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A comparison of old and new definitions of the geometric measure of entanglement

Several inequivalent definitions of the geometric measure of entanglement (GM) have been introduced and studied in the past. Here we review several known and new definitions, with the qualifying criterion being that for pure states the measure is a linear or logarithmic function of the maximal fidelity with product states. The entanglement axioms and properties of the measures are studied, and qualitative and quantitative comparisons are made between all definitions. Streltsov et al. [New J. Phys 12 123004 (2010)] proved the equivalence of two linear definitions of GM, whereas we show that the corresponding logarithmic definitions are distinct. Certain classes of states such as"maximally correlated states"and isotropic states are particularly valuable for this analysis. A little-known GM definition is found to be the first one to be both normalized and weakly monotonous, thus being a prime candidate for future studies of multipartite entanglement. We also find that a large class of graph states, which includes all cluster states, have a"universal"closest separable state that minimizes the quantum relative entropy, the Bures distance and the trace distance.


Introduction
Entanglement measures lie at the heart of quantum information theory, because they assess the usefulness of quantum states for tasks such as quantum teleportation, quantum computation and cryptography protocols [1,2]. Numerous entanglement measures have been defined in the past, each of which may capture different properties of a state as a resource for certain tasks. One well-known measure is the geometric measure of entanglement (GM). Originally introduced for pure bipartite states [3,4], the GM was subsequently generalized to multipartite and to mixed states [5][6][7]. Two key benefits of GM are that it is an inherently multipartite entanglement measure and that it is comparatively easy to compute for many states [6,8,9].
The GM has a variety of operational interpretations: It assesses the usability of initial states for Grover's algorithm [10,11], the discrimination of quantum states under LOCC [12], the additivity and output purity of quantum channels [13] and the usefulness of states as resources for one-way quantum computation [8,[14][15][16]. Further uses of GM include the construction and study of entanglement witnesses [6,17], the derivation of a generalized Schmidt decomposition [18] and the study of condensed matter systems, such as ground state characterization and detection of phase transitions [19][20][21].
Several inequivalent definitions of GM have surfaced in the literature [5-7, 9, 10, 12, 22, 23]. Regarding pure states, GM is expressed either as a linear or logarithmic function of the maximal fidelity with product states. Regarding mixed states, the pioneering papers did not agree on a unique definition, which led to the emergence of several inequivalent extensions of GM to mixed states. Although some of the GM definitions have been compared to other entanglement measures [22], a detailed comparison of all the different definitions of GM to each other has not been done before. An important milestone towards this goal was achieved by Streltsov et al. [24] who proved the equivalency of two frequently used definitions of GM.
In this paper we compare and characterize several known and new definitions of GM. The only qualifying criterion for an entanglement quantity to be regarded a GM definition is that on the subset of pure states it coincides with the welldefined linear or logarithmic GM.
Five known definitions, one little-known and one new definition of GM are studied in this paper, first with respect to their entanglement axioms. This is followed by a quantitative and qualitative comparison of the definitions to each other. The "maximally correlated states" (as defined in Definition 1) turn out to be a particularly helpful class of states for this purpose. We also discover that a large class of graph states, including all cluster states, have a "universal" closest separable state that minimizes the quantum relative entropy, the Bures distance and the trace distance.
The paper is organized as follows. Section 2 reviews some basic concepts of quantum information theory for later usage. In Section 3 the definitions of GM are introduced, and some preliminary results, e.g. with regard to entanglement axioms, are obtained. The subsequent Section 4 closely examines the relationship between all six distinct definitions of GM from a variety of perspectives, and a hierarchy that allows for a partitioning of state space is obtained. A common closest separable state with respect to three different distance measures is presented for a large class of graph states in Section 5. The concluding Section 6 summarizes our results. For convenience, Table 1, Table 2, Figure 1 and Figure  2 list some of our findings in compact form.

Preliminaries
First, we review some basic concepts for later usage, in particular axiomatic entanglement measures and distance measures. For more comprehensive reviews we refer to [1,2,25,26].

Axiomatic entanglement measures.
Operationally motivated entanglement measures such as the Entanglement Cost and the Distillable Entanglement have clear physical meanings, but tend to be difficult to study from a mathematical viewpoint, especially for multipartite systems. On the other hand, axiomatically motivated entanglement measures may not have operational implications.
Considering n parties A 1 , . . . , A n with joint Hilbert space H = ⊗ n j=1 H j , a general n-partite state shared over the parties is described by a density matrix ρ ∈ S(H) acting on H. Such a state is considered separable if it can be written in the form ρ = i p i ρ 1 i ⊗ · · · ⊗ ρ n i , with i p i = 1, and where ρ j i is a single-particle state of the j-th party. In the axiomatic approach, an entanglement measure is a functional E : S(H) → R + that satisfies two fundamental axioms: 1. E(ρ) = 0 if ρ is separable. 2. E does not increase on average under local operations and classical communication (LOCC). Depending on which quantum operations are considered, this is defined as (a) weak monotonicity: Tr Ei(ρ) with probability p i = Tr E i (ρ) .
Here, the maps E and E i stand for LOCC, and the elements {P i } form a complete POVM, i.e. i P † i P i = 1. Weak monotonicity corresponds to trace-preserving quantum operations where the measurement outcome is unknown or discarded. Strong monotonicity corresponds to selective quantum operations, also known as measuring quantum operations [1]. Weak monotonicity is a special case of strong monotonicity that follows when a single outcome is obtained with probability 1, i.e. E = E 1 . We refer to measures satisfying Axioms 1 and 2(a) as weak entanglement measures, and measures that additionally satisfy Axiom 2(b) as strong entanglement measures.
Historically, strong monotonicity was required for entanglement measures [1,27,28], but there is now a consensus that weak monotonicity suffices [2]. Weak entanglement measures can thus be considered proper entanglement measures. Another historic requirement is that entanglement measures should coincide with the entropy of entanglement for pure bipartite states [1,28]. However, many popular measures fail this property, and the property cannot be easily extended to multipartite states, so it is not considered essential anymore. Invariance under local unitary (LU) operations is clearly important for entanglement measures, but it does not need to be stated as a separate axiom, because it automatically follows from weak monotonicity [1].
Apart from the axioms discussed above, many more desirable properties could be specified. Some common ones are the following 1 : -Normalization: E(|Φ ⊗n ) = n , for 2 qubit Bell states |Φ -Convexity: E(ρ) ≤ i p i E(ρ i ) , for all ρ = i p i ρ i -Additivity: E(ρ ⊗2 ) = 2 E(ρ) , for all ρ ∈ S(H) -Strong Additivity: E(ρ ⊗ σ) = E(ρ) + E(σ) , for all ρ, σ ∈ S(H) The desirability of normalization is clear from the perception that Bell states carry 1 ebit of entanglement each. Convexity is motivated by the notion that entanglement should not increase under loss of information, namely when a selection of identifiable states ρ i (r.h.s.) is transformed into a mixture ρ (l.h.s.) [1]. One may assume that this process can be physically realized by standard quantum operations, and thus strong monotonicity implies convexity. However, some additional properties (such as continuity) need to be satisfied, and the logarithmic negativity constitutes a counterexample by being a strong entanglement measure that is not convex [29]. An important consequence of convexity is that the measure can be maximized on the subset of pure states, i.e. there exist maximally entangled states (MES) that can be cast as pure states ρ = |ψ ψ|.
If f and g are two convex functions and g is non-decreasing, then g • f is also convex. For example, if f (ρ) ≥ 0 is a convex measure, then f 2 (ρ) is also convex, using g(x) = x 2 . In analogy to convexity, concavity is defined as Regarding the additivity axioms, the tensor products in their definition have a specific physical meaning: Instead of enlarging the Hilbert space, ρ ⊗ σ refers to two states acting on the same Hilbert space [8]. If ρ and σ are both states of n d-level subsystems, then ρ ⊗ σ is a state of n d 2 -level subsystems (instead of 2n d-level subsystems). Obviously, strong additivity implies additivity.
From a mathematical viewpoint, two entanglement measures E 1 and E 2 are equivalent, if E 1 (ρ) = E 2 (ρ) holds for all ρ ∈ S(H). A less restrictive, yet physically sound criterion is the property of ordering: E 1 and E 2 have the same entanglement ordering, if the same order is obtained when sorting all states by their amount of entanglement. This is the case if for all ρ, σ ∈ S(H) the two statements E 1 (ρ) > E 1 (σ) and E 2 (ρ) > E 2 (σ) are equivalent, i.e. they are either both true or both false. Entanglement measures that are equivalent will be denoted as E 1 ≡ E 2 , and measures with the same ordering as Many different entanglement measures have been proposed, but here we only consider measures that are based on the distance to the set of separable states. The relative entropy of entanglement (REE) measures the minimum distance in terms of relative entropy between the given state ρ and the set of separable states (SEP): where is the quantum relative entropy [28]. Any state σ minimizing S(ρ|σ) is called a closest separable state of ρ. Since the definition involves the minimization over all separable states, REE is known only for a few examples, such as bipartite pure states [28,30,31], Bell diagonal states [30,32,33], Werner states [34][35][36], maximally correlated states, isotropic states [32,33], generalized Dicke states [17,22,37], antisymmetric basis states [17,22], some graph states [12,38], the Smolin state, and Dür's multipartite entangled states [37,39]. A numeric method for computing REE of bipartite states has been proposed [28]. The REE can be applied to arbitrary multipartite states, pure or mixed. The central measure of this paper, the geometric measure of entanglement (GM) -to be discussed in Section 3 -is also an inherently multipartite measure, although its definition for mixed states is not unique.

