Additivity and non-additivity of multipartite entanglement measures

We study the additivity property of three multipartite entanglement measures, i.e. the geometric measure of entanglement (GM), the relative entropy of entanglement and the logarithmic global robustness. First, we show the additivity of GM of multipartite states with real and non-negative entries in the computational basis. Many states of experimental and theoretical interests have this property, e.g. Bell diagonal states, maximally correlated generalized Bell diagonal states, generalized Dicke states, the Smolin state, and the generalization of D\"{u}r's multipartite bound entangled states. We also prove the additivity of other two measures for some of these examples. Second, we show the non-additivity of GM of all antisymmetric states of three or more parties, and provide a unified explanation of the non-additivity of the three measures of the antisymmetric projector states. In particular, we derive analytical formulae of the three measures of one copy and two copies of the antisymmetric projector states respectively. Third, we show, with a statistical approach, that almost all multipartite pure states with sufficiently large number of parties are nearly maximally entangled with respect to GM and relative entropy of entanglement. However, their GM is not strong additive; what's more surprising, for generic pure states with real entries in the computational basis, GM of one copy and two copies, respectively, are almost equal. Hence, more states may be suitable for universal quantum computation, if measurements can be performed on two copies of the resource states. We also show that almost all multipartite pure states cannot be produced reversibly with the combination multipartite GHZ states under asymptotic LOCC, unless relative entropy of entanglement is non-additive for generic multipartite pure states.


Introduction
Quantum entanglement has attracted intensive attention due to its intriguing properties and potential applications in quantum information processing [1,2], [3] (Chapter 8). Some geometrically motivated entanglement measures have been providing us new insights on quantum entanglement, e.g. entanglement of formation [4], relative entropy of entanglement (REE) [5,6], geometric measure of entanglement (GM) [7,8], global robustness (GR) [9,10], and squashed entanglement [11]. Besides providing a simple geometric picture, they are closely related to some operationally motivated entanglement measures, e.g. entanglement of distillation [4] and entanglement cost [12]. Their additivity property for the bipartite case has been studied by many researchers as a central issue in quantum information theory, because this property is closely related to operational meanings [11,[13][14][15][16][17][18][19]. However, concerning the multipartite setting, only the additivity of squashed entanglement has been proved [20], while the additivity problem on other measures has largely remained open.
In this paper, we focus on the additivity property of three main entanglement measures in the multipartite case, i.e. REE, GM, and logarithmic global robustness (LGR). These entanglement measures and their additivity property are closely related to operational concepts in the multipartite case as mentioned below. Our results may improve the understanding on multipartite entanglement and stimulate more research work on the three entanglement measures as well as others, such as the tangle [21] and generalized concurrence [22].
REE is a lower bound for entanglement of formation and an upper bound for entanglement of distillation in the bipartite case. It has a clear statistical meaning as the minimal error rate of mistaking an entangled state for a closest separable state [5,6]. It has also been employed by Linden et al [23] to study the conditions for reversible state transformation, and by Acín et al [24] to study the structure of reversible entanglement generating sets [25] in the tripartite scenario. In addition, Brandão and Plenio [26] have shown that the asymptotic REE equals an asymptotic smooth modification of LGR and a modified version of entanglement of distillation and entanglement cost, which means that the asymptotic REE quantifies the entanglement resources under asymptotic non-entangling operations. In condensed matter physics, REE is also useful for characterizing multipartite thermal correlations [27] and macroscopic entanglement, such as that in high-temperature superconductors [28].
GM is closely related to the construction of optimal entanglement witnesses [8], and discrimination of quantum states under LOCC [29][30][31]. GM of tripartite pure states is closely related to the maximum output purity of the quantum channels corresponding to these states [32]. Recently, GM has been utilized to determine the universality of resource states for one-way quantum computation [33,34]. It has also been applied to show that most entangled states are too entangled to be useful as computational resources [35]. Furthermore, the connection between GM defined via the convex roof and a distance like measure has also been pointed out [36]. In condensed matter physics, GM is useful for studying quantum many-body systems, such as characterizing ground state properties and detecting phase transitions [37][38][39].
GR is closely related to state discrimination under LOCC [29][30][31] and entanglement quantification with witness operators [40]. It is best suited to study the survival of entanglement in thermal states, and to determine the noise thresholds in the generation of resource states for measurement-based quantum computation [41].
On the other hand, the additivity property of the three measures REE, GM and LGR greatly affect the utility of multipartite states. For example, in state discrimination under LOCC [29,30], the additivity property of these measures may affect the advantage offered by joint measurements on multiple copies of input states over separate measurements. The additivity property of GM of generic multipartite states is closely related to their universality as resource states for one-way quantum computation, as we shall see in section 5.3. The additivity property of the three measures REE, GM and LGR is also closely related to the calculation of their asymptotic or regularized entanglement measures, which are the asymptotic limits of the regularized quantities with the n-copy state. These asymptotic measures will be referred to as asymptotic GM, REE and LGR, and are abbreviated to AGM, AREE and ALGR respectively. They are useful in the study of classical capacity of quantum multi-terminal channels [31]. The AREE can be used as an invariant when we build the minimal reversible entanglement generating set (MREGS) under asymptotic LOCC. The MREGS is a finite set of pure entangled states from which all pure entangled states can be produced reversibly in the asymptotic sense, which is an essential open problem in quantum information theory [24,25]. The AREE also determines the rate of state transformation under asymptotic nonentangling operations [26]. In the bipartite case, the AREE provides a lower bound for entanglement cost and an upper bound for entanglement of distillation. So it is essential to compute the regularized entanglement measures. However, the problem is generally very difficult. One main approach for computing these asymptotic measures is to prove their additivity, which is another focus of the present paper. In this case, the asymptotic measures equal to the respective one-shot measures.
Our main approach is the following. Under some group theoretical conditions, Hayashi et al [30] showed a relation among REE, LGR and GM. Due to this relation, we can treat the additivity problem of REE and LGR from that of GM in this special case. Hence we can concentrate on the additivity problem of GM.
First, we derive a novel and general additivity theorem for GM of multipartite states with real and non-negative entries in the computational basis. Applying this theorem, we show the additivity of GM of many multipartite states of either practical or theoretical interests, such as (1) two-qubit Bell diagonal states; (2) maximally correlated generalized Bell diagonal states, which is closely related to local copying [42]; (3) isotropic states, which is closely related to depolarization channel [43]; (4) generalized Dicke states [44], which is useful for quantum communication and quantum networking, and can already be realized using current technologies [45][46][47][48]; (5) the Smolin state [49], which is useful for remote information concentration [50], super activation [51] and quantum secret sharing [52] etc; and (6) Dür's multipartite entangled states, which include bound entangled states that can violate the Bell inequality [53]. By means of the relation among the three measures GM, REE and LGR, we also show the additivity of REE of these examples, and the additivity of LGR of the generalized Dicke states and the Smolin state. As a direct application, we obtain AGM and AREE of the above-mentioned examples, and ALGR of the generalized Dicke states and the Smolin state.
Our approach is also able to provide a lower bound for AREE and ALGR for generic multipartite states with non-negative entries in the computational basis, such as isotropic states [43], mixtures of generalized Dicke states. In the bipartite scenario, our lower bound for AREE is also a lower bound for entanglement cost. For non-negative tripartite pure states, the additivity of GM implies the multiplicativity of the maximum output purity of the quantum channels related to these states according to the Werner-Holevo recipe [32].
Second, we show the non-additivity of GM of antisymmetric states shared over three or more parties and many bipartite antisymmetric states. We also quantify how additivity of GM is violated in the case of antisymmetric projector states, which include antisymmetric basis states and antisymmetric Werner states as special examples, and treat the same problem for REE and LGR. For the antisymmetric projector states, while the three one-shot entanglement measures are generally non-additive, we obtain a relation among AREE, AGM and ALGR. Generalized antisymmetric states [54] are also treated as further counterexamples to the additivity of GM.
Third, we show, with a statistical approach, that almost all multipartite pure states are nearly maximally entangled with respect to GM and REE. However their GM is not strong additive; what's more surprising, for generic pure states with real entries in the computational basis, GM of one copy and two copies, respectively, are almost equal. Our discovery has a great implication for the universality of resource states for oneway quantum computation, and for asymptotic state transformation. As a twist to the assertion of Gross et al [35] that most quantum states are too entangled to be useful as computational resources, we show that more states may be suitable for universal quantum computation, if measurements can be performed on two copies of the resource states. In addition, we show that almost all multipartite pure states cannot be prepared reversibly with multipartite GHZ states (with various numbers of parties) under LOCC even in the asymptotic sense, unless REE is non-additive for generic multipartite pure states.
For the convenience of the readers, we summarize the main results on GM, REE and LGR of the states studied in this paper in table 1 and table 2. More details can be found in the relevant sections of the main text.
The paper is organized as follows. Section 2 is devoted to reviewing the preliminary knowledge and terminology, and to showing the relations among the three measures REE, GM and LGR. In section 3, we prove a general additivity theorem for GM of multipartite states with non-negative entries in the computational basis, and apply it to many multipartite states, e.g. Bell diagonal states, maximally correlated generalized Bell diagonal states, isotropic states, generalized Dicke states, mixtures of Dicke states, the Smolin state, and Dür's multipartite entangled states. Also, we treat the additivity problem of REE and LGR of these examples, and discuss the implications of these results for state transformation. In section 4, we focus on the antisymmetric subspace, and show the non-additivity of GM of states in this subspace when there are three or more parties. We also establish a simple relation among the three measures for the  [5,16]. REE of the maximally correlated generalized Bell diagonal states and isotropic states as well as their additivity were obtained in [16]. GM of the generalized Dicke states was calculated in [8], REE and LGR of the generalized Dicke states were calculated in [30,55,56]. REE of the Smolin state was calculated in [50,57], REE of the Dür's multipartite entangled states was calculated in [56,57].
Dür's multipartite entangled states x - Table 2. Non-additive cases: GM, REE, and LGR of some antisymmetric and generalized antisymmetric states. All states listed satisfy E R (ρ) = R L (ρ) = G(ρ) − S(ρ), except two copies of generalized antisymmetric states, where it is not known. When N = 2, the antisymmetric projector state reduces to the antisymmetric Werner state. GM of single copy of the antisymmetric basis state and generalized antisymmetric state was calculated in [30,54]. REE and LGR of single copy of the antisymmetric basis state and generalized antisymmetric state were calculated in [30,55,56].
tensor product of antisymmetric projector states, and compute GM, REE and LGR for one copy and two copies of antisymmetric projector states, respectively. Generalized antisymmetric states are also treated as further counterexamples to the additivity of GM. In section 5, we show that GM is not strong additive for almost all multipartite pure states, and that it is non-additive for almost all multipartite pure states with real entries in the computational basis. We then discuss the implications of these results for the universality of resource states in one-way quantum computation and for asymptotic state transformation. We conclude with a summary and some open problems.

