A striking result from nonrelativistic quantum mechanics is the monogamy of
entanglement, which states that a particle can be maximally entangled only with
one other party, not with several ones. While there is the exact quantitative
relation for three qubits and also several inequalities describing monogamy
properties it is not clear to what extent exact monogamy relations are a
general feature of quantum mechanics. We prove that in all many-qubit systems
there exist strict monogamy laws for quantum correlations. They come about
through the curious relation between the nonrelativistic quantum mechanics of
qubits and Minkowski space. We elucidate the origin of entanglement monogamy
from this symmetry perspective and provide recipes to construct new families of
such equalities.Comment: 4 pages, 3 figure

Eltschka, Christopher

Siewert, Jens

English

arXiv

A striking result from nonrelativistic quantum mechanics is the monogamy of entanglement, which states that a particle can be maximally entangled only with one other party, not with several ones. While there is the exact quantitative relation for three qubits and also several inequalities describing monogamy properties, it is not clear to what extent exact monogamy relations are a general feature of quantum mechanics. We prove that in all many-qubit systems there exist strict monogamy laws for quantum correlations. They come about through the curious relationship between the nonrelativistic quantum mechanics of qubits and Minkowski space. We elucidate the origin of entanglement monogamy from this symmetry perspective and provide recipes to construct new families of such equalities

