Phirotopes, Super p-branes and Qubit Theory

The phirotope is a complex generalization of the concept of chirotope in oriented matroid theory. Our main goal in this work is to establish a link between phirotopes, super p-branes and qubit theory. For this purpose we first discuss maximally supersymmetric solutions of 11-dimensional supergravity from the point of view of the oriented matroid theory. We also clarify a possible connection between oriented matroid theory and supersymmetry via the Grassmann-Pl\"ucker relations. These links are in turn useful for explaining how our approach can be connected with qubit theory.


1.-Introduction
Oriented matroid theory [1] is a combinatorial structure that has been proposed as the underlying mathematical framework for M-theory [2]. There are a number of evidences that suggests that this may be the case, including the following connections with oriented matroid theory: p-branes, qubit theory, Chern-Simons theory, supergravity and string theory, among others (see Refs. [3]- [10] and references therein). The key concept to realize this developments is the so called chirotope notion which provides with one of the possible axiomatization for oriented matroid theory (see Ref. [1] and references therein). Since supersymmetry is part of M-theory one may extend such analysis to include complex structure. It turns out that when the chirotopes are combined with a complex structure one is lead to the phirotope concept [11]- [13]. Thus, one should expect that when the complex structure is considered, the link between chirotopes and p-branes may be generalized to a connection between phirotopes and super p-branes.
In order to achieve our goal we first explain how phirotopes can be linked to supersymmetry (see Ref. [9]). In this case, we explain how maximally supersymmetric solutions of 11-dimensional supergravities [14]- [15] may be the key route to construct such a link. This is because the 4-form F = dA or Fμναβ, withμ,ν = 0, ..., 10, of 11-dimensional supergravity satisfies the Grassmann-Plücker relations (see Ref. [9] and references therein) which in turn are used to define both chirotopes and phirotopes. In order to clarify this constructions we briefly review of maximally supersymmetric solution. In particular, we focus in the algebraic identities of Englert solution [16] of 11-dimensional supergravity. We mention that no only in the case of the Freund-Rubin solution [17] of 11-dimensional supergravity the 4-form field Fμναβ admits an interpretation of a chirotope, but also the Englert solution [16]. In fact, if one assumes that the only non-vanishing components of Fμναβ are proportional to the completely antisymmetric symbol ε µναβ , with µ, ν, α, β = 0, ..., 3, then the Freund-Rubin solution arises from the bosonic sector of 11-dimensional supergravity field equations. While, if in addition, one assumes non-vanishing values for F ijkl , with ijkl = 4, ..., 10, one obtains a the Englert solution. From this perspective it becomes evident that it is important to study, deeply, the algebraic properties of Fμναβ. One observes, for instance, that since in the case of maximally supersymmetric solutions Fμναβ is totally decomposable, it must be possible to relate Fμναβ with the chirotope concept via the Grassmann-Plücker relations (see Ref. [18] for details).
It turns out that a natural generalization of the concept of chirotope is the so called phirotopes (see Ref. [11]- [13]). The main difference is that while the chirotope take values in the set {−1, 0, 1} the phirotopes take values in the set {e iθ | 0 < θ < 2π}. This means that the phirotopes describe a complex structure. Thus, in principle one can use phirotopes to introduce grassmann variables and in this way to define the concept of superphirotope which in turn can be used to establish a link with super p-branes.
The above scenario can be linked with class of N-qubits (see Refs. [4]- [5] and references therein), with the Hilbert space in the form C 2 N = C L ⊗ C l , with L = 2 N −n and l = 2 n . In fact, such a partition allows a geometric interpretation in terms of the complex Grassmannian variety Gr(L, l) of lplanes in C L via the Plücker embedding [19]. In the case of N-rebits one can set a L × l matrix variable b µ a , µ = 1, 2, ..., L, a = 1, 2..., l, of 2 N = L × l associated with the variable b a 1 a 2 ...a N , with a 1 , a 2,... etc taking values in the set {1, 2}. Moreover, one can consider that the first N − n terms in b a 1 a 2 ...a N are represented by the index µ in b µ a , while the remaining n terms are label by the index a in b µ a . One of the advantage of this construction is that the Plücker coordinates associated with the real Grassmannians b µ a are natural invariants of the theory. Since oriented matroid theory leads to the chirotope concept which is also defined in terms Plücker coordinates these developments establishes a possible link between chirotopes and p-branes.