Distance measures and fidelity.
A good measure of distance D(ρ, σ) : S(H)× S(H) → R + between two quantum states should be symmetric, zero iff ρ = σ, and observe weak monotonicity, which in this context means D(ρ, σ) ≥ D(E(ρ), E(σ)) for any trace-preserving quantum operation E [2,25,30]. The last property, also known as contractivity under quantum operations, guarantees LU invariance: D(ρ, σ) = D(U ρU † , U σU † ). Any distance function with these properties is called a distance measure. One such distance measure is the trace distance [25], where the λ i are the eigenvalues of the matrix (ρ − σ). For qubits D T (ρ, σ) is equal to half the Euclidean distance between the corresponding Bloch vectors. The trace distance is convex in both arguments, and furthermore is jointly convex [25]: Another distance measure is the Bures distance and the closely related fidelity [25,40,41]. The Bures distance is where F (ρ, σ) is the fidelity between two states, defined as Alternative definitions in the literature are D T := D 2 T for the trace distance, D B := D 2 B for the Bures distance, and F := F 2 for the fidelity 2 . The three necessary properties of distance measures outlined above remain invariant under exponentiation, and therefore D T and D B are also distance measures. Although closely related to the Bures distance, the fidelity is not a distance measure itself. The fidelity is symmetric, unitarily invariant, concave in both arguments, jointly concave, and has codomain F ∈ [0, 1], with F = 1 iff ρ = σ [25]. According to Uhlmann's theorem, the fidelity has a clear physical interpretation as the maximal overlap between all purifications of the input states [25,40].
If at least one of the two arguments of the fidelity is pure, then (6) simplifies to F 2 (ρ, σ) = Tr(ρσ), thus yielding In particular, for pure states the fidelity coincides with the Fubini-Study metric, the natural geometry on H.
The fidelity also provides upper and lower bounds on the trace distance, with the lower bound increasing with the purity of the input states [25]: Vedral et al. [30] found that from every distance measure D(ρ, σ) a weak entanglement measure E(ρ) can be constructed as This construction directly yields weak entanglement measures from the trace distance (3) and Bures distance (5), which we refer to as trace entanglement (TE) and Bures entanglement (BE), respectively. In analogy to D T and D B , we define E T := E 2 T and E B := E 2 B , and note that they are also weak entanglement measures. From (5) and the l.h.s. of (8a), it follows that The quantum relative entropy (2) is not symmetric, and therefore not a proper distance measure. Nevertheless, by means of (9), it gives rise to the REE, which is a strong entanglement measure [28,43]. One could also ask whether the Hilbert-Schmidt distance D HS (ρ, σ) = Tr[(ρ − σ) 2 ], a metric in the mathematical sense, gives rise to an entanglement measure. However, this metric does not satisfy weak monotonicity, and it is an open question whether inserting D HS in (9) yields an entanglement measure [44].
A simple, but important mathematical inequality for this paper is where e denotes the base of the natural logarithm. We call (11) the elementary inequality. To demonstrate its usefulness, consider the two most common entropic quantities in quantum information theory, the linear entropy M(ρ) = 1 − Tr(ρ 2 ), and the von Neumann entropy S(ρ) = − Tr(ρ log ρ). The linear entropy can be understood as an approximation of von Neumann entropy, obtained by the Taylor series log(ρ) ≈ ρ − 1, where 1 has the same range as ρ. For the commonly used logarithm bases 2 and e, (11)