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Monogamy Equalities for Qubit Entanglement from Lorentz InvarianceChristopher EltschkaInstitut für Theoretische Physik, Universität Regensburg, D-93040 Regensburg, GermanyJens SiewertDepartamento de Química Física, Universidad del País Vasco–Euskal Herriko Unibertsitatea, 48080 Bilbao, Spainand IKERBASQUE, Basque Foundation for Science, 48011 Bilbao, Spain(Received 29 July 2014; published 7 April 2015)A striking result from nonrelativistic quantum mechanics is the monogamy of entanglement,which states that a particle can be maximally entangled only with one other party, not with severalones. While there is the exact quantitative relation for three qubits and also several inequalities describingmonogamy properties, it is not clear to what extent exact monogamy relations are a general featureof quantum mechanics. We prove that in all many-qubit systems there exist strict monogamy laws forquantum correlations. They come about through the curious relationship between the nonrelativisticquantum mechanics of qubits and Minkowski space. We elucidate the origin of entanglement monogamyfrom this symmetry perspective and provide recipes to construct new families of such equalities.DOI: 10.1103/PhysRevLett.114.140402 PACS numbers: 03.65.Ud, 03.65.Aa, 03.65.Fd, 03.67.MnIntroduction.—Monogamy of entanglement as a quali-tative concept has been discussed for almost two decades[1,2]. Conceivably, this intuition can be cast into amathematical framework of inequalities for certain typesof quantum correlations. An influential result is the strongsubadditivity of the von Neumann entropy [3], among otherimportant examples for monogamy inequalities [4–9].Applications of entanglement monogamy pervade manyareas of physics, such as quantum information and thefoundations of quantum mechanics [10–12], condensed-matter physics [13–15], statistical mechanics [10], and evenblack-hole physics [16,17].This has to be contrasted with the possibility ofmonogamy equalities, i.e., exact relations for differenttypes of correlations in arbitrary pure quantum states.Although several such equalities are known (we mentionthem explicitly below), the only widely recognized—and atthe same time perhaps the most famous—result is the three-qubit monogamy relation discovered by Coffman, Kundu,and Wootters (CKW) [18]; cf. Fig. 1.The existence of monogamy inequalities appears notentirely unexpected, because intuitionally one would asso-ciate them with convexity properties of the quantum-mechanical state space. As opposed to this, rigorousmonogamy expressed in terms of equalities for pure statesrepresent a much stronger constraint and hint at a fine-tuning of the mathematical properties of quantum states. Infact, one might be tempted to use different names forequalities versus inequalities, instead of terming them both“monogamy relations.”In this Letter, we show for qubit systems that suchrelations are not coincidental, but represent a universalfeature of single-copy entanglement that is deeply rooted inthe algebraic structure of quantum theory. Since exactmonogamy relations link those properties to local SLð2;CÞinvariants [19,20], they confirm the central importance ofthese quantities for entanglement theory. Note that in suchrelations regarding single-copy entanglement, one wouldnot expect entropy-related quantities, such as entanglementof formation encountered in Refs. [3,4,7,8].The quantum-mechanical state ρABC shared betweenparties A, B, C,… contains all available information aboutthe correlations between the individual parties, whereas thereduced state, for example, ρACðBÞ ≡ ρAC, describes thestate of the subset fACg of the partners and “forgets” aboutthe information regarding the other parties. For a multi-partite system there are many ways to form subsets ofparties, and the corresponding reduced states. We mayquantify the entanglement contained in each reduced stateFIG. 1 (color online). Qualitative sketch of the CKWmonogamy equality for pure states of three qubits A, B, C.The Borromean rings on the left-hand side adequately illustratethe tripartite entanglement quantified by τres. The first term on theright-hand side stands for the linear entropy τðρAÞ≡ τðρAðBCÞÞ ofqubit A that measures the entanglement between A and thecomposite system fBCg. The remaining two terms represent theamount of bipartite entanglement of A with B or C quantified bythe concurrences CðρABÞ2 and CðρACÞ2, respectively, thereby“forgetting” (tracing out) the third party. While τres is a globalproperty of the state, the quantities on the right-hand side refer todifferent reduced states.PRL 114, 140402 (2015) P HY S I CA L R EV I EW LE T T ER Sweek ending10 APRIL 20150031-9007=15=114(14)=140402(5) 140402-1 © 2015 American Physical Societyby an appropriate entanglement measure. A monogamyrelation is nothing but a mathematical constraint for theentanglement quantifiers of different reduced states of acomposite quantum system (cf. Fig. 2).Bloch representation and Lorentz invariance.