Graph States and the Variety of Principal Minors

In Quantum Information theory, graph states are quantum states defined by graphs. In this work we exhibit a correspondence between graph states and the variety of binary symmetric principal minors, in particular their corresponding orbits under the action of $SL(2,\mathbb F_2)^{\times n}\rtimes \mathfrak S_n$. We start by approaching the topic more widely, that is by studying the orbits of maximal abelian subgroups of the $n$-fold Pauli group under the action of $\mathcal C_n^{\text{loc}}\rtimes \mathfrak S_n$, where $\mathcal C_n^{\text{loc}}$ is the $n$-fold local Clifford group: we show that this action corresponds to the natural action of $SL(2,\mathbb F_2)^{\times n}\rtimes \mathfrak S_n$ on the variety $\mathcal Z_n\subset \mathbb P(\mathbb F_2^{2^n})$ of principal minors of binary symmetric $n\times n$ matrices. The crucial step in this correspondence is in translating the action of $SL(2,\mathbb F_2)^{\times n}$ into an action of the local symplectic group $Sp_{2n}^{\text{loc}}(\mathbb F_2)$ on the Lagrangian Grassmannian $LG_{\mathbb F_2}(n,2n)$. We conclude by studying how the former action restricts onto stabilizer groups and stabilizer states, and finally what happens in the case of graph states.

In Quantum Information, stabilizer states are quantum states known in particular for quantum error correcting codes [8].In the stabilizer formalism, a stabilizer state is described by a maximal n-fold abelian subgroup S M 1 ,...,Mn of the n-fold Pauli group P n (see Sec.2 for definitions) that stabilizes it.Graph states are a special class of quantum stabilizer states elegantly described by a graph G = (V, E) that encodes its stablizer group.Graph states have many applications in quantum information processing [12]: they are in particular useful for Measured Based Quantum Computation (MBQC) [12,3,21], for quantum error correcting codes [8] and for secret sharing [20,1].As a resource for quantum information, it is interesting to propose classification of graph states.A natural framework to classify quantum states is to consider the group of local unitary operations LU.However for stabilizer states (and graph states), one usually restricts to considering the group of local unitaries within the Clifford group [26,27].We will denote by C loc n ⊂ LU the group of local Clifford acting on n qubit states.Under the action of C loc n ⋊ S n , graph states have been classified up to n = 12 qubits [12,5,4].The variety Z n of principal minors for n×n symmetric matrices over a field K [23] is an algebraic variety of P(K 2 n ) introduced by Holtz and Sturmfels [13] in order to study relations among principal minors of symmetric matrices.The existence problem of a matrix statisfying predefinite conditions on its principal minors has many applications to matrix theory, probability, statistical physics and computer vision [9,15,24].The goal of this paper is to show another potential application of the study of this variety over the two-elements field K = F 2 by establishing a bijection between classes of graph states and orbits of the variety Z n .More precisely, the main result of this paper is the following theorem.
Theorem 1.The Lagrangian mapping induces a bijection between the C loc n ⋊ S norbits of maximal abelian subgroups of P n and the (SL(2, F 2 ) ×n ⋊ S n )-orbits of Z n ⊂ P(F 2 n 2 ).In particular, there is a one to one correspondence between representatives of the graph states classification, up to C loc n ⋊ S n -action, and the representatives of the (SL(2, F 2 ) ×n ⋊ S n )-orbits of Z n .
Regarding the cardinality of the orbits, the Lagrangian mapping (Sec.2) shows that, if O i is a (SL(2, F 2 ) ×n ⋊ S n )-orbit of Z n , then the number of corresponding stabilizer states is 4 n |O i |.
Maximal (n-fold) abelian subgroups of P n correspond to subspaces of maximal dimension in the symplectic polar space W(2n − 1, 2) of rank n and order 2 (see Sec.2).The bijection induced by the Lagrangian mapping between subspaces of maximal dimension in W(2n − 1, 2) and points of Z n was already established in [14] in order to generalize observations made in [19] regarding the case n = 3 and its connection with the so-called black-holes/qubits correspondence.More recently, that same bijection was also considered in [28] with motivating examples from supergravity theory.It was proven [28] that, over F 2 , Z n is the image of the Spinor variety and thus a Spin(2n + 1)-orbit.However, the correspondence of orbits as established in Theorem 1 was not proven in the former papers, neither was the connection with graph states classification.