This article is organized as follows. In section 2, we present a proof that a p-form is totally decomposable if and only if satisfies the Grassmann-Plücker relations. In section 3, Figueroa-O'Farrill-Papadopoulos formalism of 11-supergravity is revisited. In section 4, Englert solution of 11-dimensional supergravity is reviewed. In section 5, the chirotope concept of oriented matroid theory is related to supergravity. In section 6, the generalization of chirotopes to phirotopes it is discussed. In section 7, we comment about the relation between maximally supersymmetric solutions of 11-dimensional supergravity and the chirotope concept. In section 8, we develop the idea of superphirotopes. In section 9, we focus on the a possible relation between qubit theory and oriented matroid theory. Finally in section 10, we make some final remarks.

2.-Grassmann-Plücker relations and decomposable p-forms
It is known that the Grassmann-Plücker relation [20] is one of the key concepts in oriented matroid theory [1]. In order to better understand this notion it is first convenient to recall the mathematical definition of a Grassmannian Gr(p, n) (Grassmann variety) over the real R (or any other field K). Let V a vector space of dimensions n. The space Gr(p, n) over R is defined as the set of all p-dimensional subspaces of V .
Here, we are interested in considering the Plücker embedding of Gr(p, n) into the projective space P (Λ p V ). Given a subspace W ∈ Gr(p, n) with basis {F 1 , F 2 , ..., F 3 } let a map f be given by where the symbol ∧ denotes wedge product. It not difficult to show that up to scalar multiplication, this map (called Plücker map) is injective and unique. It is worth mentioning that, when one is classifying oriented bundles, the Grassmannian Gr(p, n) can also be denoted by the coset space [20] Gr(p, n) = SO(n) SO(n − p)SO(p) .
It is interesting to compare (2) with the definition of the (n − 1)-sphere S n−1 in terms of the orthogonal group SO(n), namely Comparing (2) and (3) one sees that Gr(p, n) is a generalization of S n−1 . Moreover, one may compute the dimension of Gr(p, n) by simply recalling that how is computed the dimension of any coset space G H , with H a subgroup of G. One has dim G H = dimG − dimH. Since dimSO(n) = n(n−1) 2 one finds the result dimGr(p, n) = p(n − p).
A p-form F µ 1 ...µ p ∈ Λ p V is totally decomposable if there exit a basis F 1 , ..., F p such that In order to connect this definition with the Grassmannian Gr(p, n) we first write The expression e µ 1 ∧ e µ 2 ∧ ... ∧ e µ p denotes a basis of Λ p V . Similarly, one has The ǫ-symbol ǫ a 1 a 2 ...ap in (6) is a completely antisymmetric tensor associated with the p-subspace. So, in this context (4) means that This result implies that the Plücker map can also be understood by the transition where F a µ ∈ Gr(p, n). Now, one may ask: when a p-form F µ 1 ...µ p is totally decomposable? There are several ways to approach this question. For instance, one may prove that F µ 1 ...µ p is totally decomposable if and only if the dimension of all the v ∈ V dividing F ∈ Λ p (V ) is p [21]. Here, however, we shall be interested to consider the Grassmann-Plücker relation Here, the symbol δ α 1 ...αpα p+1 ν 1 ...νpµ p denotes a generalized delta. The idea is now to prove that a p-form F µ 1 ...µ p is totally decomposable if and only if the Grassmann-Plücker relation (9) holds.
If F µ 1 ...µ p is totally decomposable then one sees that using (7) the combination leads to But one has ǫ a 1 a 2 ...
So, if the Grassmann-Plücker relation (9) holds then F µ 1 ...µ p is totally decomposable. Perhaps, it is more difficult to prove that (9) implies (7). This can be shown using an induction method (see [21] and references therein), but here we present an alternative prove that we are not aware of its existence in the literature. Let F A µ be an extended basis of V . We can define Considering the inverse F µ A of F A µ this expression leads to Using (13), it is not difficult to see that (9) implies Assume that (15) holds. Let us apply (15) to the particular case with C p = a and a and b running in the dimension of the p-subspace. One can show that (16) Since in general with Λ an arbitrary constant, one finds that Now, considering the next particular case and using (19) one obtains Following similar procedure one ends up with that the result that the only non vanishing components of F A 1 ...Ap are given by But, one knows that F a 1 a 2 ...ap = Λǫ a 1 a 2 ...ap . Therefore, using (14) we obtain Up to constant, this expression corresponds to (7) meaning that F µ 1 µ 2 ...µ p is decomposable. The expression (23) will be very useful in the next sections.