Geometric measure of entanglement
In this section we review the two two common definitions of GM for pure states, and introduce the known and unknown extensions to mixed states.
3.1. GM for pure states. The fundamental quantity for GM of pure states is where PRO denotes the set of fully product pure states of H, henceforth referred to as product states. Comparing (13) with (7), we see that Λ 2 (|ψ ) is the maximum fidelity between |ψ and the set PRO. Furthermore, it is clear from (7) that the maximal value of F (|ψ , ·) can be found among pure states: Any product state or separable state that maximizes the corresponding fidelity expression in (14) is called closest product state (CPS) or closest separable state (CSS), respectively. Note that the CPS or CSS is in general not unique. The relationship between the CSSs and CPSs of a given state |ψ ∈ H is seen from (7): If {|φ i } is the set of CPSs, then any superposition σ = i p i |φ i φ i | with i p i = 1 is a CSS. Conversely, if σ is a CSS, then there must exist a decomposition σ = i p i |φ i φ i | such that all |φ i are CPSs.
The two common definitions of GM for pure states are which we refer to as the linear GM and logarithmic GM, respectively. Unless denoted otherwise, the base 2 logarithm is used in this paper. Thanks to the elementary inequality (11), the results of this paper are also valid for any other logarithm base up to, and including, the base of the natural logarithm. For larger bases this is not the case, because (11) then no longer holds. Both G and G l increase monotonically with Λ 2 , and eliminating Λ 2 yields Due to this monotonic relationship, G and G l have the same ordering for pure states. In particular, they have the same MESs. For bipartite states the MES (up to LU) is |ii , yielding G(|Ψ ) = 1 − 1 d and G l (|Ψ ) = log d. For the simplest multipartite case of three qubits, the W state has been analytically determined as the MES for the GM [45]. For general multipartite systems, however, W states only yield low entanglement in terms of GM, and the identification of the MESs is an open problem. For the subset of permutationsymmetric states the MES are better understood [46,47], because the CPSs of symmetric states are symmetric themselves [48], thus considerably simplifying the optimization problem.
From (16) and the elementary inequality (11) it follows that Since Λ 2 (|ψ ) = | ψ|ψ | 2 = 1 holds for all |ψ ∈ PRO, entanglement Axiom 1 is satisfied for pure states for both definitions in (15). Regarding Axiom 2, an extension of G to mixed states that satisfies strong monotonicity is known [6], which automatically implies strong monotonicity of G on the subset of pure states. In contrast to this, an explicit counterexample ruling out strong monotonicity is known for G l , and since this counterexample considers pure states only [22], no extension of G l to mixed states can be strongly monotonous. However, we will later see that extensions of G l with weak monotonicity do exist (cf. G f l defined in Section 3.2.3), and therefore G l is weakly monotonous on the subset of pure states. The axiomatic properties of G and G l are summarized on the left of Table 2.
Regarding the optional axioms, it is easy to verify that G l is normalized whereas G is not. This makes G l the natural choice for quantitative studies of entanglement, such as scaling laws or comparison with other measures. The MES entanglement of n qubits (n ≥ 3) scales linearly as n − 2 log 2 (n) − O(1) < G l (|Ψ ) < n − 1 [8,14,49]. Restricting the computational coefficients to real values does not affect this scaling [8], but for positive states (i.e. states whose coefficients are all positive in the computational basis) the n-qubit MES are bounded by G l (|Ψ ) ≤ n 2 , and this bound is strict for even n (a trivial example being n/2 Bell pairs) [47]. On the other hand, symmetric n-qubit MESs scale logarithmically as log 2 (n + 1) − O(1) < G l (|Ψ ) < log 2 (n + 1) [47,50]. These scaling laws readily generalize to qudits, leading to the conclusion that the MESs of sufficiently large multipartite systems are neither positive nor symmetric. Furthermore, since generic states are nearly maximally entangled with respect to GM [14], the above scaling laws can also be applied to random states.
Regarding the additivity axioms, neither G nor G l are additive in general. For G this is obvious from its codomain [0, 1], and for G l it has been shown that states with a high amount of entanglement are not additive [8]. Nevertheless, many states of interest are additive or even strongly additive under G l . In particular, it has been shown that positive states are strongly additive [8]: Lemma 1. Let |ψ ∈ H be a positive state. Then |ψ is strongly additive, i.e. Λ 2 (|ψ ⊗ |φ ) = Λ 2 (|ψ )Λ 2 (|φ ) and G l (|ψ ⊗ |φ ) = G l (|ψ ) + G l (|φ ) holds for all |φ ∈ H.
Examples of positive states are multipartite Dicke states of arbitrary dimension, and all bipartite pure states (by means of the Schmidt decomposition). Lemma 1 can be readily generalized to mixed states for an extension of GM that will be discussed in Subsection 3.2.2.
Note that the definition of the linear GM (15a) coincides with the Groverian entanglement measure E Gr (|ψ ) = G(|ψ ) 1/2 , which has a tangible operational interpretation by means of a quantum algorithm [7,10].

GM for mixed states.
With the GM defined for pure states |ψ ∈ H, we now consider the possible extensions to mixed states ρ ∈ S(H). Extensions of the linear GM will be labelled G x (ρ), and extensions of the logarithmic GM as G x l (ρ), where x stands for a label to denote the extension. Any valid extension must coincide with (15a) or (15b) on the subset of pure states ρ = |ψ ψ|. In other words, G x (|ψ ψ|) = G(|ψ ) and G x l (|ψ ψ|) = G l (|ψ ) must hold for all |ψ ∈ H.
Since the expressions "pure" and "mixed" can be ambiguous, we briefly clarify their usage. From a mathematical viewpoint, σ = |ψ ψ| ∈ S(H) is a mixed state, but physically it is equivalent to the pure state |ψ ∈ H. Therefore, we refer to both |ψ and σ = |ψ ψ| as pure states. On the other hand, we refer to all states ρ ∈ S(H) as mixed states, so σ = |ψ ψ| can be regarded as pure and mixed. Mixed states that are not pure will be called genuinely mixed. Mathematically, a state ρ ∈ S(H) is genuinely mixed, iff rank ρ ≥ 2.
One strategy to extend (15) to mixed states is to extend (13) to mixed states, i.e. to introduce a function Λ 2 x (ρ) : S(H) → R + with the property that Λ 2 x (|ψ ψ|) = Λ 2 (|ψ ) for all |ψ ∈ H. The following lemma asserts the properties of extensions defined in that manner.
x (ρ) : S(H) → R + be an extension of (13) to the set of all density matrices. Then the following holds for G x (ρ) : Proof. 1. This directly follows from the elementary inequality (11).
2. The relationships between G x (ρ) and G x l (ρ) are obtained by eliminating With regard to items 5. and 6. of Lemma 2, it should be noted that convexity of G x does not follow from convexity of G x l , and that concavity of G x l does not follow from concavity of G x . A counterexample for the latter case are the measures G m and G m l introduced in Section 3.2.2. Several extensions of GM have been proposed in the past, and below we will introduce these as well as new ones. The first approach is based on a convex roof construction, akin to the entanglement of formation [1]. The second and third approach are based on extending the definition (13) to mixed states by means of the fidelity between the given state and the set of all pure or mixed states, respectively. Consequently, Lemma 2 applies to these two approaches. The fourth approach is to extend the linear GM by means of the trace distance (3).

Extension by convex roof: G
Based on definitions (13) and (15), the convex roofs of the linear and logarithmic GM are where the minimum runs over all decompositions of ρ = i p i |ψ i ψ i |. Decompositions that maximize (18a) or (18b) will be called optimal decompositions, and labeled {P i , |Ψ i }. It is natural to ask whether for a given ρ the two functionals are minimized for the same decomposition. We will later show that for many states G c and G c l have the same optimal decompositions (e.g. for all isotropic states and two qubit states), but that there also exist states for which G c and G c l do not have a common optimal decomposition (e.g. for some maximally correlated states). Another open question is how many pure components |ψ i are necessary for an optimal decomposition of G c or G c l . At least for G c it is known that (dim H) 2 pure components suffice [24].
Mathematically, any two decompositions holds for all i [25,51]. This identity will later be used in some proofs.
Regarding the entanglement axioms, Axiom 1 and convexity directly follow from the convex roof definitions for both G c and G c l . The quantity G c was first studied in detail in the seminal paper of Wei et al. [6], and found to be a strong entanglement measure. On the other hand, G c l cannot be a strong entanglement measure [22]. However, it is an open question whether G c l is weakly monotonous, i.e. whether G c l (ρ) ≥ G c l (E(ρ)) holds for all channels E and all ρ ∈ S(H). For ρ ∈ SEP this is satisfied because of Axiom 1, and for all bipartite pure MES In the two qubit case G c l is a weak entanglement measure. In Appendix A we present a direct proof employing the concurrence and the optimal decomposition of the entanglement of formation [52]. The same result will later follow from a different line of argumentation.
Since the definitions (18) are not based on extending (13) to mixed states, Lemma 2 does not apply to G c and G c l . In particular, we do not know if there exists an exact analytic relation between G c and G c l , or if the two quantities have at least the same ordering (i.e. G c ∼ = G c l ). The following lemma provides an analytic relation in the form of an inequality.
Lemma 3. For every state ρ the following holds.
where the inequalities follow from that fact that f (y) = − log 2 (y) is a convex and monotonically decreasing function. An analogous derivation yields Regarding the ordering of G c and G c l , it will later be shown that the two measures do not have the same ordering in general (cf. Corollary 4).

