A Notable Relation between $N$-Qubit and $2^{N-1}$-Qubit Pauli Groups via Binary ${\rm LGr}(N,2N)$

Employing the fact that the geometry of the $N$-qubit ($N \geq 2$) Pauli group is embodied in the structure of the symplectic polar space $\mathcal{W}(2N-1,2)$ and using properties of the Lagrangian Grassmannian ${\rm LGr}(N,2N)$ defined over the smallest Galois field, it is demonstrated that there exists a bijection between the set of maximum sets of mutually commuting elements of the $N$-qubit Pauli group and a certain subset of elements of the $2^{N-1}$-qubit Pauli group. In order to reveal finer traits of this correspondence, the cases $N=3$ (also addressed recently by L\'evay, Planat and Saniga [J. High Energy Phys. 2013 (2013), no. 9, 037, 35 pages, arXiv:1305.5689]) and $N=4$ are discussed in detail. As an apt application of our findings, we use the stratification of the ambient projective space ${\rm PG}(2^N-1,2)$ of the $2^{N-1}$-qubit Pauli group in terms of $G$-orbits, where $G \equiv {\rm SL}(2,2)\times {\rm SL}(2,2)\times\cdots\times {\rm SL}(2,2)\rtimes S_N$, to decompose $\underline{\pi}({\rm LGr}(N,2N))$ into non-equivalent orbits. This leads to a partition of ${\rm LGr}(N,2N)$ into distinguished classes that can be labeled by elements of the above-mentioned Pauli groups.


Introduction
Generalized Pauli groups (also known as Weyl-Heisenberg groups) associated with finite-dimensional Hilbert spaces play an important role in quantum information theory, in particular in quantum tomography, dense coding, teleportation, error correction/cryptography, and the blackhole-qubit correspondence. A special class of these groups are the so-called N -qubit Pauli groups, N being a positive integer, whose elements are simply N -fold tensor products of the famous Pauli matrices and the two-by-two unit matrix. A remarkable property of these particular groups is that their structure can be completely recast in the language of symplectic polar spaces of rank N and order 2, W(2N −1, 2) (see, for example, [7,20,21,22,23,24,26] and references therein). The elements of the group (discarding the identity) answer to the points of W(2N −1, 2), a maximum set of pairwise commuting elements has its representative in a maximal subspace (also called a generator) PG(N − 1, 2), the projective space of dimension N − 1 over the Galois field of order 2, of W(2N − 1, 2) and, finally, commuting translates into collinear (or, perpendicular). In the case of the real N -qubit Pauli group, the structure of the corresponding W(2N − 1, 2) can arXiv:1311.2408v2 [math-ph] 8 Apr 2014 be refined in terms of the orthogonal polar space Q + (2N − 1, 2), that is, a hyperbolic quadric of the ambient projective space PG(2N − 1, 2), which is the locus accommodating all symmetric elements of the group [7]. Given this finite-geometrical picture of (real) N -qubit Pauli groups, one can invoke properties of the Lagrangian Grassmannian LGr(N, 2N ) defined over the Galois field of two elements, GF (2), to establish a very interesting bijection between the generators of W(2N −1, 2) and points lying on a sub-configuration of W(2 N −1, 2) defined by a set of quadratic equations. This furnishes an intriguing mapping of maximum sets of mutually commuting Nqubit observables into observables of 2 N −1 -qubits. For N = 3, all essential technicalities of this relation have recently been worked out in detail in [14]. In this paper, we shall first give a short rigorous proof that this bijection holds for any N . Then, after a brief addressing of a rather trivial N = 2 case, we shall again discuss in detail the N =3 case using, however, a more "projective-slanted" view to be compared with an "affine" approach of the latter reference, as well as the N = 4 case to see some novelties and get a feeling of the kind of problems one can envisage/encounter when addressing higher rank cases.