3.-Figueroa-O'Farrill-Papadopoulos formalism revisited
Consider the 4-form F µ 1 µ 2 µ 3 µ 4 . We shall assume that this form satisfies the Grassmann Plücker relation It turns out that (24) holds if any only if the following two the relations are satisfied and It is worth mentioning that (25) and (26) play a crucial role in maximally supersymmetric 11-dimensional supergravity [14]- [15]. Let us prove that in fact this result holds. First, one observes that in general the bracket [, ] in (24)- (26) can be written as The quantity G α 1 ...α d+1 is any d + 1-rank tensor. Considering the fact that whereμ k means omitting this index, one finds that (25) follows if and only if one has which means Properly applying again (28) one gets Thus, considering the fact that the (26) holds the first term in (31) vanishes, that is Similar technique it leads us to the identity which in turn gives, This expression implies that the right hand side of (34) is antisymmetric in the indices µ 1 and µ 3 .
On the other hand one obtains From (26) one sees that the left hand side of (35) vanishes and therefore we obtain This means that F µ 1 µ 2 [µ 4 ν 1 F ν 2 ν 3 ν 4 ]µ 3 is symmetric in the indices µ 1 and µ 3 which contradicts the conclusion below (34). Thus, we have found that the only consistent possibility is to set which implies (24) via (34). Summarizing, we have shown that (25) and (26) imply (24) which is the Grassmann-Plücker relation. Conversely, using once again the properties of the generalized delta δ α 1 ...α d+1 and therefore (24) implies (25) and (26). This means that the expression (24) is equivalent to the two formulae (25) and (26). Thus, we have complete an alternative proof of such a equivalence.
It turns out convenient to briefly mention how the above result is linked to maximally supersymmetric solution of 11-dimensional supergravity. In fact, Figueroa-O'Farrill and Papadopoulos proved that such a solution must be isometric to either AdS 4 × S 7 or AdS 7 × S 4 . Their starting point in this result is the vanishing of the curvature R of the supercovariant connection D living in (M 11 , g, F ). In fact, demanding the vanishing of the curvature R they found that (M 11 , g, F ) is maximally supersymmetric solution if and only if (M 11 , g) is locally symmetric space and F is parallel and decomposable.
Let us clarify further this theorem. In the non-degenerate case, spontaneous compactification allows to assume that the only nonvanishing components of F µ a are F µ a ∼ δ µ a , with µ = 0, 1, 2, 3 or Fμ a ∼ Fμ a , withμ = 8, 9, 10, 11 leading to the two possible solutions AdS 4 × S 7 or AdS 7 × S 4 , respectively. In fact, in the first case one gets that the only nonvanishing components of F µ 1 µ 2 µ 3 µ 4 are F µναβ ∼ ε µναβ . Thus, as seen from the 11-dimensional field equations one obtains the Freund-Rubin solution AdS 4 × S 7 . While in the second case one assumes the solution Fμναβ ∼ εμναβ and the field equations (39) lead to the solution AdS 7 × S 4 . Perhaps, it is also convenient to write the three equations (24)-(26) in abstract notation. From the formula R = 0 one can essentially derive two algebraic formulae for F (25) and (26) which in abstract notation become and respectively. Here ι X and ι Y denote an inner product for the two arbitrary vectors X and Y , respectively. From (25) and (26) we proved that F satisfies (24) which in abstract notation is written as It is interesting to mention the way that Figueroa-O'Farrill and Papadopoulos prove that (25) and (26) imply (24). They first observe that contracting (25) with respect to the three vectors X, Y and Z one obtains While, contracting equation (26) with a third vector field one gets Thus, comparing (43) and (44) one sees that whereas (42) implies that the expression ι Y ι X F ∧ ι Z F is symmetric in X and Z, (44) means that it is skewsymmetric. This means that the term ι Y ι X F ∧ ι Z F must vanish and therefore (42) follows (see Refs. [14] and [15] for details).

4.-Englert solution revisited
Consider the octonionic identity [22], with the indices i, j, ...etc running from 4 to 11. Here, f ijkl is a self dual object. Furthermore, f ijkl is defined in terms of the octonionic structure constants ψ ijk and its dual ϕ ijkl through the relations and From (45) it is not difficult to see that This expression can be understood as a solution for which remains us the formula (26) reduced to seven dimensions. In fact, introducing a sieben-bein h i k one can make this identification more transparent [22]. In fact, one has and therefore (49) leads to Starting from (45) and following similar arguments we may establish that and Thus, according to the discussion of previous sections (52) and (53) imply that F ijkl satisfies the relation which means that F ijkl is decomposable.
