The invariant-comb approach and its relation to the balancedness of multipartite entangled states

The invariant-comb approach is a method to construct entanglement measures for multipartite systems of qubits. The essential step is the construction of an antilinear operator that we call {\em comb} in reference to the {\em hairy-ball theorem}. An appealing feature of this approach is that for qubits (or spins 1/2) the combs are automatically invariant under $SL(2,\CC)$, which implies that the obtained invariants are entanglement monotones by construction. By asking which property of a state determines whether or not it is detected by a polynomial $SL(2,\CC)$ invariant we find that it is the presence of a {\em balanced part} that persists under local unitary transformations. We present a detailed analysis for the maximally entangled states detected by such polynomial invariants, which leads to the concept of {\em irreducibly balanced} states. The latter indicates a tight connection with SLOCC classifications of qubit entanglement. \\ Combs may also help to define measures for multipartite entanglement of higher-dimensional subsystems. However, for higher spins there are many independent combs such that it is non-trivial to find an invariant one. By restricting the allowed local operations to rotations of the coordinate system (i.e. again to the $SL(2,\CC)$) we manage to define a unique extension of the concurrence to general half-integer spin with an analytic convex-roof expression for mixed states.


I. INTRODUCTION
Entanglement is one the most counterintuitive features of quantum mechanics [1] of which there is only rather incomplete knowledge until now. We will define a quantum mechanical state of distinguishable particles as having no global entanglement with respect to a given partition P of the system if and only if (iff) it can be written as a tensor product of the parts of some subpartition of it; a state of indistinguishable particles we call not globally entangled iff it can be written as the proper symmetrization due to the particles' statistics of such a tensor product [2]. The many different ways of partitioning a physical system already imply that there are many families of entanglement in multipartite systems or even bipartite systems with many inner degrees of freedom. The concept of entanglement instead remains unaltered. Having agreed upon how to decompose the physical system such that every quantum state can be expressed as a superposition of tensor products of states of its parts, the entanglement of its components follows the definition given above.
In order to be more specific, let H i be the ith local Hilbert space of some partition of the total Hilbert space H = ⊗ i∈I H i . In this case the partition would be P := { H i ; i ∈ I }; If I 1 ∪ I 2 = I, then P sub := { ⊗ i∈I1 H i , ⊗ i∈I2 H i } is a two-elemental subpartition of P. We call an operator on H P-local iff it is a tensor product with respect to the partition P. When it is clear from the context what the partition is, we will just use the term local.
While for two qubits there is only one type of entanglement, it has been noticed rather early that starting from the three-qubit case there is more than one class of entanglement [3]. That is, for more than two parties there are different classes of states which are not interconvertible using only Stochastic Local Operations and Classical Communication (SLOCC) [3][4][5]. Due to this complication, a key question which has not been answered yet in the frame of a general theory is (despite considerable efforts, see e.g., Refs [3,[6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]) how to classify, to detect, and to quantify multipartite pure-state entanglement in a sensible and physically justified way. Nice overviews of the state of the art are given in Refs. [21,22]. Further, an initial analysis of systems of two and three qutrits can be found in Refs. [13,23], and the entanglement sharing properties for qudits have been studied in Ref. [24]. An interesting account on activities with respect to higher local dimension is given in Ref [25]. Several collective multipartite measures for entanglement have been proposed for pure states [26][27][28][29][30][31][32]. Since these approaches have no control about how various classes of entanglement will be weighted in such a measure, the decision in favor of one specific collective entanglement measure is arbitrary unless we gain significantly better understanding of the struture of entanglement itself.
In order to get additional insight, class-specific entanglement measures provide one research line to pursue. As an example of such a measure for three qubits, the three-tangle has been derived [33] as the unique measure that discriminates the two distinct classes of entanglement in three-qubit systems. It separates W states from the genuine class of three-qubit entanglement represented by the Greenberger-Horne-Zeilinger (GHZ) state. As well as its twoqubit counterpart, the concurrence [34][35][36], the three-tangle is a polynomial SL(2, C) invariant. A procedure for the construction of similar class-specific entanglement quantifiers has been developed in Refs. [37,38]. It has been systematically analyzed for four and five qubit systems in Ref. [18].
These measures combine a variety of desirable properties of class-specific quantifiers of global entanglement. Given a specific partition P, a measure for genuine n-tanglement E g : H −→ [0, 1] should satisfy the following requirements (i) E g [Π] = 0 for all pure product states Π, relative to the partition P (ii) invariance under P-local unitary transformations (iii) the entanglement monotone property [4]; i.e. the measure must not increase (on average) under Stochastic Local Operations and Classical Communication (SLOCC) [5] (iv) invariance under permutations of the P-local Hilbert spaces [33] is desirable.
