =

1. In this case,by assuming a , a ≥ a , we have G ( | ψ D (cid:105) ) = a = − a − a ( ) LC σ ( | ψ D (cid:105) ) = a a ( )Some algebra then allows us to end up with the relationbetween G and the localizable concurrence as LC σ = G ( | ψ D (cid:105) ) + ( a a ) − a . The above relation shows thelinear dependence of LC σ on G as depicted in Fig. .Since a , a ≥ a , we have ( a a ) − a ≥ LC σ ≤ LC on G also shows that thelocalizable concurrence easily exceeds the GGM.As discussed qualitatively and also in Proposition ,the connection between GGM (monogamy score) and LQ does not have any deﬁnite structure. To make theircomparison more quantitative, we consider two situa-tions — . For a ﬁxed N , we ﬁnd minimum and maxi-mum localizable QC that can be achieved and the corre-sponding genuine multipartite entanglement content of δ D α G FIG. . Scattered diagram of localizable discord against δ D α (black) and GGM (green). The choices of α are . (bottomleft), . (bottom right), . (top left) and . (top right) wherefour-qubit Haar uniformly simulated random states are con-sidered. N LCGCLDGDLNGN

FIG. . For a ﬁxed N , we study the minimum localizableQC (red) that can be obtained and its corresponding GGM(blue) are plotted. G x , x = C , N , D denote the GGMs whenminimum localizable concurrence, localizable negativity andlocalizable discord are achieved. and The y -axis is in ebits andthe x -axis is dimensionless. a state; . for a given range of GGM values, minimumand maximum QC that can be localized are focussedon. In particular, we report that the GGM at which theminimum of the correlation occurs increases with num-ber of parties for random states, as shown in Fig. .It is due to the fact that among random states, averageGGM also increases with N . As seen from Table IV, tolocalize nonvanishing QC, a very small amount of GGMis required. Moreover, we observe that to localize min-imum amount of QC in the ﬁrst and the second qubits,the GGM required is always higher than the amount ofLQC, i.e., LQ min < G . Secondly, if we ﬁx GGM in a cer-tain range, the minimum localizable QC can also follow the similar trend, i.e. to localize QC minimally, the cor-responding GGM required for that is substantial. Onthe other hand, for a ﬁxed GGM value, the localizableQCs can always reach their corresponding maximumvalue. Effects of exponents on QCs in localizable quantity.

Wenow investigate the effect of varying α introduced inthe localizable QCs and we consider the same α inmonogamy score. The trend that we observed in Propo-sition for arbitrary states or Dicke state remains sameby varying α . In particular, when we have low α < α ,low amount of entanglement can be localized. Such ob-servation is possibly artifact of the functional form ofthe QC measures as also the case of monogamy scores(see Fig. ).Similar to Fig. , we now ask a question: for a givenlocalizable QC, what is critical exponent above which,the monogamy score is always nonnegative? We ﬁndthat unlike GGM, no such universal picture emergesas we hinted by the relation between localizable QCand GGM. In this case, states with high localizable QCcan require high α to make the states monogamous (seeFig. ). Only low α is required for high N in case of δ Q , which one expects from the behavior of monogamyscore itself. This observation possibly indicates that lo-calizable QC measure has some component which aredue to multipartite state but is qualitatively differentthan multipartite entanglement monotones (cf. [ ]).Notice also that such conclusion may be changed if wealter the deﬁnition of localizable QC (cf. [ ] and refer-ences thereto). LNLD

FIG. . Critical exponent α C is plotted against LN (bluecircles) and LD (red crosses) for three-, four-, ﬁve- and six-qubit random states. All other coordinates are same as in Fig. . TABLE IV. G for the corresponding minimum localizable QCs.N LC L N LD .

083 0 . . .

237 0 . . .

33 0 . . .

36 0 .