On the other hand, in four dimensions as we already mentioned, we can take where Λ is an arbitrary function. Since ǫ µναβ is a maximally completely antisymmetric object in four dimensions we get the formula which implies Thus, F µναβ is also decomposable.
Our main observation is that despite both F ijkl and F µναβ are both decomposable, the 11-dimensional components F AναD are not. The reason comes from the fact that in spite that F ijkl and F µναβ are decomposable the components of F AναD not necessarily satisfies the relation or So, it turns out that full F ABCD is not decomposable. In fact, since ε µναβ and f ijkm take values in the set {−1, 0, 1} in general we have that or In turn this means that Consequently we no longer have maximally supersymmetric solution. Nevertheless, as Englert showed, although the right hand side of the first field equation in (39) is not vanishing the field equations still admit the solution AdS 4 × S 7 . This means that maximally supersymmetric solutions can be considered as a broken symmetry (see Ref. [18] and references therein).

5.-Connection with chirotopes
The aim of this section is to discuss part of the formalism described in section 2, 3 and 4 from the point of view of the oriented matroid theory. Indeed, our discussion will focus on the chirotope concept which provides one possible definition of an oriented matroid [1]. In fact, chirotopes has been a major subject of investigation in mathematics during the last 25 years [1]. Roughly speaking a chirotope is a combinatorial abstraction of subdeterminants of a given matrix. More formally, a realizable p-rank chirotope is an alternating function χ : {1, ..., n} p → {−1, 0, 1} satisfying the Grassmann-Plücker relation while nonrealizable p-rank chirotope corresponds to the case It is worth mentioning that there is a close connection between chirotopes and Grassmann variety. In fact, the Grassmann-Plücker relations describe a projective embedding of the Grassmannian of planes via decomposable p-forms (see Ref. [1] for details). Thanks to our revisited review of Freund-Rubin and Englert solutions given in the previous sections we find that the link between these solutions and the chirotope is straightforward. In fact, our first observation is that any ε-symbol is in fact a realizable chirotope (see Ref. [23]), since it is always true that From this perspective we recognize that the formula (24) indicates that in the case of maximally supersymmetric solutions, in 11-dimensional supergravity, the 4-form F ABCD is a realizable 4-rank chirotope. While in the case of Freund-Rubin-Englert solution, from (39) and (42) one discovers that according to our discussion of section 4 one may identify F ABCD with a nonrealizable 4-rank chirotope. From this connections one may expect that there may be many possible 4-rank chirotopes in 11-dimensions and therefore there must be many new and unexpected solutions for 11-dimensional supergravity.
One of our key tools in our formalism is the octonionic structure. This division algebra was already related to the Fano matroid and therefore, a possible connection with supergravity was established (see Ref. [2] and references therein). Here, we have been more specific and through the chirotope concept we established the relation between the Freund-Rubin-Englert solution and oriented matroid theory. However, it may be interesting to understand the possible role of the Fano matroid in this scenario.
Moreover, here we focused on 11-dimensional supergravity but, in principle, one may expect to apply similar procedure in the case of 10-dimensional supergravity and other higher dimensional supergravities such as Type I supergravity and massive IIA supergravity.
An important property in the oriented matroid theory is that one can associate any chirotopes with its dual. Thus, working on the framework of oriented matroids we can assure that any possible solution for 11-dimensional supergravity in terms of chirotopes will have a dual solution. This means that this kind of solution contains automatically a dual symmetry.
It is worth mentioning that using the idea of matroid bundle [24]- [28], Guha [29] has observed that chirotopes can be related to Nambu-Poisson structure. It may be interesting to see whether this Nambu-Poisson structure is related to 11-dimensional supergravity.
which can also be written as Using the ε-symbol property It is not difficult to prove that Σ i 1 ...i d satisfies the Grassmann-Plücker relations, namely We recall that the brackets in the indices of (68) and (69) mean completely antisymmetrized.
