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Monogamy of the Entanglement of Formation

We show that any measure of entanglement that on pure bipartite states is given by a strictly concave function of the reduced density matrix is monogamous on pure tripartite states. This includes the important class of bipartite measures of entanglement that reduce to the (von Neumann) entropy of entanglement. Moreover, we show that the convex roof extension of such measures (e.g., entanglement of formation) are monogamous also on \emph{mixed} tripartite states. To prove our results, we use the definition of monogamy without inequalities, recently put forward[Gour and Guo, Quantum \textbf{2}, 81 (2018)]. Our results promote the theme that monogamy of entanglement is a property of quantum entanglement and not an attribute of some particular measures of entanglement.

Let H A ⊗ H B ⊗ H C ≡ H ABC be a tripartite Hilbert space with finite dimension, where A, B, C are three subsystems of a composite quantum system, and S(H ABC ) ≡ S ABC be the set of density matrices acting on H ABC . Recall that the original monogamy relation of entanglement measure E is quantitatively displayed as an inequality of the following form: where the vertical bar indicates the bipartite split across which the (bipartite) entanglement is measured. However, Eq. (1) is not valid for many entanglement measures [7,[9][10][11][12][13][14][15][16][17]. This may give the impression that monogamy of entanglement is not a property of entanglement itself but of the function that is used to quantify it. Moreover, in Ref. [47] the problem of faithfulness versus monogamy was raised by showing that many measures of entanglement, such as the entanglement of formation, cannot satisfy any relation of the form where f : R + × R + → R + is some fixed function that is independent on the dimension of the underlying Hilbert space, and that it is continuous, and satisfies f (x, y) ≥ max{x, y} with strict inequality at some ranges of x and y. Despite this remarkable result, we show here that the entanglement of formation, and many other measures of entanglement that are defined in terms of convex roof extensions, are monogamous according to the definition recently put forward in Ref. [46]. According to the definition in Ref. [46] of monogamy (without inequalities), a measure of entanglement E is monogamous if for any ρ ABC ∈ S ABC that satisfies the disentangling condition, i.e., we have that E(ρ AC ) = 0. With respect to this definition, if the entanglement between system A and the composite system BC is as much as the entanglement that system A shares just with B, then there is no entanglement left for A to share just with C. Clearly, this definition captures the spirit of monogamy of entanglement, and perhaps not surprisingly, can yield a family of monogamy relations similar to (1) by replacing E with E α for some α > 0 [46]. More precisely, a continuous measure E is monogamous according to this definition if and only if there exists 0 < α < ∞ such that for all ρ ABC acting on the state space H ABC with fixed dim H ABC = d < ∞ (see Theorem 1 in Ref. [46]). Note that (4) can be expressed in a similar form as in (2) with f (x, y) = (x α + y α ) 1/α . However, we stress here that (4) is not a special case of (2) since the exponent factor α depends on the underlying dimension of the Hilbert space. Note that there is no a priori physical reason to assume that the exponent factor is universal and independent on the dimension.
In this paper, we show that almost all entanglement monotones are monogamous on pure tripartite states, and furthermore, those that are based on convex roof extension [e.g., entanglement of formation, see also Eq. (5)], are also monogamous on mixed tripartite states according to our definition in terms of Eq. (3) [or equivalently Eq. (4)]. Our results indicate that monogamy is indeed a property of entanglement and not a consequence of a particular measure of entanglement.
A function E : S AB → R + is called a measure of entanglement if (1) E(σ AB ) = 0 for any separable density matrix σ AB ∈ S AB , and (2) E behaves monotonically under local operations and classical communications (LOCC). That is, for any given LOCC map Φ we have Moreover, convex measures of entanglement that do not increase on average under LOCC are called entanglement monotones [48].
Let E be a measure of entanglement on bipartite states. The entanglement of formation E F associated with E is defined by where the minimum is taken over all pure state decompositions of ρ AB = n j=1 p j |ψ j ψ j | AB . That is, E F is the convex roof extension of E. Vidal [48,Theorem 2] showed that E F above is an entanglement monotone on mixed bipartite states if the following concavity condition holds. For a pure state |ψ AB ∈ H AB , ρ A = Tr B |ψ ψ| AB , define the function h : Note that since E is invariant under local unitaries we must have for any unitary operator U acting on H A . If h is also concave, i.e.
Theorem. Let E be an entanglement monotone for which h, as defined in (6), is strictly concave; i.e., h satisfies (8) with strict inequality whenever ρ 1 = ρ 2 , and 0 < λ < 1. Let also E F be as in (5). Then, where |φ AB1 ∈ H AB1 and |η B2C ∈ H B2C are pure states. In particular, ρ AC is a product state (and consequently E(ρ AC ) = 0), so that E is monogamous on pure tripartite states.

