Monogamy Inequality for Entanglement and Local Contextuality S

Monogamy inequality for entanglement and local contextuality S. Camalet

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S. Camalet. Monogamy inequality for entanglement and local contextuality. Physical Review A, American Physical Society, 2017, 95 (6), pp.062329. 10.1103/PhysRevA.95.062329. hal-01990905

HAL Id: hal-01990905 https://hal.sorbonne-universite.fr/hal-01990905 Submitted on 23 Jan 2019

HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. arXiv:1612.03427v2 [quant-ph] 25 Jun 2017 olwn w xrm ae.I si uesae A state, pure a in the is consider A it, If see and To cases. A extreme between other. purity two entanglement each following the The influence and clearly inequality. A, B, of this a state violate than the cannot 23], of also [22, inequality, noncontextuality it majorization a then of satisfies is always sense state that a the state if in Moreover, pure, deter- non- [18–21]. less eigenvalues is a its disobey observables, by to few mined state with a particu- inequality of not ability contextuality system, The local global observables. considered the lar of the states of the and only contextuality.system, involve both must local relation in a and monogamy Such A a entanglement is on between there whether out wonder relation thus carried can One are tests. measurements same the ny antb iltdtgehrwt h HHlo- the CHSH or the with [16], toghether inequality violated cality be cannot for only, [15], A inequality noncontextuality (KCBS) Shumovski their on state. only en- quantum depends of common systems, amount two the between since tanglement observables, specific do 14]. same involve entanglement, [13, the for not tests if inequalities both monogamy C, in contrast, and A In A on for performed satisfied are A necessarily measurements for is violated it is inequality B, this and When 12]. (CHSH) [11, Clauser-Horne-Shimony-Holt inequality the on nonlocality of and based consider measures tests [1], works different Related using concurrence [5–10]. and squared entanglement for systems, of larger first for terms derived, then, in been qubits, have [2– inequalities entanglement three two of Monogamy the measure a quantitative between 4]. specify in trade-off to In it requires a Expressing terms, vanishes. is entanglement. C, there of and amounts case, B, A and between no general In A that is the C. between and there say entanglement maximum, maximally system, the pure, is third a case, is any extreme in state and this B, this A between and Since correlation A state. say entangled 2]. systems, [1, two monogamy entanglement Consider as known is tanglement, thsbe hw htteKlyachko-Can-Binicio˘glu- the that shown been has It en- quantum of property important most the of One ASnmes 36.d 36.a 03.67.Mn 03.65.Ta, 03.65.Ud, numbers: PACS entan too states global are there ine that inequality. monogamy is found inequality, nonconte a this the state-dependent of of satisfies given purity that a the monotone violate between entanglement to and ability state, its and global the of entanglement iesse.I setal eut rmterltosbetw relations the from results essentially It system. tite edrv ooayieult o nageetadlclc local and entanglement for inequality monogamy a derive We ooayieult o nageetadlclcontextual local and entanglement for inequality Monogamy .INTRODUCTION I. aoaor ePyiu herqed aMatière Condens´ la Théorique de Physique de Laboratoire obneUieste,UM nvPrs0,F705 Paris, F-75005, 06, Paris Univ UPMC Universités, Sorbonne I 3322 nqaiy[7,when [17], inequality .Camalet