A realizable chirotope χ is defined as From the point of view of exterior algebra one finds that there is a close connection between Grassmann algebra and a chirotope. Let us denote by ∧ d R n the ( n d )-dimensional real vector space of alternating d-forms on R n . We recall that an element Σ in ∧ d R n is said to be decomposable if for some v 1 , v 2 , ..., .v d ∈ R n . It is not difficult to see that (71) can also be written as where e i 1 , e i 2 , ..., e i d are 1-form bases in R n and Σ i 1 ...i d is given in (66). This shows that Σ i 1 ...i d can be identified with an alternating decomposable d-form.
In order to define non-realizable chirotopes it is convenient to write the expression (69) in the alternative form where Here, j d+1 = i d and k establish the notation for omitting this index. Thus, for a general definition one defines a d-rank chirotope χ : with and k = 1, ..., d + 1. It is evident that (73) is a particular case of (76). Therefore, there are chirotopes that may be non-realizable. Moreover, this definition of a chirotope is equivalent to various others (see Refs. [11]- [13] for details), but it seems that the present one is more convenient for a generalization to the complex structure setting. The generalization of a chirotope to a phirotope is straightforward. A function ϕ : E d → S 1 ∪ {0} on all d-tuples of E = {1, ..., n} is called a d-rank phirotope if (a) ϕ is alternating and (b) for for k = 1, ..., d + 1 there exist r 1 , ..., r d+1 ∈ R + such that d+1 k=1 r k ω k = 0.
In the case of a realizable phirotope we have where ω(z) ∈ S 1 ∪ {0} and (u i 1 ...u i d ) are a set of complex vectors in C d . We observe that one of the main differences between a chirotope and a phirotope is that the image of a phirotope is no longer a discrete set (see Refs. [11]- [13] for details).

7-Supergravity and phirotopes
As we mentioned in section 3, maximally supersymmetric solution of 11dimensional supergravity leads to the two conditions and for the 4-form field strength F = dA which are equivalent to the Grassmann-Plücker relations meaning that F is decomposable. Thus, according to the discussion of the previous one discovers that (82) establishes that F is a realizable 4-rank chirotope with a ground set E = {1, ..., 11}. This in turn means that maximal supersymmetry in 11-dimensional supergravity is related to oriented matroid theory. Similar conclusion can be obtained for the case of 10-dimensional supergravity. Hence, one may understand the chirotope concept as the bridge between supersymmetry and the oriented matroid theory. Thus, one should expect a generalization of oriented matroid theory which would include supersymmetry. But in order to develop this idea it turns out more convenient to consider a complex structure, and this means that we need to focus on the superphirotope notion rather than on the superchirotope concept which must arise as a particular case of the former.

Superphirotope
The main goal of this section is to outline a possible supersymmetrization of a phirotope. By convenience we shall call superphirotope such a supersymmetric phirotope. Inspired in super p-brane theory one finds that one way to define a superphirotope, which assures supersymmetry, is as follows. First, we need to locally consider the expressions (77)-(79) in the sense that ϕ i 1 ....j d (ξ) is a local phirotope if for k = 1, ..., d + 1 there exist r 1 , ..., r d+1 ∈ R + such that d+1 k=1 r k ω k (ξ) = 0.
In the case of a realizable local phirotope we have where ξ = (ξ 1 , ..., ξ d ) are local coordinates of some d-dimensional manifold B.
The vectors v i 1 (ξ), ..., v i d (ξ) can be thought as vectors in the tangent space T ξ (B) at ξ. One can assume that the possibility of considering the expressions (77)-(79) in a local context may be justified in principle by the so-called matroid bundle notion (see Refs. [24]- [28]). Let us recall that the projective variety of decomposable forms is isomorphic to the Grassmann variety of ddimensional linear subspaces in R n . In turn, the Grassmann variety is the classifying space for vector bundle structures. Taking these ideas as a motivation, MacPherson developed the combinatorial differential manifold concept.
The matroid bundle notion arises as a generalization of the MacPherson proposal. Roughly speaking, a matroid bundle is a structure in which at each point of the differentiable manifold an oriented matroid is attached as a fiber (see [24]- [28] for details). Now, let us consider a supermanifold B parametrized by the local coordinates (ξ, θ) where θ are elements of the odd Grassmann algebra (anticommuting variables). We shall now consider the supersymmetric prescription Here, γ i are elements of a Clifford algebra. Using (86) one can generalize (85) in the form The symbol det means the superdeterminant. One should expect that (87) satisfies a kind of supersymmetric Grassmann-Plücker relations. It is not difficult to see that up to total derivative (87) is invariant under the global supersymmetric transformations and where ǫ is a constant complex spinor parameter. Similarly, one can generalize the superphirotope to the non-representable case by assuming that if for k = 1, ..., d + 1 there exist r 1 , ..., r d+1 ∈ R + such that Of course, in the case that the complex structure is projected to the real structure one should expect that the superphirotope is reduced to the superchirotope.