—Usually,nonrelativistic quantum mechanics is constructed startingfrom pure states in Hilbert space, which later are gener-alized to mixed states representing ensembles of purestates. In our work, we emphasize the opposite point ofview, that is, that states are positive Hermitian operators (or“density matrices”) and pure states are special statescharacterized by additional conditions. It is those condi-tions that imply peculiar properties, such as strictmonogamy of entanglement. In order to see this, it isessential to choose a representation for the density matricesthat adequately encodes the relevant state properties, theBloch representation [21–24].Consider the state ρ of a single qubit, which can bewritten as [21–24]ρ ¼ 12ðr012 þ r1σ1 þ r2σ2 þ r3σ3Þ; ð1Þwhere σj are the Pauli matrices, 1≡ σ0 is the identityoperator for qubits, and rj are real numbers. Usually, ρ isnormalized so that r0 ¼ 1. An important quantity is thedeterminant [22],4 det ρ ¼ 2trρσ2ρTσ2 ¼ r20 − r21 − r22 − r23 ≡ rμrμ; ð2Þbecause it does not change under determinant-one oper-ations F ∈ SLð2;CÞ∶ρ → FρF†. We have introduced theEinstein summation convention rμrμ ≡Pμ;νημνrμrν andthe Minkowski metric ημν ¼ diagð1;−1;−1;−1Þ. Note thatthis Minkowski metric acts on Bloch space, not on physicalspacetime. Also, the terms “spacelike” and “timelike” usedbelow refer to the Lorentz group structure found in Blochspace, not to physical space and time. As is known from Lietheory, local SL invariance translates into Lorentz invari-ance in the Bloch coefficients rμ. This representation can begeneralized to any number N of qubitsρ ¼ 12NXj1;…;jNrj1j2jNσj1 ⊗    ⊗ σjN ; ð3Þas well astrR≡ trρσ⊗N2 ρTσ⊗N2 ¼ 12N rμ1μN rμ1μN ð4Þ(using the transposed density matrix ρT). This expression isinvariant under local determinant-one operations implyingLorentz invariance independently on each qubit index [22].It is also non-negative since R can be rearranged under thetrace as a positive operator [25] trR ¼ tr ffiffiffiρp σ⊗N2 ρTσ⊗N2ffiffiffiρp.The aforementioned relations (2) and (4) are valid forarbitrary states, but we may ask whether for pure statesmore elaborate predictions are possible.Origin of exact monogamy.—The space of all states ρ is aconvex set, with the pure states πψ as extreme points. Theycan be characterized as projectorsπψ ¼ π2ψ : ð5ÞHere, ψ refers to the usual bra-ket notation, that is,πψ ¼ jψihψ j. Inserting Eq. (3) into Eq. (5) yields anoperator identity that has to be satisfied termwise. Themost prominent among these conditions is the normaliza-tion of π2ψ ,trπ2ψ ¼ 1 ¼12NXr2μ1μN : ð6ÞSurprisingly, this is enough to explain the origin ofmonogamy equalities. To this end, we write theMinkowskian and Euclidean sums (4) and (6) explicitly,separating timelike and spacelike indices (we show anexample for two qubits),22trRψ ¼ r200 −X3j¼1ðr20j þ r2j0Þ þX3j;k¼1r2jk ð7Þ≡ S0 − S1 þ S2; ð8Þ22trπ2ψ ¼ S0 þ S1 þ S2: ð9ÞThe symbol Sk denotes the sum of all terms r2 with kspacelike indices. Each quantity Sk is invariant under localunitaries. The coefficients r with a timelike index 0 atposition q are components of a reduced state trqπψ . Finally,for pure states we have trRψ ¼ jHðψÞj2 with the well-known polynomial invariant [19,26,27] HðψÞ, i.e., aquantity that characterizes and quantifies global entangle-ment in the pure state πψ .It is evident from Eqs. (8) and (9) that the term S2 withonly spacelike indices can be eliminated, so that we are leftwith an equation that contains only quantities that eithercharacterize global entanglement in the state or describeelements of reduced states. This is the general monogamyprinciple that obviously works for any number of qubits,since the corresponding relations for purity and jHðψÞj2 arealways independent. The question is whether and how theterms of the reduced states can be related to entanglementmeasures in a simple manner.Degree-2 monogamy relations.—In the following, weshow how specific monogamy relations can be deducedfrom this principle. By either subtracting (even N) oradding (odd N) the equations for trR and trρ2, the term withonly spacelike indices is eliminated and we obtain anequality for all N-qubit mixed states,PRL 114, 140402 (2015) P HY S I CA L R EV I EW LE T T ER Sweek ending10 APRIL 2015140402-2ð−1ÞN trR ¼ trρ2 − 12N−1X⌊N=2⌋k¼aNS2k−aN ; ð10Þwhere ⌊N=2⌋ denotes the largest integer not exceedingN=2and aN ¼ 12 ½1þ ð−1ÞN . The sums Sk can be expressed interms of the purities trρ2fjg of the reduced density matricesρfjg with a set of j qubits traced out. The purities are relatedto the linear entropy τðρÞ ¼ 2½ðtrρÞ2 − trρ2, so thatð−1ÞN trR ¼ −1þ trρ2 þ 12XN−1j¼1ð−1Þjþ1τðjÞ: ð11ÞHere, τðjÞ ≡PfjgτðtrfjgρÞ is the sum of all linear entropiesthat are obtained by tracing out j qubits from the state ρ.For pure states ψ the linear entropy equals the squaredconcurrence (for each bipartition AjB)τðtrBπψ Þ ¼ τðtrAπψÞ ¼ C2AjBðψÞ; ð12Þand we can straightforwardly derive an exact monogamyrelation for all integers N≧2,2jHðψÞj2 ¼Xf1gCf1gjfN−1gðψÞ2 −Xf2gCf2gjfN−2gðψÞ2þXf3gCf3gjfN−3gðψÞ2 −þ   þ ð−1ÞNXfN−1gCfN−1gjf1gðψÞ2; ð13Þwhere CfjgjfN−jg ≡ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiτðtrfjgρÞqandPfjg is taken over allcombinations fjg of j indices. Note that the right-hand sideof Eq. (13) is a combination of entanglement monotones onthe respective bipartitions. For N ¼ 4, this relation wasfound in Ref. [28].For odd qubit number, trRψ ¼ jHðψÞj2 ≡ 0, which leadsto an identity becausetrBðtrAπψÞ2 ¼ trAðtrBπψ Þ2 ð14Þ[and, hence, CAjBðψÞ ¼ CBjAðψÞ] for any bipartition AjB ina pure state.Thus, we have found the simplest monogamyrelations which contain, in a sense, averaged quantities(cf. Fig. 3). For example, for two qubits A and B theinvariant polynomial equals Wootters’ concurrence [25]jHðψÞj ¼ CðψÞ, so that 2CðψÞ2 ¼Pf1gC2f1gjf1g ¼C2AjB þ C2BjA, while obviously CðψÞ¼CAjBðψÞ¼CBjAðψÞ.In order to obtain these sharper relations, one can useEq. (5). Nonetheless, Eq. (13) is remarkable: On the right-hand side, all terms are invariant under local unitaries whilethe left-hand side is SLð2;CÞ invariant on all parties. This isone of the hallmarks of equalities for entanglementmonogamy. Notably, the relation is homogeneous in thestate (more precisely, of degree 2 in the density matrix).Degree-4 monogamy relations.