S. n ,i oiie n aihsif vanishes and positive, systems local is the B, of and consisting A system, global the of state n w nt ytm.Tevalue The systems. finite two any u oa prtos n omnct lsial.Suchically, classically. communicate function carry and a operators two operations, when local increase out not does it Moreover, flcloperations local of form faraiycmual uniy htdtrie such determines V. that Sec. quantity, in computable terms states, in readily condition, a simple of a the non- obtain and local we systems the inequality, CHSH four-level violate For to there entangled inequality. that contextuality too is states inequality, global monogamy are found conse- the important of An an quence function. build entropy we particular bounded this IV, upper by is Sec. which monotone, in can entanglement Finally, state explicit its inequality. wether A, the of size disobey measure, specific a entropic for dictates, from an which define, observables, inequality, we dichotomic noncontextuality III, involving state-dependent Sec. state, given in global Then, any the of other. entanglement each the purity and constrain the state, how local entropy quantitatively a an of expresses the of result properties of essential This function the a [24]. has exceed that entangle- A, cannot Sec. of the B, in state and monotone, show, A entanglement between first any ment We for that, mentioned purity. II, above local the between and and exploit contextuality, entanglement we and so, purity between do bipar- relations To finite any system. for tite contextuality, local and one. tanglement other any by majorized maximally is the is which entan- A state, maximally for are mixed operator B density and reduced A the If gled, uncorrelated. are B and a ytmonly, system cal I EAINBTENENTANGLEMENT BETWEEN RELATION II. nti ae,w eieamngm nqaiyfren- for inequality monogamy a derive we paper, this In ecnie measure a consider We ρ ldt ilt h oa noncontextuality local the violate to gled e h nrp falclsaeadthe and state local a of entropy the een taiyieult.W ul nexplicit an build We inequality. xtuality E 7→ [Λ( P E ult.A motn consequence important An quality. tt,i h es fmajorization, of sense the in state, ρ k )] netaiy o n nt bipar- finite any for ontextuality, sa nageetmntn.Mr specif- More monotone. entanglement an is M N OA ENTROPY LOCAL AND ≤ k P ρM E e M 7600, UMR ee, k ( ρ M ρ k † o rnfrain composed Λ transformations for ) 7→ | ⊗ k † M France k E P k ih seult h corresponding the to equal is fteetnlmn between entanglement the of k k | M where , k ρM E k † ity ( ρ n aso the of maps and , M ρ ,where ), sntentangled. not is k cso n lo- one on acts ρ sthe is 2 identity operator, and k are orthonormal states of an with Schmidt coefficients √pi, there are unitary opera- | i ancilla close to the other system [2]. We reiterate that E tors UA and UB, acting on A and B, respectively, such ′ ′ ′ is defined for local systems of any sizes. Clearly, the last that Ψ = UA UB Ψ . Thus, Ψ Ψ and Ψ Ψ can map above transforms states of a system, into states of a be transformed| i ⊗ into each| i other by| localih | operations.| ih | Con- ′ ′ different system. Moreover, Mk can be a linear operator sequently, E( Ψ Ψ ) = s, and hence, s is a function from the Hilbert space of a system, to that of one of its of p only. Since| ihρ is| an arbitrary state of (ρ ), and C A subsystems, or to that of a local larger system [3]. Ψ Ψ (ρA), Sd(ρA)= s. We are interested in the constraint on the reduced den- | Ifih p|∈C= 1, Ψ is a product state, and so s = 0. | i sity operator of a local system, set by the entanglement Consider Φ = √qi i χi with q majorizing p. We | i i | i| i E(ρ). To express it, we define, for states ρA of a d-level have s(p) = E( Ψ Ψ ) and s(q) = E( Φ Φ ). Since system A, Ψ Ψ can be changed| Pih | into Φ Φ by local| ih operations| |andih classical| communication [29],| ih s|(p) s(q). ≥ Sd(ρA) max E(ρ), (1) ≡ ρ∈C(ρA) Relation (2) means not only that