Preliminary knowledge and terminology
In this section, we recall the definitions and basic properties of the three main multipartite entanglement measures, that is, the relative entropy of entanglement, the geometric measure of entanglement and the global robustness of entanglement, and introduce the additivity problem on these entanglement measures. We also present a few known results concerning the relations among these measures, which will play an important role later. The impact of permutation symmetry on GM and the connection between GM of tripartite pure states and the maximum output purity of quantum channels are also discussed briefly.

Geometric measure, relative entropy and global robustness of entanglement
Consider an N-partite state ρ shared over the parties A 1 , . . . , A N with joint Hilbert space ⊗ N j=1 H j . REE measures the minimum distance in terms of relative entropy between the given state ρ and the set of separable states, and is defined as [6] where S(ρ||σ) = tr ρ(log ρ − log σ) is the quantum relative entropy, and the logarithm has base 2 throughout this paper. Here "SEP" denotes the set of fully separable states, which are of the form σ = j σ 1 j ⊗ · · · ⊗ σ N j , such that σ k j is a single-particle state of the kth party. For a pure state ρ = |ψ ψ|, E R (|ψ ) is used to denote E R (ρ) through this paper, similarly for other entanglement measures to be introduced. Any state σ minimizing (1) is a closest separable state of ρ. As its definition involves the minimization over all separable states, REE is known only for a few examples, such as bipartite pure states [5,6,58], Bell diagonal states [5,16], some two-qubit states [19], Werner states [17,18,59], maximally correlated states, isotropic states [16], generalized Dicke states [30,55,56], antisymmetric basis states [30,55], some graph states [31], the Smolin state, and Dür's multipartite entangled states [56,57]. A numeric method for computing REE of bipartite states has been proposed in [6].
REE with respect to the set of states with positive partial transpose (PPT) E R,PPT , which is obtained by replacing the set of separable states in (1) with the set of PPT states, has also received much attention [16,17,60]. However, in this paper, we shall follow the definition in (1).
GM measures the closest distance in terms of overlap between a given state and the set of separable states, or equivalently, the set of pure product states, and is defined as [8] Here "PRO" denotes the set of fully product pure states in the Hilbert space ⊗ N j=1 H j . Any pure product state maximizing (2) is a closest product state of ρ. It should be emphasized that, for mixed states, the GM defined in (3) is not an entanglement measure proper, and there are alternative definitions of GM through the convex roof construction [8]. However, GM of ρ defined in (3) is closely related to GM of the purification of ρ [61], and also to REE and LGR of ρ, as we shall see later. Meanwhile, this definition is useful in the construction of optimal entanglement witnesses [8], and in the study of state discrimination under LOCC [29,30]. Thus we shall follow the definition in (3) in this paper. GM is known only for a few examples too, such as bipartite pure states, GHZ type states, generalized Dicke states [8], antisymmetric basis states [30,54], pure symmetric three-qubit states [62][63][64], some other pure three-qubit states [8,62,65], and some graph states [31]. Several numerical methods for computing GM of multipartite states have been proposed in [66,67]. Different from the above two entanglement measures, GR [9,10] measures how sensitive an entangled state is to the mixture of noise, and is defined as follows, The logarithmic global robustness of entanglement (LGR) is defined as LGR is known for even fewer examples, such as bipartite pure states [9,10], generalized Dicke states, antisymmetric basis states [30,56], some graph states [31]. A numerical method for computing LGR has been proposed in [68,69].

Additivity problem on multipartite entanglement measures
In quantum information processing, it is generally more efficient to process a family of quantum states together rather than process each one individually. In this case, entanglement measures can still serve as invariants under reversible LOCC transformation, provided that we consider the family of states as a whole. A fundamental problem in entanglement theory is whether the entanglement of the tensor product of states is the sum of that of each individual. First we need to make it clear what the entanglement of the tensor product of states means. Take two states as an example, let ρ be an N-partite sate shared over the parties A 1 1 , . . . , A 1 N , and σ be another N-partite state shared over the parties A 2 1 , . . . , A 2 N , where we have added superscripts to the names of the parties to distinguish the two states. Now there are 2N parties involved in the tensor product state ρ ⊗ σ, however, in most scenarios that we are concerned, the pair of parties A 1 j , A 2 j for each j = 1, . . . , N are in the same lab, and can be taken as a single party A j . In this sense, ρ ⊗ σ can be seen as an N-partite state shared over the parties A 1 , . . . , A N . The definition of any entanglement measure, such as GM, REE and LGR of the tensor product state ρ⊗σ follows this convention throughout this paper; similarly for the tensor product of more than two states, except when stated otherwise.
A particularly important case is the entanglement of the tensor product of multiple copies of the same state. In the limit of large number of copies, we obtain the regularized or asymptotic entanglement measure, which reads where E is the entanglement measure under consideration. When E is taken as E R , G and R L , respectively, the resulting regularized measures are referred to as asymptotic REE (AREE) E ∞ R , asymptotic GM (AGM) G ∞ , and asymptotic LGR (ALGR) R ∞ L , respectively.
The entanglement E of an N-partite state ρ is called additive if E ∞ (ρ) = E(ρ), and strong additive if the equality E(ρ ⊗ σ) = E(ρ) + E(σ) holds for any N-partite state σ. Obviously, strong additivity implies additivity. An entanglement measure itself is called (strong) additive if it is (strong) additive for any state. Similarly, the entanglement of the two states ρ, σ is called additive if the equality E(ρ ⊗ σ) = E(ρ) + E(σ) holds.
Historically, both GM and REE had been conjectured to be additive, until counterexamples were found. The first counterexample to the additivity of REE is the antisymmetric Werner state found by Vollbrecht and Werner [18]. The first counterexample to the additivity of GM is the tripartite antisymmetric basis state found by Werner and Holevo [32]. Coincidentally, the two counterexamples are both antisymmetric states, and the tripartite antisymmetric basis state is exactly a purification of the two-qutrit antisymmetric Werner state. We shall reveal the reason behind this coincidence in section 4.
For bipartite pure states, REE is equal to the Von Neumann entropy of each reduced density matrix [5,6,58]; GM is equal to the logarithm of the inverse of the largest eigenvalue of each reduced density matrix [8]; and LGR is equal to one half the logarithm of the trace of the positive square root of each reduced density matrix [9,10]; thus REE, GM and LGR are all additive. GM and REE are also additive for any multipartite pure states with generalized Schmidt decomposition, such as the GHZ state. More generally, REE (GM, LGR) of a multipartite pure state is additive if it is equal to the same measure under some bipartite cut. For example, some graph states have additive REE, GM and LGR for this reason [31]. In general, it is very difficult to prove the additivity or non-additivity of GM, REE and LGR of a given state, or to compute AGM, AREE and ALGR. The additivity of REE is known to hold for a few other examples, such as maximally correlated states, isotropic states [16], two-qubit Werner states [17,59], and some other two-qubit states [16,19]. Little is known about the additivity property of GM and LGR.