With the superphirotope Ψ i 1 ...i d (x, θ) at hand one may consider a possible partition function where is a Schild type action for a superphirotope. Here, λ is a Lagrange multiplier and T d is the (d − 1)-phirotope tension. Moreover, in a more general context the action may have the form The advantage of the actions (93) and (94) is that duality is automatically assured. In fact, in the oriented matroid theory duality is a main subject in the sense that any chirotope has an associated dual chirotope. This means that a theory described in the context of an oriented matroid automatically contains a duality symmetry. Therefore, with our prescription one is assuring not only the supersymmetry for the action (93) or (94) but also the duality symmetry.
The action (93) can be related to an ordinary super p-brane by assuming that Ψ i 1 ...i d (ξ, θ) is a closed d-form because in that case we can write The coordinates x i are the p-brane bosonic coordinates. It is worth mentioning that the bosonic sector of Ψ i 1 ...i d (ξ, θ) is a constraint of Nambu-Poisson geometry which has been related to oriented matroid theory (see Ref. [29] for details). It may be interesting for further research to consider the action (93) from the point of view of a superfield formalism instead of using the prescription (95). In this case one may consider a supersymmetrization in the form π i a (ξ, θ) = ∂ a X i , with X i as a scalar superfield admitting a finite expansion in terms of θ. For instance, in four dimensions one may have The state ψ i denotes a Majorana spinor field, while B i refers to an auxiliary field. By substituting (96) into (93) one should expect a splitting of (93) in several terms containing the variables x i (ξ), ψ i (ξ) and B i (ξ). The important thing is that using the prescription (96) supersymmetry becomes evident in the sense that the algebra of supersymmetry transformations is closed off the mass-shell. Although in section 3 we focused on 11-dimensional supergravity similar arguments can be applied to the case of 10-dimensional supergravity. Specifically, as we already mentioning by studying maximal supersymmetry in IIB supergravity Figueroa-O'Farril and Papadopoulos [14]- [15] used the vanishing of the curvature of the supercovariant derivative to derive the analogue Grassmann-Plücker formula for the five-form F LP 1 P 2 P 3 P 4 . Moreover, in Refs. [14]- [15] is proved that (97) implies that where G is a decomposable 5-form and * G denotes the 10-dimensional dual of G. This means that G and * G satisfy the Grassmann-Plücker relations and therefore can be identified with a 5-rank chirotope.

Connection with qubit theory
A connection between 4-rebits (real qubits) and the Nambu-Goto action with target 'spacetime' of four time and four space dimensions ((4 + 4)dimensions)) was proposed in Ref. [5]. The motivation for this proposal came three observations. The first one is that a 4-rebit contains exactly the same number of degree of freedom as a complex 3-qubit and therefore 4-rebits are special in the sense of division algebras. Secondly, the (4 + 4)-dimensions can be splitted as (4 + 4) = (3 + 1) + (1 + 3) and therefore they are connected with an ordinary (1 + 3)-spacetime and with changed signature associated with (3 + 1)-spacetime [30]. Moreover it was shown how geometric aspects of 4-rebits can be related to the chirotope concept of oriented matroid theory (see Ref. [4]).
It is worth mentioning that the discovery of new hidden discrete symmetries of the Nambu-Goto action (through the identification of the coordinates x µ of a bosonic string, in target space of (2 + 2)-signature, with a 2 × 2 matrix x ab ) [31] leads to increase the interest in qubit theory. It turns out that the key mathematical tool in this development is the Cayley hyperdeterminant Det(b) [32] of the hypermatrix b bc a = ∂ a x bc . A striking result is that Det(b) can also be associated with the four electric charges and four magnetic charges of a STU black hole in four dimensional string theory [33]. Even more surprising is the fact that Det(b) makes also its appearance in quantum information theory by identifying b bc a with a complex 3-qubit system a bc a [34]. These coincidences, among others, have increased the interest on the qubit/black hole correspondence [35].
It turns out that in information theory 4-qubit is just subclass of N-qubit entanglement. In fact, the Hilbert space can be broken into the form C 2 N = C L ⊗ C l , with L = 2 N −1 and l = 2. Such a partition it allows a geometric interpretation in terms of the complex Grassmannian variety Gr(L, l) of 2planes in C L via the Plücker embedding. In this case, the Plücker coordinates of Grassmannians Gr(L, l) are natural invariants of the theory (see Refs [19] and [37] for details).
