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A resource theory of entanglement with a unique multipartite maximally entangled state

Entanglement theory is formulated as a quantum resource theory in which the free operations are local operations and classical communication (LOCC). This defines a partial order among bipartite pure states that makes it possible to identify a maximally entangled state, which turns out to be the most relevant state in applications. However, the situation changes drastically in the multipartite regime. Not only do there exist inequivalent forms of entanglement forbidding the existence of a unique maximally entangled state, but recent results have shown that LOCC induces a trivial ordering: almost all pure entangled multipartite states are incomparable (i.e. LOCC transformations among them are almost never possible). In order to cope with this problem we consider alternative resource theories in which we relax the class of LOCC to operations that do not create entanglement. We consider two possible theories depending on whether resources correspond to multipartite entangled or genuinely multipartite entangled (GME) states and we show that they are both non-trivial: no inequivalent forms of entanglement exist in them and they induce a meaningful partial order (i.e. every pure state is transformable to more weakly entangled pure states). Moreover, we prove that the resource theory of GME that we formulate here has a unique maximally entangled state, the generalized GHZ state, which can be transformed to any other state by the allowed free operations.

Introduction. Entanglement is a striking feature of quantum theory with no classical analogue. Although initially studied to address foundational issues [1], the development of quantum information theory [2] in the last few decades has elevated it to a resource that allows tasks to be implemented which are impossible with systems governed by the laws of classical physics. The resource theory of entanglement [3] aims at providing a rigorous framework in order to qualify and quantify entanglement and, ultimately, to understand fully its capabilities and limitations within the realm of quantum technologies. However, this theory is much more firmly developed in the bipartite than in the multipartite case. In fact, although a few applications have been proposed within the latter setting such as secret sharing [4], the one-way quantum computer [5] and metrology [6], a deeper understanding of the complex structure of multipartite entangled states might inspire further protocols in the context of quantum information science and better tools for the study of condensed-matter systems.
The wide applicability of the formulation of entanglement theory as a resource theory has motivated an active line of work [7] that studies different quantum effects from this point of view such as coherence [8], reference frame alignment [9], thermodynamics [10], non-locality [11] or steering [12]. The main question a resource theory addresses is to order the set of states and provide means to quantify their nature as a resource. The so-called free operations play a crucial role in this task. This is a subset of transformations, which the physical setting dictates can be implemented at no cost. Thus, all states that can be prepared with these operations are free states. On the other hand, non-free states acquire the status of a resource: granted such states, the limitations of the corresponding scenario might be overcome. Moreover, the concept of free operations allows an order relation to be defined. If a state ρ can be transformed into σ by some free operation, then ρ cannot be less resourceful than σ since any task achievable by σ is also achievable by ρ as the corresponding transformation can be freely implemented. However, this is not necessarily true the other way around. Furthermore, one can introduce resource quantifiers as functionals that preserve this order.
Since entanglement is a property of systems with many constituents which may be far away, the natural choice for free operations in this resource theory is local operations and classical communication (LOCC). Indeed, parties bound to LOCC can only prepare separable states, and entangled states become a resource to overcome the constraints imposed by LOCC manipulation. Nielsen characterized in [13] the possible LOCC conversions among pure bipartite states, which revealed that the LOCC ordering boils down to majorization [14] and, remarkably, that there is a unique maximally entangled state for a fixed local dimension. This is because this state can be transformed by LOCC into any other state of that dimension but no other state of that dimension can be transformed into it. This state is then regarded as a gold standard to measure entanglement and, unsurprisingly, it turns out to be the most useful state for bipartite entanglement applications such as teleportation. Importantly, the situation changes drastically in the multipartite case. Reference [15] and subsequent work [16] have shown that in this case there exist inequivalent forms of entanglement: the state space is divided into classes, the so-called stochastic LOCC (SLOCC) classes, of states which can be interconverted into each other with nonzero probability by LOCC but cannot be transformed outside the class by LOCC, not even probabilistically. This in particular shows that there cannot be a maximally entangled state for multipartite states. Still, one could in principle study the ordering induced by LOCC within each SLOCC class. Recent work [17] in this direction has revealed, however, an extreme feature that culminates with the result of Ref. [18]: almost all pure states of more than three parties are isolated, i.e. they cannot be obtained from nor transformed to another inequivalent pure state of the same local dimensions by LOCC. This means that almost all pure states are incomparable by LOCC, hence inducing a trivial ordering and a meaningless arbitrariness in the construction of entanglement measures. In this sense, one may say that the resource theory of multipartite entanglement with LOCC is generically trivial.