The paper is organized as follows.In Sec.2 we recall the basic definitions regarding the n-qubit Pauli group, the symplectic polar space and the Lagrangian mapping.In Sec.3 and 4 we show how the C loc n ⋊ S n action on the symplectic polar space translates into an action on the variety of principal minors, proving the first part of Theorem 1.In Sec.5 and 6 we recall the definitions and basic properties of stabilizer states and graph states, and we complete the proof of our Theorem.Finally Sec.7 is dedicated to applications of our correspondence.

Preliminaries
In this section we recall the definitions of the n-fold Pauli group and the Clifford group, we introduce the symplectic polar space of rank n and order 2, encoding the commutation relations of P n , and finally we describe the Lagrangian mapping.The group of the elementary Pauli matrices is P 1 = iX, iZ, iY ⊂ U(2, C) where We notice that X 2 = Z 2 = Y 2 = I and the following commutation rules hold Moreover, #P 1 = 16 and its center is Definition.The n-fold Pauli group is From the commutation rules above it is clear that In particular, we can exhibit the following generators Notice that #P n = 4 • 4 n and its center is Clearly, this correspondence is not unique, but it depends on the coordinates we choose in F 2n 2 : for instance, another one-to-one correspondence is given by Definition.The n-fold Clifford group is the normalizer where U † = t U is the hermitian (i.e.conjugated transposed) of U .
It is known that , while CN OT st acts as CN OT on the s-th and t-th factors of P n ⊂ U(2, C) ⊗n and as the identity on the remaining ones.More precisely, there is the following action of CN OT st on the generators of P n : Since not all matrices in C n are decomposable (i.e. of the form U 1 ⊗ . . .⊗ U n with U i ∈ U(2, C)), it makes sense to give the next definition.
Definition.The n-fold local Clifford group is the subgroup By definition, the local Clifford group C loc n acts on the Pauli group P n by conjugacy It is known that Let us explicit the action of C 1 on the elementary Pauli matrices: 2.2 The symplectic polar space W(2n − 1, 2) and I n By Remark 2.1 we know that the Pauli group quotient V n is in one-to-one correspondence with the vector space F 2n 2 .Let us denote the binary projective space P(F 2n 2 ) by P 2n−1 2 and let us fix the coordinates (3).Thus we have where the arrow is dashed since the above map is actually not defined in αI ⊗n .Up to the coefficients α's (which are considered as global phases), this association completely describes P n by projective points as a set, but it losts the commutation information we have in P n .Next we want to recover such information [11,18].Let M 1 , M 2 ∈ P n and let P 1 , P 2 ∈ P 2n−1 2 be the corresponding points.By a simple count it comes out that Consider the symplectic bilinear form •, • J on F 2n 2 (hence on P 2n−1

2
) where that is commuting Pauli group elements correspond to isotropic points with respect to •, • J .
Definition.The symplectic polar space of rank n over F 2 (with respect to •, • J ) is the set of (fully) isotropic subspaces of Let M ∈ P n and let P ∈ P 2n−1 2 be its associated point: then P defines the hyperplane in P 2n−1 2 Clearly, this hyperplane is not fully isotropic.However, we can extend this construction to any set of Pauli group elements with the only condition that they mutually commute: The condition for the M i 's to be mutually commuting implies that the P i 's are two-bytwo isotropic, but the subspace H P 1 ,...,P k is not fully isotropic in general.We know that, for any , H P is a hyperplane, hence it has (projective) dimension 2n − 2. Generally, given P 1 , . . ., P k ∈ P 2n−1 2 two-by-two isotropic, the subspace H P 1 ,...,P k does not have dimension 2n − 1 − k, but it holds so if we start from k mutually commuting Pauli group elements M 1 , . . ., M k with the additional condition to be independent, in the sense is the subspace generated by the columns of S and Col(S) ⊥ is its orthogonal for the symplectic form , J .
Proof.(i) ⇔ (ii) follows from the definitions: the condition of being independent is equivalent to require that the matrix S has rank k while the condition of being mutually commuting translates to t SJS = 0. (ii) ⇔ (iii): The condition on the dimension is equivalent to the fact that S has rank k.By definition, H P 1 ,...,Pn = Col(S) ⊥ and Col(S) ⊂ Col(S) ⊥ is equivalent to the condition t SJS = 0.