Extension by trace inner product
Apart from the convex roof, the most widely studied extension of GM to mixed states is obtained by extending (13) to mixed states via the Hilbert-Schmidt inner product Tr(A † B), also known as trace inner product [25]. As proved below, this is equivalent to maximizing the fidelity between the input state and set of pure product states.
Proof. The last equality is clear from (7a), and the ≥ part of the middle equality follows from the fact that the set of fully separable states contains the pure product states. The ≤ part of the middle equation follows as where σ m = i p i |φ i φ i |, with |φ i ∈ PRO for all i, is the separable state that maximizes Tr(ρσ).
For pure states Λ 2 m (ρ) obviously coincides with Λ 2 (|ψ ) from (13). Therefore, the functionals (15a) and (15b) are extended to mixed states as Since Lemma 2 applies to these measures, there is some interdependence in their entanglement axioms. Indeed, neither G m nor G m l is an entanglement measure. This can be readily seen from that fact that the two measures attain their maximum for the maximally mixed state 1 dim(H) , a separable state [23], thus violating Axiom 1 and Axiom 2(a).
With regard to convexity, the maximally mixed state is also a counterexample, because it can be decomposed into pure product states, 1 = i |i i|, with G(|i ) = G l (|i ) = 0. To check whether G m or G m l are concave, consider an arbitrary decomposition ρ = i p i ρ i . We have Therefore, Λ 2 m (ρ) is convex. From this it directly follows that G m (ρ) = 1−Λ 2 m (ρ) is concave. It remains to investigate whether G m l is also concave. Let us consider the isotropic state |ii [53]. We easily see that , and the inequality is strict for p ∈ (0, 1). So, the isotropic state is a counterexample to the concavity of G m l . To conclude, G m l is neither convex nor concave. Although not entanglement measures, the quantities G m and G m l are easier to calculate than other definitions of GM, and have received a considerable amount of attention [8,12,17,23,23,37,49]. The measure G m l has been found to be closely related to the relative entropy of entanglement and the logarithmic global robustness of entanglement [8,17,22,47]. Furthermore, G m l has been employed for the construction of optimal entanglement witnesses [17] and for the study of state discrimination under LOCC [17,23]. Zhu et al. [8] calculated Λ 2 m (ρ) for many states of interest, and Jung et al. [49] found that tracing out one subsystem of an n-partite pure state does not change this quantity, i.e. Λ 2 m (|ψ ) = Λ 2 m (ρ), with ρ = Tr i (|ψ ψ|), holds for all |ψ ∈ H and all 1 ≤ i ≤ n.
The measure G m l allows to generalize Lemma 1 to mixed states [8]. A density matrix is called positive if all its entries in the computational basis are positive.
. This lemma has been employed to show the strong additivity of many mixed states, such as mixtures of Dicke states, Bell diagonal states, isotropic states, multiqubit Dür states and the Smolin state [8]. The additivity problem of G m l is closely related to that of the relative entropy and the logarithmic global robustness [17], which facilitated the study of additivity under these two entanglement measures, as well [8].

Extension by fidelity
We can also extend Λ 2 to mixed states by means of the fidelity: This quantity has been previously studied [7,9,10,24,42,54], and has been described as fidelity of separability [24]. In the bipartite case (22) is equivalent to the so-called maximum k-extendible fidelity of a state in the limit k → ∞ [54]. The maximum k-extendible fidelity has an operational interpretation as the maximum probability with which one party can convince another party that ρ is separable in a specific protocol [54]. It is easy to verify that for pure states Λ 2 f coincides with Λ 2 : The corresponding extensions of the linear and logarithmic GM are As seen for G m and G m l , Lemma 2 applies to these measures. G f is intimately related to the Groverian entanglement measure [7,10], thus giving it an operational interpretation by a quantum algorithm. G f has been shown to be a weak entanglement measure [7], and has also been studied in [42]. On the other hand, little is known about G f l . It has been touched upon in the context of additivity in [9], but to our knowledge its properties have not been studied before.
Intriguingly, it was discovered that G f (ρ) is equivalent to its convex roof [24], and since the convex roof is precisely G c (ρ), the definitions (23a) and (18a) are equivalent: We will jointly refer to these two definitions as G f/c in the following, except in situations where the emphasis is on their formal definitions (i.e. fidelity-based vs. convex roof-based). The relationship between σ f , the CSS of ρ in terms of (23a), and the optimal decomposition {P i , |Ψ i } of ρ in terms of (18a) is also fully understood, and outlined in [24]. Since G c is known to be a strong entanglement measure with convexity, the same is true for G f . In particular, the convexity of (23a) implies that Λ 2 f (ρ) is concave. There are many states for which G c (ρ) has been computed [6], and from Proposition 1 and Lemma 2 the values of G f (ρ) and G f l (ρ) directly follow. With the known properties of G f , it follows from Lemma 2 that G f l is a weak entanglement measure with convexity. From the convexity of G f l it then follows that G f l (ρ) ≤ G c l (ρ) holds for all ρ: where {P i , |Ψ i } is an optimal decomposition for G c l (ρ). The question whether G f l (ρ) ≤ G c l (ρ) is strict for some states will be extensively studied in Section 4.
3.2.4. Extension by trace distance: G t . It is tempting to introduce another mixed extension of GM, based on the trace distance defined in (3). From (8c) we obtain D 2 T (|ψ , |φ ) = 1 − F 2 (|ψ , |φ ) = 1 − | ψ|φ | 2 , an expression with the form of (15a). We therefore define and call this measure the trace extension of GM. For pure input states G t obviously coincides with (15a), so (25) is an extension of the linear GM. Note that the related definition, was already introduced (up to a square operation) as the trace entanglement in (10a) and shown to be a weak entanglement measure. From (25) and (26) it immediately follows that E T (ρ) ≤ G t (ρ) for all ρ. To see whether E T also coincides with (15a) for the subset of pure states, we need to answer the question whether for pure input states the closest separable state σ t ∈ S(H) in terms of the trace distance can always be chosen to be pure, i.e. σ t = |φ φ|. The cluster states provide a counterexample for this: There exist pure states |ψ for which E T (|ψ ) < G t (|ψ ) holds.
Proof. From Theorem 9 and the succeeding paragraph, together with (8c) it holds for all n qubit cluster states |C n with even n.
As a consequence, E T is not an extension of the linear GM, because D T (|ψ , ·) is in general not minimized by pure states. However, E T is an interesting quantity on its own, because it is a weak entanglement measure and a lower bound to G t . Furthermore, we will see in Section 4.3 that E T is also a lower bound to G f/c , which makes it a joint lower bound to all the GM definitions discussed in this paper.
The convexity of E T , and thus the convexity of E T = E 2 T can be proved with the joint convexity of the trace distance. For any ρ = i p i ρ i we have where the inequality follows from (4).
On the other hand, G t is not an entanglement measure. For this, note that G t (|ϕ ) = 0 holds for all |ϕ ∈ PRO, and that G t (ρ) > 0 holds for all genuinely mixed ρ ∈ SEP. From this it is not only clear that Axiom 1 is violated, but one can also immediately construct counterexamples for the concavity and the weak monotonicity (e.g. with the depolarizing channel [25]). Using the isotropic state as a counterexample, it is shown in Appendix B that G t is not concave either.