We believe this calls for a critical reexamination of the resource theory of entanglement and, in particular, for the notion of LOCC as the ordering-defining relation. Indeed, although LOCC transformations have a clear operational interpretation, it turns out that this is not the most general class of transformations that maps the set of separable states into itself. In other words, LOCC is strictly included in the class of non-entangling operations. Thus, if one is willing to pay the price of lifting the physical interpretation behind the allowed class of free operations (as happens in other scenarios [19]), from the abstract point of view of resource theories other consistent theories of entanglement (i.e. with separable states being the set of free states) are possible in which the set of free operations is larger than LOCC. Hence, in principle, these could allow for a more meaningful ordering and revealing structure in the set of multipartite entangled states. To study such possibility is precisely the goal of this Letter. Since we seek whether a nontrivial theory is at all possible, we consider the resource theory of entanglement under the largest possible class of free operations: non-entangling operations. However, there exist two different forms of entanglement in the multipartite scenario. Following standard usage, we will call entangled those states that contain some entanglement among its constituents but not necessarily all, i.e. states which are not fully separable (FS), while we will call genuinely multipartite entangled (GME) those states in which all parties are entangled, i.e. states which are not biseparable (BS). Thus, one can formulate two theories: one in which entangled states are considered a resource and where the free operations are full separabilitypreserving (FSP) and the analogous with GME states and biseparability-preserving (BSP) operations. Interestingly, our first result is that both formalisms lead to non-trivial theories: there are no inequivalent forms of entanglement and no resource state is isolated in any of these scenarios. Then, we consider whether there exists a unique multipartite maximally entangled state in these theories as happens in the bipartite case. While we find that the answer is no (at least in the simplest non-trivial case of 3-qubit states) in the case of FSP operations, our main result is that the question is answered in the affirmative in the case of the resource theory of GME under BSP operations. The maximally GME state turns out to be the generalized Greenberger-Horne-Zeilinger (GHZ) state. Our approach is reminiscent of that of [20], which shows that shifting the paradigm from LOCC to asymptotic non-entangling operations provides a much cleaner theory of asymptotic entanglement interconversion.
Definitions and preliminaries. We will consider npartite systems with local dimension d, i.e. states in the Hilbert space H = H 1 ⊗ · · · ⊗ H n = (C d ) ⊗n . Given a subset M of [n] = {1, . . . , n} and its complementM , we denote by H M the tensor product of the Hilbert spaces corresponding to the parties in M and analogously with HM . A pure state |ψ ∈ H is FS (otherwise entangled) if |ψ = |ψ 1 ⊗ |ψ 2 ⊗ · · ·⊗ |ψ n for some states |ψ i ∈ H i ∀i, while it is BS (otherwise GME) if |ψ = |ψ M ⊗ |ψM for some states |ψ M ∈ H M and |ψM ∈ HM and M [n]. These notions are extended to mixed states by the convex hull and we define the sets of FS and BS states by where here and throughout the paper we use the notation ψ = |ψ ψ| whenever a state is specified as pure. Transformations in quantum theory are given by completely positive and trace preserving (CPTP) maps and we say that such a map Λ (from and to operators on H) is FSP (BSP) if Λ(ρ) ∈ F S ∀ρ ∈ F S (Λ(ρ) ∈ BS ∀ρ ∈ BS). We will say that a functional E taking operators on H to non-negative real numbers is an FSP-measure (BSPmeasure) if E(ρ) ≥ E(Λ(ρ)) for every state ρ and FSP (BSP) map Λ. This is completely analogous to entanglement measures, which are required to be non-increasing under LOCC maps. Although LOCC is a strict subset of the FSP and BSP maps, some well-known entanglement measures are still FSP-or BSP-measures and this will play an important role in order to assess which transformations are possible within the two formalisms that we consider here. Indeed, measures of the form where X stands for either F S or BS, have the corresponding monotonicity property as long as the distinguishability measure E(ρ||σ) is contractive, i.e. E(Λ(ρ)||Λ(σ)) ≤ E(ρ||σ) for every CPTP map Λ. This includes the relative entropy of entanglement [21] for E(ρ||σ) = tr(ρ log ρ) − tr(ρ log σ) and the robustness (R X ) [22] for E(ρ||σ) = R(ρ||σ) = min{s : (ρ + sσ)/(1 + s) ∈ X }.