† where (ρA) is the set of all states ρ of all composite Sd(UρAU )= Sd(ρA), (3) systemsC consisting of A, and another system, such that where U is any unitary operator of A, but also that the reduced density operator for A is ρA. For any system B, and any state ρ of the global system AB, consisting of d d A and B, S p ˜ı ˜ı = S p i i , (4) d+1 i| ih | d i| ih | i=1 ! i=1 ! Sd(trB ρ) E(ρ), X X ≥ where i d and ˜ı d+1 are orthonormal bases of the {| i}i=1 {| i}i=1 where trB denotes the partial trace over B. As we will considered Hilbert spaces, and the probabilities pi obey see, the equality is reached when ρ is pure. We show be- d i=1 pi = 1. The classical form of equation (4) is known low that the functions (1) have the essential properties as the expansibility property, and is an essential require- of the familiar entropies (von Neumann, Rényi, Tsallis, mentP for an entropic measure [24, 30]. . . . ) [24]. Thus, the above inequality expresses how the purity of the local state trB ρ, and the entanglement of the global state ρ, constrain each other. We remark that III. ENTROPIES FROM this inequality is not necessarily satisfied if Sd is replaced NONCONTEXTUALITY INEQUALITIES by an arbitrary entropy function. For distillable entanglement, entanglement cost, entanglement of formation, Our aim is to study the influence of the entanglement and relative entropy of entanglement, eq.(1) gives the von between systems A and B, on contextuality tests involv- Neumann entropy [2, 25, 26]. For robustness and nega- ing only A. This local contextuality can be revealed by tivity, Sd(ρA) is simply related to the 1/2-Rényi entropy considering N dichotomic observables Ak of A, such that [27, 28]. each observable is compatible with some other ones, but Proposition 1. The functions (1) satisfy not with all. We restrict ourselves to the usual case of projective measurements with two outcomes. When Sd(ρA)= s(p), (2) evaluated with a noncontextual hidden variable theory, the correlations of the compatible observables, satisfy where p is the vector made up of the nonzero eigenvalues inequalities, which can be violated by quantum states. of ρA, in decreasing order, and s does not depend on Such a noncontextuality inequality reads d, vanishes for p = 1, and obeys s(q) s(p) when q ≤ p xn Ak 1, (5) majorizes . ≤ n k∈En X Y Proof. Consider any system B’, and any state ρ (ρ ) A where are subsets of 1,...,N , of any possible size, of the composite system AB’. Denote its eigenvalues∈C by n and ...E = tr(ρ ...) is{ the average} with respect to the λ , and its eigenstates by ψ . Let us introduce a third A m m densityh matrixi ρ . The observables A and A commute system, say B”, which constitutes,| i together with B’, sys- A k l with each other when k,l . The coefficients x are tem B. Provided the Hilbert space dimension of B” is n n such that the maximum value∈ E of the left-hand side of large enough, ρ can be written as ρ = tr ′′ Ψ Ψ , where B | ih | eq.(5), is 1 for noncontextual hidden-variable models, i.e., Ψ = m √λm ψm φm is a pure state of system AB, | i | i| i there are ak = 1, such that n xn k∈En ak = 1. The with orthonormal states φm of B”. As trB′′ is a local ± P | i familiar CHSH and KCBS inequalities [11, 12, 15], for operation, on B, E(ρ) s where s = E( Ψ Ψ ). P Q Since tr Ψ Ψ =≤ p i i , where| ih p| are the example, can be cast into the form (5). Let us define B | ih | i i| ih | i nonzero eigenvalues of ρA, and i are the correspond- P | i ing eigenstates, Ψ = i √pi i χi , where χi are or- Cd(ρA) sup tr ρA xn Ak , | i | i| i | ′ i ≡ A∈Ad thonormal states of B. For any pure state Ψ of AB, n k∈En ! P | i X Y 3