Our main motivation for having a detailed look at the above-outlined 'Lagrangian Grassmannian' relationship between different multi-qubit Pauli groups stems from an important role of the maximum sets of mutually commuting N -qubit observables in the quantum information theory. On the one hand, such sets are vital for simple demonstrations of quantum contextuality. Every such set can be regarded as a context and various 'magic' collections of such contexts are intimately linked with sub-geometries of the associated symplectic polar space W(2N − 1, 2). The simplest such configuration can already be found in the N = 2 case, being known as a Mermin magic square [16]. It represents a set of nine observables placed at the vertices of a 3×3 grid and forming six maximum sets of pairwise commuting elements that lie along three horizontal and three vertical lines, each observable thus pertaining to two such sets. The observables are selected in such a way that the product of their triples in five of the six sets is +I, whilst in the remaining set it is −I, I being the identity matrix. Geometrically, each Mermin square is isomorphic to the smallest slim generalized quadrangle, GQ(2, 1), or to a hyperbolic quadric Q + (3,2). A number of other magic configurations, exhibited by higher-order Pauli groups and featuring a varying degree of complexity, can be found in Waegell's preprint [27]. On the other hand, existence of these sets is intricately related to the existence of mutually unbiased bases (MUBs) of the associated Hilbert space. In particular, W(2N − 1, 2) possesses spreads [25], that is sets of generators of W(2N − 1, 2) partitioning its point-set, whose cardinality is equal to the maximum number of MUBs, d + 1, in the associated d = 2 N -dimensional Hilbert space. Thus, for example, spreads of W(3, 2) feature five elements each, and the associated 4-dimensional Hilbert space is indeed found to be endowed with sets of 4 + 1 = 5 MUBs [21].
The paper is organized as follow. In Section 2, we recall the definition of the symplectic polar space W(2N − 1, 2) and how this space encodes the geometry of the N -Pauli group. In Section 3, we prove our main result by establishing the existence of a projection which maps bijectively the aggregate of maximum sets of mutually commuting N -qubit observables into a distinguished subset of 2 N −1 -qubit observables. Then, in Section 4, we illustrate our construction for N = 2, 3, and 4 by explicitly computing the equations defining the image of the projection in PG(2 N − 1, 2). In Section 5, one shows how our findings can be used to partition the set of generators of W(2N − 1, 2). Finally, in Section 6 we point out a relation between our construction and similar ones done over the field of complex numbers.
Notation. In what follows, we will denote by K the Galois field GF (2) and, if V is a K-vector space, we will use the symbol P(V ) to represent the corresponding projective space over K; thus, P(K N ) will be an alternative expression for PG(N −1, 2), the projective space of dimension N −1 over GF (2). Given a nonzero vector v ∈ V , we will denote by [v] ∈ P(V ) the corresponding point in the associated projective space. On the other hand, for any X ⊂ P(V ), we define the cone over X, X ⊂ V , to be the pre-image of X in V , i.e. the set of all vectors x ∈ V such that [x] ∈ X.
A tensorial basis of (K 2 ) ⊗n ≡ K 2 ⊗ · · · ⊗ K 2 (n factors) will be denoted by i } is a basis of (K 2 ) i . Let A = (a ij ) be an n × n matrix with coefficients in K and let I = {i 1 , . . . , i k } and J = {j 1 , . . . , j k } be subsets of {1, . . . , n}. The symbol ∆ I,J will stand for the corresponding k × k minor of A, i.e. ∆ I,J (A) = det((a i,j ) i∈I,j∈J ); when I = J, ∆ I,I (A) will be called a principal minor of A and simply referred to as ∆ I (A).