Clearly, the most basic requirement appears to be the condition (i). This is accomplished by what has been termed a filter in Refs. [18,37,38], that is, an operator that "filters out" all product states, in the sense that it has zero expectation value for them. In other words, the filter image of any product state must be orthogonal to the original product state. The filters are built from so-called comb operators for one-and two-copy single qubit states (see below). This approach appears appealing since, interestingly, for qubits it automatically implies SL invariance and thus the monotone property [12]. Consequently, also (ii) and (iii) are satisfied. Nevertheless, already for five qubits there is a large number of such measures (even after imposing condition (iv) [18]) such that one would like to understand their essence more deeply, and to reduce their number on physical grounds. In this paper we analyze what is common to the states that are detected by the SL(2, C) invariant operators. We find in general that only the balanced part of a state is measured, a property that had already been noticed for the three-qubit case in Ref. [33], in terms of a geometrical interpretation. On the other hand it is known that the modulus of a polynomial invariant assumes its maximum on the set of stochastic states [12]. The combination of these characteristics leads us to the concept of irreducibly balanced states. (We would like to mention that Klyachko et al. have discussed maximum multipartite entanglement and its relation to quantification of multipartite entanglement from a different, very interesting perspective, see Refs. [20,39] and references therein.) We study various interesting properties of irreducibly balanced states as we are convinced that their investigation might give further insight into the nature of multipartite entanglement.
As the basis of all this discussion is the invariant-comb approach, it is interesting to ask whether there is any possibility to extend the method to systems with local Hilbert space dimension larger than 2. We discuss some basic aspects of such an extension.
The structure of the paper is as follows. The invariant-comb approach for qubits [37,38] is summarized in section II; section II A introduces the main concepts, notations, and the elements that eventually build up the (filter) invariants. In section II B we exemplarily write down filters for up to six qubits and discuss some elementary properties of them. Section II C is devoted to the notion of states with maximal n-qubit entanglement. Interestingly, a central prerequisite of such maximally entangled states can be connected with the concept of balancedness in section III. In section III A we show that it is precisely the balanced part that is detected by the SL(2, C) invariant (filter) operators. In section III B, this leads us to the definition of irreducibly balanced states and the investigation of their properties. Finally, we discuss the possibilities for an application of the invariant-comb approach to non-qubit systems and general partitions in section IV. First we focus on the entanglement of blocks of qubits (part IV A), while in part IV B a measure for bipartite entanglement, subject to a certain restricted class of local operations, is derived for general half-integer spin pure and mixed states. In the last section, we present our conclusions. The fundamental concept of the method and the basis of the construction of polynomial invariants is the comb. A comb A is a local antilinear operator with zero expectation value for all states of the local Hilbert spaces, that is where C is the complex conjugation in the computational basis |ψ * := C |ψ ≡ C 1 j1,...,jq=1 ψ j1,...,jq |j 1 , . . . , j q = 1 j1,...,jq=1 ψ * j1,...,jq |j 1 , . . . , j q .
We call L i the linear operator associated to the comb A i . For simplicity we abbreviate (throughout this article, there will be no ambiguities whether we mean linear or antilinear expectation values). The requirement to have vanishing expectation values for an arbitrary single qubit state clearly cannot be accomplished by any linear operator (it would be identically zero); but it is amenable to antilinear operators. The idea is to identify a sufficiently large set of combs in order to construct the desired filter operators that satisfy all the requirements (i) -(iv) listed above. It is worth noticing that a filter constructed exclusively from combs automatically is invariant under P-local unitary transformations if the combs are. Even more, it is invariant with respect to the complex extension of the corresponding unitary group, which is isomorphic to the corresponding special linear group. Since the latter mathematically represents the non-projective LOCC operations [3,6] [12].) It is evident that any polynomial invariant can be turned into a linearly homogeneous function of |ψ ψ| by applying the appropriate inverse power. In order to avoid misunderstandings we emphasize that not every function of SL-invariants (which still is an SL invariant) can be an entanglement monotone; it is not even clear whether homogeneity of arbitrary degree implies the monotone property. The main part of this work focuses on multipartite registers of n qubits, i.e., n spins 1/2. Then, the local Hilbert space is H i = C 2 =: h for all i ∈ {1, . . . , n}. We will need the Pauli matrices Since here the local unitary group is SU (2) ⊗n , we only need fully local combs and hence can we restrict ourselves to SU (2) combs. We mention that any tensor product f ({σ µ }) := σ µ1 ⊗ · · · ⊗ σ µn with an odd number N y of σ y is an n-qubit comb. This can be seen immediately from In particular, is a comb acting on a single qubit. We do not know, whether combs acting on more than a single site might be needed to some extent. As yet there is no evidence that they need to be included in order to classify multipartite qubit entanglement. In what follows, comb operators are to be understood as acting on a single site only. We will call A (1) : h → h a comb of order 1. In general we will call a (single-qubit) comb A (n) : h ⊗n → h ⊗n to be of order n. It is worthwhile noticing that the n-fold tensor product h ⊗n on which an n'th order comb acts symbolizes n-fold copies of one single qubit state. In order to distinguish this merely technical introduction of a tensor product of copies of states from the physically motivated tensor product of different qubits we will denote the tensor product of copies with the symbol •, and hence write A (n) : h •n → h •n . When we say expectation value of A (n) = L (n) C we mean the following ψ| •n A (n) |ψ •n =: ((L (n) )) .
Strictly speaking, this is a linear combination of products of expectation values: if ..,jn σ j1 • · · · • σ jn C then the expectation value of A (n) would be There is a unique (up to rescaling) single site comb of order 2, which is orthogonal to the trivial one σ y • σ y (with respect to the Hilbert-Schmidt scalar product): one can verify that for an arbitrary single qubit state (notice the similarity with the Minkowski metric). We will denote this second order comb by For both combs one can demonstrate that they are SL(2, C) invariant [18,38], and hence they satisfy the basic requirements for the construction of filter operators. Please note that any linear combination of combs is again a comb. The two combs A