d where d is the Hilbert space dimension of A, A stands Proof. Consider any orthonormal bases ˜ı i=1 and d0 d1 {| i} for (A1,...,AN ), and d is the set of all A consisting of i , and define Ω = ˜ı ρ ˜ i j , where d = A i=1 i,j=1 A 1 dichotomic observables Ak, such that [Ak, Al] = 0 for {| i} h |−1| i| ih | min d, d0 , and the state ω = t Ω, where t = tr Ω. k,l n. By construction, for a state ρA such that { } P ∈ E It has been shown that Cd0 (ω) = supµ∈Λ[µ λ(ω)], Cd(ρA) > 1, there are observables Ak with which in- · where a b = d0 a b [20]. Since tλ(ω) = λ(Ω), equality (5) is violated. · i=1 i i tCd0 (ω) = supµ [µ λ(Ω)]. It has been shown that Cd(ρA)= cd(p), where p is the ∈PΛ · We denote λ(ρA) by p. For d > d0, the matrix rep- vector made up of the eigenvalues of ρA, in decreasing resentation of Ω, in the basis i , is a diagonal block order, and cd satisfies cd(q) cd(p) when q majorizes p {| i} ≥ of that of ρA, in the basis ˜ı . Thus, p weakly sub- [20]. However, the functions Cd do not obey the expan- {| i} majorizes λ(Ω) [23], and so, for j = 1,...,d0, Rj sibility condition (4), and really depend on the dimen- j ≡ sion d. Due to the above-mentioned property of cd, Cd i=1[λi(Ω) pi] is negative. Consequently, for any µ max − [d0] d0−1 ∈ reaches its maximum, Cd cd(1), for pure states. We Λ, µ [λ(Ω) p ]= j=1 (µj µj+1)Rj +µd0 Rd0 0, ≡ max P · − − ≤ assume that there are dimensions d for which C > 1. [d0] d where p is made up of the d0 largest pi, in decreasing For these values of d, inequality (5) constitutes a proper P [d0] order. Hence, tCd0 (ω) supµ∈Λ(µ p ). For d d0, contextuality test, since it is not always satisfied. Note ≤ · ≤ this inequality becomes an equality, with p[d0] made up that, for some state-dependent noncontextuality inequal- of the p , in decreasing order, followed by d d zeros, ities, Cmax does not depend on d, provided it is larger i 0 d since λ(Ω) = p[d0]. − than some value [31]. For CHSH inequality, for example, d For any d, when ˜ı i=1 is such that ρA = it is equal to √2, for d 4. We also remark that the d [d{|0] i} ≥ i=1 pi ˜ı ˜ı , λ(Ω) = p , which finishes the proof. operators Ak = akA, where A is any dichotomic observ- | ih | able, and ak = 1 are such that n xn k∈En ak = 1, P obviously fulfill the± above-stated commutation relations. P Q IV. MONOGAMY OF ENTANGLEMENT AND Such a case describes a set-up that consists of N measure- LOCAL CONTEXTUALITY ment apparatuses corresponding to the same observable A. As a consequence, if all products in eq.(5), have an The functions (6) have all the required characteristics even number of terms, C 1. d to satisfy eq.(1) with an entanglement monotone E. It To study the impact of≥ the entanglement between A remains to show that there is indeed such a measure E. and B, on the local contextuality test (5), we define This can be achieved, by using the convex roof method lc d1 [2], since, due to the convexity of Cd0 [20], Sd , given by lc max Sd (ρA) Cd0 max tCd0 ˜ı ρA ˜ i j /t , (6) eq.