If one uses the fidelity E(ρ||σ) = 1 − F (ρ||σ) = 1 − tr 2 √ ρσ √ ρ, for pure states Eq. (2) boils down to the geometric measure [23], which we will denote by G X and which is then seen to be a measure under maps that preserve X [43]. Notice, however, that, as has been recently shown in the bipartite case in [24], not all LOCCmeasures remain monotonic under non-entangling maps since the latter formalism allows for state conversions that are impossible in the former. In the following, in order to understand the ordering of resources induced by these theories, we study which transformations are possible among pure states under FSP and BSP maps. However, first one should point out that whenever there exist maps Λ and Λ ′ in the corresponding class of free operations such that Λ(ψ) = φ and Λ ′ (φ) = ψ, then the states ψ and φ are equally resourceful and should be regarded as equivalent in the corresponding theory. This is moreover necessary so as to have a well-defined partial order. Hence, although for simplicity we will talk about properties of states, one should have in mind that one is actually speaking about equivalence classes. Specifically, it is known that two pure states are interconvertible by LOCC if and only if they are related by local unitary transformations [25]. Interestingly, we will see that the equivalence classes are wider in the resource theory of GME under BSP. It should be stressed that, to our knowledge, this is the first time that a resource theory of GME is formulated. Notice that the restriction to LOCC can only have FS states as free states. Furthermore, allowing a strict subset of parties to act jointly and classical communication does not fit the bill either as BS is not closed under these operations. Non-triviality of the theories. Our first two results are valid in both the FSP and BSP regimes. Thus, following the notation introduced before, the two possible classes of maps will be referred to as X -preserving.
Theorem 1 (collapse of the SLOCC classes). In a resource theory of entanglement where the free operations are X -preserving maps, all resource states are interconvertible with non-zero probability, i.e. given any pure ψ 1 , ψ 2 / ∈ X , there exists a completely positive and trace non-increasing X -preserving map Λ such that Λ(ψ 1 ) = pψ 2 with p ∈ (0, 1]. Theorem 2 (no isolation). In a resource theory of entanglement where the free operations are X -preserving maps, no resource state is isolated, i.e. given any pure ψ 1 / ∈ X on H, there exists an inequivalent pure ψ 2 / ∈ X on H and a CPTP X -preserving map Λ such that Λ(ψ 1 ) = ψ 2 .
The full proof of these two results can be found in [26]. The idea behind the proof of Theorem 1 is that we can explicitly construct a completely positive and trace nonincreasing X -preserving map Λ such that Λ(ψ 1 ) = pψ 2 whenever it holds that Since it can be guaranteed that R X (ψ 2 ) > 0 and 0 < G X (ψ 1 ) < 1 when ψ 1 , ψ 2 / ∈ X , there always exists p ∈ (0, 1] such that Eq. (6) is fulfilled. Theorem 2 then arises as a corollary as, given any ψ 1 / ∈ X , by continuity arguments one can prove that there always exists an inequivalent ψ 2 / ∈ X with R X (ψ 2 ) small enough so that one can take p = 1 in Eq. (6) and construct a CPTP map.
Theorem 1 proves that in our case there are no inequivalent forms of entanglement. This is in sharp contrast to LOCC where, leaving aside the case H = (C 2 ) ⊗3 , the state space splits into a cumbersome zoology of infinitely many different SLOCC classes of unrelated entangled states. Theorem 2 provides the non-triviality of our theories. While almost all states turn out to be isolated under LOCC [18], our classes of free operations induce a meaningful partial order structure where, as in the case of bipartite entanglement, every pure state can be transformed into a more weakly entangled pure state. It is important to mention that the result of [18] proves generic isolation when transformations are restricted among GME states fully supported on H (i.e. with the rank of all n single-particle reduced density matrices equal to d). However, Theorem 2 still holds under this restriction. Its proof can be adapted to show that both BSP and FSP operations allow transforming any GME state fully supported on H into inequivalent states of the same kind.
