On the Monogamy of Entanglement of Formation
It is well known that a particle cannot freely share entanglement with two or more particles. This restriction is generally called monogamy. However the formal quantification of such restriction is only known for some measures of entanglement and for two-level systems. The first and broadly known monogamy relation was established by Coffman, Kundu and Wootters for the square of the concurrence. Since then, it is usually said that the entanglement of formation is not monogamous, as it does not obey the same relation. We show here that despite that, the entanglement of formation can not be freely shared and therefore should be said to be monogamous. Furthermore, the square of the entanglement of formation does obey the same relation of the squared concurrence, a fact recently noted for three particles and extend here for particles. Therefore the entanglement of formation is as monogamous as the concurrence. We also numerically study how the entanglement is distributed in pure states of three qubits and the relation between the sum of the bipartite entanglement and the classical correlation.
In the last decades, we have seen many advances in the theory of entanglement HorodeckiReview () and, more generally, in the theory of quantum correlation rmp (). Nonetheless, the setting with more than two parts is still a challenge even from a conceptual point of view, not to mention the quantification. In this scene monogamy relations are important, as they may indicate a structure for correlation in the multipartite setting. Monogamy has also been found to be the essential feature allowing for security in quantum key distribution Pawlowski (). In the literature, the concept of monogamy for a entanglement measure becomes synonymous with satisfying the inequality
The first monogamy relation established was due to Coffman, Kundu, and Wootters (CKW) Kundu () for three qubits and latter generalized for qubits Osborne (). It relates the squared concurrence between bipartitions as follows:
Such relation tells us that the one particle (particle 1) cannot freely share entanglement with other qubits, thus the name monogamy. This monogamous relation has also been studied for other entanglement measures Koashi2004 (); Cornelio2010 (); Bai () and also for the discord Fanchini11 (); Giorgi11 (); Prabhu12 (); Braga (). It is well known that the entanglement of formation, when not squared, does not obey the inequality given by Eq. (1). Thus it is usually said that the entanglement of formation is not monogamous.
In this work, we discuss the concept of monogamy beyond inequality given by Eq. (1), focusing on the entanglement of formation (EF). We argue that not satisfying the inequality given by Eq. (1) does not mean that a measure is not monogamous and can be freely shared. In fact, we numerically found an upper bound for using the entanglement of formation, which is considerably smaller than 2. Besides, very recently it was shown by Bai et al Bai () that, when squared, EF does satisfy a monogamy relation for three-qubit systems like the concurrence. We also generalize this result for qubits. Therefore there is no reason at all to say the the entanglement of formation is not monogamous. Finally we analyze the distribution of the bipartite entanglement in three-qubit pure states and the relation between it and the classical correlation.
The text is organized as follows: We begin by presenting a detailed study exploring the distribution of quantum correlations, considering EF and concurrence. First, we study the squared EF for tripartite and multipartite systems. We point out that the EF is as monogamous exactly as the concurrence, since the squared EF does obey the CKW relation. Following, we consider a system composed of three qubits and, numerically, obtain a bound for and a similar result for the discord. Moreover, we show how these monogamy relations behave for random states.
Ii Monogamy Relations
The most basic and drastic monogamy relation happens for maximally entangled pure states: two particles and maximally entangled can not be entangled with a third particle. That happens because to be maximally entangled they have to be in a pure state, in an singlet, for example, . But a pure state has null entropy and therefore null correlation with any other system: cannot be entangled, or even classically correlated, with a third particle and thus neither can or . However, this argument only applies when the first two parts are in a pure state. When they are in a mixed state the restriction is not so drastic: one particle may be entangled with two others at the same time.
We divide our work in two parts: first we study the monogamy considering squared measures of entanglement and, in sequence, we explore the monogamy for each measure per se, i.e., the measure up to 1.
ii.1 Monogamy for the squared version
For three qubits, the amount of total entanglement that can be shared is restricted by the CKW inequality:
Note that here is the concurrence between qubit and the joint qubits , which can be analytically obtained for pure states and it is at most . In Fig. 1 we show a histogram of the value of for random pure states of three qubits sampled uniformly (Haar measure 111All the figures were made using random states to make the files smaller. But we checked that the same behavior holds using random states ). It can be seen that few states are close to the upper bound . As mentioned before the monogamy relation is also true for particles as shown in Osborne ().
Concurrence was actually introduced as an intermediate measure to obtain the entanglement of formation, but, as it is a monotonic function of EF, it is usually used as the entanglement measure Bennett96 (); Wooters98 (). However, contrary to EF, it has no clear operational meaning. Therefore, it is natural to ask whether the EF also obeys the relation above. Already in the original paper, CKW have shown that EF does not obey the monogamy relation, since it is a concave function of . It is important to emphasize, however, that the authors even mention that this is not a paradox, since they have only shown that EF does not obey this particular kind of monogamy relation, given by Eq. (3). Since then, it is usually said that EF is not monogamous. It is curious that the authors analyzed EF and not its square, since they were working with and not .
In fact it is easy to show that does obey the monogamy relation for qubits, and this fact was first noticed by Bai et al Bai (). Their proof is for 3 qubits, but their argument is equally valid for qubits. Actually, in the first paper of CKW Kundu (), it was already noticed that any monotonic convex function of would also be a monogamous measure of entanglement. Indeed, that is exactly the difference between the EF and the squared EF: EF is a concave function of while the squared EF is a convex function of . Given that the general proof cannot be found either in Ref. Kundu () or in Ref. Bai (), we reproduce their arguments here for the general case. We start with the fact that the is a monotonic function of and, using the CKW monogamy for qubits, we have
Now we consider the expression
We have to prove that it is greater than zero. For that, we use the fact that is a convex function of . This implies that the inclination of the straight lines from the origen to the points increases monotonically with . That is, we have, for
that for all . Replacing the expressions for the ’s in Eq. (4) and using that we have that is always positive.