(6), is concave. ≡ − {|˜ıi}  h | | i| ih |  i,j=1 X Proposition 3. Consider, for any composite system AB,   and any state ρ of AB, where d0 is a specific dimension, d1 = min d, d0 , t = d1 d0 { } i=1 ˜ı ρA ˜ı , i i=1 is an orthonormal basis, and the cr h | | i {| i} d E (ρ) inf PmSd (trB Ψm Ψm ) , (7) maximum is taken over the orthonormal bases ˜ı i=1 ≡ {Pm,|Ψmi}∈D(ρ) | ih | P {| i} m of A. Since Cd0 obeys eq.(3), the definition (6) does not X depend on any particular basis. For d = d , it reduces 0 where (ρ) is the set of all ensembles Pm, Ψm such to Slc (ρ ) = Cmax C (ρ ), but, for d = d , Slc and D { | i} d0 A d0 d0 A 0 d that Pm Ψm Ψm = ρ, d is the Hilbert space dimen- C are not simply related− to each other. As6 Cmax is the m | ih | d d0 sion of A, and Sd are positive concave functions obeying maximum value of Cd0 , the functions (6) are positive. eq.(2),P and vanishing for pure states. As a consequence of the result below, they also fulfill The function Ecr is an entanglement monotone, and the properties enumerated in proposition 1. Note that satisfies eq.(1) with Sd. there are state-independent noncontextuality inequalities [32–35] for which the definition (6) gives zero for any Proof. We first consider that ρ is not entangled. Then, state, and is thus of no use. In this case, no meaningful by definition, ρ is a mixture of pure product states Ψm . lc | i entanglement monotone E can obey eq.(1) with Sd , since The corresponding states trB Ψm Ψm are pure, and the only possibility is E = 0. In the following, we use the hence Ecr(ρ) = 0. | ih | notation λ(M) for the vector made up of the eigenvalues Let us now prove that interchanging A and B of the Hermitian operator M, in decreasing order. does not modify expression (7). The reduced den- Proposition 2. The functions (6) satisfy sity operators trB Ψm Ψm and trA Ψm Ψm , have the same nonvanishing| ih eigenvalues.| | Thus,ih | due to d0 lc max eq.(2), Sd(trB Ψm Ψm ) in eq.(7), can be replaced by | ih | ′ Sd (ρA)= Cd0 sup µipi , S ′ (tr Ψ Ψ ), where d is the Hilbert space dimen- − µ d A m m ∈Λ i=1 ! | ih | X sion of B. It follows from eq.(7) that Ecr is convex [3]. For op- where Λ is the set of all vectors λ( n xn k∈En Ak), with (A ,...,A ) , p = λ (ρ ) for i d, and p = 0 erators B of system B, such that B†B is equal 1 N ∈ Ad0 i i A P ≤Q i k k k k for i > d. to its identity operator, the concavity of Sd leads to P 4

Ecr(ρ) p Ecr(ρ ), where p = tr(B†B ρ) and map, which is not greater than 1 when ρ is not entan- ≥ k k k k k k ρ = B ρB†/p [3]. This inequality and the convexity of gled [38–40]. In ref.[41], a lower bound is derived for the k k kP k Ecr ensure that Ecr does not increase under local oper- entanglement of formation, in terms of ations on B. With expression (7) rewritten as explained x max ρΓ , (ρ) , (9) above, the same proof shows that this is also the case for ≡ {k k kR k} local operations on A. Since ρk andρ ˜k = ρk k k , which is readily computable. We show below that a sim- where k is a pure state of an ancilla close⊗ | toih A,| | i ilar bound can be obtained for any entanglement mono- can be transformed into each other by local operations, tone of the form (7). Ecr(˜ρ ) = Ecr(ρ ). Thus, Ecr( p ρ˜ ) Ecr(ρ), k k k k k Proposition 4. Consider an entanglement monotone Ecr which finishes the proof that Ecr is an entanglement≤ given by eq.(7), two systems, A and B, of Hilbert space monotone. P dimensions d and d′, respectively, and the function f de- Consider a given state ρ of A, and any state ρ A ∈ fined, for y [1, d∗], where d∗ = min d, d′ , by (ρA). The definition (7) and the concavity of Sd give ∈ { } C cr ′ E (ρ) Sd(ρA). If d d, there are pure states Ψ of f(y) inf s(p), (10) ≤ ≥ cr | i p∈F(y) AB such that trB Ψ Ψ = ρA, and hence E ( Ψ Ψ )= ≡ S (ρ ). Consequently,| ih max| Ecr(ρ)= S |(ρih). | d A ρ∈C(ρA) d A where s is given by eq.