Relations among the three measures
There is a simple inequality among the three measures REE, GM and LGR [29,56], where S(ρ) is the von Neumann entropy. So the inequality R L (ρ) ≥ E R (ρ) ≥ G(ρ) holds when ρ is a pure state. The same is true if the three measures are replaced by their respective regularized measures. This inequality and its equality condition are crucial in translating our results on GM to that on REE and LGR in the later sections. A sufficient condition for the equality is given as lemma 9 in Appendix C of [30]. For convenience, we reproduce it in the following proposition, Proposition 1 Assume that a projector state P trP satisfies the following. There exist a compact group H, its unitary representation U, and a product state |ϕ N such that (1) U(g) is a local unitary for all g ∈ H.
(2) U(g)P U(g) † = P . (3) The state |ϕ N is one of the closest product states of P .
where µ is the invariant probability measure on H. Then, Under condition (1), conditions (2)-(4) are satisfied if (5) the range of P is an irreducible representation of H whose multiplicity is one in the representation U.
For example, generalized Dicke states, antisymmetric basis states [30,55], and some graph states [30,31] satisfy the conditions (1)-(4), so they satisfy (8). In this case, if GM is additive, then both LGR and REE are additive, which follows from proposition 2 below. If in addition condition (5) is satisfied, then LGR, REE and GM are simultaneously additive or simultaneously non-additive, which follows from proposition 3 below.
Proposition 2 Assume that two multipartite states ρ, σ satisfy , then the following relations hold, Proof.
Let H j and U j be the group and the local unitary representation satisfying the conditions (1) and (5) of proposition 1 concerning the projector state P j tr(P j ) for j = 1, . . . , n. Define the representation n j=1 ×U j of the direct product group n j=1 ×G j by ( n j=1 ×U j )(g 1 , . . . , g n ) := n j=1 ⊗U j (g j ). This satisfies the conditions (1) and (5) of proposition 1 concerning the projector state n j=1 P j n j=1 trP j , which implies (11).
⊓ ⊔ Next, we present two known results concerning the relation between a given entanglement measure of a pure multipartite state and that of its reduced states after tracing out one party. Let |ψ be an N-partite pure state, and ρ one of its (N −1)-partite reduced states. First, Jung et al [61] have proved that the following equality holds: So the additivity problem on an N-partite pure state is equivalent to that on its (N −1)partite reduced states. Second, Plenio and Vedral [58] have proved a useful inequality concerning REE, which means that the reduction in entanglement is no less than the increase in entropy due to deletion of a subsystem. If G(|ψ ) = E R (|ψ ) (this is true if, for example, proposition 1 is satisfied), combining (7), (12) and (13), we obtain an interesting equality, In this case, the total entanglement E R (|ψ ) is the sum of the remaining entanglement E R (ρ) after losing a subsystem and the increase in entropy S(ρ). Moreover, if GM of |ψ is additive, then GM of ρ, REE of |ψ and that of ρ are all additive.

Geometric measure and permutation symmetry
Permutation symmetry plays an important role in the study of multipartite entanglement. A multipartite state is called (permutation) symmetric (antisymmetric) if its support is contained in the symmetric (antisymmetric) subspace, and permutation invariant if it is invariant under permutation of the parties. Note that both symmetric states and antisymmetric states are permutation invariant. Hayashi et al [70] and Wei et al [71] have shown that the closest product state to a symmetric pure state with non-negative amplitudes in the computational basis can be chosen to be symmetric. Hübener et al [72] have shown this fact for general symmetric states (corollary 5). In addition, if ρ is a pure state shared over three or more parties, the closest product state is necessarily symmetric (lemma 1). Here we present a stronger result on general symmetric states shared over three or more parties.

Proposition 4
The closest product state to any N-partite pure or mixed symmetric state with N ≥ 3 is necessarily symmetric.

Proof.
Let ρ be an N-partite symmetric state with N ≥ 3. Assume that ρ is mixed, otherwise the proposition is already proved as lemma 1 in [72]. Suppose |ψ is a purification of ρ, and |ϕ N a closest product state to ρ. According to theorem 1 in [61], there exists a single-particle state |a , such that |ϕ N ⊗ |a is a closest product state to |ψ ; thus |ϕ N is a closest product state to the unnormalized state a|ψ . Since the purification has the form |ψ = j |ψ j ⊗ |j with each |ψ j a symmetric N-partite state, a|ψ is an unnormalized N-partite pure symmetric state with N ≥ 3. According to lemma 1 in [72], |ϕ N is necessarily symmetric too. ⊓ ⊔ We shall prove an analog of proposition 4 for antisymmetric states in section 4.1.

Geometric measure of tripartite pure states and maximum output purity of quantum channels
Finally, we mention a interesting connection between GM of tripartite pure states and the maximum output purity of quantum channels established by Werner and Holevo [32]. Let Φ be a CP map with the Kraus form Φ(ρ) = k A k ρA † k . The maximum output purity of the map Φ is defined as where ||ρ|| p = (trρ p ) 1/p , and the maximum is taken over all quantum states. From the Kraus representation of the map Φ, one can construct a tripartite state |Φ (not necessarily normalized) with components h j |A k |e l and vice versa, where |h j s and |e j s are orthonormal bases in the appropriate Hilbert spaces, respectively. Note that, as far as entanglement measures are concerned, it does not matter which Kraus representation of the map Φ is chosen, because different representations lead to tripartite states which are equivalent under local unitary transformations. It should be emphasized that the map constructed from a generic tripartite pure state according to the above correspondence may not be trace preserving.
The maximum output purity of the channel Φ and GM of the tripartite state |Φ is related to each other through the following simple formula [32]: According to this result, we can get GM of a tripartite pure state by computing the maximum output purity ν ∞ of the corresponding map and vice versa. Generally speaking, the computation of the maximum output purity involves far fewer optimization parameters. Moreover, we can translate the multiplicativity property about the maximum output purity to the additivity property about GM and vice versa. Actually, the non-additivity of GM of the tripartite antisymmetric basis state corresponds exactly to the non-multiplicativity of the maximum output purity ν ∞ of the Werner-Holevo channel [32].

Additivity of geometric measure of non-negative multipartite states
A density matrix is called non-negative if all its entries in the computational basis are non-negative. Many states of either theoretical or practical interests can be written as non-negative states, with an appropriate choice of basis, such as (1)  In this section, we prove a general theorem on the strong additivity of GM of nonnegative states, and show the additivity of REE and LGR for many states mentioned in the last paragraph. For general non-negative states, our additivity result on GM can provide a lower bound for AREE and ALGR. These results can be used to study state discrimination under LOCC [29,30], and the classical capacity of quantum multiterminal channels [31]. The result on AREE can be utilized to determine the possibility of reversible transformation among certain multipartite states under asymptotic LOCC, and determine the transformation rate under asymptotic non-entangling operations. For non-negative bipartite states, our results also provide a lower bound for entanglement of formation and entanglement cost. For non-negative pure tripartite states, the additivity of GM implies the multiplicativity of the maximum output purity of the quantum channels related to these states according to the Werner-Holevo recipe [32].
In section 3.1, we prove the strong additivity of GM of arbitrary non-negative states, and provide a nontrivial lower bound for AREE and ALGR, which translates to a lower bound for entanglement of formation and entanglement cost in the bipartite case. In section 3.2, we prove the strong additivity of GM of Bell diagonal states, maximally correlated generalized Bell diagonal states, isotropic states, and the additivity of REE of Bell diagonal states, maximally correlated generalized Bell diagonal states. In section 3.3, we prove the strong additivity of GM and additivity of REE of generalized Dicke states and their reduced states after tracing out one party, as well as the additivity of LGR of generalized Dicke states. The implications of these results for asymptotic state transformation are also discussed briefly. In section 3.4, we give a lower bound for AREE of mixtures of Dicke states. In section 3.5, we prove the strong additivity of GM, and the additivity of REE and LGR of the Smolin state. In section 3.6, we prove the strong additivity of GM and additivity of REE of Dür's multipartite entangled states.

General additivity theorem for geometric measure of non-negative states
We start by proving our main theorem of this section.
Theorem 5 GM of any non-negative N-partite state ρ is strong additive; that is, for any other N-partite state σ, the following equalities hold: Proof. Assume that |ϕ N is a closest product state to ρ ⊗ σ, we can write it in the following form: where |j l A 1 l s for given l form an orthonormal basis, |c lj l A 2 l s are normalized states, and a lj l ≥ 0, In the above derivation, the next to last inequality is due to the assumption that ρ is non-negative, and the following inequality: which follows from the Schwarz inequality and the definition of Evidently, the closest product state to ρ ⊗ σ can be chosen as the tensor product of the closest product states to ρ and σ, respectively. ⊓ ⊔ Theorem 5 provides a new way to compute GM of the tensor product of multipartite states, when GM of each member is known. In particular, it enables us to calculate AGM of non-negative states, which are a large family of multipartite states.
For a non-negative pure tripartite state, the additivity of GM translates immediately to the multiplicativity of the maximum output purity ν ∞ of the corresponding quantum channel constructed according to the Werner-Holevo recipe [32]. Thus, theorem 5 may also be useful in the study of the additivity problem concerning quantum channels.
In addition, theorem 5 gives a lower bound for AREE and ALGR for non-negative states. This lower bound is often nontrivial as we shall see later. According to (7), for non-negative states ρ j s, where we have employed the additivity of Von Neumann entropy.
, where E F and E c denote entanglement of formation and entanglement cost, respectively. Therefore, when ρ is non-negative, G(ρ)−S(ρ) also gives a lower bound for entanglement of formation and entanglement cost. ‡ Tzu-Chieh Wei showed an alternative proof of the inequality in (19) in his comment to our manuscript (private communication).
Theorem 7 Both ALGR and AREE of any non-negative state ρ are lower bounded by the difference between GM and the Von Neumann entropy of the state, is also a lower bound for entanglement of formation and entanglement cost, Next, we prove a useful lemma concerning the closest product states of non-negative states.
Lemma 8 The closest product state to any non-negative state ρ can be chosen to be non-negative.
Proof. Represent ρ in the computational basis, where ρ k 1 ,...,k N ;j 1 ,...,j N ≥ 0. Assume that |ϕ N is a closest product state to ρ which reads the inequality is saturated when b j l ,l s are all non-negative, that is |ϕ N is non-negative. ⊓ ⊔ In the rest of this section, we illustrate the power of theorems 5, 7 and lemma 8 with many concrete examples. In particular, we prove the strong additivity of GM of the following states: Bell diagonal states, maximally correlated generalized Bell diagonal states, isotropic states, generalized Dicke states, mixtures of Dicke states, the Smolin state, and Dür's multipartite entangled states. Moreover, we prove the additivity of REE of Bell diagonal states, maximally correlated generalized Bell diagonal states, generalized Dicke states, generalized Dicke states with one party traced out, the Smolin state, and Dür's multipartite entangled states. The additivity of LGR of generalized Dicke states and the Smolin state is also shown. The implications of these results for state transformation under asymptotic LOCC and asymptotic non-entangling operations, respectively, are also discussed briefly.