In Section 4, computations will be handled using Maple and Macaulay2 to get the equations of the ideal of the Lagrangian Grassmannian LGr(N, 2N ) for N = 3 and N = 4. The sources of the codes are available at http://www.emis.de/journals/SIGMA/2014/041/codes.zip which contains two files: one is a Maple file to compute all equations defining the ideal of LGr (4,8), the other is a Macaulay2 script to compute the ideal of the projection of LGr (4,8) by elimination theory based on the equations stemming from the previous code. In what follows we shall only be concerned with W(2N − 1, 2); this space features |PG(2N − 1, 2)| = 2 2N − 1 = 4 N − 1 points and the number of its generators amounts to (2 + 1)(2 2 + 1) · · · (2 N + 1). The generalized real N -qubit Pauli group, denoted by P N , is generated by N -fold tensor products of the matrices Explicitly, The associated factor group P N ≡ P N /Z(P N ), where the center Z(P N ) consists of ±I (1) ⊗I (2) ⊗ · · · ⊗ I (N ) , features 4 N elements. The elements of P N \{I (1) ⊗ I (2) ⊗ · · · ⊗ I (N ) } can be bijectively identified with the same number of points of W(2N − 1, 2) in such a way that two commuting elements of the group will lie on the same totally isotropic line of this polar space. If one selects a basis of W(2N − 1, 2) in which the symplectic form σ(x, y) is given by then this bijection acquires the form: with the understanding that thus, for example, in W(7, 2) the point having coordinates (0, 1, 1, 0, 0, 1, 0, 1) corresponds to the element I ⊗ Y ⊗ Z ⊗ X ≡ IY ZX. The elements of the group P N whose square is +I (1) I (2) · · · I (N ) (i.e., symmetric elements) lie on a certain Q + (2N − 1, 2) of the ambient space PG(2N − 1, 2). It follows from the definition of the bijection that the equation of the Q + (2N − 1, 2) accommodating all symmetric elements must have the following standard form This can readily be inspected using the fact that the matrix Y is the only skew-symmetric element in the set {I, X, Y, Z} and, so, any symmetric element of the group must contain an even number of Y 's. It should also be added that generators, of both W(2N − 1, 2) and Q + (2N − 1, 2), correspond to maximal sets of mutually commuting elements of the group (see [7] for a proof of this fact).

Mapping
LGr(N, 2N ) to PG(2 N − 1, 2) Recently, Lévay, Planat and Saniga [14] found and analyzed in detail an explicit bijection between the set of 135 maximum sets of mutually commuting elements of the three-qubit Pauli group (that is, the set of generators of W(5, 2)) and the set of 135 symmetric operators of the four-qubit Pauli group (that is, the set of points lying on a particular Q + (7, 2) of W(7, 2)). Following the spirit of this work, we will generalize this physically important result and prove the existence of a similar bijection between any N -qubit and 2 N −1 -qubit Pauli groups. This will be done by considering first the Grassmaniann variety Gr(N, 2N ), then its associated Lagrangian Grassmannian 1 LGr(N, 2N ) and, finally, using a crucial fact that we work in characteristic 2.
To this end in view, let us first recall the definition of the variety of N -planes in K 2N , i.e. the Grassmannian variety Gr (N, 2N ). An N -plane (respectively an (N − 1)-projective-plane) P , spanned by N non-zero vectors u 1 , u 2 , . . . , u N of K 2N (respectively by N points . The embedding of the Grassmannian variety is given by the so-called Plücker map: In other words, the Grassmanian variety is the set of all skew symmetric tensors that can be factorized (i.e., are separable). The algebraic equations defining Gr(N, 2N ) are known as the Plücker equations. Let (e i ) 1≤i≤2N be a basis of the vector space K 2N and let P ∈ P(∧ N K 2N ), i.e.
The Lagrangian variety is thus the variety of all the generators PG(N − 1, 2) of W(2N − 1, 2).