Relationships between the GM definitions
In the previous section we introduced and discussed seven different definitions of GM, of which two are equivalent. In the following we analyze the relationship between these different definitions.
For an arbitrary mixed state ρ ∈ S(H) the quantities G m (ρ) and G m l (ρ) correspond to the same closest product state |φ m ∈ PRO, and the quantities G f (ρ) and G f l (ρ) correspond to the same closest separable state σ f ∈ SEP. In contrast to this, G c (ρ) and G c l (ρ) correspond to optimal decompositions {P i , |Ψ i } that may be different for the two measures. The quantity G t (ρ) corresponds to a closest product state |φ t ∈ PRO. In total, with the exception of the convex roof-based measures, the different values of GM for a given state are determined by two product states |φ m , |φ t , and one separable state σ f .

4.1.
Comparison between G f/c , G c l and G f l . From the previous discussion we already know that G f ≡ G c , G f/c ∼ = G f l , and that G c l (ρ) ≥ G f l (ρ) = − log 2 1 − G f/c (ρ) . In this subsection, we study the connection between the fidelity-based and convex roof-based extensions in further detail by addressing some open problems. For example, it is neither obvious nor known whether G c l and G f l are equivalent (G c l ≡ G f l ), and if not, whether they have at least the same ordering (G c l ∼ = G f l ). For this purpose, we will first derive necessary and sufficient conditions for G c l (ρ) = G f l (ρ), and then investigate optimal decompositions for specific classes of states (e.g. maximally correlated states, isotropic states, 2 qubit states).
Theorem 1. For any state ρ the following four conditions are equivalent: Here, the meaning of 3(b) and 4(b) is that Λ 2 (|Ψ i ) = Λ 2 (|Ψ j ) holds for all i, j.
Proof. Let {P i , |Ψ i } be some optimal decomposition of G c l (ρ). Using Lemma 2, Proposition 1 and (18), we have where the second inequality follows from the concavity of the logarithm.  (27), this inequality becomes an equality iff {P i , |Ψ i } is also an optimal decomposition for G c (ρ). Regarding the second inequality in (27), the strict concavity of the logarithm implies that this inequality becomes an equality iff Λ 2 (|Ψ i ) = Λ 2 (|Ψ j ) holds for all i, j. Together, this yields that condition 1. holds if and only if condition 4. holds. Note that for states ρ that fall under Theorem 1 any optimal decomposition for G c l (ρ) is also optimal for G c (ρ), but the converse is not necessarily true. In other words, for states ρ that satisfy G c l (ρ) = G f l (ρ), the set of optimal decompositions for G c l (ρ) is a non-empty subset of the set of optimal decompositions for G c (ρ). It is an open question whether states ρ with G c l (ρ) = G f l (ρ) exist, for which the set of optimal decompositions of G c (ρ) is strictly larger than that of G c l (ρ). Without the condition G c l (ρ) = G f l (ρ), G c (ρ) and G c l (ρ) may not even have any common optimal decomposition, as shown later in Corollary 3.
The following corollary helps to understand the relationship between optimal decompositions of G c (ρ) and G c l (ρ). Corollary 2. Let {P i , |Ψ i } be some decomposition of ρ. The following two conditions are equivalent: 1. {P i , |Ψ i } is an optimal decomposition for G c (ρ), and the |Ψ i are all equally entangled. 2. {P i , |Ψ i } is an optimal decomposition for G c l (ρ), and G c l (ρ) = G f l (ρ) holds.
where the |Ψ i are all equally entangled. In analogy to (27), we have where the third equality follows from the fact that the |Ψ i are all equally entangled. As shown in (24), G f l (ρ) ≤ G c l (ρ) holds for all ρ, so the inequality in (28) must be an equality. Therefore, {P i , |Ψ i } is also an optimal decomposition for G c l (ρ), and G f l (ρ) = G c l (ρ) holds.
Following the derivation of general results, we next investigate classes of states for whom G f l (ρ) and G c l (ρ) coincide. We will see that for two-qubit systems G f l ≡ G c l holds, and that for general bipartite systems G f l (ρ) = G c l (ρ) holds for the subset of isotropic states.
|ii . The parametrization employed in [6] translates to ours as , ρ is separable [6], which implies G f l (ρ) = G c l (ρ) = 0. Next, consider the entangled region F ∈ ( 1 d , 1]. From [6] (Eq. (36), (54)) it follows that E sin 2 = G f/c and G f/c (ρ iso (F )) = 1 − 1 d ( To compute G c l (ρ iso (F )), we follow the idea of [6] to obtain the first and second equalities of the following equation: Here C conv is the convex hull. The last equality in (29) follows from the fact that G f l (ρ iso (F )) is convex. This completes the proof.
Next, we investigate maximally correlated states, because necessary and sufficient conditions for G c l (ρ) = G f l (ρ) can be derived for these states. The maximally correlated states have been extensively studied in terms of the entanglement of formation and distillable entanglement [36,55]. In this paper we focus on a special type of maximally correlated states defined as follows: with r < d be a set of integers with 0 = n 0 < · · · < n r = d, and let From now on, we refer to maximally correlated states as the states defined above. The integer r can be readily identified as the rank of the maximally correlated state, i.e. rank ρ = r.
Lemma 5. Let ρ = r i=1 q i |Θ i Θ i | be a d × d maximally correlated state. Then, 1. The unique optimal decomposition for G c (ρ), up to overall phases, is iff the |Θ i are all equally entangled. 5. G c l (ρ) = G f l (ρ) iff n i − n i−1 is the same for all i = 1, . . . , r . 6. If G c l (ρ) = G f l (ρ), then G c l (ρ) has the same optimal decompositions as G c (ρ).
Proof. First we prove item 1. Let {P i , |Ψ i } be some optimal decomposition of G c (ρ). Then According to (19), the relationship between the decompositions {P i , |Ψ i } and By summing over all i, we obtain i P i Λ 2 (|Ψ i ) ≤ r j=1 qj nj −nj−1 . Comparing this inequality to (30), we see that the inequalities must become equalities, and therefore {q j , |Θ j } is also optimal for G c (ρ). Since the inequality in (31) becomes an equality, all but one u i1 , . . . , u ir are zero. Hence, the state |Ψ i is identical to one of the states |Θ j , j = 1, . . . , r, up to an overall phase. Therefore, {q i , |Θ i } is the unique optimal decomposition for G c (ρ), up to overall phases.
With item 1 proved, the other items easily follow: -Item 2 and 3: These follow directly from item 1 and Λ 2 (|Θ i ) = , then it follows from item 1 and Theorem 1 that every optimal decomposition of G c l (ρ) must be of the form {q i , |Θ i }, up to overall phases. Since overall phases do not change the value of (18a) or (18b), G c (ρ) and G c l (ρ) have the same optimal decompositions.