Bipartite mixed states and tripartite pure states
In the bipartite scenario, for any pure states, REE, GM and LGR can be easily calculated and their additivity has been shown [5,6,[8][9][10]. Note that any bipartite pure state is nonnegative in the Schmidt basis; hence, its GM is strong additive according to theorem 5. The same is true for any multipartite state with a generalized Schmidt decomposition.
However, even in the bipartite scenario, the calculation of REE, GM and LGR is not so trivial for mixed states. Moreover, the additivity problem on generic mixed states is notoriously difficult. Due to (12), the difficulty in GM for bipartite mixed states is equivalent to that for tripartite pure states.
As one of the most simple examples of bipartite mixed states, we focus on maximally correlated generalized Bell diagonal states. Maximally correlated states are known as a typical example where REE is known to be additive [16]. By applying a suitable local unitary transformation, any maximally correlated generalized Bell diagonal state can be transformed into the following form, where p = (p 0 , . . . , p d−1 ) is a probability distribution. It's easy to see that Λ 2 (ρ MCB (p)) = max |ϕ ϕ, ϕ|ρ MCB (p)|ϕ, ϕ ≤ max |ϕ ,k ϕ, ϕ|Ψ k Ψ k |ϕ, ϕ ≤ 1 d , and the upper bound is achievable by setting |ϕ = |j , ∀j. In addition, the state ρ MCB (p) can be converted into a non-negative state via a suitable local unitary transformation, such as the simultaneous local Fourier transformation. According to theorem 5, we get Proposition 9 The maximally correlated generalized Bell diagonal state in (23) has strong additive GM, and thus G ∞ (ρ MCB (p)) = G(ρ MCB (p)) = log d.
is the Shannon entropy of the distribution p. Applying the inequality (7) and its asymptotic version to the maximally correlated generalized Bell diagonal state ρ MCB (p), we obtain the additivity of REE for ρ MCB (p): The same result has been obtained by Rains [16] with a different method.
In the two-qubit system, any rank-two Bell diagonal state, a mixture of two orthogonal Bell states, can always be converted into the form in (23), with a suitable local unitary transformation. So, any two-qubit rank-two Bell diagonal state has strong additive GM and additive REE. Actually, this is true for all Bell diagonal states. Let ρ BD be any Bell diagonal state, where p = (p 0 , p 1 , p 2 , p 3 ) is a probability distribution, and |Ψ j s are the standard Bell basis. |Ψ 0 , |Ψ 1 are already defined in (23), the other two states are defined as Since local unitary transformations can realize all 24 permutations of the four Bell states, with out loss of generality, we may assume p 0 ≥ p 1 ≥ p 2 ≥ p 3 . Then ρ BD is clearly a non-negative state, and its GM is strong additive according to theorem 5. Meanwhile, its closest product state can be chosen to be non-negative according to lemma 8. Let The maximum in the above equation can be obtained at θ 1 = θ 2 = 0, that is |ϕ 2 = |00 . REE of Bell diagonal states have been computed by Vedral et al [5] and by Rains [16], with the result, , except for rank-two Bell diagonal states, REE of Bell diagonal states is also additive. This can be shown as follows, with a suitable local unitary transformation and twirling, ρ BD (p) can be turned into a Werner state with the same maximal eigenvalue p 0 , and thus with the same REE according to (28). Recall that REE of any two-qubit Werner state is additive [17,59], it follows from the monotonicity of AREE under LOCC that REE of any Bell diagonal state is also additive.

Proposition 10
The Bell diagonal state in (25) has strong additive GM, and additive REE, thus To compute LGR of the Bell diagonal state ρ BD (p), let ρ ′ be an unnormalized separable state with the minimal trace such that In addition, ρ ′ can also be chosen to be a Bell diagonal state. Since a Bell diagonal state is separable if and only if its largest eigenvalue is no larger than one half of its trace, ρ ′ can be chosen to be Next, we consider the isotropic state ρ I,λ : It is easy to see that Λ 2 (ρ I,λ ) = λd+1 d(d+1) , and that the state |jj for each j = 0, 1, . . . , d − 1 is a closest product state. Since ρ I,λ is a non-negative state, its GM is strong additive according to theorem 5. So we obtain Proposition 11 The isotropic state ρ I,λ with 1 d 2 ≤ λ ≤ 1 has strong additive GM, and The REE and AREE of the isotropic state were calculated by Rains [16] with the result, To compute LGR of the isotropic state ρ I,λ , let ρ ′ be an unnormalized separable state with the minimal trace such that ρ ′ ≥ ρ I,λ , then R L (ρ I,λ ) = log[tr(ρ ′ )]. In addition, ρ ′ can also be chosen to be an isotropic state. Since the isotropic state ρ I,λ is separable if 0 ≤ λ ≤ 1 d , and entangled otherwise, ρ ′ can be chosen to be Now, we focus on pure three-qubit states as the most simple multipartite pure states. Recall that any pure three-qubit state can be turned into the following form via a suitable local unitary transformation [73], If φ = 0, the resulting four-parameter family of states are all non-negative. In that case, according to theorems 5 and 7, their GM is strong additive and gives a lower bound for their AREE and ALGR. The bound for AREE and ALGR is tight for the W state as we shall see in section 3.3. For generic two-qubit states, previous numerical calculation in [6] found no counterexample to the additivity of REE, while our numerical calculation found no counterexample to the additivity of GM. We thus conjecture that both REE and GM are additive for generic two-qubit states. Note that each bipartite reduced state of a pure three-qubit state is a rank-two two-qubit state. According to (12), GM of pure three-qubit states would be additive if GM of general two-qubit states were additive.

Generalized Dicke states
Generalized Dicke states are also called symmetric basis states; they are defined in H = (C d ) ⊗N as follows [8,44], Here {P } denotes the set of all distinct permutations of the spins, and C N, 0) is sometimes referred to as the totally symmetric basis state and written as |ψ N + [30]. When d = 2, |N, (k 0 , k 1 ) is called a Dicke state and denoted as |N, k 0 . Dicke states are useful for quantum communication and quantum networking [45,46]. Some typical Dicke states have been realized in trapped atomic ions [47]. Recently, the multiqubit Dicke state with half excitations |N, N/2 has been employed to implement a scalable quantum search based on Grover's algorithm by using adiabatic techniques [48]. In view of the fast progress made in experiments, further theoretical study is required to explore the full potential of Dicke states. GM, REE and LGR of the generalized Dicke states have been computed in [8,30,56] with the result, In addition, the generalized Dicke states have been proved to satisfy the conditions (1)-(4) of proposition 1 in section III B of [30]. Since the generalized Dicke states have non-negative amplitudes, theorem 5 and proposition 2 imply that In particular when all states |N, k α are identical, we get Proposition 12 Generalized Dicke states have strong additive GM, additive REE and LGR, hence Let ρ N, k be the (N − 1)-partite reduced state of the N-partite generalized Dicke state |N, k . Since E R (|N, k ) = G(|N, k ), equation (14) implies that is the Shannon entropy. This equality has already been proved in [56] with explicit calculation. In contrast, our derivation is much simpler and more general. Finally, since REE of ρ N, k is also additive, we get the AREE as follows, In the case N = 3, the above result gives a lower bound for the entanglement cost of the following two states, respectively: the two-qubit state 1 3 (|01 + |10 )( 01| + 10|) + 1 3 |00 00| and the two-qutrit state 1 6 (|01 +|10 )( 01|+ 10|)+ 1 6 (|02 +|20 )( 02|+ 20|)+ 1 6 (|21 + |12 )( 21| + 12|).
Another application of our result is to help determine whether two multipartite pure states can be inter-converted reversibly under asymptotic LOCC, and help solve the long standing problem about MREGS [24,25]. Consider two tripartite states |ψ 1 , |ψ 2 over the three parties A 1 , A 2 , A 3 . According to the result of Linden et al [23], reversible transformation between the two states under asymptotic LOCC would mean the ratio of the AREE E ∞ R (A 1 : Table 3 shows the bipartite and tripartite AREE of the GHZ state, W state, tripartite totally symmetric and antisymmetric basis states |ψ 3± (|ψ 3− is defined in (55) in section 4.1) respectively. The inequality E ∞ R (ψ 3− ) ≥ log 5 in the table follows from (13) . With these results, it is immediately clear that there is no reversible transformation between any two states among the four states.  Similar argument can be used to show that the transformation between the Npartite GHZ state and any N-partite symmetric basis state is not reversible. Also, the transformation between two symmetric basis states is generally not reversible if they cannot be converted into each other by a permutation of the kets in the computational basis.