Remark 2.3.The condition Col(S) ⊂ Col(S) ⊥ imposes some restrictions on k for the maximal number of mutually communting and independent elements M 1 , . . ., M k .Indeed, Col(S) By Remark 2.3, it follows that in order to actually reach out subspaces of type H P 1 ,...,P k (with P i 's mutually commuting and independent) which are fully isotropic (i.e.H P 1 ,...P k ∈ W(2n − 1, 2)) we need to impose dim H P 1 ,...,P k = 2n − 1 − k = n − 1, that is k = n.The (n − 1)-dimensional fully isotropic subspaces of P 2n−1 2 are known as generators of W(2n − 1, 2) and we denote their set by By Proposition (2.2)(ii), it follows that every W ∈ I n is of the form H P 1 ,...,Pn where P 1 , . . ., P n come from mutually commuting and independent Pauli group elements M 1 , . . ., M n via the following correspondence: We recall that the Grassmannian Gr F 2 (n, 2n) is the set of n-dimensional subspaces of F 2n 2 : we will write Gr(n, 2n) by omitting the ground field F 2 .The Grasssmannian gains the structure of projective variety via the Plücker embedding [10] Pl : Gr(n, 2n By abuse of notation, we simply look at Gr(n, 2n) as a subset of P n F 2n 2 .It is useful to describe the Grassmannian by its parameterizations in P on the standard open subsets for any index subset {i 1 , .
a ji e n+j (11) and, by putting the basis vectors in columns, this gives the matrix is parameterized by all minors of n × n matrices with coefficients in F 2 .
Remark 2.6.The parameterization of the open subset U {1,...,n} ⊂ Gr(n, 2n) by all minors (of any size) of n × n matrices (as well as the one of any subset U I ) can be seen as induced by all maximal minors (of size n × n) of 2n × n matrices of the form In A as A ∈ Mat n (F 2 ) varies: indeed, it is enough to project onto the coordinates given by minors fully contained in A.
Let us restrict the Plücker embedding (9) to I n ⊂ W(2n − 1, 2): more formally, we have to apply the Plücker embedding to the set of n-dimensional vector subspaces of F 2n 2 whose projectivizations are in I n .
Definition.The Lagrangian Grassmannian LG F 2 (n, 2n) is the image via the Plücker embedding of Thus the Lagrangian Grassmannian is the projective variety parameterizing all (n − 1)dimensional (fully) isotropic subspaces of P 2n−1 2 : again, we write LG(n, 2n) by omitting the ground field.
Definition.Since LG(n, 2n) ⊂ Gr(n, 2n), for any standard open subset U I ⊂ Gr(n, 2n) as in (10) we denote its restriction to the Lagrangian Grassmannian by Our next goal is to find parameterizations of LG(n, 2n) in P ( 2n n )−1 2 [17].
Given (v 1 , . . ., v n ) a Plücker basis of H P 1 ,...,Pn ∈ I n ∩ U {1,...,n} , by isotropicity we have v i , v j J = 0 for all i, j = 1 : n: then from (11) we get a ki e n+k , e j + n l=1 a lj e n+l J = 0 a lj e l + e n+j I = 0 ) is parameterized by all minors of n × n symmetric matrices with coefficients in F 2 .For instance, the coordinates in the standard open subset LU {1,...,n} are Among all minors A {i 1 ...i k }{j 1 ...j k } of a n × n symmetric matrix A ∈ Sym 2 (F n 2 ), we can restrict ourselves to consider the principal ones, that are the ones such that i 1 = j 1 , . . ., i k = j k : we denote them by Clearly, the principal minors appear in the coordinates of the Lagrangian Grassmannian: in a very naïf way, we can write Thus it makes sense to consider the rational projection of the Langrangian Grassmannian LG(n, 2n) onto the coordinates represented by principal minors: since the number of principal minors of a n × n matrix is n q=0 n q = 2 n , we have that such rational projection has values in 1 : a ji : A {i,j}{s,t} : . . .: det A → 1 : a ii : A [i,j] : . . .: det A .
Definition. [22] The image of LG(n, 2n) via the rational projection π is the variety of principal minors of n × n symmetric matrices.We denote it by Moreover, for any standard open subset LU I ⊂ LG(n, 2n) we denote its image in Z n by ZU I := π(LU I ) .