We will now show that over K the variety LGr(N, 2N ) can further be projected bijectively to a subset of points of PG(2 N − 1, 2), where PG(2 N − 1, 2) is the projective space obtained by eliminating the variables involved in the linear conditions which define P(L) (i.e., the linear conditions given by the extension of σ to P(∧ N K 2N )). Let Expanding this expression, we obtain the local parametrization of Gr(N, 2N ): This shows that locally the coordinates of P can be written as where A = (a i,j ). Requiring P to be totally isotropic means that the vectors spanning P must be totally isotropic. Denoting u i = e i + j a i,j e N +j , we get σ(u s , u t ) = a st − a ts , which is zero if and only if A = (a ij ) is a symmetric matrix. Thus P will be totally isotropic if its coordinates locally correspond to minors of a symmetric matrix A over K.
The linear conditions defining P(L) correspond locally to the fact that the minors ∆ I,J (A) and ∆ J,I (A) are equal for I = J. Moreover, these conditions do not involve the coordinates corresponding to principal minors. Thus, we obtain a splitting ∧ N K 2N = V + W such that the coordinates defining V are locally given by minors of type ∆ I,J , whereas the coordinates defining W are principal minors ∆ I (A). But for symmetric matrices over K = GF (2)  Since all principal minors cannot vanish simultaneously, we obtain a well-defined projective map π : LGr(N, 2N ) → P(W ) = PG(2 N − 1, 2). The map π sends P to p ∈ P(W ), where p is defined by the coordinates of P not occurring in the equations defining V . All in all, we obtain a bijective mapping by projecting LGr(N, 2N ) to PG(2 N − 1, 2) after eliminating all the variables involved/occurring in the linear conditions.

An explicit construction of the bijection: a few examples
The above-given proof of the existence of the mapping π : LGr(N, 2N ) → PG 2 N − 1, 2 provides us with a recipe of how to obtain the equations of the image Indeed, following our reasoning one first needs to find the ideal I(Gr(N, 2N )) (i.e., a set of equations) defining Gr(N, 2N ), as well as the linear conditions J = (l 1 , . . . , l m ) induced by the associated symplectic form. These two sets of equations will then define the ideal of LGr(N, 2N ), i.e., Then we calculate the ideal of the projection π(LGr(N, 2N )) by eliminating in I(LGr (N, 2N )) all the variables appearing in J. The last step can be done by hand when cases are rather simple, or be handled with the formalism of the elimination theory [5] when calculations become more tedious. This formalisms provides algorithms to compute the ideal of the projection (more precisely, the ideal I such that the variety V (I) contains the projection). In practice, however, with increasing N we quickly face insurmountable computational difficulties, as explicitly pointed out at the end of this section. We shall now illustrate this approach on the first three cases in the sequence.   We are only interested in the Lagrangian grassmannian LGr (3,6), that is in those planes of PG(5, 2) that are totally isotropic with respect to a given symplectic polarity. Choosing the latter to have again the 'canonical' form (equation (1)), These last four equations are, however, not independent, as each of them is equal to the sum of the remaining three. Moreover, summing the first of them with the third one, or the second with the fourth, yields p 123 p 456 + p 126 p 345 + p 135 p 246 + p 156 p 234 = 0, which after relabeling the variables as x 1 = p 123 , x 2 = p 126 , x 3 = p 135 , x 4 = p 156 , x 5 = p 456 , x 6 = p 345 , x 7 = p 246 and x 8 = p 234 reads and is readily recognized to represent a hyperbolic quadric Q + (7, 2) in a particular subspace PG(7, 2) of the ambient projective space PG(19, 2) of Gr (3,6). More precisely, the quadric defined by equation (8)  LGr(3, 6) ⊂ K 14 as a graph over the quadric Q + (7, 2) defined by equation (8) in K 8 , i.e., LGr(3, 6) = (x, g(x)) ∈ K 14 , x ∈ Q + (7, 2) ⊂ K 8 .