Mixture of Dicke states
Next, we consider the mixture of Dicke states REE of these states has been derived by Wei [55,56]. We shall give a lower bound for AREE of these states based on the relation between REE and GM. Similar techniques can also be applied to the mixture of generalized Dicke states. The lower bound can often be improved if the convexity of AREE is taken into account, as we shall see shortly. For simplicity, we illustrate our method with the mixture of two Dicke states.  Figure 1. REE and lower bound for AREE given by G − S of three families of states, ρ 2;0,1 (s) (left plot), ρ 3;1,2 (s) (middle plot), and ρ 3;0,2 (s) (right plot), respectively, see (41) for the definitions of these states. In the left plot, entanglement of formation E F is also plotted for comparison; the dotted line is the improved lower bound for AREE after taking the convexity into account. In the right plot, REE is obtained by convex roof construction from the dotted curve [56]. After taking the convexity of AREE into account, the lower bound for AREE derived from G − S is almost equal to REE.
Following [55,56], define Since the mixture of Dicke states is both symmetrical and non-negative, corollary 5 in [72] (see also proposition 4) and lemma 8 implies that the closest product state to ρ N ;k 1 ,k 2 (s) can be chosen to be of the form |ϕ N = (cos θ|0 +sin θ|1 ) ⊗N with 0 ≤ θ ≤ π 2 . Λ 2 (ρ N ;k 1 ,k 2 (s)) = max The maximization over θ is easy to carry out; for example, let x = cos 2 θ, the extremal condition leads to a (k 2 − k 1 + 1)-order polynomial equation in x, which can be solved straightforwardly . In particular, this equation can be solved analytically if k 2 − k 1 ≤ 3. Since ρ N ;k 1 ,k 2 (s) is non-negative, according to theorems 5 and 7, G(ρ N ;k 1 ,k 2 (s)) is strong additive, and E ∞ R (ρ N ;k 1 ,k 2 (s)) is lower bounded by G(ρ N ;k 1 ,k 2 (s)) − S(ρ N ;k 1 ,k 2 (s)). Figure 1 illustrates E R (REE is given by theorem 1 of Wei [56]) and G − S for the following three families of states: where |Ψ 2 = 1 3 . In addition, G − S is a lower bound for entanglement cost. For ρ 3;1,2 (s) (middle plot), the bound is very good in the whole parameter region. The bound is tight at s = 1 2 , since ρ 3;1,2 ( 1 2 ) is the tripartite reduced state of the Dicke state |4, 2 4, 2|. For ρ 3;0,2 (s) (right plot), REE is obtained by convex roof construction from the dotted curve as described in [56]. The lower bound for AREE given by G(s) − S(s) does not look very good at first glance. However, taking the convexity of AREE into account, we can obtain a lower bound for AREE which is very close to REE for almost entire family of states ρ 3;0,2 (s).

The Smolin state
The Smolin state is a four-qubit unlockable bound entangled state, from which no pure entanglement can be distilled under LOCC. However, if any two of the four parties come together, they can create a singlet between the other two parties [49]. The Smolin state can be expressed in several equivalent forms, one of which is where |Ψ j s are the four Bell states 1 √ 2 (|00 ± |11 ) and 1 √ 2 (|01 ± |10 ). It can also be written in a more symmetric form which clearly shows that it is permutation invariant and non-negative. Since its discovery, the Smolin state has found many applications, such as remote information concentration [50], superactivation [51], and multiparty secret sharing [52]. It can maximally violate a two-setting Bell inequality similar to the CHSH inequality [74]. It was also used to show that four orthogonal Bell states cannot be discriminated locally even probabilistically [75]. Recently, Amselem and Bourennane have realized the Smolin state in experiments with polarized photons and characterized its entanglement properties [76]. Similar experiments were performed later by several other groups [77,78]. Hence, it is desirable to quantify the amount of entanglement in the Smolin state.
The multipartite REE of the Smolin state has been derived by Murao and Vedral [50] and by Wei et al [57], with the result E R (ρ ABCD ) = 1. The derivation in [57] relies on the following alternative representation of the Smolin state, which again shows that it is non-negative, with They also give a closest separable state to ρ ABCD , which reads Note that ρ sep = 1 2 (ρ ABCD +ρ ⊥ ), where ρ ⊥ is orthogonal to ρ ABCD , hence R L (ρ ABCD ) ≤ 1 according to (4) and (5). Since R L (ρ ABCD ) ≥ E R (ρ ABCD ) = 1, we get R L (ρ ABCD ) = 1.
To compute GM of the Smolin state, note that the closest product state to ρ ABCD can be chosen to be non-negative, according to lemma 8. Suppose |ϕ 4 = 4 j=1 (c j |0 + s j |1 ) is a closest product state, where c j = cos θ j , s j = sin θ j with 0 ≤ θ j ≤ π 2 for j = 1, 2, 3, 4.
where the last inequality was derived in [57]. The same result can also be obtained with the approach presented in [77]. Since Λ 2 (ρ ABCD ) ≥ 0000|ρ ABCD |0000 = 1 8 , we thus obtain Λ 2 (ρ ABCD ) = 1 8 and G(ρ ABCD ) = 3. Note that S(ρ ABCD ) = 2, R L (ρ ABCD ) = E R (ρ ABCD ) = G(ρ ABCD ) − S(ρ ABCD ), and ρ ABCD is non-negative. According to theorems 5 and 7, we have Proposition 13 The Smolin state has strong additive GM, additive REE and LGR, and thus The additivity of REE of the Smolin state can also be derived in an alternative way by first considering REE under the bipartite cut A : BCD [57]. Since every pure state in the support of ρ A:BCD is maximally entangled, the entanglement of formation of the state is given by E F (ρ A:BCD ) = 1. On the other hand E D (ρ A:BCD ) ≥ 1, where E D denotes entanglement of distillation, because a singlet can be distilled from the Smolin state when any two of the four parties come together. From the chain of inequalities, The additivity of REE then follows from the following chain of inequalities: Recall that, under asymptotic non-entangling operations, state transformation can be made reversible, and AREE determines the transformation rate [26]. Hence, the Smolin state and the four-qubit GHZ state can be converted into each other reversibly under these operations.

Dür's multipartite entangled states
Dür's multipartite bound entangled state ρ N was found in search of the relation between distillability of multipartite entangled states and violation of Bell's inequality [53].
where |Ψ G = 1 √ 2 (|0 ⊗N + e iα N |1 ⊗N ) is the N-partite GHZ state, P k is the projector onto the product state |u k = |0 A 1 |0 A 2 · · · |1 A k · · · |0 A N , andP k is the projector onto the product state |v k = |1 A 1 |1 A 2 · · · |0 A k · · · |1 A N . Dür has shown that, for N ≥ 4, the state in (48) is bound entangled and, for N ≥ 8, it violates two-setting Mermin-Klyshko-Bell inequality [53]. Since the phase factor e iα N can be absorbed by redefining the computational basis, we may assume e iα N = 1 without loss of generality. It is then clear that ρ N is non-negative. In the following discussion, we assume N ≥ 4.
Wei et al [57] have generalized Dür's multipartite bound entangled state to the following family of states: and shown that the state is bound entangled if 0 ≤ x ≤ 1 N +1 and free entangled if 1 N +1 < x ≤ 1. Moreover, they had conjectured REE of this state to be which was later proved in [56]. We shall show that REE of ρ N (x) is additive by first showing that REE of ρ N = ρ N ( 1 N +1 ) is additive, and then extending the result to the whole family of states via the convexity of AREE. Note that ρ N (x) is a convex combination of ρ N (0) and ρ N (1), that is, ρ N (x) = xρ N (1) + (1 − x)ρ N (0).

Non-additivity of geometric measure of antisymmetric states
In this section, we turn to the antisymmetric subspace, and explore the connection between the permutation symmetry and the additivity property of multipartite entanglement measures. Starting from a simple observation on the closest product states to antisymmetric states and that to symmetric states, we show that GM is non-additive for all antisymmetric states shared over three or more parties, and provide a unified explanation of the non-additivity of the three measures GM, REE and LGR of the antisymmetric projector states. In particular, we establish a simple equality among the three measures GM, REE and LGR of the tensor product of antisymmetric projector states, and derive analytical formulae of the three measures in the case of one copy and two copies, respectively. Our results may be found useful in the study of fermion systems, which are described by antisymmetric states due to the super-selection rule. In section 4.1, we introduce Slater determinant states, which are analog of product states in the antisymmetric subspace, and give a simple criterion on when an antisymmetric state is a Slater determinant state. Then we prove that the N one-particle reduced states of each closest product state to any N-partite antisymmetric state are mutually orthogonal, and derive a lower bound for the three measures GM, REE and LGR based on this observation. In section 4.2, we show that GM of antisymmetric states shared over three or more parties is non-additive. In section 4.3, we establish a simple equality among the three measures GM, REE and LGR of the tensor product of antisymmetric projector states, and compute the three measures in the case of one copy and two copies respectively. REE and LGR of the mixture of Slater determinant states are also derived. In section 4.4, we treat generalized antisymmetric states [54] as further counterexamples to the additivity of GM.