(2), and (y) is the set of the d∗-component probability vectorsF p, such that We have thus, for a d0-level system A, the monogamy ∗ d 2 inequality ( i=1 √pi) = y. For any state ρ of AB, Ecr(ρ) co(f)(x), where co(f) max P ≥ E(ρ)+ Cd0 (ρA) C 0 , (8) is the convex hull of f, and x is given by eq.(9). ≤ d where E is given by eq.(7) with the functions (6). Thus, Proof. Let us first show that co(f) exists and is nonde- the entanglement of A with B, as quantified by E(ρ), creasing. Since s is positive, f 0, and thus, f has a restricts the value of the left side of inequality (5). In convex hull [42]. It is the maximum≥ of the convex func- max particular, for a state ρ such that E(ρ) Cd0 1, this tions not larger than f. As f 0, co(f) is positive. noncontextuality inequality cannot be≥ violated.− Equa- The only element of (1) is p =≥ 1. Thus, f(1) = 0, F tion (8) can also be read as an upper bound on the entan- and hence, co(f)(1) = 0. Consider y1 and y2 such that ∗ glement E(ρ). In the extreme case of maximal violation 1 y1 y2 d . We have y1 = τ + (1 τ)y2 with max ≤ ≤ ≤ − of eq.(5), i.e., Cd0 (ρA)= Cd0 , it gives E(ρ) = 0. There τ [0, 1]. So, using the convexity and positivity of co(f), may be other entanglement monotones that coincide with and∈ co(f)(1) = 0, we get co(f)(y ) co(f)(y ). 1 ≤ 2 the functions (6) when ρ is pure, and so satisfy inequality Consider any ensemble Pm, Ψm (ρ), and de- (8). But, there is no entanglement monotone, for which note by p(m) the d∗-component{ | vectori} ∈ made D up of the eq.(8) is always an equality, since Cd0 [Λ(ρA)] Cd0 (ρA) squared Schmidt coefficients of Ψm , in decreasing or- for some local operations Λ on A. Some noncontextuality≤ der, possibly completed with zeros.| i By definition of inequalities (5) are violated for any state ρ, which, in this f, PmSd(trB ρm) Pmf(ym), where ρm = m ∗ m max ≥ d (m) 1/2 2 case, necessarily satisfies E(ρ) < Cd0 1. If the cor- Ψ Ψ , and y = [ (p ) ] . The right side − | mPih m| m i=1P i responding function Cd0 is constant, E = 0, and eq.(8) of this inequality is not smaller than co(f)(y) where y = is trivially obeyed, and of no relevance. This is not sur- P Γ m Pmym. Since ym = ρm = (ρm) [28, 38, 43], prising, since such a state-independent noncontextuality and the trace norm is convex,k kx kRy. Usingk this inequal- ≤ inequality is always maximally violated [32–35]. If Cd0 is Pity, and the monotonicity of co(f), leads to the result. larger than unity, but not constant, eq.(8) still gives an upperbound, that depends on the entanglement beween The above proposition, and the monogamy inequality A and B, for the left side of eq.(5). (8), give, for a d0-level system A, C (ρ ) Cmax co(f)(x), (11) d0 A ≤ d0 − V. COMPUTABLE MEASURES OF where f is given by eq.(10) with the functions (6), and ENTANGLEMENT ∗ ′ ′ d = min d0, d with d the Hilbert space dimension of { } max B. For states ρ such that co(f)(x) Cd0 1, eq.(5) can- The monogamy inequality (8) involves an unusual en- not be violated. As noted above, if≥ all products− in eq.(5), tanglement monotone, defined from the considered non- have an even number of terms, Cd 1, and so, co(f) can contextuality inequality. Moreover, even familiar entan- max ∗ ≥ reach Cd0 1 only for x = d , i.e., for maximally en- glement monotones are difficult to evaluate for an arbi- tangled states.− However, even in this case, eq.(11) can trary density matrix ρ [36]. An exception is the negativ- be useful to determine states too entangled to violate Γ √ † ∗ ity ( ρ 1)/2, where M = tr MM denotes the inequality (5). As an example, consider d = d0 = 4, k k− k k Γ trace norm of operator M, and ρ is a partial transpose and the CHSH inequality, for which, as shown below, ′ of ρ [2, 28, 37]. There are entangled states with vanishing C4(ρA) = max 1, C4(ρA) , where negativity. Other quantities can be used to detect entan- { } glement, e.g., (ρ) , where is a matrix realignment C′ (ρ )= 2 [(p p )2 + (p p )2], (12) kR k R 4 A 1 − 4 2 − 3 p 5