1/2 and A
(2) 1/2 have the above-mentioned additional important property of being mutually orthogonal with respect to the Hilbert-Schmidt scalar product. Filter invariants for n qubits are obtained as antilinear expectation values of filter operators; the latter are constructed from combs so as to have vanishing expecation value for arbitrary product states. We will use the word "filter" for both the filter operator and its filter invariant. For n-qubit filters we will use the symbol F (n) .
We start by writing down some filters for two qubits Both forms are explicitly permutation invariant, and they are filters. Indeed, if the state is a product, the combs lead to a vanishing expectation value. We obtain the pure state concurrence from them in two different, equivalent forms: Now we go ahead to three qubits and write down a selection of three-qubit filters •σ y σ ν σ y • σ y σ y σ λ )) The last two are evidently permutation invariant, but also the first filter is invariant under permutations. All coincide with the three-tangle [33] (or powers thereof) Interestingly, all two-qubit filters are homogeneous polynomials of the concurrence and in the same way all three-qubit filters coincide with polynomials of the three-tangle. This is due to the fact that both concurrence and three-tangle generate the whole algebra of polynomial SL(2, C) ⊗2 and SL(2, C) ⊗3 invariants, respectively (see e.g. Ref. [18] and references therein).

B. Filters for 4 and more qubits
In this section we will present explicitly a list of filters for systems of four and five qubits. By means of the sixqubit example we sketch a straightforward procedure to construct filters for larger systems. In order to get compact formulas, the tensor product symbol ⊗ will be omitted, as in eq. (8).
For 4 qubits, the whole filter ideal in the ring of polynomial SL invariants is generated by [18] For five qubits, we mention where . . . s;a means that the object between brackets is symmetrized/antisymmetrized. Double indices indicate that both symmetrization and antisymmetrization lead to independent generators. These filters can be found in Ref. [18].
Together with[49] P 2 − 5 j=1 D 3 j and the square of an antisymmetric invariant F that is constructed from an Ω-process (see Ref. [40]), they generate the filter invariants for five qubits up to polynomial degree 12.
The following examples for six-qubit filters provide the opportunity to highlight a way to construct filters for higher qubit numbers where in the latter formula all the µ • are to be contracted properly; in the σ • the "•" either have to be substituted by σ y , or by indices which then have to be contracted properly. This suggests that for an n-qubit system the filter property requires at least h •(n−1) , leading to polynomial degree of at least 2(n − 1) for the corresponding polynomial invariant.
We emphasize again that every filter is an SL invariant because the local elements it is constructed from (i.e., the combs) are SL invariant. It is clear that linear combinations and in fact any function of invariants is an invariant (but not necessarily an entanglement monotone, cf. Ref. [12]). By noticing the consequences of including global phases of the states we see that only homogeneous functions of the same degree can be combined linearly.