Existence of a maximally resourceful state. Theorems 1 and 2 show that limitations of the resource theory of multipartite entanglement under LOCC can be overcome if one considers instead FSP or BSP operations. These positive results raise the question of whether the induced structure is powerful enough so as to have a unique multipartite maximally entangled state. If this were the case, our theories would point out to a relevant class of states that should be at the heart of the applications of multipartite entanglement playing a similar role to that of the maximally entangled state in the bipartite case. In order to answer this question, we provide first an unambiguous definition of a maximally resourceful state which, on the analogy of the bipartite case, depends on the number of parties n and local dimension d: a state ψ on H is the maximally resourceful state on H if it can be transformed by means of the free operations into any other state on H [44]. We analyze first the case of FSP operations, where we find a negative answer to the above question. Although the details of the proof are given in [26], we outline here its structure. First, we use that if a maxi-mally entangled state in this case existed, it would need to be the W state |W = (|001 + |010 + |100 )/ √ 3. This is because it has been shown in [27] that the W state is the unique state in this Hilbert space that achieves the maximal possible value of G F S , which we have shown above to be a FSP-measure. Thus, if there existed a maximally entangled state, it would be necessary that the W state could be transformed by FSP into any other state. However, we show that there exists no FSP map transforming the W state into the GHZ state (|GHZ (3,2) in Eq. (5) below). To verify this last claim, it suffices to find an FSP-measure E such that E(GHZ) > E(W ). However, as discussed above, not many FSP-measures are known and, as in the case of the geometric measure, it is also known that the relative entropy of entanglement of the W is larger than that of the GHZ state [28]. This leaves us then with the robustness measure R F S , whose computation is not at all straightforward. Nevertheless, using a result of [29] that provides a dual characterization of the robustness, we are able to show that R F S (W ) = R F S (GHZ) = 2. This alone does not forbid that W → F SP GHZ but from the insight developed in order to compute the robustness for these states, an obstruction to such transformation can be found even though they are equally robust. It is worth mentioning that, to our knowledge, this is the first time that the robustness is computed for multipartite states and we have reasons to conjecture that the W and GHZ states attain its maximal value on H, being the only states that do so.
Theorem 3 forbids then the existence of a multipartite maximally entangled state under FSP in the simplest case of H = (C 2 ) ⊗3 . However, it is instructive to compare with the LOCC scenario since for these values of n and d it is the only case where no state is isolated in the latter formalism (in addition to the bipartite case). We show in [26] that the W and GHZ states can be transformed by FSP operations into states that are not obtainable from any other 3-qubit states by LOCC. These states might be chosen to lie in different SLOCC classes, so, in addition, this provides an explicit example of deterministic FSP conversions among states in different SLOCC classes.
Finally, we study the resource theory under BSP operations where, remarkably, we find the existence of a unique maximally GME state for any value of n and d, which is given by the generalized GHZ state Theorem 4. In the resource theory of entanglement where the free operations are BSP maps, there exists a maximally GME state on every H. Namely, ∀|ψ ∈ (C d ) ⊗n , there exists a CPTP BSP map Λ such that Λ(GHZ(n, d)) = ψ.
The complete proof of this result is given in [26]. The main idea behind it is to use again the construction of the proof of Theorems 1 and 2, that shows that there is a CPTP BSP map Λ such that Λ(GHZ(n, d)) = ψ if R BS (ψ) ≤ G BS (GHZ(n, d))/(1 − G BS (GHZ(n, d))) (cf. Eq. (6)). However, in contrast to the FS case, G BS is straightforward to compute [30] in terms of the Schmidt decomposition across every possible bipartite splitting of where λ M|M 1 is the largest Schmidt coefficient of ψ in the corresponding splitting. This immediately shows that the generalized GHZ state has maximal value of the geometric measure, G BS (GHZ(n, d)) = (d − 1)/d. Finally, a simple estimate shows that R BS (ψ) ≤ d − 1 ∀|ψ ∈ (C d ) ⊗n , which leads to the desired result.