It is interesting to make the following observations. First, one notes that a a 1 a 2 a 3 has 8 complex degrees of freedom, that is 16 real degrees of freedom, while a a 1 a 2 a 3 a 4 contains 16 complex degrees of freedom, that is 32 real degrees of freedom. Let us denote N-rebit system (real N-qubit ) by b a 1 a 2 ...a N . So we shall denote the corresponding 3-rebit, 4-rebit by b a 1 a 2 a 3 and b a 1 a 2 a 3 a 4 , respectively. One observes that b a 1 a 2 a 3 has 8 real degrees of freedom, while b a 1 a 2 a 3 a 4 has 16 real degrees of freedom. Thus, by this simple (degree of freedom) counting one note that it seems more natural to associate the 4-rebit b a 1 a 2 a 3 a 4 with the complex 3-qubit, a a 1 a 2 a 3 , than with the complex 4-qubit, a a 1 a 2 a 3 a 4 . Of course, by imposing some constraints one can always reduce the 32 real degrees of freedom of a a 1 a 2 a 3 a 4 to 16, and this is the kind of embedding discussed in Ref. [19]. The main idea in reference [5] was to make sense out of a 4-rebit in the Nambu-Goto context without loosing the important connection with a division algebra via the Hopf map S 15 S 7 −→ S 8 . Let us first show the formalism concerning the Nambu-Goto action/qubits correspondence in a spacetime of (2 + 2)-signature. In the (2 + 2)-dimensions one may introduce the matrix Using (102) the line element can also be written as where is a flat metric corresponding to (2 + 2)-signature and ε ab is the completely antisymmetric symbol (ε-symbol) with ε 12 = 1.
On the other hand in a target space of (4 + 4)-signature one may introduce the matrices and At first sight one may consider the line element as the analogue of (104). But this vanishes identically because is a symmetric quantity, while ε cf is antisymmetric. In fact, the correct line element in (4 + 4)-dimensions turns out to be Notice that we have changed the last ε-symbol in (110) for the η-symbol. Here, η cf = diag(−1, 1). Moreover, one can prove that (103), with η µν = (−1, −1, −1, −1, +1, +1, +1, +1), follows from (110). Similarly, the world sheet metric in (2 + 2)-dimensions can be written as While in (4 + 4)-dimensions, one has In (2 + 2)-dimensions one can write the determinant of γ ab , in the hyperdeterminant form Thus, this proves that the Nambu-Goto action for a flat target "spacetime" with (2 + 2)-signature can also be written as [31] S = 1 2 While in (4 + 4)-dimensions the determinant can be written as where One recognizes in (121) the hyperdeterminant of the hypermatrix c ef rs . So, we can write This proves that the Nambu-Goto action in (4 + 4)-dimensions can also be written as [5] S = 1 2 Moreover, one may connect qubits with the chirotope concept in oriented matroid theory. In space of (2 + 2)-signature one writes where Here, we have used the definition Since σ µν satisfies the identity σ µ[ν σ αβ] ≡ 0, one can verify that χ µν = signσ µν satisfies the Grassmann-Plücker relation and therefore χ µν is a realizable chirotope (see Ref. [22] and references therein). The Grassmann-Plücker relation (130) implies that the ground set is and the alternating map determine a 2-rank realizable oriented matroid M = (E, χ µν ). The collection of bases for this oriented matroid is which can be obtained by just given values to the indices µ and ν in χ µν . Indeed, the pair (E, B) determines a 2-rank uniform non-oriented ordinary matroid.
In the case of qubits, one may introduce the underlying ground bitset (from bit and set) and the pre-ground set It turns out that E 0 and E can be related by establishing the identification (1, 1) ↔ 1, (1, 2) ↔ 2, Observe that (136) Using the definition one can show that This establishes a link between the hyperdeterminant and "chirotope" structure. In (4 + 4)-dimensions one may introduce the quantity Here, one has considered the definitions and b µ2 a = ∂ a y µ .
In turn this means that we can write One recognizes in this expression the Plücker coordinates for both cases u µ a = ∂ a x µ and v µ a = ∂ a y µ .