C. Maximally Entangled States
We will now define our notion of a multipartite state with maximal genuine multipartite entanglement.
Definition II.1 A pure q-qubit state |ψ q has maximal multipartite entanglement, i.e. q-tangle, iff (ia) The state is not a product, i.e. the minimal rank of any reduced density matrix of |ψ q is 2.
(ib) All reduced density matrices of |ψ q with rank 2 (this includes all (q − 1)-site and single-site ones) are maximally mixed within their range.
Further, there is a list of desirable features for maximally multipartite entangled states: (ii) all p-site reduced density matrices of |ψ q , have zero p-tangle; 1 < p < q. This is clearly an implicit requirement in that a check of it would require the knowledge of convex-roof extensions of the relevant multipartite entanglement measures. Furthermore, it is not even clear which q-qubit entanglement families possess a representative for which all tangles for less than q qubits vanish. The q qubit GHZ state is an example that satisfies condition (ii).
(iii) there is a canonical form of any maximally q-tangled state, for which properties (ia) and (ib) are unaffected by relative phases in the amplitudes, i.e. their quality of being maximally entangled is phase insensitive.
All above requirements are invariant under local SL transformations. We briefly discuss the implications of each single requirement. Condition (ia) excludes product states, which are certainly not even globally q-tangled. (ib) implies maximal gain of information when a bit of information is read out. This condition contains the definition of stochastic states in Ref. [12], where it is also proved that every entanglement monotone assumes its maximum on the set of stochastic states. An even more stringent condition has been imposed in Ref. [41], where all reduced density matrices are required to being maximally mixed within their range. Requirements (ia) and (ib) are therefore wellestablished. Constraint (ii) is intriguing by itself: it excludes hybrids of various types of entanglement and thereby follows the idea of entanglement as a resource whose total amount has to be distributed among the possibly different types of entanglement; see e.g. Ref. [33,42]. To our knowledge, it is not clear whether this condition can be regarded as being fundamental, since up to date no extended monogamy relation is found that would substantiate the idea of entanglement distribution (see e.g. [42,43]). We have no striking argument in favor of (iii), except that maximally entangled states for two and three qubits have such a canonical form. We mention, however, that according to Ref. [3] entangled states have a representation with a minimal number of product components; it appears that in this representation the entanglement is not "sensitive" to changes in the relative phases between the components (consider, e.g., the GHZ state). It could be promising to analyze a possible connection to the concept of envariance put forward in Ref. [44].
In order to illustrate the above conditions and to check the existence of such states, we give some examples: The Bell states (|σ, σ ′ ± σ,σ ′ )/ √ 2 are the canonical form of maximally 2-tangled states. By tracing out one qubit one obtains 1 2 1l as the reduced density matrix of the remaining qubit. The 2-tangle is indeed robust against multiplication of the components with arbitrary phases: (|σ, σ ′ + e iφ σ,σ ′ )/ √ 2 is maximally entangled for arbitrary real φ. Condition (ii) is meaningless here. For two qubits, these are all maximally entangled states, and the class of maximally entangled states is represented by |GHZ 2 = 1 √ 2 (|11 + |00 ), which is like the GHZ state but for two qubits. Also the generalized GHZ-state for q qubits, 1 √ 2 (|1 . . . 1 + |0 . . . 0 ), satisfies all the above requirements. It is straightforward to see that the GHZ state is detected by every filter constructed in the way described in the preceding sections. Having a pure state of three qubits, there are two other classes of entangled states: the class represented by 1 , do not contain 3-tangle at all. Indeed, they violate the requirements (ib) and (ii) in Def. (II.1). An apparently different class of maximally 3-tangled states instead can be read off directly from the coordinate expression for the three-tangle [33]: its representative is and satisfies all the above conditions for a maximally 3-tangled state. It is interesting to note that all its reduced twosite density matrices are an equal mixture of two orthogonal Bell states, which thus have zero concurrence. However, this state is unitarily equivalent to a GHZ state by the transformation Summarizing the above examples, we conclude that the set of states that satisfy the conditions in Def. (II.1) is not empty for any number q of qubits and we have one 2-tangled and 3-tangled representative (actually two, which are equivalent for three qubits, though). In what follows we analyze the above conditions and prove that there are at least q − 1 inequivalent q-tangled representatives.
for the first order comb. The second order comb is the sum of the following outcomes The remaining indices of the quantum state are kept fixed for the moment. Summing up these three terms and performing the sum over j, we get It is seen from this result that in order to give a non-zero outcome, every component i must come with the flipped componentī. This is what we will call a balanced qubit component. Since the above consideration has to be extended to all qubits, we conclude that the filter has contributions only from the balanced part of a state. We can say even more: the homogenous degree of the filter must fit with the length of the balanced parts in the state, i.e. the number of product states in the computational basis needed for this balanced part. As a consequence, the way the filter is constructed already implies valuable information about which type of state the filter can possibly detect. This further underpins the relevance of polynomial SL(2, C) ⊗q invariants as far as entanglement classification and quantification of multipartite qubit states are concerned. In particular, we see that a state which can be locally transformed into a normal form without balanced part has zero expectation value for all filters operators (the |W state is a prominent example). This analysis suggests in-depth investigation of states with balanced parts in their pure-state decomposition into the computational (product) basis. It is worth noticing, that it is not conclusive to look at some given pure state and see whether it has a balanced part or not. In fact, every pure state has a balanced part after a suitable choice of local basis. The concept becomes useful only modulo local unitary transformations. Then, two qualitatively different classes of states occur: • states that are unitarily equivalent to a form without a balanced part, • states for which arbitrary local unitaries lead to a state with a balanced part.
The latter case naturally splits up into two sub-classes. One is characterized as the reducibly balanced case in the sense that distinct smaller balanced parts always exist. The complementary situation is the irreducibly balanced case.
It is clear that maximal entanglement as measured by some polynomial SL-invariant is achieved when no unbalanced residue is present, i.e. when the state is balanced as a whole. Indeed, we will show that stochasticity of a state implies balancedness; against the background of the finding in Ref. [12] that every entanglement monotone assumes its maximum on the set of stochastic states, this underpins a tight connection between balanced states and the notion of maximal (multipartite) entanglement.

B. Irreducibly Balanced States
For analyzing the first two conditions (ia) and (ib) in Def. II.1 it is convenient to express a pure state i w i |i as an array; the first row of this array contains the amplitudes w i , p i := |w i | 2 . The column below each amplitude is the binary sequence of the corresponding product basis state. For example For the moment, we will not pay much attention to the weights p i := |w i | 2 , they will become important later on (cf. Theorem III.3). In the following, we define two types of matrices which are based on this array representation of a state. It will turn out that these matrices are quite helpful in the discussion of the properties of balanced states. The proofs of some of the theorems will be rather straightforward by using this representation.

Definition III.1 (alternating and binary matrix)
We call binary matrix B |ψ of the state |ψ the matrix of binary sequences below the amplitude vector and equivalently we call alternating matrix A |ψ of the state |ψ the matrix obtained from its binary matrix, when all zeros are replaced by −1. It will be useful to allow for multiple repetition of certain columns. This means of course that the alternating and binary matrix will not be unique. The minimal form without repetitions is unique modulo permutations of the columns and qubits. We define the length of a state as the minimal number of elements of the standard product basis that occur in the state (without repetition of columns), i.e. the number of columns of the minimal form.
In the above example we have and the length of this state is 4. A |ψ and B |ψ are q × L matrices where L is the number of basis states required for its representation.
Definition III.2 (irreducibly balanced states) 1. We call a pure state |ψ (entirely) balanced iff in each row of B |ψ there are as many ones as zeros (allowing for multiple occurrence of some of its columns), or equivalently, iff the elements of each row of A |ψ sum up to zero (including multiplicities as for the binary matrix). This can be expressed as where A |ψ ∈ Z Z q×L , i.e. qubit number q and length L. We furthermore call a balanced state irreducible or irreducibly balanced, iff it cannot be split into different smaller balanced parts (i.e., iff there is no subset of less than L columns that is already balanced). In contrast, a balanced state which can be split into different smaller balanced parts, will be called reducible. 2. We call a pure state |ψ partly balanced (i.e. it has a balanced part) if (32) is satisfied for n j ∈ IN only if some n j = 0 (but not all of them). A partly balanced state is called reducible/irreducible iff its balanced part is reducible/irreducible.
As an example, we give the B |ψ matrix for a reducibly balanced 3-qubit state