Geometric measure of antisymmetric states
We shall be concerned with antisymmetric states in the multipartite Hilbert space Given N orthonormal single-particle states, |a 1 , . . . , |a N , a Slater determinant state can be constructed by anti-symmetrization, a procedure routinely used in the study of fermion systems, i.e.
where S N is the symmetry group of N letters, sgn(σ) is the signature of σ [79], and 1 √ N ! is the normalization factor. Apparently, all Slater determinant states are locally unitarily equivalent to each other. In particular, they are locally unitarily equivalent to antisymmetric basis states, |j 1 ∧ · · · ∧ |j N with 0 ≤ j 1 < · · · < j N ≤ d − 1, which form an orthonormal basis in the antisymmetric subspace. When d = N, there is only one antisymmetric basis state, For the convenience of the following discussion, we summarize a few useful properties of Slater determinant states; see [80] for some mathematical background. If the N single-particle states |a 1 , . . . , |a N are linearly dependent, then |a 1 ∧ · · · ∧ |a N vanishes. If they are linearly independent but not mutually orthogonal, |a 1 ∧ · · · ∧ |a N is a subnormalized Slater determinant state. In that case, we can choose N orthonormal states |a ′ 1 , . . . , |a ′ N from the span of |a 1 , . . . , |a N , such that |a 1 ∧ · · · ∧ |a N = c|a ′ 1 ∧· · ·∧|a ′ N , where c is a constant with modulus between 0 and 1. The projection of a generic pure product state onto the antisymmetric subspace is a subnormalized Slater determinant state, that is, Suppose |b 1 , . . . , |b N are another N normalized single-particle states. Then |a 1 ∧ · · · ∧ |a N and |b 1 ∧· · ·∧|b N are linearly independent if and only if the subspaces spanned by |a 1 , . . . , |a N and by |b 1 , . . . , |b N , respectively, are different but of the same dimension N. In other words, up to overall phase factors, there is a one-to-one correspondence between N-partite Slater determinant states and N-dimensional subspaces of the singleparticle Hilbert space. Slater determinant states play a similar role in the antisymmetric subspace as product states do in the full Hilbert space [81]. Given an N-partite antisymmetric state |ψ N , a basic task is to determine whether it is a Slater determinant state. Note that the one-particle reduced state of any N-partite Slater determinant state is a subnormalized projector with rank N. On the other hand, if the one-particle reduced state of an antisymmetric state is of rank N, then there is only one linearly independent Slater determinant state that can be constructed from the one-particle states in the support of this one-particle reduced state. Obviously, the rank of the one-particle reduced state can not be less than N; otherwise, no Slater determinant state can be constructed. So we obtain Proposition 15 The one-particle reduced state of any N-partite antisymmetric state has rank at least N. Moreover, an antisymmetric state is a Slater determinant state if and only if its one-particle reduced state has rank N.
We are now ready to study GM of antisymmetric states. Suppose ρ N is an N-partite antisymmetric state and thus P d,N ρ N P d,N = ρ N . Let ϕ N = |a 1 ⊗ · · · ⊗ |a N , Recall that |a 1 ∧ · · · ∧ |a N is in general a subnormalized Slater determinant state, and that it is normalized if and only if the N single-particle states |a 1 , . . . , |a N are orthonormal, which is also a necessary condition for |ϕ N to be a closest product state.
Proposition 16 The N one-particle reduced states of any closest product state to an N-partite antisymmetric state are mutually orthogonal.
Thus searching for the closest product state of ρ N is equivalent to searching for its closest Slater determinant state. A peculiar feature of an antisymmetric state ρ N is the high degeneracy of its closest product states. If |a 1 ⊗ · · · ⊗ |a N is a closest product state, then the tensor product of any N orthonormal states from the span of the N single-particle states |a 1 , . . . , |a N is also a closest product state. Recall that there is a one-to-one correspondence between N-partite Slater determinant states and Ndimensional subspaces of the single-particle Hilbert space. Proposition 16 is in a sense the analog of proposition 4 for antisymmetric states. It is crucial to computing GM of antisymmetric states and to proving the non-additivity of GM of antisymmetric states shared over three or more parties in section 4.2.
Suppose λ max is the largest eigenvalue of ρ N , then Λ 2 (ρ N ) ≤ λmax N ! according to (57), and the inequality is saturated if and only if there is a Slater determinant state in the eigenspace corresponding to λ max . So we obtain A typical example where all the inequalities are saturated is the antisymmetric projector state, as we shall see in section 4.3.

Non-additivity theorem for geometric measure of antisymmetric states
The permutation symmetry of multipartite states plays a crucial role in determining the properties of their closest product states, as demonstrated in propositions 4 and 16. It is also closely related to the non-additivity of GM of antisymmetric states Σ.
Theorem 17 When N ≥ 3, GM is non-additive for any two N-partite antisymmetric states ρ N and ρ ′ N , that is, Suppose there exists a closest product state of ρ N ⊗ ρ ′ N which is of the tensor-product form |ϕ N ⊗ |ϕ ′ N , then |ϕ N and |ϕ ′ N are closest product states of ρ N and ρ ′ N , respectively. Since the set of one-particle reduced states of |ϕ N (|ϕ ′ N ) are mutually orthogonal according to proposition 16, |ϕ N ⊗ |ϕ ′ N cannot be symmetric. On the other hand, ρ N ⊗ ρ ′ N is a symmetric state and, if N ≥ 3, its closest product states are necessarily symmetric according to proposition 4, hence a contradiction would arise. In other words, no closest product state of ρ N ⊗ ρ ′ N can be written as a tensor product of the closest product states of ρ N and ρ ′ N , respectively, which implies that . ⊓ ⊔ The non-additivity of GM of antisymmetric states can be understood as follows. Antisymmetric states are generally more entangled than symmetric states as noticed in [30]. However, two copies of antisymmetric states turn to be a symmetric state. Theorem 17 establishes a simple connection between permutation symmetry and the additivity property of GM of multipartite states. In some special cases, this connection carries over to other multipartite entanglement measures, such as REE and LGR, as we shall see in section 4.3.
For a pure tripartite antisymmetric state, the non-additivity of GM translates immediately to the non-multiplicativity of the maximum output purity ν ∞ of the corresponding quantum channel constructed according to the Werner-Holevo recipe. For example, the non-multiplicativity of the maximum output purity of the Werner-Holevo channel is equivalent to the non-additivity of GM of the tripartite antisymmetric basis state [32].
Theorem 17 can be generalized to cover the situation where the two states are not fully antisymmetric.
Corollary 18 GM is non-additive for two N-partite states, if there exists a subsystem of three parties such that the respective tripartite reduced states of the two N-partite states are both antisymmetric.
Proof. Assume N > 3, suppose σ N and σ ′ N are two N-partite states whose respective tripartite reduced states σ A 1 ,A 2 ,A 3 N and σ ′ N A 1 ,A 2 ,A 3 are antisymmetric. Let |a 1 ⊗· · ·⊗|a N and |a ′ 1 ⊗ · · · ⊗ |a ′ N be the closest product states to σ N and σ ′ N , respectively; then ( a 4 | ⊗ · · · ⊗ a N |)σ N (|a 4 ⊗ · · · ⊗ |a N ) and ( a ′ 4 | ⊗ · · · ⊗ a ′ N |)σ ′ N (|a ′ 4 ⊗ · · · ⊗ |a ′ N ) are both antisymmetric. Theorem 17 applied to the two subnormalized antisymmetric states shows that G(σ N ⊗ σ ′ N ) < G(σ N ) + G(σ ′ N ). ⊓ ⊔ In the bipartite scenario, if either ρ 2 or ρ ′ 2 is pure, then G(ρ 2 ⊗ ρ ′ 2 ) = G(ρ 2 ) + G(ρ ′ 2 ), since GM of bipartite pure states is strong additive, as shown in section 3.2. On the other hand, the closest product state to ρ 2 ⊗ ρ ′ 2 cannot be of tensor-product form if it is symmetric and vice versa, according to the same reasoning as that in the proof of theorem 17. The additivity of GM of ρ 2 and that of ρ ′ 2 is related to the existence of closest product states of ρ 2 ⊗ ρ ′ 2 which are not symmetric. This in turn is due to the degeneracy of Schmidt coefficients of ρ 2 or ρ ′ 2 [79]. Indeed, every Schmidt coefficient of a bipartite pure antisymmetric state is at least doubly degenerate [81].
For generic bipartite antisymmetric states, we suspect that the non-additivity of GM is a rule rather than an exception, which is supported by the following observation. If both ρ 2 and ρ ′ 2 admit purifications that are antisymmetric, then their GM is nonadditive, due to theorem 17 and (12).
Theorem 17 can also be derived in a slightly different way, which offers a new perspective. According to corollary 5 in [72] (see also proposition 4 of this paper), the closest product state to ρ N ⊗ ρ ′ N can be chosen to be symmetric. Let According to (98) in the Appendix, = max where V ∧N = P d,N V ⊗N P d,N is the restriction of V ⊗N onto the antisymmetric subspace, which does not vanish if and only if the rank of V is at least N. Since the rank of V is exactly the Schmidt rank of |a V , the Schmidt rank of |a V must be at least N, if |a V ⊗N is a closest product state. Recall that the closest product state to ρ N ⊗ ρ ′ N is necessarily symmetric if N ≥ 3, according to proposition 4. It follows that each closest product state to ρ N ⊗ ρ ′ N must be entangled across the cut A 1 1 , . . . , A 1 N : . In addition to providing an alternative proof of theorem 17, the second approach also enables us to compute GM of the antisymmetric projector states in section 4.3, and to derive an upper bound for GM of multipartite states of tensor-product form in section 5.2.