If ρ ABC is a mixed tripartite state and
where {p x } is some probability distribution, and for each x the Hilbert space H B has a subspace isomor- such that up to local unitary on system B, each pure state |ψ x ABC is given by In particular, the marginal state ρ AC is separable so that E F is monogamous (on mixed tripartite states).
Remark 1. The condition that E in the theorem above is an entanglement monotone can be replaced with a weaker condition that the measure of entanglement E satisfies E ≤ E F on all bipartite density matrices. This is due to the fact that both Theorem 4 and Corollary 5 in Ref. [46] are still true if we assume only that E satisfies E ≤ E F and E is not necessarily an entanglement monotone.
in such a case. Corollary 5 in Ref. [46] proved only that if Proof. Part 1. In Ref. [46] it was shown that if (3) holds for a pure tripartite state ρ ABC ≡ |ψ ψ| ABC then all pure state decompositions of ρ AB must have the same average entanglement. Let ρ AB = n j=1 p j |ψ j ψ j | AB be an arbitrary pure state decomposition of ρ AB with n = Rank(ρ AB ). Then, where the inequality follows from the convexity of E, and the equality holds since all pure state decompositions of ρ AB have the same average entanglement. Moreover, since E F is an entanglement monotone, we must have Given that ρ A = n j=1 p j ρ A j and h is strictly concave we must have Set r ≡ Rank(ρ A ) ≤ dim H B , and let H B1 be an rdimensional subspace of H B such that there exists a pure state |φ AB1 ∈ H AB1 with marginal on part A being ρ A . Since all the reduced density matrices of {|ψ j AB } have the same marginal on system A they must be related via local isometry on Bob's side, to a purification |φ AB1 of ρ A . Therefore, there exists isometries {V B1→B j } such that Now, let ρ AB = n k=1 q k |φ k φ k | AB be another pure state decomposition of ρ with the same number of elements n. For the same reasons leading to (13), there exists isometries W B1→B k such that On the other hand, since both decompositions {p j , |ψ j AB } and {q k , |φ k AB } correspond to the same density matrix ρ AB , they must be related by a unitary matrix U = (u kj ) in the following way: Denoting by we have Now, multiplying both sides of the equation above by (ρ A ) −1/2 (the inverse is understood to be on the support of ρ A ), we get and we therefore conclude that X B1→B . This means that X B1→B k is an isometry for any choice of unitary matrix U = (u kj ). But since U = (u jk ) is an arbitrary unitary matrix, we can take its first row, {u 1j } j to be an arbitrary normalized vector. Hence, we conclude that any linear combination of the isometric matrices {V B1→B j } is proportional to an isometric matrix. We now discuss the consequence of this property on the form of {V B1→B j }. The isometries V B1→B j can be expressed as where {|k } is an orthonormal basis of H B1 , and for each j, {|v kj } k are some orthonormal vectors in H B . Consider the arbitrary linear combination j c j V j . It can be expressed as Therefore, j c j V j is proportional to an isometry if and only if for all k = k ′ , u k ′ |u k = 0 and u k = u k ′ . Observe that Now, for a fixed k and k ′ , the above equation can be viewed as an inner product between a vector v kk ′ , whose components are v k ′ j ′ |v kj , and a vectorc ⊗ c, whose components are c * j c j ′ . Since v kk ′ is orthogonal to any vector of the formc ⊗ c whenever k = k ′ , it must be equal to the zero vector. We therefore conclude that v k ′ j ′ |v kj = 0, k = k ′ and v kj ′ |v kj = d jj ′ = 1 for some d jj ′ which are independent of k. Note that we can assume with no loss of generality that ρ AB = j p j |ψ j ψ j | AB is the spectral decomposition of ρ AB . In such a case, we have holds for any given k. We now obtain that Hence, Therefore, where σ B2 is some density matrix given by σ B2 = j p j |j j| B2 . Therefore, ρ AB has a purification of the form where |η B2C ≡ j p j |j B2 |j C . Since all purifications of ρ AB in H ABC are related via a local unitary on C, we conclude that |ψ ABC has the form (9) up to local unitary. Therefore ρ AC is a product state and consequently E(ρ AC ) = 0. This completes the proof of part 1.