Definition III.3 (completely unbalanced states)
We call a state completely unbalanced if it is locally unitarily equivalent to a state without balanced part.
Please note that the maximally q-tangled states for q = 2, 3 are irreducibly balanced, and it can be straightforwardly verified that they are the only ones for these cases. The W-states are completely unbalanced. Further examples of completely unbalanced states are fully factorizing states. Therefore it is clear that complete unbalancedness can occur for both globally entangled states and completely disentangled states; it therefore is an indicator only as far as genuine multipartite entanglement (i.e., q-qubit entanglement in q-qubit states) is concerned.
Theorem III.1 Product states are not irreducibly balanced.

Proof:
First we observe that a product state is balanced iff its factors are. Let the state be |Φ ⊗ |Ψ which have n and m components, respectively, i.e. |Φ = with length mn divided into n blocks, n > m without loss of generality. Note that m, n are even because |Φ and |Ψ are assumed to be balanced. Consequently, the smallest common multiple of m and n is smaller than or equal to mn 2 . That is, there do exist f, g ∈ N relatively prime such that f m = gn and g ≤ m 2 , f ≤ n 2 . Now we choose from each of the n blocks ( corresponding to the state |Φ i ⊗ |Ψ , i = 1, . . . , n ) g states such that from the first f blocks we choose the first state, from blocks f + 1 up to 2f we choose the 2nd state, ..., from blocks (m − 1)f + 1 modulo n up to m · f modulo n we choose the mth state. This state is balanced and has length m · f ≤ mn/2. This proves that any product state -if balanced -is reducible. q.e.d.
It is important to emphasize that every state can be made balanced by local unitary transformations, raising in general the number of components in the state.
Theorem III.2 Every balanced q-qubit state with length larger than q + 1 is reducible.

Proof:
Balancedness of the state implies the existence of n 1 , . . . , n L such that L j=1 n j (A |ψ ) ij = 0. Irreducibility implies that no subset K of L := {1, . . . , L} exists with K ∩ L = L such that j∈K m j (A |ψ ) ij = 0 for some positive integers m j . Without loss of generality A |ψ has rank q. In order to fix the idea of the proof, we insert a vertical cut in the matrix A |ψ such that both parts contain at least (q + 1)/2 colums. This means to introduce two disjunct non-empty sets K and K ′ := L \ K with |K|, |K ′ | ≥ q+1 2 . We define α K =: (α K 1 , . . . , α K q ) such that α K i = j∈K n j (A |ψ ) ij . Irreducibility implies α K = 0. We now split nonempty sets κ ⊂ K and κ ′ ⊂ K ′ off the subsets K and K ′ and defineK := (K \ κ) ∪ κ ′ . Then, including arbitrary non-negative integers m j , j ∈ κ ′ , and defining m j = n j for j ∈ K, we find Irreducibility then implies that for all such subsets K and κ no integer numbersm j (m j can be also negative or zero) do exist such that m ∈ Z Z |κ|+|κ ′ | satisfies the condition j∈κ∪κ ′mj (A |ψ ) ij = α K i for all i ∈ {1, . . . , q}. Without loss of generality we can assume that (A |ψ ) i∈{1,...,q};j∈κ∪κ ′ has rank q (a suitable choice of K and κ guarantees that). This implies that the condition can be satisfied for every integer vector α K even in Z Z q , hence contradicting our assumption of irreducibility. q.e.d.
A comment is in order here. It must be stressed that the integers entering the balancedness condition must be positive. Therefore, linear dependence of the column vectors does not imply balancedness. In fact, not every q qubit state with more than q + 1 product state components is balanced. The reason is that the set of positive integers is not a field. Our proof however nicely highlights that the balancedness condition itself bridges this gap and provides a mapping onto a set of linear equations over the field Z Z. Thus, for balanced states the argument of linear independence can indeed be used. The state being irreducibly balanced thus implies that the rank of its corresponding alternating (q × L)-matrix (q rows and L columns) is equal to L − 1. Since its maximal rank is min{q, L}, this implies L ≤ q + 1. A canonical form of such an irreducibly balanced state thus becomes ¿From this canonical form, further such states (except the GHZ state) can be generated by duplicating rows, NOToperations on certain bits, and permutation of rows, i.e. of bits. We mention that in Ref. [42], the procedure of duplicating rows has been termed telescoping; it was used to generate certain multipartite entangled states that obey a monogamy relation.