It follows from the proof that it suffices to have maximal value of G BS to be convertible to any other state by BSP operations. Thus, any state fulfilling that G BS = (d − 1)/d must automatically maximize any other BSPmeasure. More importantly, this also shows that any two states achieving this value of the geometric measure are deterministically interconvertible by BSP operations and, therefore, belong to the same equivalence class of GME despite potentially not being related by local unitary transformations. An example of such class when d = 2 are GME graph states for which it is known that G BS = 1/2 [31]. Hence, all graph states like the generalized GHZ state are in the equivalence class of the maximally GME state in this theory. It is remarkable to find that this very relevant family of states [32] in quantum computation and error correction have this feature in a resource theory of GME and we believe that this is worth further research. Another previously considered family of states that belongs to this equivalence class is that of absolutely maximally entangled (AME) states [33], which is defined as those states for which the reduced density matrices for all parties are proportional to the identity in the maximum possible dimensions. It follows from Eq. (6) that for all AME states it holds that G BS = (d − 1)/d (for those values of n and d for which they exist). Equation (6) also tells us that a necessary condition for a state to be in the equivalence class of the maximally GME state is that all single-particle reduced density matrices must be proportional to the d-dimensional identity. However, this condition is not sufficient: the state in (C 2 ) ⊗4 |φ = √ p|φ + 12 |φ + 34 Conclusions. The advantages of relaxing the set of free operations in the resource theory of multipartite entanglement are manifest: inequivalent forms of entanglement and isolation are done away with. More importantly, the paradigm of BSP operations makes it possi-ble to introduce a resource theory of GME with a maximally resourceful state. Given this conceptually satisfying structure, it would be interesting to study further the features of this theory, which might be relevant not only for the recent experiments aimed at creating large-n GME states [34] but also for the quest of new applications of multipartite entanglement. While FSP operations do not allow for the existence of a maximally entangled state for 3-qubit states, future work should study whether this no-go result generalizes to other values of n and d.

Supplemental material
We prove the theorems introduced in the main text. For the reader's convenience, we provide the necessary definitions and restate the theorems.
Throughout the proof we will use repeatedly that, if ρ 1 and ρ 2 are density matrices, the map is CPTP if 0 ≤ A ≤ ½ (see e.g. [1]).
We say that a functional E taking operators on H to non-negative real numbers is an FSP-measure (BSPmeasure) if E(ρ) ≥ E(Λ(ρ)) for every state ρ and FSP (BSP) map Λ. Measures of the form where X stands for either F S or BS, have the corresponding monotonicity property as long as the distinguishability measure E(ρ||σ) is contractive, i.e. E(Λ(ρ)||Λ(σ)) ≤ E(ρ||σ) for every ρ, σ ∈ H and every CPTP map Λ. As already explained in the main text, two FSP-(BSP-)measures that will play a key role in developing the resource theory of FSP (BSP) operations are the geometric measure and the robustness. For the reader's convenience, we recall their definitions. The robustness is given by where R(ρ||σ) = min s : and the geometric measure, which we only need to consider here for pure states, boils down to I. NON-TRIVIALITY OF THE THEORIES Theorem 1. In a resource theory of entanglement where the free operations are X -preserving maps, all resource states are interconvertible with non-zero probability, i.e. given any pure ψ 1 , ψ 2 / ∈ X , there exists a completely positive and trace non-increasing X -preserving map Λ such that Λ(ψ 1 ) = pψ 2 with p ∈ (0, 1] .
Proof. Consider the map (7) from the proof of Theorem 1. This map can be made deterministic if R X (ψ 2 ) is sufficiently smaller than G X (ψ 1 ) . Indeed, if then we can pick p = 1 in the map (7) so Λ is CPTP (see equation (1)). Since robustness is a continuous function of the input state [3], it can be arbitrarily close to zero and so there exists ψ 2 such that the above condition is fulfilled for any ψ 1 . Further, ψ 1 , ψ 2 are inequivalent if they have different robustness, but R(ψ 2 ) can always be picked to be different from R(ψ 1 ) and still satisfying equation (9).
The generic isolation result proven in Ref. [4] holds when transformations are restricted among GME states fully supported on H (i.e. such that all n single-particle reduced density matrices have rank d). Importantly, ψ 1 and ψ 2 in Theorem 2 may both be fully supported on H , in contrast to the LOCC scenario. It suffices to consider as an example of a GME fully supported state on H which meets the requirements for small enough ε .
II. FSP REGIME Theorem 3. In the resource theory of entanglement where the free operations are FSP maps, there exists no maximally entangled state on H = (C 2 ) ⊗3 .
To prove this theorem, it is useful to introduce the following two lemmas in order to compute the robustness of the W and GHZ states.
