Are general quantum correlations monogamous?

Are general quantum correlations monogamous? Alexander Streltsov, 1 Gerardo Adesso, 2 Marco Piani, 3 and Dagmar Bruß 1 1 Heinrich-Heine-Universität Düssel dorf, Institut für Theor etische Physik III, D-40225 Düsseldorf, Germany 2 School of Mathematical Sciences, University of Nottingham, University P ark, Nottingham NG7 2RD, United Kingdom 3 Institute for Quantum Computing and Department of Physics and Astr onomy , University of W aterloo, W aterloo ON N2L 3G1, Canada Quantum entanglement and quantum non-locality are known to exhibit monogamy , that is, they obey strong constraints on how they can be distributed among multipartite systems. Quantum correlations that comprise and go beyond entanglement are quantified by , e.g., quantum discord. It was observed recently that for some states quantum discord is not monogamous. W e prov e in general that any measure of correlations that is monogamous for all states and satisfies reasonable basic properties must vanish for all separable states: only entanglement measures can be strictly monogamous. Monogamy of other than entanglement measures can still be satisfied for special, restricted cases: we prove that the geometric measure of discord satisfies the monogamy inequality on all pure states of three qubits. Entanglement, nonclassical correlations, and nonlocal cor- relations, are all forms of correlations between two or more subsystems of a composite quantum system that are different from strictly classical correlations, and in general dif ferent from each other . One of the characteristic traits of classical correlations is that they can be freely shared. A party A can hav e maximal classical correlations with tw o parties B and C simultaneously . This is no longer the case if quantum entan- glement or nonlocal correlations are concerned [1]. The limits on the shareability of those types of nonclassical correlations are known as monogamy constraints, see Fig. 1 for illustra- tion. Strict monog amy inequalities ha v e been pro v en that con- strain the distribution of particular measures of entanglement and nonlocal correlations (the latter expressed in terms of vio- lation of some Bell-type inequality [2]) among the subsystems of a multipartite system [3–11]. These relations can be seen as a particular case of trade-off relations that in general may re- late and constrain different quantifiers of correlations [10, 12]. Monogamy is the crucial property of correlations that makes quantum ke y distrib ution secure [1, 13], e v en in no-signalling theories more general than quantum mechanics. Nonclassical correlations that go beyond entanglement, of- ten quantified e.g. via the quantum discord [14, 15], ha ve re- cently attracted considerable attention [16, 17]. While en- tanglement captures the non-separability of two subsystems [18, 19], quantum discord detects nonclassical properties even in separable states. Different attempts were presented to con- nect the new concept of quantum discord to quantum entan- glement [20 – 26], and to broadcasting [27–29]. Sev eral ex- perimental results hav e been reported in [30 – 33]. Quantum discord, as well as related quantifiers of quantum correla- tions [17, 22, 23, 34–44], hav e also been linked to better-than- classical performance in quantum computation and commu- nication tasks, e v en in the presence of limited or strictly van- ishing entanglement [30, 45 – 53]. An important question to understand the role of quantum correlations as signatures of genuine nonclassical behavior is whether they distrib ute in a monogamous way among multipartite systems. A bipartite measure of correlations Q satisfies monogamy Figure 1: [Color online] Entanglement is monogamous: for a fixed amount of entanglement between A and BC , the more entanglement exists between A and B , the less can exist between A and C . Quan- titativ ely this is expressed using the monogamy relation, see Eq. (1) in the main te xt. In particular, the latter implies—for a monogamous measure of entanglement E —that E A | C = 0 if E A | B C = E A | B . In