One thus automatically gets a bijection between
LGr(3, 6) and the Q + (7, 2) by taking the projection to the base of the graph K 8 ⊕ K 6 → K 8 . This procedure can be rephrased in algebraic terms in the framework of elimination theory [5]. Given an ideal I ⊂ K[x 1 , . . . , x n ], the l-th elimination ideal, with 1 ≤ l ≤ n, is the ideal of K[x l+1 , . . . , x n ] defined by Let π l be the projection K n → K n−l defined by π l (a 1 , . . . , a n ) = (a l+1 , . . . , a n ). If V (I) = {(a 1 , . . . , a n ) ∈ K n , f (a 1 , . . . , a n ) = 0, ∀ f ∈ I} is an affine variety corresponding to the ideal I, then π(V (I)) ⊂ V (I l ), i.e., the projection of V (I) is contained in the algebraic variety defined by the elimination ideal (which is, in fact, the smallest affine variety containing π(V )). Using the notion of Groebner basis, one can compute the elimination ideal I from the fact that where G is the Groebner basis of I and G l that of I l . To perform this calculation, it suffices to choose a monomial order adapted to eliminate the first l variables.
In both examples (N = 3 and N = 4) the set of equations obtained provides not only a set of equations which cuts out the variety π(LGr (N, 2N )), but also the ideal I of the variety. This is obvious for N = 3, because the ideal is principal and generated by an irreducible polynomial.
For N = 4 we can directly check with Macaulay2 that the ideal I of equation (9) is prime, being thus also the ideal of π (LGr(4, 8)). We shall return to this point in Section 6. As N increases, the calculations are more and more time-consuming and put also big demand on memory resources. The case N = 5 was already out of reach for our computers.
5 Stratif ication of PG(3, 2), PG(7, 2) and PG (15,2) Let us consider the natural action of the group G = SL(2, 2) × SL(2, 2) × · · · × SL(2, 2) S N on PG(2 N − 1, 2). This action partitions the set of points of the projective space in terms of Gorbits and allows us, thanks to the bijection described in Section 3, to partition LGr (N, 2N ) in terms of G-equivalent classes. Knowing a representative of each orbit, we can use the equations obtained in the previous section to check whether a particular orbit does or does not belong to π(LGr (N, 2N )). The G-action preserves also different notions of rank. First, it preserves the tensor rank (T -rank), which can be defined as follows: if p ∈ PG(2 N − 1, 2) we will say that p where v i j ∈ (K 2 ) i and k is the minimal integer such that this property holds (see [1,2] for recent work on tensor rank over GF (2)). Another notion of rank, more interesting in our situation, is that of exclusive rank, E-rank, as proposed in [18]. The E-rank is only defined for points p of π (LGr(N, 2N )). Given p ∈ π (LGr(N, 2N )) there exists a unique P ∈ LGr(N, 2N ) which is in local coordinates defined by a symmetric matrix A (Section 3 and equation (6)). Then we will say that p is of E-rank k if, and only if, all (k + 1) × (k + 1) exclusive minors of A, i.e. minors ∆ I,J (A) with I ∩ J = ∅, are zero.
In the following examples, we use the classification of G-orbits of points of PG(7, 2) and PG (15,2) obtained by Bremner and Stavrou [2] to partition the sets of maximal sets of mutually commuting N -qubits operators (for N = 2, 3 and 4) and provide information on sizes, representatives in PG(2 N − 1, 2), corresponding observables as well as ranks of these sets.