Over a generic field K, the projection π is just surjective, but over F 2 it is injective too.

Proposition 2.7 ([14]
).The projection π : LG Fix an open subset LU I ⊂ LG(n, 2n): then any subspace in it is parameterized by a certain A ∈ Sym 2 (F n 2 ).The off-diagonal entries of A are determined by the 2 × 2 principal minors: indeed Let us recap all the objects we have worked with so far in a unique diagram: 3 Orbits in Z n induced by the action C loc n P n In this section we discuss the correspondence between the action of the group C loc n on W(2n − 1, 2) and the action of the group SL(2, F 2 ) ×n on Z n .
3.1 Sp loc 2n (F 2 ) acting on W(2n − 1, 2) and I n We recall the conjugacy action (4) of the local Clifford group on the Pauli group and its induced action on the maximal abelian subgroups We are interested in understanding how this action translates onto the symplectic polar space W(2n − 1, 2), in particular onto I n , via the correspondence (7).
Remark 3.1.We have to underline that the translated action on W(2n − 1, 2) will depend on the choice of coordinates we take in P 2n−1

2
. From now on we will work with the choice of coordinates (3) together with the symplectic form J, but all the results also hold with respect to the choice of coordinates in (2) and the symplectic form over Fix the coordinates (3) in P 2n−1 2 and the symplectic form •, • J .The first natural step is to translate the conjugacy action C loc n P n into linear transformations of P 2n−1 2 .Remark 3.2.By definition, any element of the Clifford group C n induces an automorphism of P n , hence an automorphism of V n = P n /Z(P n ): but automorphisms of V n ≃ F 2n 2 are linear maps and they preserve commutators, hence also the symplectic form J on F 2n 2 is preserved.It follows that there exists a well-defined homomorphism 2 ) is surjective, since the symplectic group is spanned by symplectic transvections [25, Sec.II.B], but it is not injective: its kernel is exactly the Pauli group P n [16], thus one has the isomorphism C n /P n ≃ Sp(F 2n 2 ).
The homomorphism C n → Sp(F 2n 2 ) in Remark 3.2 restricts to an homomorphism The elements in the image of the above restriction are of the form as one can check by looking at the action of C loc n on the Pauli elements: by applying a given U = U 1 ⊗ . . .⊗ U n ∈ C loc n to the generators of P n in (1), we get which in coordinates corresponds to Ũ (0, . . ., µ k , . . ., ν k , . . ., 0) = (0, . . ., (18) depend on the matrices ), but they are not the same (the former have coefficients in F 2 , the latter in C).Moreover, since Ũ is symplectic, it holds det Ũi = a i d i − b i c i = 0 for any i = 1 : n.
Definition.We define the local symplectic group Sp loc 2n (F 2 ) to be the image of the homomorphism (17), i.e.
Remark 3.4.The surjective homomorphism C loc n → Sp loc 2n (F 2 ) is not injective since its kernel is the Pauli group P n .Indeed, for any two Pauli elements U, A ∈ P n it holds U AU † = βA for a suitable phase β ∈ {±1, ±i}, thus in the quotient space 2 ).In particular, we get the isomorphism From now on, we will denote by Ũ ∈ Sp loc 2n (F 2 ) the symplectic matrix corresponding to U ∈ C loc n .By (19) we can explicit the projective coordinates in Next we translate the action onto W(2n−1, 2): it immediately follows from the relation (even for non-isotropic subspaces) Moreover, the local symplectic transformations preserve the dimensions of the (fully) isotropic subspaces in , hence the subspace We can update the correspondence (7) to the following one: Remark 3.5.By definition and by Remark 3.3, the local symplectic group Sp loc 2n (F 2 ) is isomorphic to SL(2, F 2 ) ×n ≃ S ×n 3 : 18) → ( Ũ1 , . . ., Ũn ) .
Remark 3.6.The same arguments and results of this section hold if we fix the coordinates (2) in F 2n 2 and the symplectic form •, • J ′ .In this case a local Clifford element However, we will keep working only in the coordinates setting ((3), J). ).Consider a subspace H P 1 ,...,Pn ∈ I n ∩ U {1,...,n} (with Plücker basis as in (11)) and a transformation Ũ ∈ Sp loc 2n (F 2 ) such that where S = (s ji ) ∈ Sym 2 (F n 2 ) and a i d i − b i c i = 0 for all i = 1 : n.By applying Ũ to the Plücker basis we obtain (24) where δ kj is the Kronecker symbol: this is equivalent to the matrix product We conclude that the action ( 22) Sp loc 2n Remark 3.7.We must pay attention to the fact that the coordinates of ( 26) are given by all maximal (i.e.n × n) minors of the Plücker matrix of ( 25) with respect to a suitable open subset LU I .