One may wonder whether the optimal decomposition for G c (ρ) in Lemma 5 is also optimal for G c l (ρ) when G c l (ρ) = G f l (ρ). In the following we show that this is the case for all maximally correlated qutrit states. Only rank-2 states need to be considered, because for d = 3 this is the only non-trivial case.
Proof. Let {P i , |Ψ i } be some optimal decomposition of ρ for G c l (ρ). According to (19), there exists a unitary u ij , so that Without loss of generality we assume 2x i ≥ y i for i ∈ [1, k] and 2x i ≤ y i for i ∈ [k + 1, r]. We define Y := k i=1 y i ≥ 0 and X := The first inequality follows by applying Lemma 8 from Appendix C to the first sum, and using x j ≥ 0, y j > 0 in the second sum. The application of Lemma 8 is possible despite the restriction 2x i ≥ y i , because the minimum in (44) cannot be smaller with additional restrictions than without. The second inequality follows from inf x>0 (1 + 1 x ) log 2 (1 + x) = 1/ ln 2. On the other hand, the decomposition {q, |00 ; 1−q, |ψ } yields G c l (ρ) ≤ 1−q. This completes the proof.
One may conjecture that this proposition can be generalized to higher dimensions, with |ψ = 1 √ n n i=1 |ii being a pure MES of any dimension, i.e. that {q, |00 ; 1 − q, |ψ } is an optimal decomposition of ρ = q|00 00| + (1 − q)|ψ ψ| for G c l (ρ), yielding G c l (ρ) = (1 − q) log n. From Proposition 5 it will follow that this is not the case for n > 2. We will see that -compared to the qutrit case -the optimal decomposition for G c l (ρ) of higher-dimensional maximally correlated states is more complex, even in the comparatively easy rank-2 case. In the following, e denotes the base of the natural logarithm. Proposition 5. Let m, n ∈ N with m n ≤ 1 and q ∈ (0, 1) be constants that define the rank-two maximally correlated qudit state ρ = q|ψ m ψ m | + (1 − q)|ψ n ψ n |, with |ψ m = 1 √ m m i=1 |ii and |ψ n = 1 √ n m+n j=m+1 |jj . Depending on the constants m, n and q, an optimal decomposition and the value of G c l (ρ) are m n ≥ 1 e : {q, |ψ m ; 1 − q, |ψ n }, yielding G c l (ρ) = q log m + (1 − q) log n. -m n < 1 e and q ≥ em n : { 1 2 , √ q|ψ m ± √ 1 − q|ψ n }, yielding G c l (ρ) = log( m q ). -m n < 1 e and q < em n : {1− nq em , |ψ n ; nq 2em , em n |ψ m ± 1 − em n |ψ n }, yielding G c l (ρ) = log n − q n log e me . Proof. The above decompositions provide trivial upper bounds (note that in the second case Λ 2 ( √ q|ψ m ± √ 1 − q|ψ n ) = q m follows from q m > q(1−q) m ≥ e(1−q) n > 1−q n , and in the third case Λ 2 ( em n |ψ m ± 1 − em n |ψ n ) = e n follows from e n > 1 n ≥ 1 n (1 − em n )). Below we show that these are also lower bounds. Let {P i , |Ψ i } be some optimal decomposition for G c l (ρ). According to (19), there exists a unitary u ij , so that First, we rule out nx i = my i ∀i by showing that G c l (ρ) = i P i log(mP i /x i ) = i P i log(m + n) = log(m + n) surpasses the upper bounds outlined above: For all m n ≤ 1 we have log(m + n) > log n ≥ q log m + (1 − q) log n, as well as log(m + n) > log n ≥ log n − q n log e me . Furthermore, for m n < 1 e and q ≥ em n we have log(m + n) > log n > log( mn me ) ≥ log( m q ). In the following we therefore assume that nx i = my i holds for at least one i ∈ [1, r].
Without loss of generality we assume where the inequality follows from applying Lemma 8 to each of the two sums. In Lemma 9 of Appendix C we show that the last line of (33) is an upper bound to the values outlined in the proposition. Hence, the upper and lower bounds coincide. This completes the proof.
Note that for qutrits (m + n = 3) Proposition 5 simplifies to Proposition 4, because 1 e < 1 2 ≤ m n . From the symmetry of ρ = q|ψ m ψ m | + (1 − q)|ψ n ψ n |, it can be seen that Proposition 5 can be extended to the case m n > 1 simply by swapping q and 1 − q. Importantly, Proposition 5 yields necessary and sufficient conditions for rank-2 maximally correlated qudit states to have common optimal decompositions for G c (ρ) and G c l (ρ): Corollary 3. Rank-2 maximally correlated states ρ = q|ψ m ψ m |+(1−q)|ψ n ψ n | have common optimal decompositions for G c (ρ) and G c l (ρ) iff 1 e ≤ m n ≤ e. Proof. According to Lemma 5, the unique optimal decomposition for G c (ρ) is {q, |ψ m ; 1 − q, |ψ n }, up to overall phases. For symmetry reasons it suffices to consider m n ≤ 1. For m n ≥ 1 e the statement immediately follows from Proposition 5. For m n < 1 e it is seen in the proof of Lemma 9 that for all q ∈ (0, 1) the minimum of f (h, s) is strictly smaller than f (0, 0) = q log m + (1 − q) log n. Therefore, {q, |ψ m ; 1 − q, |ψ n } cannot be an optimal decomposition of G c l (ρ) for m n < 1 e . Let us sum up the preceding findings. Theorem 1 gives necessary and sufficient conditions for G c l (ρ) = G f l (ρ). Apart from the trivial classes of pure states and separable states, this includes all two-qubit states (Proposition 2), and all isotropic states (Proposition 3). Further examples are the 4-qubit Smolin state, and multiqubit Dür states, for whom G c l (ρ) = − log 2 (1 − G c (ρ)) can be easily verified from [39].
According to Lemma 5, G c l (ρ) > G f l (ρ) holds for maximally correlated states iff there are two states |Θ i and |Θ j in the optimal decomposition of G c (ρ) that are not equally entangled. This is the case for the two-qutrit states of Proposition 4. Despite this, Proposition 4 shows that G c (ρ) and G c l (ρ) still have common optimal decompositions, i.e. item 3(a) of Theorem 1 can be true, while item 1 is false. In this case, the first inequality of (27) is an equality, while the second inequality is strict.
Optimal decompositions of two-qutrit isotropic states were found in [56] for the entanglement of formation, a convex roof-based entanglement measure. In all of these optimal decompositions some of the pure states are not equally entangled (although it is unknown whether [56] exhausts all optimal decompositions for two-qutrit isotropic states). Using Lemma 5, one can easily construct a state ρ in whose optimal decomposition {P i , |Ψ i } for G c (ρ) any two states |Ψ i and |Ψ j are not equally entangled. In some sense, this is a stronger result than the one of [56].
Next, we address the question whether G c l has the same ordering as G f/c or The two-qubit case is trivial, because of G c l ≡ G f l (cf. Proposition 2), so we need to consider higher-dimensional systems. 3 , but G c l (ρ) < G c l (σ). We remark that Corollary 4 can be easily verified for a much wider range of systems, e.g. all bipartite d-level systems with d ≥ 4, by considering any rank-2 maximally correlated state ρ belonging to the second class outlined in Proposition 5, together with a suitably chosen isotropic state σ, yielding G f/c (ρ) < G f/c (σ) and G c l (ρ) > G c l (σ).