Proof. The robustness can be bounded from above from the definition (equations (3), (4)), as any fully separable state which is a convex combination of the GHZ state with a fully separable state will give an upper bound to the robustness. Ref. [5] provides a dual characterization in terms of entanglement witnesses which we use to bound the robustness from below: A witness for a state ρ is an operator W such that tr(Wσ) ≥ 0 for all σ ∈ F S and tr(Wρ) < 0 . If the witness also satisfies tr(Wσ) ≤ 1 for all σ ∈ F S (which defines the set M above), then − tr(Wρ) is a lower bound to the robustness. First, we show R F S (GHZ) ≤ 2 . We will use the following notation as a means to characterize full separability of certain states (this is a simplified version of the separability criterion in [6, §2.1]): a state of the form where |GHZ − = (|000 − |111 )/ √ 2 and the summation index i ranges from 001 to 110 in binary, is fully separable iff We must also have λ + + λ − + λ = 1 for normalization, and λ ± , λ ≥ 0 for ρ(λ + , λ − , λ) to be positive. Thus, the set of fully separable states of the form (12) is a polytope, and this property will be used later. Consider the following state: It is straightforward to check that both ρ 0, 1 4 , 3 4 and ρ 1 3 , 1 6 , 1 2 satisfy (13) with equality, so R F S (GHZ) ≤ 2 . Next, we show R F S (GHZ) ≥ 2 . Let be a candidate witness for this purpose. To show 0 ≤ tr(Wσ) ≤ 1 for all fully separable states σ, it is enough to restrict to states σ of the form (12), as can be shown by considering the twirling map T GHZ onto the GHZsymmetric subspace. This map is defined in [7], but we will only need the following properties: it is FSP and self-dual, it maps all states onto states of the form (12), i.e.
Proof. The strategy is similar to the proof of Lemma 1. First, we prove R F S (W ) ≤ 2 . We will show that As shown in Theorem 6.2 of [10], if a symmetric 3-qubit state remains positive after partial transposition (PPT), then it is FS. Since both η and τ are symmetric 3-qubit states, it is enough to check that they are PPT, which is readily done, to conclude that they are fully separable. Another way to see this is by writing η and τ as a convex combination of fully separable states using a result from [11]. Observe that η = 5 9 |000 000| where, in each case, the first term is clearly fully separable. As we shall see, the second term is of the form for some qubit state φ. Ref. [11] shows that all states of this form are fully separable. Writing |φ = cos α |0 + e iβ sin α |1 .
and inserting it into equation (25), the parameter β cancels in all terms and the state in equation (25) can be written in terms of α alone with α = π/3 for η and α = π/6 for τ . Next, we prove R F S (W ) ≥ 2 . We will show that is a witness for the state |W W | such that and 0 ≤ tr(Aσ) ≤ 1 for all σ ∈ F S .
Let σ ∈ F S . Without loss of generality, to prove (29) we can assume σ = |ψ ψ| is pure. So we want to show Notice that A is permutationally invariant, and that we can express A in the basis of Pauli matrices as for some λ ijk ∈ R , so that has no identity component in the basis of Pauli matrices. That is, A ′ contains only full correlation terms, and it is still permutationally invariant so it satisfies the conditions of Corollary 5 (ii) in [12]. In particular, A ′ can be viewed as a symmetric three-linear form acting on R 3 . This means that can be attained by a symmetric state |ψ = |a |a |a ≡ |aaa . The qubit |a can be expressed in terms of two real parameters as and so But this completes the proof, since, by linearity, to show (which is equivalent to (30)) it suffices to show This can be seen by viewing tr(A ′ |ψ ψ|) as a symmetric three-linear form in R 3 . If the maximum absolute value is attained by some state |a * , then the state |ã * which flips the sign of the vector which the three-linear form acts on will give a minimum of the expression equal to minus the maximum. Hence, Therefore (30) holds true and hence the witness A gives the stated lower bound for the FS robustness of W .
We note that the values obtained for the robustness R F S of the W and GHZ states show that, unlike in the bipartite case, the robustness can be strictly larger than the generalized robustness. The generalized robustness, R G (·) , is defined as where, this time, τ may be separable or entangled. Hence R G (·) ≤ R(·) but, in addition, it was shown in [13] that R G (·) = R(·) for bipartite pure states. However, the generalized robustness of the W state has been computed in [11] to be 5/4, and that of the GHZ state was shown to be 1 in [7], so they are both strictly less than the robustness of these states. To the best of our knowledge, this is the first time that states such that R G (·) < R(·) have been found.