Two distinguished classes of mutually commuting two-qubit operators
It is well known (see, e.g., [8]) that the projective space PG (3,2) is the union of two G-orbits, PG(3, 2) = O 1 ∪ O 2 , with O 1 = 9 and O 2 = 6. The orbit O 1 , comprising the points lying on a hyperbolic quadric Q + (3, 2), corresponds to the G-orbit of any separable vector in the tensorial basis (for example, the orbit of x 1 1 ⊗ x 1 2 ). On the other hand, O 2 , consisting of six off-quadric points, is the orbit corresponding to non-separable tensors (for example, the orbit of x 0 1 ⊗ x 0 2 + x 1 1 ⊗ x 1 2 ). Our bijection associates the two orbits of PG(3, 2) with two distinguished classes of maximal sets of mutually commuting two-qubit operators, as described in Table 1. The projective line PG(1, 2) a , spanned by XI, IX , is obviously mapped to [0 : 0 : 1 : 0] by our construction. Indeed, according to equations (2) and (3), the observables XI and IX correspond to the points of W(3, 2) having the coordinates (0, 0, 1, 0) and (0, 0, 0, 1) and these two points define the line represented by the matrix The partition of PG(3, 2) into two orbits O 1 and O 2 tells us that we can similarly partition LGr(2, 4) into two classes of lines; a class of cardinality 9, which is the G-orbit of PG(1, 2) a , and a class of cardinality 6, which is the G-orbit of PG(1, 2) b .

Three distinguished classes of mutually commuting three-qubit operators
The projective space PG (7,2) is the union of five G-orbits (see [2,8,13]), PG (7,2) It is also well known (see, e.g., [8]) that Q + (7, 2) = O 1 ∪ O 2 ∪ O 4 . Hence, in light of our main result of Section 4.2, the variety π(LGr(3, 6)) is partitioned into three different G-orbits whose properties are summarized in Table 2; here, we used the explicit expression of π given in [14] and the representatives of the orbits O i were taken from [2] (transformed, of course, into our adopted system of coordinates). To illustrate how we assign a projective plane of order two to a representative of O i , let us detail the calculation for the orbit O 2 . A representative of the second non-trivial orbit in the classification of [2] is, in the tensorial basis, x 0 1 ⊗ x 1 2 ⊗ x 1 3 + x 1 1 ⊗ x 0 2 ⊗ x 1 3 , which in our notation corresponds to x 4 = x 7 = 1. Using the labeling of the Plücker coordinates given in Section 4.2 this means that p 156 = p 246 = 1, the remaining coordinates being zero. The 3 × 6 matrix satisfying these conditions is of the form  LGr (3,6) into three different classes is also obvious.

Six distinguished classes of mutually commuting four-qubit operators
The stratification of PG (15,2) in terms of 29 G-orbits was also established in [2]. In order to identify the orbits which partition the variety π (LGr(4, 8)), we checked the representative of each orbit, taken from Table 5 of [2], and found out that six of them annihilate the polynomials of the ideal I (see equation (9)). The results of our calculations are portrayed in Table 3 (here the first non-trivial orbits is denoted O 2 to be compatible with the numbering of [2], which also takes into account the trivial orbit). Table 3. Classes of mutually commuting 4-qubits operators; here, PG(3, 2) a = XIII, IXII, IIXI, IIIX , PG (3,2)  To identify the PG(3, 2)'s that correspond to the representatives of the orbits we proceed similarly as in the previous two cases, that is, we create the 4 × 8 matrix whose minors satisfy the conditions implied by the corresponding representative.

The Lagrangian Grassmannian and the variety of principal minors
At this point is is worth mentioning several papers [15,17,19] that deal with similar problems over the field of complex numbers and which are deeply related to the construction over GF (2) considered in this paper.
Let K = C and let Z N ⊂ P(C 2 ⊗ · · · ⊗ C 2 N times ) be the image of the following rational map [19]: with A being a symmetric complex matrix and X I = x i 1 1 ⊗ · · · ⊗ x i N N , i j = 0 if j / ∈ I, 1 if j ∈ I, a tensorial basis of (C 2 ) ⊗N . Z N is an algebraic variety, called the variety of principal minors of symmetric matrices, corresponding to the linear projection of the Lagrangian Grassmannian (over the complex numbers). The linear projection means that from the set of minors of cardinality 2N N we only keep the set of principal minors of cardinality 2 N , i.e. it is the type of the projection π defined in Section 3 over GF (2). However, in the complex case this projection is no longer a bijection, that is, the principal minors do not contain all the information on the Lagrangian Grassmannian. In particular, over C, as well as over any algebraically closed field of characteristic different from two, the off-diagonal entries of the symmetric matrices (Section 3) are determined by the principal minors (a 2 ij = ∆ i (A)∆ j (A) − ∆ i,j (A)) only up to the sign and, thus, the projection is generically two to one.