this Letter we show that the monogamy relation does not hold in general for any quantum correlation measure beyond entanglement, i.e. for any measure that does not v anish on separable states. if [3, 19] Q A | B C ( ρ AB C ) ≥ Q A | B ( ρ AB ) + Q A | C ( ρ AC ) (1) holds for all states ρ AB C . Here ρ AB = T r C ( ρ AB C ) denotes the reduced state of parties A and B , and analogously for ρ AC . The vertical bar is the familiar notation for the bipartite split. The concept of monogamy is visualized in Fig. 1. If Q denotes in particular an entanglement measure [18, 19], then there are a number of choices that satisfy monogamy for pure states of qubits, including the squared concurrence [3], and the squared negativity [54], as well as their con- tinuous variable counterparts for multimode Gaussian states [5, 6]. The only kno wn measure that is monogamous in all dimensions is the squashed entanglement [10, 55]. Other en- tanglement measures such as, e.g., the entanglement of for- mation do not satisfy the monogamy relation [3]. There is no known a priori rule about whether a giv en entanglement measure is monogamous or not. It is natural to ask whether a gi ven measure for general quantum correlations is monog- amous. Certain measures of general quantum correlations, such as quantum discord, were shown to violate monogamy by finding explicit e xamples of states for which the inequality 2 (1) does not hold [56 – 61]. Those examples, ho we ver , do not exclude the possibility that other measures of quantum corre- lations, akin to the quantum discord, could exist that do satisfy a monogamy inequality . In this Letter we address the issue of whether monogamy , in general, can extend to general quantum correlations be- yond entanglement. Quantitativ ely this question can be for- mulated as follows: Does there exist a measure of correlations Q which obe ys the monogamy relation (1) and is nonzer o on a separable state? W e will put this question to rest by pro ving that all measures for quantum correlations beyond entangle- ment (i.e., that are non-v anishing on at least some separable state) and that respect some basic properties are not monog- amous in general. These basic properties of the correlation measure Q are the follo wing: • positi vity , i.e. Q A | B ( ρ AB ) ≥ 0; (2) • in v ariance under local unitaries U A ⊗ V B , i.e. Q A | B ( ρ AB ) = Q A | B  U A ⊗ V B ρ AB U † A ⊗ V † B  ; (3) • no-increase upon attaching a local ancilla, i.e. Q A | B ( ρ AB ) ≥ Q A | B C ( ρ AB ⊗ | 0 i h 0 | C ) . (4) These properties are valid for se veral measures of correla- tions kno wn in the literature, including all entanglement mea- sures [18, 19]. In particular , positi vity and in variance un- der local unitaries are standard requirements [62]. F or the quantum discord defined in Ref. [14, 15], which is an asym- metric quantity , Eq. (4) can be verified by inspection, and is valid independently of whether the ancilla is attached on the side where the measurement entering the definition of dis- cord is to be performed, or on the unmeasured side. In a more general scenario, quantum correlations can be defined as the minimal distance to the set of classically correlated states [23, 38, 39, 41]. In this case Eq. (4) follows from the f act that any "reasonable" distance does not change upon attaching an ancilla: D ( ρ, σ ) = D ( ρ ⊗ | 0 i h 0 | , σ ⊗ | 0 i h 0 | ) . The same arguments can be applied to measures which are defined via measurements on local subsystems [36]. Alternati vely , quan- tum correlations may be inv estigated and quantified in terms of the minimal amount of entanglement necessarily created between the system and a measurement apparatus realizing a complete projectiv e measurement [22, 23, 26, 63]. Eq. (4) also holds in this case, which can be seen solely using the properties of entanglement measures. W e are no w in position to prov e the follo wing theorem. Theorem 1. A measure of corr elations Q that r espects Eqs. (2) , (3) , and (4) , and is also monogamous accor ding to (1) , must vanish for all separable states. Pr oof. Consider a measure