4.2.
Comparison between G m /G m l and G c /G c l . In contrast to the convex roofbased extensions, G m and G m l are demonstrably not convex, and they attain their maximum for the maximally mixed state. It is therefore intuitive to expect that G m (ρ) ≥ G c (ρ) and G m l (ρ) ≥ G c l (ρ) hold for all ρ. To prove these statements, we need the following lemma. Lemma 6. Let ρ, |ϕ be two arbitrary states, and ϕ|ρ|ϕ = g. Then, there exists a decomposition ρ = r i=1 p i |ψ i ψ i |, such that r = rank ρ, and | ϕ|ψ i | 2 = g for all i.
Denoting the corresponding {q i , |φ i } as {Q i , |Φ i }, we see that the state is a state of rank r that satisfies ϕ|ρ 1 |ϕ = g. Using the induction assumption on This completes the proof. Proof. Let |ϕ ∈ PRO be a closest product state of ρ in accordance with (20), and let rank ρ = r. By virtue of Lemma 6, there exists a decomposition ρ = where the third equality follows from the fact that the | ϕ|Ψ i | have the same value for all i.
Proof. Using Lemma 2, Theorem 2 and Lemma 3, we obtain G m (ρ) The following Theorem 3 establishes necessary and sufficient conditions for G m l (ρ) = G c l (ρ) in form of a straightforward relationship between the CPS for Λ 2 m (ρ) and the optimal decomposition of G c l (ρ). Hence, this theorem bears resemblance to Theorem 1, as well as to Proposition 5 of [24].
Theorem 3. For any state ρ the following two conditions are equivalent: Proof. -1. ⇒ 2.: Because of G c l (ρ) = G m l (ρ), the two inequalities in (34) must become equalities, from which it follows that for every CPS |φ m there exists a decomposition {P i , |Ψ i } of ρ which is optimal for G c l (ρ) and for which In Section 4.5 this theorem will be demonstrated by means of the maximally correlated states. With regard to the linear measures, it will be shown in Theorem 7 that G m (ρ) = G f/c (ρ) holds iff ρ is pure.

Comparison between G t and E T .
Theorem 5. G t (ρ) ≤ G m (ρ) holds for all states ρ.
Proof. Let ρ = i p i |ψ i ψ i | be an arbitrary decomposition of ρ. Then, where the inequality follows from the convexity of D 2 T .

Inequalities and hierarchies.
Using the results from the preceding sections, we find the following inequality chains that include all the GM definitions considered in this paper. These inequality hierarchies are summarized and visualized in Figure 1.
Theorem 6. The following inequalities hold for all states ρ: Proof. -1: The first inequality was shown in Section 3.2.4, and the second one in Theorem 5.
-2: The first inequality was shown in Theorem 4, and the second one in Corollary 5.
-3: The first inequality follows from Lemma 2, and the second one was shown in [22,23].
-4: The first inequality follows from Lemma 2, the second was shown in (24), and the third in Theorem 2.
G f/c (ρ 1 ) = 0, and from item 2 of Lemma 5 it follows that G f/c (ρ 2 ) = 1−q 2 . By choosing |φ = |00 in (25), we have G t (ρ 2 ) ≤ (1 − q) 2 . So, G t (ρ 2 ) < G f/c (ρ 2 ) for q ∈ ( 1 2 , 1). Hence, G t ≷ G f/c . From Theorem 6 and Figure 1 we see that G m and G m l are upper bounds for all linear and logarithmic GM definitions, respectively. Since Λ 2 m (ρ) can be computed for many prominent states (see e.g. [8]), these upper bounds are readily accessible. For pure states the bounds are strict, and the more mixed a given state ρ is, the weaker the bounds are.
4.5. Partitioning of state space. As shown in the previous subsection, the inequalities hold for all states. Here we will see that these inequalities provide a physically meaningful partitioning of state space. For this we first prove that the measures in the above inequalities coincide iff ρ is pure.
According to Theorem 7, G m (or G m l ) coincides with the proper entanglement measure G f/c (or G f l ) only for pure states, thus further reinforcing the observation that Λ 2 m (ρ) assesses the entanglement as well as the mixedness of a state ρ. Together with the known fact that Λ 2 f (ρ) < 1 iff ρ is entangled, we can use (35) to partition the state space into four subsets, S(H) = A∪B∪C∪D, corresponding to pure separable, pure entangled, mixed separable and mixed entangled states, respectively. As shown in Table 1 and Figure 2, this partitioning is done by determining whether the inequalities in (35) are strict or become equalities. The inequalities (35) between the logarithmic measures can also be used for partitioning S(H), and in that case, the subset of mixed entangled states is further divided into three subsets, . Since G f l (ρ) = G c l (ρ) holds for isotropic states and all 2 qubit states, these states belong to the set D 2 . In particular, for the special case of 2 qubits (H = C 2 ⊗ C 2 ) we have D = D 2 , i.e. D 1 and D 3 are empty. From this one could conjecture that generic mixed entangled states belong to D 2 . However, for maximally correlated states, it is clear from item 5 of Lemma 5 that most states do not belong to D 2 . The following theorem elucidates the relationship between the parameters of rank-two maximally correlated states and the subgroups D 1 , The set a state ρ belongs to is uniquely determined by whether the inequalities in (35) are strict or equalities. By means of Λ 2 f (ρ) and Λ 2 m (ρ), the same partitioning can be facilitated by the logarithmic quantities G f l and G m l . When taking G c l into account, as well, the set of genuinely mixed entangled states D = D 1 ∪ D 2 ∪ D 3 is further subdivided into three pairwise disjoint sets. D 2 and D 3 . For this we note that Λ 2 m (ρ) can be easily calculated for maximally correlated states of the form in Proposition 5 as Λ 2 m (ρ) = max{ q m , 1−q n }. Theorem 8. Depending on the value of the parameters m, n ∈ N with m n ≤ 1 and q ∈ (0, 1), the rank-two maximally correlated states of Proposition 5 belong to either of the three subsets of genuinely mixed entangled states: : for all other parameter values.
Proof. From item 5 of Lemma 5 it follows that G f l (ρ) = G c l (ρ) iff m = n. Hence, states belong to D 2 iff m n = 1. For m n < 1 one can distinguish between ρ ∈ D 1 and ρ ∈ D 3 by determining whether G c l (ρ) ≤ G m l (ρ) is strict or not. Recall that G m l (ρ) = min{log( m q ), log( n 1−q )}. If q ≥ em n , then m q ≤ n e < n 1−q , yielding G m l (ρ) = log( m q ). Therefore, if m n < 1 e and q ≥ em n (second case of Proposition 5), then G c l (ρ) = G m l (ρ) = log( m q ), yielding ρ ∈ D 3 . Regarding the first and third case of Proposition 5, it is seen from its proof (including Lemma 9 and its proof) that for parameter values in the interior of the domain (i.e. excluding m n = 1) the value of G c l (ρ) is strictly smaller than both log( m q ) and log( n 1−q ). Therefore, G c l (ρ) < G m l (ρ) = min{log( m q ), log( n 1−q )} holds, so ρ ∈ D 1 .
From Theorem 8 we see that the states belonging to D 3 precisely coincide with those outlined in the second case of Proposition 5. This allows us to demonstrate Theorem 3: Let ρ ∈ D 3 be a rank-two maximally correlated state. Then every CPS |φ m of Λ 2 m (ρ) necessarily has the form |φ m = ( Using this decomposition, we obtain Λ 2 m (ρ) = Λ 2 (|Ψ ± ) = | φ m |Ψ ± | 2 = q m , thus verifying Theorem 3.
For maximally correlated qutrit states the only possible value of m n is 1 2 , so all states lie in D 1 . In contrast to this, for four levels (d = 4), there are rank-two maximally correlated states in each of the three sets D 1 , D 2 and D 3 .
In the following we determine whether the various subsets of the genuinely mixed states, C ∪ D = C ∪ D 1 ∪ D 2 ∪ D 3 , are convex or not. C and C ∪ D are clearly convex sets, while D is not. All other subsets are investigated in the following lemma.
Proposition 7. The following sets are not convex: and C ∪ D 2 ∪ D 3 , at least one of the two sets is not convex.
Although partially answered by Proposition 7, it is still unknown whether C ∪ D 1 ∪ D 2 or C ∪ D 2 ∪ D 3 are convex or not.