—Our strategy forobtaining monogamy relations so far was as follows:consider a local SL invariant expression for the state,reveal its Lorentz-invariant structure, and rewrite it interms of the reduced states, thereby eliminating the exclu-sively spacelike parts. Until now we have considered onlythe invariant HðψÞ, which is of homogeneous degree 2 inthe coefficients of the Hilbert vector ψ. In the next step, weinvestigate degree-4 invariants, which will lead us also tothe CKW monogamy relation.A local SL invariant of degree 4 deriving from the BðNÞinvariants for odd N qubits in Ref. [27] isðtrρΣγ1ρTΣγ2ÞðtrρΣγ1ρTΣγ2Þ≡ Bð3ÞC ðρÞ; ð15Þwith Σγ ≡ σ2 ⊗ σ2 ⊗ σγ . Pure states satisfy Bð3ÞC ðψÞ ¼jBð3ÞC ðψÞj2. Here, we also define the BðNÞ invariants withthe full Minkowski metric (as opposed to Ref. [27]).We obtainFIG. 2 (color online). Why is the CKW equality in Fig. 1termed a “monogamy relation”? The linear entropy τðρAÞ can beviewed as the total amount of Alice’s social connectivity (firstpicture on the left), while the residual tangle τres representsconnectivity in a group that Alice, Bob, and Charlie sharesymmetrically (second picture on the left). On the other hand,the concurrences CðρABÞ2, CðρACÞ2 characterize Alice’s personalrelation with Bob or Charlie, respectively. As the equality shows,if Alice’s total connectivity and the group connectivity are fixed,so is the sum of her personal relations. That is, Alice may sharerelations with Bob and Charlie, but she cannot dedicate maximalattention to both Bob and Charlie at the same time.FIG. 3 (color online). Illustration of the degree-2 monogamyrelation for pure four-qubit states. The big square on the leftrepresents jHðψÞj2 while the other items stand for the sixpossibilities of two-qubit concurrences. Correspondingly, onthe right-hand side, the sums of all concurrences related tosingle-qubit and three-qubit bipartitions are shown. Note that theterms on the right are pairwise identical according to Eq. (14). Forexample, the single-qubit concurrence of the “green qubit” equalsthat of the other three qubits (lower right entry of the three-qubitcontributions).PRL 114, 140402 (2015) P HY S I CA L R EV I EW LE T T ER Sweek ending10 APRIL 2015140402-3Bð3ÞC ðψÞ ¼124rαβγrαβνrλμνrλμγ ¼ τresðψÞ2: ð16ÞIn order to eliminate the spacelike indices on qubit C, weuse the following pure-state identities, which can beverified using the Schmidt decomposition. Abbreviating~πψ ≡ Σ2πTψΣ2, we haveðtrRABÞ2 ¼ tr½πψðtrC ~πψ trAB ~πψÞπψ ;trR2AB ¼ tr½ðtrCπψÞ ~πψ trABðtrCπψÞ ~πψ ;where the rightmost trace is taken first and traces have to beread such that their argument extends all the way to theright, except for the C traces, which remain within theirparentheses. Thus,jBð3ÞC ðψÞj2 ¼ 8½ðtrRABÞ2 − trR2AB¼ 4½trRAB − CðρABÞ22;whereCðρABÞ is the concurrence of the rank-2 state ρAB andthe calculation follows the spirit of Ref. [18]. For N > 3, allderivations are completely analogous and we obtainjBðNÞj ðψÞj ¼ 2½trR½j − jHðtrjπψ Þj2; ð17Þwhere trR½j ≡ trRðtrjπψÞ and jHðtrjπψÞj is the convexroof of jHj for the rank-2 state trjπψ according to theWootters-Uhlmann method [25,29]. Further, trR½j may bereplaced using Eq. (11). The simplest way to get a quarticmonogamy equality like Eq. (13) is to add the relations (17)for the jBðNÞj j of all qubits,XNj¼1½jBðNÞj ðψÞj þ 2jHðtrjπψÞj2¼ ð−1ÞNXN−1j¼1ð−1Þjþ1jXfjgCfjgjfN−jgðψÞ2: ð18ÞWe mention that CKW equality is obtained by adding therelations for Bð3ÞC and Bð3ÞB only.Curiously, for even N the quartic monogamy equality isnot strictly of degree 4 because jBð2mÞj ðψÞj ¼ jHðψÞj2, andby using Eq. (14) we get back to Eq. (13), since all termsjHðtrjπψÞj vanish. Yet, there do exist degree-4 monogamyequalities also for N ¼ 4 (cf. Ref. [27]); e.g.,jBð4Þ12 ðψÞ − Bð4Þ13 ðψÞj2 ¼ 482 detðtr14πψÞ:Discussion.—We have derived degree-2 and degree-4monogamy relations of pure-state qubit entanglement, thecentral results being Eqs. (13) and (18). We emphasizethat on both sides of these equalities, there appear onlypolynomial functions of the Bloch coefficients.Consequently, these relations can contain only entangle-ment measures based on polynomial functions, and there-fore describe single-copy entanglement properties. Theycan be interpreted as follows. Tracing out B in a bipartitionAjB removes the coherences between A and B, thusconverting their mutual quantum correlations into classicalcorrelations of A alone. The global accounting of all thesecorrelations, which are quantified by the concurrences, isgiven by the SLð2;CÞ⊗N invariants on the left-hand side ofEqs. (13) and (18). 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Monogamy equalities for qubit entanglement from Lorentz invariance

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