Proof:
For the proof, let us consider an arbitrary q-qubit state |ψ that satisfies the conditions (ia), (ib) -and maybe (iii) -in Def. (II.1) and trace out everything but the first qubit. We can write the array of the state |ψ as follows Now assume that some of the states |Φ i coincide with some of the |Φ ′ i , and call |Ψ their superposition with corresponding weights of the right hand side; the corresponding superposition of the Φ ′ i can be written as Ψ = α |Ψ + β |Ψ ⊥ , |α| 2 + |β| 2 = 1; Ψ |Ψ =:x. Note that these states are not normalized to one.

Note that balancedness means
where 1 := (1, . . . , 1). Equation (36) has a unique solution with all weights equal iff the state is irreducible [50]. Otherwise the state is reducible and all states in each irreducible block B b have the same weight p b . This corresponds to a superposition of irreducibly balanced states. Phase insensitivity however turns out to be incompatible with more than one block except when all the states |Φ i , Φ ′ j were perpendicular to each other. This means that the superposed irreducibly balanced states must be orthogonal to each other. Tracing out only one qubit (including possible telescope copies of it) gives exactly the same condition (35). q.e.d.
It is worth noticing at this point that with local operations on q qubits, the maximum number of free phases is q + 1 (including a global phase), which coincides with the maximum length of an irreducibly balanced block. Therefore the only remaining reducible states which could be maximally entangled by virtue of the demanded phase insensitivity are to be superpositions of irreducible ones with total length not larger than that of the irreducible state of maximal length.
We want to remark that we can -without loss of generality -shrink all states such that no telescope bits occur; the shrinking does not affect the reducibility.
The above observations eventually lead to the following set of maximally entangled q-qubit states of maximal length where |i denotes the state with all bits zero except the i-th, which is one. The maximally entangled state of minimal length is always the GHZ state. States of intermediate length are obtained from those with maximal length for p qubits (p < q) by means of telescoping[51]. It is worth mentioning that irreducibility does not trivially imply the form (33). An example for such a state of five qubits is Theorem III.4 Every irreducibly balanced state is equivalent under local filtering operations SL ⊗q (LFO) to a stochastic state.

Proof:
The proof goes by construction. Let a j , j = 1, . . . , L be the amplitudes of the product state written in the i-th column of B |ψ , and let us consider the LFO's with t i =: t zi for some real positive t and complex z i ; i = 1, . . . , q. Without loss of generality let the multiplicities n j = 1 for all (at most q + 1) j (differring multiplicities could be absorbed in the p i ). We must then show that after suitable LFO's all amplitudes are equal. After this transformation, the amplitude of the product states (i.e. of the column vectors) would become Balancedness implies L j=1 t j = 1. Without loss of generality let (B |ψ ) i,1 = 0 for all i = 1, . . . , q. Dividing by t 1 the amplitudes become a j t q i=1 2zi(B |ψ )ij . Stochasticity requires that all amplitudes have to be equal (up to a phase) and leads to the set of linear equations Since L ≤ q + 1 and B |ψ has rank L − 1, a solution vector (z 1 , . . . , z q ) exists for arbitray a j = 0. The resulting pure state is stochastic. q.e.d.
The fact that every irreducibly balanced state is SL-equivalent to a stochastic state, in combination with the negation of theorem III.3, leads to the following Corollary III.1 An irreducibly balanced state is locally unitarily inequivalent to every state without balanced part. In other words, irreducibly balanced states are not completely unbalanced.
In the light of the fact that the minimal number of orthogonal product states in which a pure quantum state can be represented is invariant under SL(2, C) transformations [3], the following property of irreducibly balanced states is important.
Theorem III.5 For q > 3 qubits, irreducibly balanced states descending from Eq. (38) are minimal in the sense that an irreducibly balanced state of length L can not be represented as a superposition of less than L states of a computational basis (i.e., elements of a completely factorized basis).

Proof:
The irreducibly balanced state |X q of q qubits (cf. eq. (38)) has length q + 1 and its (q − 1) qubit reduced density matrix is spanned by a generalized (q − 1) qubit GHZ and W state; for q > 3 it has no product state in its range. The minimal lengths of the (q − 1) qubit GHZ and W are 2 and (q − 1) respectively, and hence they are different for q = 3 (this implies that they are SLOCC-inequivalent [3]). It can be shown that the possibility to express |X q as a superposition of less than q + 1 computational basis states implies that there must be a product state in the range of the (q − 1) qubit reduced density matrix, which leads to a contradiction. This inductively proofs the minimality of all irreducibly balanced states as defined before. q.e.d.
It is an important step now to realize the following Theorem III.6 All irreducibly balanced states belong to the SLOCC non-zero class, i.e. they are robust against infinitely many LFO's SL ⊗q and therefore possess a finite normal form [12]. As a consequence, they are maximally entangled states according to definition II.1 (also in the sense of Ref. [12]).