Graph states and cluster states
Graph states are an important class of states for quantum information [57]. A subset of them, the cluster states, are the central ingredient for one-way quantum computation [58]. Here we show that a large class of graph states, including all cluster states, have a "universal" closest separable state that minimizes several inequivalent distance measures. This property helps to prove the previous Corollary 1.
Consider a general pure graph state |G with underlying graph G = (V, E), where V is the set of vertices and E is the set of edges. The maximum independent set α is the largest possible set of non-adjacent vertices, and the minimum vertex cover β is the complement of α, i.e. α + β = V . The minimum vertex cover can be thought of as the minimal set of qubits that needs to be measured in the computational basis to completely disentangle the graph state.
As outlined in [38], the stabilizer S of |G is generated by n generators {g j } n j=1 , and these generators stabilize a unique state, namely |G . If one or more of the generators from the generating set of S are discarded, the smaller set generates a new Abelian group S which now stabilizes a set of states {|ψ i } rather than a unique |G . Depending on the structure of the generating set of S , the states {|ψ i } may or may not be entangled. In [38] it is shown that the optimal way of discarding generators, such that the stabilized states {|ψ i } are product states, is to discard generators corresponding to the vertices of the minimum vertex cover. So, if we only keep the generators corresponding to the maximum independent set {g j |j ∈ α}, denoting the correspondingly generated Abelian group as S α , the states it stabilizes {|ψ α i } are all product states. These states form the basis vectors used below.
Ignoring possible negative amplitudes that are not important here, |G can be written as an equal superposition of the basis vectors, where D α is the number of states |ψ α i and is related to the cardinality of the minimum vertex cover as D α = 2 |β| . In other words, for each generator discarded from the generating set, the size of the set of stabilized states doubles. The decomposition (38) is of minimal rank for graph states whose underlying graphs satisfy certain conditions [38]. This is the case for all bipartite (2-colorable) graphs, which includes all cluster states of arbitrary size and dimension. However, there also exist many non-2-colorable states that satisfy the conditions.
In the following we assume that |G satisfies the conditions, i.e. the decomposition (38) is of minimal rank. Since the |ψ α i are product states, it immediately follows from (38) that any of the |ψ α i is a CPS, i.e. Λ 2 (|G ) = | G|ψ α i | 2 = D −1 α = 2 −|β| . Correspondingly, the separable state was found to be a CSS for the REE [38]. Here we show that it is also the CSS in terms of the Bures distance (5) and the trace distance (3).
Theorem 9. Let |G be a graph state of the form (38) with minimal rank. Then, (39) is a closest separable state with respect to the quantum relative entropy, the Bures distance, and the trace distance.
Next, consider the trace distance. The inequality min σ∈SEP D T (|G , σ) ≥ 1 − max σ∈SEP F 2 (|G , σ) = 1 − D −1 α follows from (8b). To show that δ minimizes the trace distance, it therefore suffices to show that D T (|G , δ) = 1 − D −1 α . From Theorem 9 applies to all cluster states. In particular, for n qubit cluster states |C n the minimum vertex cover has the size |β| = n 2 , yielding the cardinality D α = 2 n 2 . For even n, this yields Λ 2 (|C n ) = 2 − n 2 and D T (|C n , δ) = 1 − 2 − n 2 . For graph states that do not satisfy the minimal rank condition, the state (39) generally does not minimize the three distance measures, but it nevertheless yields upper bounds on the distances and on the corresponding entanglement measures, the REE, the BE, and TE.

Conclusion
In this paper we reviewed and studied seven different definitions of GM for arbitrary multipartite systems. Five of these are known (G c , G f , G c l , G m , G m l ), one has previously received only little interest (G f l ), and one has not been studied before (G t ). The entanglement axioms of the measures were investigated, and are summarized in Table 2. A remaining open question is whether G c l satisfies weak monotonicity, something we showed to be true at least for two-qubit states and isotropic states. A complete quantitative hierarchy between the measures was derived (shown in Figure 1), and it was found that this hierarchy can be employed to partition the state space into pairwise disjoint sets with clear physical properties (pure vs. mixed, and separable vs. entangled). This is summarized in Table 1 and Figure 2.
As a byproduct of Corollary 1, we found that for pure input states ρ = |ψ ψ| the trace distance D T (|ψ , ·) has in general no pure CSS. This is in stark contrast to the Bures distance, for which (14) implies that D B (|ψ , ·) always has at least one pure CSS. It is therefore not trivial to find states for whom the Bures and trace distance have a common CSS, something we did for a large class of graph states in Theorem 9.
With regard to the convex roof-based measures G c and G c l , it was found that -unlike G f and G f l , or G m and G m l -these two measures are not simple functions of each other, and in fact do not even have the same ordering. Nevertheless, some connections between the two measures and their optimal decompositions could be made (Lemma 3, Theorem 1, Corollary 2). For this, the maximally correlated states were particularly helpful, because their optimal decompositions for G c l depend qualitatively on their parameters (Lemma 5, Proposition 4, Proposition 5). This way it could be shown in Corollary 3 that for some states G c and G c Proof. Because of log[n(1 + yi xi )] = log n + log(1 + yi xi ), it suffices to consider n = 1. Eq. (44) clearly holds for k = 1, so we assume k ≥ 2. First, we consider the x i to be fixed, with their values denoted as {x 1 , . . . , x k }. Then, the function f (y 1 , . . . , y k ) := is a function of k variables. We use this function to define the k functions of k − 1 variables, g j (y 1 , . . . , y j−1 , y j+1 , . . . , y k ) := f (y 1 , . . . , y j−1 , Y j , y j+1 , . . . , y k ) , with j ∈ [1, k], and where Y j is short-hand for Y j := Y − i =j y i . Obviously, if (45) is minimized at (y 1 , . . . , y k ) under the condition k i=1 y i = Y , then (46) is minimized at (y 1 , . . . , y j−1 , y j+1 , . . . , y k ). In particular, all partial derivatives must vanish at this point, and from (45) and (46) it follows that ∂g j ∂y l = log x j x l + x j y l x j x l + Y j x l , for all j, l ∈ [1, k] with j = l .
To rule out boundary points as solutions, we calculate the second derivatives, and find that they are strictly positive. From this it follows that (47) can be zero at only one point, and that the unique minimum is reached there. Since x j y l = Y j x l results in ∂gj ∂y l = 0, we obtain x j y l = y j x l for all j, l ∈ [1, k] with j = l. From this it follows that x j ( i y i ) = y j ( i x i ), and thus y j x j = Y X for all j ∈ [1, k]. Inserting this in the l.h.s. of (44) yields the r.h.s. of (44), regardless of the initial choice of the fixed {x 1 , . . . , x k }. This completes the proof. For m n ≥ 1 e , we have f h (h) ≥ 0, hence f h attains its minimum at f h (0) = f (0, 0). For m n < 1 e and q ≥ em n , we have f h (h) ≤ 0, hence the minimum is f h ( 1−q q ) = log( m q ). On the other hand, for m n < 1 e and q < em n , then f h ( n em − 1) = 0, so the minimum of f h is f h ( n em − 1) = log n − q n log e me .