The motivation for studying Z N in the complex case comes from the principal minors assignment problem [9,10,15]. This problem asks for necessary and sufficient conditions for a collection of 2 N numbers to arise as the principal minors of an N ×N matrix. In the case of a symmetric matrix, a collection of 2 N numbers corresponds to its principal minors if and only of the corresponding point in P((C 2 ) ⊗N ) belongs to the variety Z N . This problem for symmetric matrices was solved by Oeding [17,19], who successfully described a set of degree-four polynomials which cut out the variety Z N . In particular, Oeding proved, using representation theory techniques, that a set of equations defining Z N are obtained by taking the G = SL 2 (C) × · · · × SL 2 (C) S N orbit of a certain peculiar quartic polynomial (the so-called 2 × 2 × 2 Cayley hyperdeterminant). Oeding's result also provides a set-theoretical solution to a conjecture of Holtz and Strumfels [10] which says that the G-orbit of the Cayley hyperdeterminant generates the ideal of the variety Z N .
It is, naturally, tempting to rephrase Oeding's result and the conjecture of Holtz and Strumfels into the GF (2) where k = N + k. For N = 3, Q = 0 is readily recognized to be identical to equation (8) defining π(LGr(3, 6)). The quadratic polynomial Q of equation (8) is nothing but the irreducible factor of the Cayley hyperdeterminant which can be written as Q 2 over GF(2) (see Remark 18 of [11]). Thus, for N = 3, both Oeding's result and Holtz and Strumfels' conjecture are true over GF (2). For N = 4, the distinguished polynomial Q coincides with our Q 8 appearing in the ideal defined by equation (9). It can be shown that the G-orbit of Q = Q 8 consists of the polynomials Q 1 , Q 2 , . . . , Q 8 and Q 0 . However, via the repeated action of the generators of G we did not manage to get the remaining four-term polynomials Q 9 and Q 10 , merely their sum Q 0 = Q 9 +Q 10 . This means that in the N = 4 case, the G-orbit of the Cayley hyperdeterminant over GF (2) does not generate the ideal of the variety of principal minors, i.e. the Holtz-Strumfels conjecture is not valid. However, Oeding's result remains true, for one can readily check that the set Q 1 , Q 2 , . . . , Q 9 and Q 0 indeed cuts out the variety π (LGr(4, 8)). This case thus features over GF(2) some subtle properties that have no counterpart over the field of complex numbers. Nevertheless, we are convinced that the approach developed by Oeding is a very promising one, which can be appropriately adjusted/modified to be meaningful also over the smallest Galois field.

Conclusion
In this paper, we gave, for any N ≥ 2, a rigorous proof of the existence of a bijection between the set of generators of the symplectic polar space W (2N − 1, 2) and a distinguished subset of points of W(2 N − 1, 2). Physically speaking, we established a one-to-one mapping between the maximal sets of pairwise commuting operators of the N -qubit Pauli group and a subset of the 2 N −1 qubit observables. Proving this correspondence, we also found a method how to get the defining equations of the image of the mapping within PG(2 N − 1, 2) and explicitly illustrated this method for the cases N = 2, N = 3 and N = 4.
The image of our mapping is over the complex numbers known as the variety of principal minors of symmetric matrices [17,19]. We have also pointed out that the calculations in the GF(2)-regime deserve a special treatment and are, in general, not the direct translation of the results obtained over the field of the complex numbers. Since GF(2)-settings have already acquired a firm footing in the context of Quantum Information Theory, we aim at getting deeper insights into the variety of principal minors of GF(2)-symmetric matrices employing, if possible, a more coordinate-free approach.