′ with pi = λi(ρA). The functions (6), defined with C4, of [Πk, Πk+2], where k = 1 or 2, are iνk νˆk . Since have all the necessary properties to obey eq.(11) with the these two commutators commute with± each| other,| and corresponding f. Using the method of Lagrange multi- tr R = 0, there are Ak such that λ(R)=4r(1, 1, 1, 1), pliers, we find where r ]0, 1]. For these A , tr(A A ) = 0 for− com-− ∈ k k l muting Ak and Al, and hence tr T = 0. So, λ(T ) = 3/2 1/2 f(y)= √2 (3√y+ 32 7y) ( 32 7y √y) /32. (r+, r−, r−, r+), where r± = 2√1 r. Maximising − − − − λ(T ) p−over −r, leads to eq.(12). ± This function is convex,p and is hencep equal to its convex · max hull. It increases from 0 to C4 = √2. Consequently, ′ for states ρ such that x 2.95, C4(ρA) 1, and thus, the local CHSH inequality≥ is always satisfied.≤

Proof. For CHSH inequality, Cd(ρA) = supA∈Ad T /2, where T = tr(ρ T ), and T = A (A + A )+ A h(Ai VI. CONCLUSION h i A 1 2 4 3 2 − A4). For Ak = A1, T = 2, and hence Cd 1. We are thus interested onlyh i in observables A such≥ that T k In summary, a monogamy inequality for entanglement can be larger than 2. For d = 4, A can be writtenh i k and local contextuality, has been derived. It involves as A = η (2Π I) where η = 1, I is the identity k k k k an entanglement monotone that depends on the consid- operator, and Π −is a projector of rank± not greater than k ered noncontextuality inequality, and the Hilbert space 2. Using this expression, one finds T 2 =4I 16R, where dimension of the local system. It essentially results from R = [Π , Π ][Π , Π ] [44]. If R = 0, the eigenvalues± of T 1 3 2 4 the relations between the entanglement of the global state can only be 2 and 2, and so T 2. We thus search and the entropy of the local state, and between the eigen- for the projectors Π− for which hR =i ≤ 0. If two commuting k values of the local state and its ability to disobey the Π and Π , obey Π Π =0orΠ 6 Π = Π , then R = 0. k l k l k l k noncontextuality inequality. Thus, other entanglement Consequently, the sought projectors are of rank 2, and monotones, different from the one we have built, may such that, for [Π , Π ]=0, Π Π is a rank-1 projector. k l k l satisfy the same monogamy inequality. A consequence of This gives Π = k k + k′ k′ , with k | ih | | ih | the found monogamy, is that there are global states so entangled that they cannot violate the noncontextuality 1 = 1˜ , 1′ = 2 = 2˜ , 2′ = 3˜ , 3 = ν 2˜ +ˆν 3˜ , | i | i | i | i | i | i | i | i 1| i 1| i inequality. The obtained monogamy inequality relates ′ ′ 3 = ν1 1˜ +ˆν1 4˜ , 4 = ν2 1˜ +ˆν2 2˜ , 4 =ν ˆ2 3˜ +ν2 4˜ , entanglement per se to local contextuality. It would thus | i | i | i | i | i | i | i | i | i be of interest to find out if there are global states, that 4 2 2 where ˜ı i=1 is any orthonormal basis, νˆk + νk = 1, are Bell-local [11], but still entangled enough to prevent and ν {| i}[0, 1]. For these projectors, the| | eigenvalues the violation of a local noncontextuality inequality. k ∈

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