Following, we also study how the entanglement is distributed in the states. More specifically, we are interested in the following question: do the states which are close to the bound have most of their entanglement from one pair and just a little between the other pair or does it comes from both pairs? In Fig. 2 we plot the sum of the square of the entanglement versus the entanglement of one the pairs. We can see that, for the concurrence, it seems that the saturation of the bound can come just from the entanglement of one pair, or from both pairs. On the other hand, the saturation of the square of EF comes exclusively from one of the pairs: there are no state close to the bound with . This shows that although monotonically related the concurrence and EF may have different qualitative properties.
ii.2 Monogamy for the linear version
Noting that does not obey the monogamy relation given by Eq. (3), should we say it is not monogamous? Can it be freely shared between three particles? This is the question we address here. We want to find an upper bound for
We know the sum cannot achieve 2, but how close can it be? We studied this case numerically for a system of three qubits considering a sampling of random states uniformly distributed and we found that
In Fig. 3 we plot the value of the sum of EF for random states sampled uniformly. We can note that there is an upper bound and that just a few states are close to it. Thus it is at least misleading to say that the EF is not monogamous. Indeed, as we can see in Fig. 3, there are strong constrains on how it can be shared. Furthermore, even concurrence also does not obey the usual monogamy relation as can be seen in Fig. 4. We also study the distribution of the entanglement, as we did for the square of the measures, in Fig 5. We can see that in this case the behaviors of the concurrence and the EF are similar. This indicates that the square of the measure can have different qualitative behavior from the measure itself.
We should also point out that for pure states of three qubits there is a conservation law between discord and EF, such that Fanchini11 (). So the monogamies of the two quantum correlations are tightly connected, and a violation of one would imply of the other, something already noted before by some of us Fanchini13 (). These relations are obtained from the Koashi-Winter (KW) relation Koashi2004 (), where is the one way classical correlation between and Henderson01 (); Olivier01 (). The classical correlation is defined as the condition entropy after measurements: with being the reduced state of , after a measurement made on with as the result. Additionally, a maximization over all measurements on is performed.
Is it also worth noting that using such relation, we can obtain some insight about the state that maximizes . The idea is to use the KW relation to write the bipartite EF sum as a function of the entropy . Our point is to answer an important question: what is the state and the value of entanglement between subsystems and that saturates the monogamy inequality? Is it a maximally entangled state? To answer this question we begin with an expression where the sum of the bipartite EF is written as a function of the sum of the bipartite classical correlation and the entropy . We use the KW relation twice to obtain that . The sum of the classical correlation as a function of the sum of the EF is plotted in Fig. 6. It can be seen that, as the states increase the sum of the EF, the range of possible values of the classical correlation decrease and goes to around 0.8. At this situation we note that the sum of EF tends to since . It means that the maximum value of occurs when , i.e. when subsystem is maximally entangled with subsystem . With this result we then try to determine the state that maximizes . Since we look for a state with and the sum of bipartite entanglement maximum, we discard the GHZ states (since for these states ) which leads us to the state given by
This state has and the sum of the bipartite EF is give by . This result is greater than all of our numerical results which strongly suggests that the saturation of the monogamy inequality is reached when subsystem is maximally entangled with subsytem .
We have studied how the entanglement between three particles may be shared. We numerically found un upper bound on the sum of the entanglement of formation, , showing that it can not be freely shared even though it does not obey the Coffman, Kundu, and Wootters (CKW) relation . Furthermore, the square of the entanglement of formation does obey the relation, as shown here for particles extending the proof for three particles Bai (). Thus the entanglement of formation is as monogamous as the concurrence: the square of both obeys the relation, but not the measure itself which is limited by another upper bound that we numerically found. Interestingly, the states with maximum sum of the square of the entanglement of formation have their entanglement coming mostly from one of the pairs, while for the concurrence it can come from both pairs. Finally we analyzed the relation between the sum of the entanglement of formation and the classical correlation.
As a perspective one could define a genuine tripartite entanglement measure analogous to the tangle but using the square of the entanglement of formation instead of the square of the concurrence . We already observed that is 1 for the GHZ state, but not null for the W state. Note that when using the concurrence is null for the W state and this is usually used to argue that the W state has no genuine tripartite entanglement. So one could argue that the W does have genuine tripartite entanglement or that are not really good measures of genuine tripartite entanglement. But before that one should first study with more care to see if it is really a bona fide entanglement measure: for example, is it monotonic under local operation and classical communications (LOCC)?
Note added - After submission of this manuscript we became aware of related work, arXiv:1401.3205, in which the monogamy inequality is proved also for mixed states.
Acknowledgements - We would like to thanks the organizers of "IV Quantum Information School and Workshop of Paraty" where the discussions which lead to this work were started in the very informal and fruitfulbeach atmosphere of the workshop. We are also in debt to Y. K. Bai for clarifying discussions about monogamy of the squared entanglement of formation. The authors are supported by the National Institute for Science and Technology of Quantum Information (INCT-IQ) under process number 2008/57856-6 and FFF is supported by São Paulo Research Foundation (FAPESP) under grant number 2012/50464-0, and by the National Counsel of Technological and Scientific Development (CNPq) under grant number 474592/2013-8.
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