27 0 . B. Deviation from algebraic maximum

Let us now investigate the behavior of ∑ Ni = LQ i for an N-partite state. The algebraic maximum of thisquantity is ( N − ) which can be achieved by the GHZstate. However, the actual bound turns out to be quitedifferent for Haar uniformly generated states. We ob-serve that the sum falls short of its algebraic value forall classes of states, especially for high N (see Table V) .For random three-qubit states, the actual upperbound is close to its algebraic maximum, i.e., 2 althoughthe difference between algebraic maxima and the max-imum obtained numerically increases with the numberof parties. E.g., if we consider concurrence as a quan-tum correlation measure, the gap is . for three-qubitrandom states while it rises to . in case of six-qubits(see Fig. ). In case of Dicke state, the gap turns out tobe signiﬁcant, i.e., it fails to attain the algebraic thresh-old by a big margin. In this instance too, the differenceincreases with the number of constituent qubits. E.g.,the sum reaches only about half of the algebraic max-imum for | ψ D (cid:105) . However, with more number of exci-tations in Dicke states, picture similar to random statesdevelops. LQ FIG. . f ( ∑ Ni = ( LC ) i ) (ordinate) against ∑ Ni = ( LC ) i (ab-scissa) for three-(red), four- (blue), ﬁve- (green) and six- qubit(black) random states. It also shows that although for three-qubits, sum is close to the algebraic maximum, the gap be-tween algebraic maxima and the maximum of the sum ob-tained via random states increases with the increase of thenumber of parties. The ordinate is dimensionless while theabscissa is in ebits. TABLE V. Maximum of ∑ Ni = LQ i . ∑ Ni = LN i ∑ Ni = LC i ∑ Ni = LD i Random ψ D ψ D ψ D Random ψ D ψ D ψ D Random ψ D ψ D ψ D .

997 0 .

707 1 .

993 1 .

414 1 .

995 1 . .

481 0 .

866 1 .

47 2 .

973 1 .

732 2 .

94 2 .

946 1 .

654 2 .

225 1 .

944 1 . .

917 3 .

94 2 3 .

83 3 .

845 1 .

939 2 . .

34 1 .

398 2 .

104 2 .

22 4 .

52 2 .

58 4 .

17 4 . .

14 2 .

435 2 .

94 4 . IV. CONCLUSION

In a multipartite domain, quantum correlations (QC)even for pure states cannot be characterized in a uniqueway. Over the years, several quantiﬁcations from differ-ent origins have been proposed which elucidate spe-ciﬁc features of quantum states, important for build-ing quantum technologies. In this work, we provide aconnection between three such independent quantumcorrelation measures, deﬁned from different perspec-tives, thereby bringing them under a single umbrella.In particular, we choose a geometry-based entangle-ment measure quantifying genuine multipartite entan-glement, monogamy-based quantum correlation mea-sures with different exponents and measurement-basedmeasures. Both monogamy- and measurement-basedmeasures are constructed by considering both entan-glement and other quantum correlation measures.We reported that there exists a critical content of gen-uine multipartite entanglement above which no multi-partite states violate monogamy inequality. Typically,monogamy relations for a quantum correlation are con-sidered with an exponent that can be thought of as "ad-hoc" [ ]. We ﬁnd that for a ﬁxed genuine multipar-tite entanglement, there always exists a critical expo-nent above which all measures satisfy monogamy re-lation. For Haar uniformly generated states, such acritical exponent decreases with the increase of num-ber of parties. We also proved that if an arbitrarythree-qubit state and generalized Greenberger-Horne-Zeilinger states (gGHZ) possess the same amount ofgenuine multipartite entanglement, then the entangle-ment monogamy score of the former is bounded aboveby that of the latter. Such a hierarchy between randomstates and gGHZ states does not hold for states with ahigher number of parties. The back of the envelope cal-culations also reveals that average genuine multipartiteentanglement content of random states with arbitrarynumber of parties coincides with the Dicke states hav-ing half of the sites excited.On the other hand, we showed that a state having low multipartite entanglement content can localize ahigh amount of quantum correlations in two partiesand vice-versa. This result indicates that localizablequantum correlations can have some components carry-ing multipartite characteristics of states although it alsohighlights the difference between genuine multipar-tite entanglement and localizable entanglement. No-tice that a different process of sweeping entanglementtowards two parties than the one considered in thiswork may show different characteristics. Interestingly,we observe that the monogamy score of QCs behavesmore like multipartite measures than localizable QCs.Speciﬁcally, we observed that states having high local-izable entanglement may require a high critical expo-nent to satisfy the corresponding monogamy inequal-ity, thereby showing its different nature from genuinemultipartite entanglement. Moreover, we found thatthe sum of bipartite localizable entanglement of mul-tipartite random states are bounded above by a quan-tity which is close to its algebraic maximum and thegap between algebraic and the actual bounds increaseswith the number of parties for random states while thedifference is substantial for Dicke states.Since all the quantum correlation quantiﬁers have dif-ferent kinds of importance in quantum information sci-ence, the connection established in this paper possiblygives a hint towards choosing the QC measure, depend-ing on their tasks, instead of their amount. V. ACKNOWLEDGEMENT

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