Proof:
Those transformations that go beyond SU (2) are essentially the LFO's of the form when expressed in a suitable local basis for the i-th qubit. Now assume the existence of LFO's that scale the state down to zero after infinitely many applications -we will call this the zero-class assumption. Defining a set of real numbers p i ∈ R, i = 1, . . . , q such that t i =: t pi with t > 1 (without loss of generality), the action of this single LFO rescales the weight of the j-th column of the alternating matrix A |ψ by the factor where L is the length of the state of q qubits and the negativity of all the s j expresses the zero-class assumption. [52] This is equivalent to for all j ∈ {1, . . . , L}. Now we make use of the balancedness of the state, meaning that A i,L = − L−1 j=1 A i,j for all i ∈ {1, . . . , q} and that (44) must apply to all j ∈ {1, . . . , L − 1} by virtue of our zero-class assumption. Consequently Thus, at least one positive scaling exponent must exist. This contradicts our initial assumption. Now it is crucial that for irreducibly balanced states no basis exists in which the state has no balanced part (theorem III.1). This completes the proof. q.e.d.
The same applies to q qubit states which are superpositions of orthogonal irreducibly balanced states with length smaller than q + 2.
As a consequence, there must exist independent entanglement monotones which attribute to each of these states a non-zero value, and which can distinguish between them. Equivalently, all completely unbalanced states belong to the SLOCC zero-class. An example is the class of W states for arbitrary number of qubits, but also products of states where at least one of the factors is part of the corresponding SLOCC zero-class. Therefore, every SL(2, C) ⊗q invariant entanglement monotone when applied to such states gives zero.
We now briefly discuss the requirements in definition II.1 for irreducibly balanced states with particular emphasis on condition (ii) of vanishing sub-tangles. The GHZ state satisfies all conditions; the subtangles are all trivially zero, because the reduced density matrices are mixtures of product states. Therefore, they are maximally q-tangled. The states of maximal length behaves analogously, except that tracing out the first qubit, a mixture of a generalized GHZ state and a W state is obtained. For three qubits, the resulting W state is a GHZ (or Bell-) state and the mixture has zero 2-tangle. Also for four qubits one can construct a decomposition of the density matrix whose elements all have zero 3-tangle. For growing number of qubits, the GHZ weight monotonically decreases to zero. However, it can be shown that it contains a subtangle that is detected by certain factorized filters.
It is straightforward to show that the GHZ state is detected by all simple filters (that is those SL(2, C) invariants that are directly created from invariant combs, but not e.g. linear combinations of such invariants).

A. Compound entanglement or block entanglement
The invariant-comb approach also provides suggestions how to possibly extend such an ansatz towards entanglement measures for blocks of spins of variable size. To this end we exploit the fact that each operator with an odd number of σ y is a comb. Furthermore, if two q-qubit filters are identical for pure q-qubit states, but not identical as operators, their difference is a comb. Examples are σ µ σ i • σ µ σ i − σ µ σ j • σ µ σ j , for i = j fixed, and σ µ σ ν • σ µ σ ν − 3σ y σ y • σ y σ y for two qubits and σ µ σ ν σ τ • σ µ σ ν σ τ − 3σ µ σ y σ y • σ µ σ y σ y for three qubits. However, this constitutes just a starting point as this will typically lead to a set of combs on which the local unitary group acts in a non-trivial way; in order to guarantee that a constructed filter is an entanglement monotone, we need an invariant comb. Clearly, abandoning the requirement for the monotone property would open up a vast variety of possible "measures" or "indicators" for entanglement. This, however, is not what we have in mind.
In order to be invariant, it is necessary that the combs are regular and all their eigenvalues must have equal modulus. This is a clear criterion for designing an approach we have in mind. The approach pursued e.g. in [45] has some overlap with concurrence vector approaches (see Refs. [25,46]), which for bipartite systems coincides with the universal state inversion (see Ref. [25] and Ref. [45]). The local antilinear operators used there are not regular and therefore can not be invariant under local SL operations in higher local dimensions.
This opens up a rich and promising field for future investigation. Some insight into the intrications and consequences involved with this requirement is given in the next section on general spin S. It is worth noticing that the concept of balancedness introduced above is tailor-made for qubit systems; it is not appropriate for higher local dimension. The notion of maximal entanglement would need to be modified correspondingly, once such invariant combs have been found.
The operator S y = −i(S + − S − )C is a comb for arbitrary spin S. The crucial difference to the spin-1/2 case is that there are more first-degree combs for S > 1/2 due to the fact that there are non-trivial powers of spin operators up to order 2S > 1 -since (S + ) 2S+1 = 0. It turns out that for S = 1, there is a three-parameter variety of first-degree combs A 1 [a, b, c] := (aS y + (bS x S y + cS y S z + h.c.)) C and a six-parameter variety for spin 3 2 As in the qubit case, every product of spin operators containing an odd number of S y (plus its hermitean conjugate) is a comb. A generalization to general spin S is therefore straight forward: We have S(2S + 1) independent off-diagonal (pure imaginary) entries, which is the dimension of the variety. The corresponding operators are those appearing in (S x + S z ) m ; m = 0, . . . , 2S − 1.
Unfortunately, for integer spin, i.e. for 2S + 1 odd, these combs are not regular. This follows from the hairy ball theorem, stating that in order to have a continuous map from the surface of a d-dimensional sphere onto itself, d has to be even. In our case the surface corresponds to the real part of the normalized Hilbertspace (due to the antilinearity, every comb on the real Hilbert space is a comb on all the Hilbert space). Therefore, for integer spin, one has to look out for a comb of higher order. Unfortuantely, also for half-integer spin no first order SL(2S + 1, C) invariant combs do exist.
In order to make a first step towards higher spins in the spirit of the invariant-comb approach, let us first consider a simplified scenario, where only local rotations are accessible in the laboratory. Then, the group of local operations is the complex extension of the 2S + 1 dimensional representation of SU (2), hence still SL(2, C). We want to stress that this situation differs considerably from that of an arbitrary 2S +1 level system, where the most general local operations are out of the complexified SU (2S + 1), which is the SL(2S + 1, C). For half-integer spin S, the SL(2, C)-invariant comb is obtained as Here, antidiag{λ 1 , . . . , λ n } indicates the n × n matrix with λ 1 , . . . , λ n on the anti-diagonal, e.g. σ y = antidiag{−i, i}. With these combs, we can immediately construct an analogue for the concurrence for arbitrary half-integer spin for which the convex-roof extension procedure from Ref. [47] can be applied, and hence the SL(2, C)-concurrence for general half-integer spin S is It must be stressed that this concurrence is a measure of entanglement under restricted local operations, namely to local rotations of the cartesian axis. The notion of SLOCC is modified correspondingly. Each restricted entanglement class will be subdivided into classes with respect to the full group of local transformations SL(2S + 1, C). We therefore are confident that an analysis of the SL(2, C) invariant concurrence (48) will nevertheless give interesting insight into the entanglement classes for higher local dimensions. It could be interesting to compare these combs with further existing proposals as the universal state inversion [48], which however are constructed for general d-state systems. We leave this investigation for future studies.