Antisymmetric projector states
In this section, we focus on the antisymmetric projector states, which are typical examples of antisymmetric states, and include antisymmetric basis states and antisymmetric Werner states as special cases. In particular, we establish a simple equality among the three measures GM, REE and LGR of the tensor product of antisymmetric projector states, and compute the three measures in the case of one copy and two copies, respectively. Our study provides a unified explanation of the non-additivity of the three measures of the antisymmetric projector states.
The antisymmetric projector P d,N is invariant under the action of the unitary group U(d) with the representation U → U ⊗N for U ∈ U(d). The range of P d,N is an irreducible representation with multiplicity one [30]. In other words, it satisfies the conditions (1) and (5) of proposition 1. Moreover, the tensor product of the antisymmetric projector states n j=1 ρ d j ,N satisfies the conditions of proposition 3. So we obtain Proposition 19 GM, REE and LGR of antisymmetric projector states satisfy the following equalities: Combining the above result with that on symmetric basis states presented in section 3.3, we obtain Proposition 20 The three measures GM, REE and LGR are equal for the tensor product of any number of symmetric basis states and antisymmetric basis states, so are AGM, AREE and ALGR.
For the single copy antisymmetric projector state ρ d,N , all eigenvalues are equal to 1/tr(P d,N ), and the eigenspace corresponding to the largest eigenvalue of ρ d,N is exactly the antisymmetric subspace. Hence, all the inequalities in (58) are saturated, which implies that Interestingly, REE and LGR of the antisymmetric projector state ρ d,N do not depend on the dimension of the single-particle Hilbert space. When d = N, the antisymmetric projector state turns to be an antisymmetric basis state. The result on GM reduces to that found in [30,54], and the result on REE and LGR reduces to that found in [30,55,56]. When N = 2, there is a lower bound for AREE of ρ d,N found by Christandl et al [82] which reads E ∞ R (ρ d,2 ) ≥ log 4 3 , from which we can get a lower bound for ALGR, In general, none of the three measures is easy to compute for the tensor product of antisymmetric projector states.
We now focus on two copies of antisymmetric projector states. Note that all entries of P d,N in the computational basis are real. Let |ϕ N be as defined in (59), according to (98) in the Appendix, where in deriving the last equality, we have used the fact that V ⊗N and P d,N commutes due to the Weyl reciprocity, (see also [80]). The trace in (64) is exactly the Nth symmetric polynomial of the set of eigenvalues µ 0 , . . . , Recall that elementary symmetric polynomials are Schur concave functions [83], so the maximum in (65) is obtained if and only if µ 0 = µ 1 = · · · = µ d−1 = 1, that is, V is unitary, or equivalently, |a V is maximally entangled. So we obtain In conjunction with (62), (63) and proposition 4 (see also corollary 5 in [72]), we get Proposition 21 GM, REE and LGR of one copy and two copies of the antisymmetric projector states are respectively given by For ρ d,N , a state is a closest product state if and only if it is a tensor product of orthonormal single-particle states. For ρ ⊗2 d,N , any tensor product of identical maximally entangled states across the cut A 1 j : A 2 j for j = 1, . . . , N, respectively, is a closest product state, and each closest product state must be of this form if N ≥ 3.
GM, REE and LGR of ρ d,N are all non-additive if d ≥ 3 and 2 ≤ N ≤ d. Moreover, Compared with (63) Recall that GM, REE and LGR are all equal to log N! for the antisymmetric basis state |ψ N − according to (63), and they are all equal to N log N − log N! for the symmetric basis state |ψ N + according to (34). Since |ψ N + is non-negative, theorem 5 and proposition 2 imply that Surprisingly, GM, REE and LGR are all equal to N log N for both |ψ N − ⊗2 and |ψ N + ⊗ |ψ N − . It is not known whether this is just a coincidence, or there is a deep reason.
When N = 2, ρ d,N is an antisymmetric Werner state, and (67) reduces to REE of two copies of antisymmetric Werner states was derived by Vollbrecht and Werner [18], who discovered the Werner state as the first counterexample to the additivity of REE. In the case of two-qutrit antisymmetric Werner state, the non-additivity of the three measures is in contrast with the additivity of entanglement of formation [84]. Equation (67) can also be generalized to the tensor product of two antisymmetric projector states whose respective single-particle Hilbert spaces have different dimensions, say d 1 , d 2 , respectively. Suppose N ≤ d 1 ≤ d 2 , with a similar reasoning that leads to (67), one can show that Interestingly, REE and LGR of ρ d 1 ,N ⊗ ρ d 2 ,N are independent of d 2 , as long as The antisymmetric projector state can be seen as a uniform mixture of Slater determinant states. The above results on REE and LGR can also be generalized to an arbitrary mixture of Slater determinant states. Let ρ N = j p j |ψ j ψ j |, where |ψ j s are N-partite Slater determinant states, and {p j } is a probability distribution. Due to the convexity of REE and LGR, On the other hand, since ρ N can be turned into the antisymmetric projector state by twirling, Combining (72), (73) and proposition 21, we obtain Proposition 22 REE and LGR of any convex mixture ρ N of N-partite Slater determinant states satisfy the following equations: If N ≥ 3, GM, REE and LGR of any convex mixture of Slater determinant states are all non-additive.
For each triple d, p, k, define an N-partite state with N = kp as follows, |ψ d,p,k can be seen as a k-partite antisymmetric basis state with single-particle Hilbert space of dimension d p , if we divide the kp parties into k blocks each with p parties, and view each block as a single party. The state |ψ d,p,d p is exactly the generalized antisymmetric state introduced by Bravyi [54]. By definition, Λ 2 (|ψ d,p,k ) ≤ Λ 2 (|ψ k− ) = 1 k! , and since | ψ d,p,k |φ(1), . . . , φ(k) When k = d p , this result reduces to that found by Bravyi [54].

Non-additivity of geometric measure of generic multipartite states
Many examples and counterexamples to the additivity of GM presented in the previous sections invite the following question: What is the typical behavior concerning the additivity property of GM of multipartite states, additive or non-additive? In this section, we show that if the number of parties is sufficiently large, and the dimensions of the local Hilbert spaces are comparable, then GM is not strong additive for almost all pure multipartite states. What's more surprising, for generic pure states with real entries in the computational basis, GM for one copy and two copies, respectively, are almost equal. This conclusion follows from the following two observations which are of independent interest: First, almost all multipartite pure states are nearly maximally entangled with respect to GM and REE; second, there is a nontrivial universal upper bound for GM of multipartite states with tensor-product form. Our results have significant implications for universal one-way quantum computation and to asymptotic state transformation under LOCC.

Universal upper bound for the geometric measure of multipartite states with tensor-product form
In this section, we derive a universal upper bound for GM of the tensor product of two multipartite states, and discuss its implications.

Proposition 23
Suppose ρ N and ρ ′ N are two N-partite states on the Hilbert space |kk is a maximally entangled state (also a pure isotropic state) across the two copies of the jth party. According to (97) in the Appendix, . In other words, GM cannot be strong additive if the states are too entangled with respect to GM. This intuition will be made more rigorous in theorem 24. For states with real entries in the computational basis (real states for short), proposition 23 sets a universal upper bound for G(ρ ⊗2 N ) and According to a similar reasoning as in the proof of proposition 23, the same upper bound also applies to any state that is equivalent to its complex conjugate under local unitary transformations. Hence, GM cannot be additive if such states are too entangled with respect to GM.
If d j = d, ∀j and d ≥ N, the universal upper bound for G(ρ ⊗2 N ) of real states ρ N given in proposition 23 is saturated for the antisymmetric projector states (see section 4.3). If d j = d, ∀j and N is even, there is a simple scheme for constructing a pure state whose GM saturates the upper bound: Divide the parties into N 2 pairs, and choose a maximally entangled state for each pair of parties, then the tensor product of the N 2 maximally entangled states (note that all the entries of the state can be made real by a suitable local unitary transformation) is such a candidate. Moreover, GM of the state so constructed is additive, so are REE and LGR. A more attractive example which saturates the upper bound is the cluster state with even number of qubits, whose GM, REE and LGR are all equal to N/2 and are additive [31]. Hence, proposition 23 implies that, in any multipartite Hilbert space with even number of parties and equal local dimension, any pure state with real entries in the computational basis cannot be more entangled with respect to AGM than the tensor product of bipartite maximally entangled states, or the cluster state, for a multiqubit system.
If N is odd, however, there may exists no pure state (even with complex entries in the computational basis) that can saturate the upper bound given in proposition 23. For example, W state has been shown to be the maximally entangled state with respect to GM among pure three-qubit states [62,63], while its AGM, which equals to its GM log 9 4 , is strictly smaller than the upper bound 3 2 log 2 given in the proposition.
It is interesting to know whether the same bound is true for states with arbitrary entries and whether there is a similar universal upper bound for REE and LGR; in particular, whether AREE or ALGR is upper bounded by 1 2 log d T . It is also not clear whether REE and LGR are not strong additive for generic multipartite states. We have shown in section 4.3 that AREE is upper bounded by 1 2 log d T for antisymmetric basis states. The same is true for all symmetric basis states, according to (36). However, a complete picture is still missing. We hope that our results can stimulate more progress along this direction.