This proves that both quantities σ ef rs and c ef rs (qubitopes), belongs to an underlying structure Q = (E, E 0 , B 0 ) called qubitoid. The word "qubitoid" is a short word for qubit-matroid.
One may now be interested to see how the 4-rebit b a 1 a 2 a 3 a 4 is connected with a a 1 a 2 a 3 . The simplest (but no the most general) possibility seems to be a a 1 a 2 a 3 = b a 1 a 2 a 3 1 + ib a 1 a 2 a 3 2 . (144) In turn this implies Hence the 3-qubit a a 2 a 3 a 1 is related to the two 2-rebits states x a 2 a 3 1 and x a 2 a 3 2 . This observation may help eventually to relate 4-rebit b a 1 a 2 a 3 a 4 with the Hopf fibration S 15 S 7 −→ S 8 . In fact, a normalization of the complex states a a 1 a 2 a 3 leads to the 15-dimensional sphere S 15 which under the Hopf map, admit parametrization of the parallelizable sphere S 7 fibration over S 8 . It turns out, that just as the norm group of quaternions is SO(4) = S 3 × S 3 , the norm group of octonions is SO(8) = S 7 × S 7 × G 2 (see Ref. [21] for details). Since in the 4 + 4-signature the relevant group is SO(4, 4). One find that 8-dimensional spinor representation associated with spin(8) can be written as The objects (r i ) k j are structure constants of octonions o i , (o i o j = (r i ) k j o k ). This means that when SO(8) decomposed under the subgroup SO(4) × SO(4) one gets irreducible representation 8 −→ (4, 1) + (1, 4).
It turns out that these two direct summands correspond to the variables x ab1 and x ab2 . This explains why dx abc , is contracted with η ab , and no with ε ab . The above scenario can be generalized for class of N-qubits, with the Hilbert space in the form C 2 N = C L ⊗ C l , with L = 2 N −n and l = 2 n . Such a partition allows a geometric interpretation in terms of the complex Grassmannian variety Gr(L, l) of l-planes in C L via the Plücker embedding [19].
In the case of N-rebits one can set a L × l matrix variable b µ a , µ = 1, 2, ..., L, a = 1, 2..., l, of 2 N = L×l associated with the variable b a 1 a 2 ...a N , with a 1 , a 2,... etc taking values in the set {1, 2}. In fact, one can take the first N − n terms in b a 1 a 2 ...a N are represented by the index µ in b µ a , while the remaining n terms are considered by the index a in b µ a . One of the advantage of this construction is that the Plücker coordinates associated with the real Grassmannians b µ a are natural invariants of the theory. Since oriented matroid theory leads to the chirotope concept which is also defined in terms Plücker coordinates these developments establishes a possible link between chirotopes, qubitoids and p-branes.

Final remarks
It is evident from the discussion of the previous sections that the Grassmann-Plücker relations play a central role on a number of links between different physical and mathematical scenarios including Grassmannian varieties, 11dimensional supergravity, qubit theory, p-branes and oriented matroid theory. If a p-form satisfies the Grassmann-Plücker relations then such a p-form is decomposable. An application of this result to maximally supersymmetric solutions of 11-dimensional supergravity opens the possibility for writing the 4-form field strength Fμ 1μ2μ3μ4 in terms of some kind of gauge field F â µ . Thus, following this kind of though one is lead to look for the analogue F â µ for of Bianchi identities dF = 0 associated with Fμ 1μ2μ3μ4 . In this case, it is found that Fμ 1μ2μ3μ4 can be written in terms of a gauge field Aμ 1μ2μ3 in the form Fμ 1μ2μ3μ4 = ∂ [μ 1 Aμ 2μ3μ4 ] . So, the question arises whether the Bianchi identities and the Grassmann-Plücker relations associated with a general p-form field strength F µ 1 ...µ p are connected. In other word, the challenge is to know what is the relation between F a 1 µ 1 and A µ 1 ...µ p−1 . Let us assume that a local p-form F µ 1 ...µ p satisfies the Grassmann-Plucker relations According to our previous discussion one knows that (149) implies that F µ 1 ...µ p is decomposable. This means that F µ 1 ...µ p can be written as F µ 1 ...µ p = ε a 1 ...ap F a 1 µ 1 ...F ap µ p .
This establishes a connection between the Bianchi identity dF = 0 and the Grassmann-Plücker relations. It is worth mentioning that the expression (156) is a key tool in the p-brane theory when one writes the Nambu-Goto action for p-branes in Schild type formalism (see Ref. [3] for details).