V. CONCLUSIONS
In the recent literature, an efficient procedure for the construction of local SL(2, C) ⊗q invariant operators for q qubit wavefunctions has emerged out of the simple requirement to create entanglement indicators that should vanish for all product states (a minimal requirement for a quantity to detect only global entanglement) [37,38]. We call this procedure the invariant-comb approach, because the local building blocks already are SL(2, C) invariant. It is interesting that some definitely globally entangled states as the W state are not detected by any of these polynomial invariants. This motivates the concept of genuine multipartite entanglement in order to distinguish globally entangled quantum states detected by some non-zero polynomial SL invariant from others. The fact that those invariants automatically lead to entanglement monotones has motivated our detailed analysis of the properties of many-qubit states that are detected by the entanglement measures created from invariant combs. We have chosen an approach from two different points of view, with significantly overlapping results: On the one hand, we find that a necessary requirement for a pure quantum state of many qubits to have finite genuine multipartite entanglement is that the state has a balanced part. This balancedness constitutes a continuation of the curious geometric picture of the three-tangle as highlighted in Ref. [33] to a higher number of qubits. On the other hand, also basic necessary requirements for maximal pure-state entanglement, namely that the state has to be stochastic [12,41], are demonstrated here to readily imply balancedness. This curious coincidence justifies a systematic analysis of balanced states. We have extracted the locally SU (2) invariant "nucleus" of balancedness, which is the irreducible balancedness. It is shown that irreducibly balanced states are locally SL invariant to stochastic states, a prerequisite for being maximally entangled. Irreducible balancedness is also shown to exclude the existence of a completely unbalanced form (as e.g. the W state has). This result is essential in that it demonstrates that irreducibly balancedness is a well-defined and valuable concept. Furthermore we could prove that irreducibly balanced states belong to the non-zero SLOCC class of states; hence, they have a non-trivial normal form after local filtering operations [12].
A canonical form for a family of irreducibly balanced states has been found, and this family has the minimal number of components in a fully factorized basis. This minimal "length" is a non-polynomial SL invariant [3,17] which according to our analysis has a tight connection with an entanglement classification using polynomial SL invariant entanglement measures. This connection consists in that the homogeneous degree of the polynomial invariant has to fit with the length of the balanced part of the minimal form. From the latter we can read off (up to a normalization factor) the value of the polynomial SL invariant.
Precise sufficient conditions have been singled out for reducibly balanced states in order to be maximally entangled. Such states clearly exist, possibly even without irreducibly balanced form. However, irreducibly balanced states provide a generating "basis" (not claiming completeness) for the costruction of such states, in the sense that reducibly balanced states are superpositions of irreducibly balanced ones.
It is worth giving reference to a collective entanglement measure proposed in [26]. For qubits, it is equivalent to the averaged one-tangle τ 1 = 4 n j det ρ j (see e.g. Ref. [28]), where ρ j is the reduced density matrix of qubit number j.
These measures are only sensitive to the requirement (ia) of Def. II.1. They assume their global maxima for all those maximally entangled states presented here (satisfying the condition (ib)). This also includes arbitrary tensor products of such maximally entangled states. So, this measure is an indicator of stochasticity of a pure state, but cannot discriminate any of the SLOCC entanglement classes present in that state. This shortcoming might be overcome to some extent by looking at maxima of suitable functions of e.g. von Neumann entropies of certain reduced density matrices. Such an analysis has been pursued among others in Ref. [32] and has singled out the four qubit "X state" in eq. (38): an irreducibly balanced state in the canonical form presented here (and before in Ref. [37,38]).
An additional advantage of the invariant-comb approach is that it suggests possible generalization to general subsystems. We have discussed to some extent generic complications encountered with such an extension. A specific analysis for bipartite entanglement of general half-integer spins is added. With a restriction of the local operations to just local rotations in the laboratory, an analogue to the concurrence is presented explicitly and its exact convex-roof extension has been constructed using a result of Ref. [47]. Its comparison with other existing proposals remains to be further investigated.