We are now ready to prove Theorem 3.
Proof. As we outlined in the main text, the only candidate for a maximally entangled state of three qubits is the W state, as it is the unique state on H = C 2 ⊗3 that achieves the maximum value of the FSP-measure G F S (among both pure and mixed states, since the convexroof extension of G F S to mixed states ensures that the maximum value will always be achieved by a pure state). So, if there existed a maximally entangled state, it would need to be possible that the W state be transformed into any other state via an FSP map. We will assume that there exists an FSP map Λ such that Λ(W ) = GHZ , and will arrive at a contradiction by showing that there exists a state η ∈ F S such that Λ(η) ∈ F S . Let Λ be an FSP map such that Λ(W ) = GHZ and let where τ, η ∈ F S , be the convex combination that gives the upper bound to R F S (W ) in equations ( (19)- (21)). Let T GHZ be the twirling map onto the GHZ-symmetric subspace (defined in [11]; see also the proof of Lemma 1). Then, the sense that it has measure zero on H (in contrast, for other values of n and d the fact that isolation is generic imposes that the MES has full measure on H). However, interestingly, the MES under FSP is smaller even in this case, given that it is strictly included in the MES under LOCC. This is because, as we will now show, the W and GHZ states can be transformed by FSP operations into inequivalent states that are in the MES under LOCC. It is worth mentioning that the target states may be chosen to lie in different SLOCC classes with respect to the initial states, and so this gives an explicit example of deterministic FSP conversions among states in different SLOCC classes. Let ψ + GHZ denote states of the form where |φ A = cos α |0 + sin α |1 , |φ B = cos β |0 + sin β |1 , α, β, γ ∈ (0, π/2] and K = (2(1 + cos α cos β cos γ)) −1 is a normalisation factor. States of the form ψ + GHZ are in the MES under LOCC, since they cannot be reached by any LOCC map regardless of the input state on H = (C 2 ) ⊗3 [14][15][16]. So the following proposition does not hold in the LOCC regime. Proposition 1. There exists an FSP map Λ such that Λ(W ) = ψ + GHZ for some state of the form ψ + GHZ . Proof. Let where τ F S ∈ F S is the state that gives the robustness of the state ψ + GHZ . Clearly, Λ(W ) = ψ + GHZ and it remains to be shown that Λ is FSP. As argued in Theorems 1 and 2, this happens when But, by continuity of the robustness, such a state ψ + GHZ can always be found by picking the parameters α, β, γ sufficiently close to zero since in this case the states ψ + GHZ approach the set of FS states. Anyway, for the sake of completeness, we provide an explicit quantitative upper bound in what follows. Consider the invertible local operations C = 1 cos γ 0 sin γ .
We will now show the converse result: there are FSP maps which take the GHZ-state to states in the W -class which are in the MES under LOCC. Such states are of the form where x 1 + x 2 + x 3 = 1. They are in the MES under LOCC, as no LOCC map can reach these states for any input state on H = (C 2 ) ⊗3 [14,15,17], but (as we will now prove) not under FSP.

Proposition 2.
There exists an FSP map Λ such that Λ(GHZ) = ψ W for some state of the form ψ W .
Proof. Since G F S (GHZ) = 1/2 , it suffices to find a state ψ W such that R F S (ψ W ) ≤ 1 , which can be done since the robustness is continuous and there are states ψ W arbitrarily close to the set of FS states. Then, where τ F S ∈ F S is the state such that that R F S (ψ W ) = R(ψ W ||τ F S ) , is the required map.
III. BSP REGIME Theorem 4. In the resource theory of entanglement where the free operations are BSP maps, there exists a maximally GME state on every H. Namely, ∀|ψ ∈ (C d ) ⊗n , there exists a CPTP BSP map Λ such that Λ(GHZ(n, d)) = ψ.
As discussed in the main text, it follows from equation (6) that G BS (GHZ(n, d)) = (d − 1)/d and, therefore, Λ is BSP iff R BS (ψ) ≤ d−1 . It is shown in [3] that for every bipartite pure state ψ A|B with Schmidt decomposition it holds that Thus where the latter inequality follows from considering the state with all eigenvalues λ i = 1/d . Hence, ∀ |ψ ∈ C d ⊗n there exists a BSP map Λ such that Λ(GHZ) = ψ .