Proof of Part 2:
Let {p x , |ψ x ABC } be the optimal pure state decomposition of ρ ABC = x p x |ψ x ψ x | ABC such that Denote by ρ AB x ≡ Tr C (|ψ x ψ x | ABC ) and note that where we used the convexity of E F . On the other hand, since E F is a measure of entanglement for each x we have E(|ψ x A|BC ) ≥ E F (ρ AB x ). Combining this with the equation above we conclude that Therefore, the rest of the proof of part 2 follows from part 1 of the theorem.
Note that if system B in the second part of the theorem above has dimension not greater than 3, then we must have for each x that either H B (x) are one dimensional. We therefore get the following corollary. Corollary. Using the same notations as in the theorem above, if E F (ρ A|BC ) = E F (ρ AB ) and dim H B ≤ 3 then ρ ABC is bi-separable, and in particular it admits the form where σ A|BC is A|BC separable, γ AB|C is AB|C separable, and t ∈ [0, 1]. In particular, if ρ ABC = |ψ ψ| ABC is a pure state, then |ψ ABC has the form |φ AB |η C or |φ A |η BC . At last we discuss the strict concavity of the entanglement measures so far. Many operational measures of entanglement such as the relative entropy of entanglement, entanglement cost, and distillable entanglement, all reduce on a bipartite pure state to the entropy of entanglement given in terms of the von Neumann entropy of the reduced state, H(ρ) ≡ −Tr(ρ log ρ). The von Neumann entropy is known to be strictly concave [49] and therefore they are all monogamous on pure tripartite states. The first part of the theorem above generalizes a similar result that was proved in Ref. [43, the disentangling theorem] for the special case in which E is taken to be the negativity. It demonstrates that many measures of entanglement are monogamous on pure tripartite states, while their convex roof extensions as defined in (5) are monogamous even on mixed tripartite states.
Any function that can be expressed as where p j are the eigenvalues of ρ is strictly concave if g ′′ (p) < 0 for all 0 < p < 1. This includes the quantum Tsallis q-entropy [50,51] with q > 0. In particular, the linear entropy (or the Tsallis 2-entropy) is strictly concave, and therefore the tangle is a monogamous measure of entanglement since it is defined in terms of the convex roof extension. Another important example is the Rényi α-entropy [52][53][54]. For the Rényi parameter α ∈ [0, 1] the Rényi entropies are strictly concave (see, e.g., Ref. [48]), but in general, for α > 1 the Rényi entropies are not even concave (although they are Schur-concave). To the authors' knowledge, except for this case of Rényi α-entropy of entanglement with α > 1, all other measures of entanglement that have been studied intensively in literature, correspond on pure bipartite state to strict concave functions of the reduced density matrix. These include the negativity, tangle, concurrence (see the Appendix), Gconcurrence, and the Tsallis entropy of entanglement.
In conclusion, we showed that many measures of entanglement, such as the entanglement of formation, that were believed not to be monogamous (irrespective of the specific monogamy relation [47]), are in fact monogamous according to a new definition of monogamy without inequalities that we put forward in Ref. [46]. This new definition is equivalent to the quantitative inequality (4), but with a key difference that the exponent factor α can depend on the underlying dimension. Therefore, the results presented here support this non-universal (i.e., dimension dependent) definition of monogamy. The fact that so many important measures of entanglement are not universally monogamous [47] may give the impression that monogamy of entanglement cannot be attributed to entanglement itself but rather is a property of the particular measure that is used to quantify entanglement. Furthermore, as was shown in Ref. [47], measures of entanglement cannot be simultaneously faithful (as defined in Ref. [47]) and universally monogamous. Here we avoided all these issues by adopting a new definition of monogamy that allows for non-universal monogamy relations, while at the same time maintaining a quantitative way [as in (4)] to express monogamy relations.
While we were not able to show that all measures of entanglement are monogamous (according to our definition), we are also not aware of any continuous measures of entanglement that are not monogamous. It may be the case that all continuous measures of entanglement are monogamous, which will support our assertion that monogamy is a property of entanglement and not of some particular functions quantifying entanglement. Moreover, many important measures of entanglement, are not defined in terms of convex roof extensions. For such measures, our theorem does not provide any information regarding their monogamy on mixed tripartite states. One example of that is the negativity. Our theorem implies that the convex roof extended negativity is monogamous but we do not know if the negativity itself is monogamous.

Appendix: Constructing monogamous measures of entanglement from other monogamous measures
For any entanglement monotone E, and any monotonically increasing function g : R + → R + with the property that g(x) = 0 iff x = 0, denote where the minimum is taken over all pure state decompositions of ρ AB = n j=1 p j |ψ j ψ j | AB . Observe that if in addition g is convex then we must have Therefore, if E g is an entanglement measure and is monogamous on pure tripartite states then E is also monogamous on pure tripartite states. To see why, note that if E(ψ A|BC ) = E(ρ AB ) we also have But since E g is a measure of entanglement we must have E g (|ψ A|BC ) ≥ E g (ρ AB ) so that we get E g (|ψ A|BC ) = E g (ρ AB ). Since we assume here that E g is monogamous, thus E g (ρ AC ) = 0, which implies that E(ρ AC ) = 0. As a simple example of this, consider the function g(x) = x 2 and take E = C be the concurrence as defined in Ref. [33]. Then, E g = C 2 is the tangle which is monogamous (it is given in terms of the linear entropy, which is strictly concave). Hence, the above analysis implies that the concurrence C is also monogamous.