In general, the action by Ũ ∈ Sp loc 2n (F 2 ) does not preserve the standard open subsets: given H P 1 ,...,Pn ∈ LU {1,...,n} , the image Ũ (H P 1 ,...,Pn ) may lie in a different standard open subsets LU I , and thus one has to consider the Plücker matrix of the latter subspace with respect to LU I .For instance, in the notations of (25), the subspace Ũ (H P 1 ,...,Pn ) lies in LU {1,...,n} if and only if det(A + BS) = 0.

Sp
The action (27) translates into an action of Sp loc 2n (F 2 ) on via the projection π : LG(n, 2n) −→ Z n .By Remark 3.5 this action is equivalent, up to isomorphism, to an already known and natural action of SL(2, F 2 ) ×n on Z n : the space P 2 n −1 is homogeneous under the natural action of SL(2, F 2 ) ×n and the following result shows that this action restricts to an action of SL(2, F 2 ) ×n on Z n ⊂ P 2 n −1 2 .Proposition 3.8 ( [22], Theorem III.14).Z n is invariant under the natural action of SL(2, F 2 ) ×n .Moreover, the action is represented by giving exactly the isomoprhism SL(2, F 2 ) ×n ≃ Sp loc 2n (F 2 ).
Remark 3.9.Actually, in his PhD thesis [22] L.Oeding proved the above result over C, but it is straightforward that then it holds over F 2 too.
Resume.We conclude this section by resuming how orbits in the Pauli group P n under the action of the local Clifford group C loc n induce orbits in the variety of symmetric principal minors Z n under the action of SL(2, F 2 ) ×n .
The local Clifford group C loc n acts on the Pauli group P n by ( 4) By fixing the setting "coordinates -symplectic form" ((3),J), the above action induces the action (22) of the local symplectic group Sp loc 2n (F 2 ) on the set I n of (n − 1)dimensional (fully) isotropic subspaces of P 2n−1 2 defined as follows: given M 1 , . . ., M n ∈ P n mutually commuting and independent such that n , and given their corresponding points for any U ∈ Sp loc 2n (F 2 ) as in (18).The action of Sp loc 2n (F 2 ) on I n induces, via the Plücker embedding, the action (27) of the local symplectic group Sp loc 2n (F 2 ) on the Lagrangian Grassmannian LG(n, 2n) Finally, the latter action translates into the action (28) of Sp loc 2n (F 2 ) on the variety of symmetric principal minors Z n , which is equivalent to the natural action of SL(2, F 2 ) ×n on Z n via the representation (29).In this section we extend the previous group actions to the semidirect product with the symmetric group S n in order to prove the first part of Theorem 1.

The actions
By definition of the n-fold Pauli group there is a natural action of the symmetric group S n on P n permuting the tensor entries: Each permutation σ ∈ S n induces a transformation σ ∈ U (2 n , C) permuting the basis vectors, so that for any σ ∈ S n and for any A 1 ⊗ . . .⊗ A n ∈ P n one gets Notice that σ † = σ−1 .In particular, the above action preserves P n , thus for any σ ∈ S n it holds σ ∈ C n = N U (2 n ,C) (P n ): it follows that there is an injective homomorphism which allows to identify S n as a subgroup of the Clifford group C n .Moreover, the symmetric group S n naturally acts on the local Clifford group C loc n by conjugacy φ : where It follows that the subgroups C loc n and S n (the second up to isomorphism) generate a subgroup in C n which is isomorphic to the semidirect product acting on P n as follows where, if Remark 4.2.We recall that the operation in the semidirect product is and the following commutation rule holds = (U, σ) = (U, id) • φ (I, σ) .