Q respecting the hypothesis, and a generic separable state ρ AC = P i p i | ψ i i h ψ i | A ⊗ | φ i i h φ i | C . In the follo wing we will concentrate on a special e xtension of ρ AC , defined as ρ AB C = X i p i | ψ i i h ψ i | A ⊗ | i i h i | B ⊗ | φ i i h φ i | C , (5) with orthogonal states {| i i B } . Observe that ρ AB C has the same amount of correlations Q A | B C as the state σ AB C = X i p i | ψ i i h ψ i | A ⊗ | i i h i | B ⊗ | 0 i h 0 | C , (6) since both states are related by a local unitary on B C . On the other hand, Eq. (4) implies that σ AB C does not hav e more cor- relations than the reduced state σ AB . T aking these two obser - vations together we obtain Q A | B ( σ AB ) ≥ Q A | B C ( ρ AB C ) . Now we in vok e the monogamy relation for the state ρ AB C , which leads us to the inequality Q A | B ( σ AB ) ≥ Q A | B ( ρ AB ) + Q A | C ( ρ AC ) . (7) The final ingredient in the proof is the fact that the two states ρ AB and σ AB are equal. From the positivity of the measure it follows immediately that Q A | C must v anish on the state ρ AC . Since the latter is a generic separable state, Q must vanish on all separable states. The power of Theorem 1 lies in its generality . Under very weak assumptions it rules out the existence of monogamous correlations beyond entanglement. Note that the arguments used in the proof of Theorem 1 are strong enough to show that the violation of monogamy appears ev en in three-qubit systems. This can be seen starting from Eq. (5), with each subsystem being a qubit. The measure Q violates monogamy , if it is nonzero on some separable tw o-qubit state of rank tw o. This is the case for quantum discord and any related measures of quantum correlations. As we ha ve argued below Eq. (4), the properties (2-4) are satisfied by all reasonable measures of quantum correlations known to the authors. Howe ver , in general it cannot be ex- cluded that the measure under study violates one of the prop- erties gi v en in Eq. (2), (3), or (4). Alternati v ely , we assume that some of these properties cannot be proven. In this situ- ation, Theorem 1 does not tell us whether Q is monogamous or not. Then, it is still possible to sho w that a monogamous measure Q must be zero on all separable states, if it remains finite for a fixed dimension of one subsystem, i.e. if Q A | B ≤ f ( d A ) < ∞ (8) for fixed d A , and some function f . T o see this we use the fact that any separable state ρ AB has a symmetric extension ρ AB 1 ··· B n such that ρ AB = ρ AB i holds for all 1 ≤ i ≤ n , where n is an arbitrary positiv e integer [64 – 67]. Eq. (8) im- plies that the measure Q A | B 1 ··· B n ( ρ AB 1 ··· B n ) is finite for all 3 n , including the limit n → ∞ . On the other hand, if Q is monogamous, it has to fulfill the follo wing inequality: Q A | B 1 ··· B n ( ρ AB 1 ··· B n ) ≥ n Q A | B ( ρ AB ) . (9) Howe v er , if the measure Q is nonzero on the separable state ρ AB , one can always choose some n which is large enough such that Eq. (9) is violated, and thus Q cannot be monoga- mous. So far we have presented two dif ferent ways to show that a giv en measure of quantum correlations Q violates monogamy , namely Theorem 1 and Eq. (8). At this stage it is natural to ask whether these two results have the same po wer , i.e. whether they allo w to draw the same conclusions about the structure of a given measure Q . As already noted above, the proof of Theorem 1 allows to rule out monogamy ev en for the sim- plest case of three qubits, as long as the measure Q does not vanish on some separable state of two qubits having rank not larger than two. On the other hand, this argument does not apply to Eq. (8) and (9). Indeed, if Q is nonzero on some separable two-qubit state ρ AB , Eq. (8) and (9) only allow the statement that the measure Q violates monogamy for some extension ρ AB 1 ...B n . In particular , if n > 2 , this result does not provide any insight about the monogamy of the measure for three-qubit states. W e move on to observe that monogamy (Eq. (1)), together with positi vity (Eq. (2)), in v ariance under local unitary (Eq. (3)) and no-increase under attaching a local ancilla (Eq. (4)), imply no-increase under local operations. This is due to the fact that any quantum operation Λ admits a Stinespring di- lation: Λ[ ρ B ] = T r C  U B C ρ B ⊗ | 0 i h 0 | C U † B C  , i.e. any quantum operation can be seen as resulting from a unitary op- eration on a larger-dimensional Hilbert space. Thus, for Q respecting Eqs. (1), (2), (3), and (4), one finds Q A | B ( ρ AB ) ≥ Q A | B C ( ρ AB ⊗ | 0 i h 0 | C ) = Q A | B C  U B C ρ AB ⊗ | 0 i h 0 | C U † B C  ≥ Q A | B  T r C  U B C ρ AB ⊗ | 0 i h 0 | C U † B C  + Q A | C  T r B  U B C ρ AB ⊗ | 0 i h 0 | C U † B C  ≥ Q A | B (Λ B [ ρ AB ]) . (10) No-increase under local operations [73], and thus, a fortiori , monogamy (the latter together with the almost tri vial proper- ties (2), (3), and (4)) imply the following Theorem 2. A measur e of correlations Q that is non- incr easing under operations on at least one side must be maximal on pur e states; that is, for any ρ AB on C d ⊗ C d ther e exists a pur e state | ψ i h ψ | AB ∈ C d ⊗ C d such that Q A | B ( | ψ i h ψ | AB ) ≥ Q A | B ( ρ AB ) . Pr oof. Immediate when one uses the fact that any state ρ AB can be seen as the result of the application of a channel Λ B ( Λ A ) on any purification | ψ i AB of ρ A ( ρ B ) (see, for exam- ple, [55]). Suppose that the measure Q is non-increasing un- der quantum operations on A . Then: Q A | B ( | ψ i h ψ | AB ) ≥ Q A | B (Λ A [ | ψ i h ψ | AB ]) = Q A | B ( ρ AB ) . (11) This simple theorem is relev ant, in particular , for the case of symmetric measures of quantum correlations. Sev eral such measures were proposed in Refs. [23, 38, 41]. Some of these measures hav e counterintuiti ve properties. In particular , in [23] it was shown that for the relative entropy of quantum- ness there e xist mixed states ρ AB which have more quantum correlations than any pure state | ψ AB i . The just pro ven the- orem can be interpreted as a signature of the fact that general quantum correlations can increase under local operations (and a fortiori as a signature of the lack of monogamy) [41]. Theorem 1 and the reasoning in its proof amount es- sentially to the follo wing insight about the violation of monogamy: if there is a separable state ρ AB with nonzero correlations Q , then there exists a mixed state ρ AB C which prov es that the measure under scrutiny is not monogamous: Q A | B C ( ρ AB C ) < Q A | B ( ρ AB ) + Q A | C ( ρ AC ) . On the other hand, crucially , a measure of correlations can still re- spect monogamy when ev aluated on pure states ρ AB C = | ψ i h ψ | AB C . As will be demonstrated in the follo wing, the ge- ometric measure of discord has exactly this property for three qubits. Before we present this result, we recall the definition of this measure. The geometric measure of discord D G was defined in Ref. [39] as the minimal Hilbert-Schmidt distance to the set of classical-quantum states (CQ): D A | B G ( ρ AB ) = min σ AB ∈ C Q k ρ AB − σ AB k 2 2 . (12) Here we used the 2-norm, also kno wn as Hilbert-Schmidt norm, k ρ − σ k 2 = q T r ( ρ − σ ) 2 , and the minimum is taken ov er all classical-quantum states σ AB . These are states which can be written as σ AB = P i p i | i i h i | A ⊗ σ i B with some local orthogonal basis {| i A i} . The geometric discord has an op- erational interpretation in terms of the av erage fidelity of the remote state preparation protocol for two-qubit systems [68]. As noted abov e, the geometric measure of discord cannot be monogamous in general, since it is nonzero on some separa- ble states. Ho we ver , this measure is monogamous for all pure states of three qubits. Theorem 3. The geometric measure of discord is monoga- mous for all pur e states | ψ i AB C of thr ee qubits: D A | B C G ( | ψ i h ψ | AB C ) ≥ D A | B G ( ρ AB ) + D A | C G ( ρ AC ) , (13) wher e ρ AB = T r C ( | ψ i h ψ | AB C ) and analogously for ρ AC . Pr oof. W e notice that for proving the inequality in Eq. (13) it is enough to show that for any pure state | ψ i AB C there exists 4 a classical-quantum state σ AB C such that D A | B C G ( | ψ i h ψ | AB C ) ≥ k ρ AB − σ AB k 2 2 + k ρ AC − σ AC k 2 2 . (14) This inequality then automatically implies inequality (13), as, due to the minimization in the geometric measure of discord, the right-hand side of (13) can only be smaller than or equal to the right-hand side of (14) . In order to sho w the existence of the mentioned classical-quantum state σ AB C we choose a specific parametrization for a pure state of three qubits [69]: | ψ AB C i = √ p | 0 i A  a | 00 i B C + p 1 − a 2 | 11 i B C  + p 1 − p | 1 i A  γ  p 1 − a 2 | 00 i B C − a | 11 i B C  + f | 01 i B C + g | 10 i B C  . (15) The real numbers p , a and f range between 0 and 1 , g is com- plex with 0 ≤ f 2 + | g | 2 ≤ 1 , and γ = q 1 − f 2 − | g | 2 is also real. W e proceed by ev aluating the left-hand side of Eq. (14), using the explicit formula for pure states [70, 71]: D A | B C G ( | ψ i h ψ | AB C ) = 2 (1 − p ) p. (16) In the next step we define the classical-quantum state σ AB C = P 1 i =0 Π i A ρ AB C Π i A with local projectors in the computational basis: Π i A = | i i h i | A . The e v aluation of the right-hand side of Eq. (14) is straightforward: k ρ AB − σ AB k 2 2 + k ρ AC − σ AC k 2 2 = 2 c (1 − p ) p (17) with c = 1 + [4 a 2  1 − a 2  − 1] γ 2 . The proof is complete, if we can show that c cannot be larger than 1 . This can be seen by noting that the term 4 a 2  1 − a 2  is maximal for a 2 = 1 2 , which leads to the maximal possible value c = 1 . Even though quantum correlations be yond entanglement cannot be monogamous in general, Theorem 3 demonstrates that for pure states of three qubits monogamy of the geomet- ric measure of discord is still preserved. T o the best of our knowledge this is the first instance of a measure of quantum correlations beyond entanglement that satisfies a restricted monogamy inequality . Certainly , this is not a property which all measures of quantum correlations ha ve in common: As shown e.g. in Ref. [56], the original quantum discord violates monogamy e ven on some pure states of three qubits. In conclusion, we hav e addressed the question of monogamy for quantum correlations beyond entanglement. Using very general arguments, we hav e proven that any mea- sure of correlations which is nonzero on some separable state unav oidably violates monogamy . Furthermore, we have shown that any monogamous measure of quantum correla- tions must be maximal on pure states. These results imply sev ere constraints on any monogamous measure of quantum correlations, and can also be used to witness the violation of monogamy . Finally we have shown that ev en though all measures of nonclassical correlations akin to quantum dis- cord cannot be monogamous for all states, they still may obey monogamy in certain restricted situations. In particular, we prov ed that the geometric measure of discord is monogamous for all pure states of three qubits. It is an open question whether there e xists a measure of general quantum correla- tions which is monogamous for tripartite pure states of arbi- trary dimensions. Another open question, which points to a possible future research direction, arises from the generaliza- tion of quantum discord to theories which are more general than quantum [72]. W e hope that the results presented in this paper are also useful for this more general scenario. - Thus, the answer to the question posed in the title is: General quan- tum correlations are in general not monogamous. Acknowledg ements: W e thank Da vide Girolami and Her - mann Kampermann for discussions. MP acknowledges sup- port by NSERC, CIF AR, Ontario Centres of Excellence. GA is supported by a Nottingham Early Career Research and Knowledge T ransfer A ward. MP and GA ackno wledge joint support by the EPSRC Research De velopment Fund (Pump Priming grant 0312/09). GA acknowledges ESF for sponsor- ing the workshop during which this work was started. DB and AS acknowledge financial support by DFG; AS was supported by ELES. [1] B. M. T erhal, IBM J. Res. & Dev . 48 , 71 (2004) [2] J. F . Clauser, M. A. 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