Non-additivity theorem for geometric measure of generic multipartite states: a statistical approach
In this section we prove the following theorem.
The fraction of pure states whose GM is strong additive is smaller than exp[− 2 For pure states with real entries in the computational basis, the fraction of pure states whose GM is additive is smaller than exp[− 1 T . Theorem 24 implies that GM is not strong additive for almost all pure multipartite states, if the number of parties is sufficiently large, and the dimensions of the local Hilbert spaces are comparable. Moreover, GM of |ψ and |ψ ⊗ |ψ * , respectively, is almost equal. If the dimensions of the local Hilbert spaces are equal, the probability that GM is strong additive decreases doubly exponentially with the number of parties N. Concerning real states, GM is non-additive for almost all pure multipartite states, and GM of one copy and two copies, respectively, is almost equal. The generalization to mixed states is immediate, since GM of any mixed state is equal to GM of its purification [61] (see also (12)).
Theorem 24 is an immediate consequence of proposition 23 in section 5.1 and proposition 25 presented below. The later proposition, which is inspired by a similar result on multiqubit pure states of [35], shows that almost all multipartite pure states are nearly maximally entangled with respect to GM.
Proposition 25 Suppose pure states are drawn according to the Haar measure from the Hilbert space N j=1 H j with N ≥ 3 and Dim H j = d j (d j ≥ 2, ∀j); define d T = N j=1 d j and d S = N j=1 d j . The fraction of pure states whose GM is smaller than log d T − log(d S ln d T ) − log 9 2 is less than d −d S T ; the fraction of pure states whose GM is smaller than 1 . For pure states with real entries in the computational basis, the fraction of pure states whose GM is smaller than log d T − log(d S ln d T ) − log 9 is less than d −d S T ; the fraction of pure states whose GM is smaller than 1 2 log d T is less than exp[− 1 By means of the relation among the three measures GM, REE, and LGR (see (7)), we obtain Corollary 26 Suppose pure states are drawn according to the Haar measure from the Hilbert space The fraction of pure states whose REE or LGR is smaller than log d T − log(d S ln d T ) − log 9 2 is less than d −d S T ; the fraction of pure states whose REE or LGR is smaller than 1 2 log d T is less than exp[− 2 . For pure states with real entries in the computational basis, the fraction of pure states whose REE or LGR is smaller than log d T − log(d S ln d T ) − log 9 is less than d −d S T ; the fraction of pure states whose REE or LGR is smaller than 1 2 log d T is less than Note that G(ρ) ≤ E R (ρ) ≤ log d T for any pure state ρ, since S(ρ I/d T ) = log d T . Proposition 25 and corollary 26 implies that almost all multipartite pure states are nearly maximally entangled with respect to GM and REE, if the number of parties is sufficiently large, and the dimensions of the local Hilbert spaces are comparable. In particular, if the dimensions of the local Hilbert spaces are equal, then the probability that GM (REE, LGR) is smaller than log d T − log(d S ln d T ) − log 9 2 decreases exponentially with the number of parties N, and the probability that GM (REE, LGR) is smaller than 1 2 log d T decreases doubly exponentially. Proof. To prove the proposition, we need the concept of ε-net. An ε-net N ε,N on the set of pure product states is a set of states that satisfy or equivalently, We shall show that there exists an ε-net with |N ε,N | ≤ (5 √ N/ε) 2d S , where |N ε,N | denotes the number of elements in the ε-net. From [85], we know that there is an ε-net M on the Hilbert space of single qudit with |M| ≤ (5/ε) 2d . Let M j be an (ε/ √ N )-net on H j with |M j | ≤ (5 √ N /ε) 2d j for j = 1, . . . , N, and N ε,N := { N j=1 |ã j : |ã j ∈ M j }. Suppose |ϕ = N j=1 |a j is an arbitrary product state, by definition of the (ε/ √ N)-net, for each j, there exists |ã j ∈ M j such that | a j |ã j | 2 ≥ 1 − ε 2 /4N. It follows that the following relation holds for |φ = N j=1 |ã j ∈ N ε,N , Hence, N ε,N is an ε-net on the set of product states with |N ε,N | ≤ (5 √ N/ε) 2d S . necessary, |Φ can be turned into the form (a + bi)|0 + ci|1 , where |0 and |1 are two basis kets within the orthonormal basis |0 , |1 , . . . , |d T − 1 , and a, b, c are real numbers satisfying a 2 + b 2 + c 2 = 1. Suppose |Ψ R = d T −1 j=0 x j |j , where x j s are real numbers According to the same reasoning that leads to (86), the probability that G(|Ψ R ) ≤ − log( 3 2 ε) is at most Let ε = 6d S ln d T /d T , the probability that G(|Ψ ) ≤ [log d T − log(d S ln d T ) − log 9] is at most Next, let ε = 2 3 d −1/2 T , the probability that G(|Ψ ) ≤ 1 2 log d T is at most Prob max The derivation of (92) and (93) is similar to that of (87) and (88). ⊓ ⊔

Implications of additivity property for one-way quantum computation and for asymptotic state transformation
Recently, Gross et al [35] (see also [86]) showed that most quantum states are too entangled to be useful as computational resources. One of the key ingredient in their proof is the observation that almost all pure multiqubit states are nearly maximally entangled with respect to GM. However, their arguments would break down, if measurements are allowed on the tensor product of the resource states, since ρ ⊗ ρ * is just moderate entangled (GM is nearly one half of the maximal possible value) for a generic pure multiqudit states ρ, according to theorem 24. In particular, two copies of ρ is moderate entangled if ρ is a real state. Hence, it is conceivable that we may realize universal quantum computation on certain family of multiqudit states if they come in pairs, even if this is impossible on a single copy. It would be very desirable to construct an explicit example of such a family of multiqudit states or disprove this possibility. However, a detailed investigation along this direction would well go beyond the scope of this paper. Corollary 26 has a significant implication for asymptotic state transformation. In particular, it implies that almost all multiqudit pure states cannot be prepared reversibly with multipartite GHZ states (of various numbers of parties) under asymptotic LOCC, unless REE is non-additive for generic multiqudit states. This can be seen as follows. According to the result of Linden et al [23], reversible transformation between two pure states under asymptotic LOCC would mean that the ratio of the bipartite AREE E ∞ R (A j :Ã j ) to the N-partite AREE E ∞ R is conserved, for j = 1, 2, . . . , N, whereÃ j denotes all the parties except A j . As a result, the ratio [ N j=1 E ∞ R (A j :Ã j )]/E ∞ R is conserved. If a state |ψ ψ| can be prepared reversibly with n k copies of k-partite GHZ states for k = 2, . . . , N, then where we have used the fact that REE of the tensor product of GHZ type states is additive. On the other hand, E ∞ R (|ψ A j :Ã j ) = E R (|ψ A j :Ã j ) ≤ log d, in addition, E R (|ψ ) > 1 2 log d T = N 2 log d for almost all multiqudit pure states, according to corollary 26. If REE of |ψ is additive, then we have which contradicts (94). Hence, almost all multiqudit pure states cannot be prepared reversibly under asymptotic LOCC, unless REE is non-additive for generic multiqudit pure states. Our observation adds to the evidence that a reversible entanglement generating set [24,25] with a finite cardinality may not exist. As a concrete example, similar reasoning has been employed by Ishizaka and Plenio [87] to show that |ψ 3− cannot be generated reversibly from the GHZ state and EPR pairs under asymptotic LOCC if its REE is additive. The same is true for |ψ N − with N ≥ 3, since E R (|ψ N − ) = log(N!) > N 2 log N [30,55,56] (see also (63) in section 4.3).

Summary
In this paper, we have studied the additivity property of three main multipartite entanglement measures, namely GM, REE and LGR.
Firstly, we proved the strong additivity of GM of non-negative states, thus simplifying the computation of GM and AGM of a large family of states of either experimental or theoretical interest. Thanks to the connection among the three measures, GM of non-negative states provides a lower bound for AREE and ALGR, and a new approach for proving the additivity of REE and LGR for states with certain group symmetry. In particular, we proved the strong additivity of GM and the additivity of REE of Bell diagonal states, maximally correlated generalized Bell diagonal states, generalized Dicke states and their reduced states after tracing out one party, the Smolin state and Dür's multipartite entangled states etc. The additivity of LGR of generalized Dicke states and the Smolin state was also shown. These results can be applied to studying state discrimination under LOCC [29,30], the classical capacity of quantum multi-terminal channels. The result on AREE is also useful in studying state transformation either under asymptotic LOCC or under asymptotic non-entangling operations. For non-negative bipartite states, the result on AREE also leads to a new lower bound for entanglement of formation and entanglement cost. The result on GM and AGM may find applications in the study of quantum channels due to the connection between pure tripartite states and quantum channels [32].
Secondly, we established a simple connection between the permutation symmetry and the additivity property of multipartite entanglement measures. In particular, we showed that GM is non-additive for antisymmetric states shared over three or more parties. Also, we gave a unified explanation of the non-additivity of the three measures GM, REE and LGR of the antisymmetric projector states, and derive analytical formulae of the three measures for one copy and tow copies of such states. Our results on antisymmetric states are expected to be useful in the study of fermion systems, which are described by antisymmetric states due to the super-selection rule.
Thirdly, we showed that almost all multipartite pure states are maximally entangled with respect to GM and REE. However, their GM is not strong additive; moreover, for generic pure states with real entries in the computational basis, GM of one copy and two copies, respectively, are almost equal. Based on these observations, we showed that more states may be suitable for universal quantum computation, if measurements can be performed on two copies of the resource states. We also showed that, for almost all multipartite pure states, the additivity of their REE implies the irreversibility in generating them from GHZ type states under LOCC, even in the asymptotic sense.
There are also quite a few open problems which can be new directions in the future study of multipartite entanglement.
(i) Are GM and REE of arbitrary two-qubit states and pure three-qubit states additive?
(ii) Are GM and REE of arbitrary symmetric states additive? We cannot find any counterexamples at the moment; however, the possibility has not been excluded.
(iii) When are GM and REE of bipartite mixed antisymmetric states additive or nonadditive?
(iv) What are AGM, AREE and ALGR of the antisymmetric projector states? It is enough to compute any one of the three measures, since they are related to each other by the simple equalities in proposition 19.
(v) Are GM, REE and LGR non-additive for generic multipartite states?
(vi) Does there exist a family of quantum states such that two copies are universal for quantum computation while one copy is not?
Suppose ρ N and ρ ′ N are two N-partite states on the Hilbert space In this appendix, we prove the following formula, where the complex conjugate is taken in the computational basis. The formula reduces to in the special case d j = d, V j = V, ∀j. Proof.