Since φ −1 σ = φ σ −1 , by equality (a) it follows (in agreement with (34)) Since the elements in P n are of the form αZ µ 1 X ν 1 ⊗ . . .⊗ Z µn X νn for α ∈ {±1, ±i}, the action (32) of the symmetric group S n on P n can be equivalently described by Finally, from (21) we know that the action of a given depends on U i and it acts on the coefficients µ i , ν i 's.In particular, in the same notation, we have and the action (34) of the semidirect product C loc n ⋊ φ S n on P n can be rephrased as The above formula shows that the action of C loc n ⋊ φ S n on P n induces an action on the vectors (µ 1 , . . ., µ n , ν 1 , . . ., ν n ) ∈ F 2n 2 : more precisely, the action (37) induces an action on the quotient V n = P n /Z(P n ) which is isomorphic to the vector space F 2n 2 in the coordinates (3).< Sp(F 2n 2 ) with kernel P n .Actually, one can also consider the restriction to the subgroup C loc n ⋊ φ S n giving an homomorphism which is again not injective having kernel P n ⋊ φ S n .
In particular, any σ ∈ S n corresponds to a S σ ∈ Sp(F 2n 2 ) such that and this allows to identify S n as a subgroup of Sp(F 2n 2 ).
Finally, the symmetric group acts by conjugacy on Sp loc 2n (F 2 ) as follows which is well-defined since More precisely, if Ũ ∈ Sp loc 2n (F 2 ) is as in (18), then . . . . . .
We conclude that the restriction of where in the last line we formally identify (U, σ) with the Clifford transformation U σ ∈ C n and ( Ũ , σ) with the symplectic transformation Ũ S σ ∈ Sp(F 2n 2 ).We recall that the kernel of ( 42) is Moreover, from equations ( 37) and (41) it follows that, although the two semidirect products are not isomorphic, the above homomorphism translates the action of C loc n ⋊ φ S n on the Pauli group P n into the action of Sp loc 2n (F 2 ) ⋊ ϕ S n on the symplectic space (F 2n 2 , J), and viceversa.Thus we get a correspondence between orbits (up to phases in The next step is to extend the orbit correspondence (43) to an orbit correspondence between the set of n-fold maximal abelian subgroups in P n and the set of maximal fully isotropic subspaces in F 2n 2 (with respect to the symplectic form J) which are in bijection via (7).From diagram (23) we already have the correspondence between the orbits Action on S (P n ).Given n ∈ P n and σ ∈ S n , we denote Then, for any σ ∈ S n , the observables σ M i 's are such that and σ( 1) where the equalities (♣) and (♠) respectively follow from the commutation and the independence of the M i 's.It follows that σ M 1 , . . ., σ M n are independent and mutually commuting too: thus we get the following well-defined action Actually, since the symmetric group S n acts on the generators of a maximal abelian subgroup, the above action coincides with the one permuting both the entries of any generators and the generators among them, that is : given N = P 1 | . . .|P n the 2n × n matrix representing the subspace H P 1 ,...,Pn , the coordinates of [P 1 ∧ . . .∧ P n ] are given by the maximal minors (i.e. of size n × n) of N , that is where N I is the minor of N given by the I-indexed rows and all the n columns.
It is straightforward that the action (46) of Sp loc 2n (F 2 ) ⋊ ϕ S n on I n is equivalent to the following action on the Lagrangian Grassmannian (which extends the action ( 27)): where Ũ S σ P σ(i) are as in (47).
Remark 4.5.From Remark 3.7 we know that the action of Sp loc 2n (F 2 ) does not preserve the open subsets LU I defined in ( 13), hence the above action does not preserve them either.
We can translate the action (49) on the Lagrangian Grassmannian into an action on the set of full-rank 2n × n matrices.By (25) we already know that Sp loc 2n (F 2 ) acts on the full-rank 2n × n matrices by left-multiplication, that is a transformation By substituting the identity matrix Ũ = I in (47) we deduce that a permutation σ ∈ S n acts on a full-rank 2n × n matrix where the n × n matrix σ F (resp.σ G) is obtained by the n × n matrix F (resp.G) by permuting both columns and rows by σ ∈ S n : more precisely, if where A σ is the n × n permutation matrix defined by σ, then the action of σ onto corresponds to the conjugacy action by A σ onto F and G separately, that is Remark 4.6.By (51) the above action preserves the full-rankness.Moreover, from the matrix setting we deduce that, analogously to Remark (3.7), the action (49) of S n on LG F 2 (n, 2n) does not preserve the open subsets LU I either.For instance, denote by E ij the n × n matrix having 1 in the entry (i, j) and zero everywhere else: then lies in the open subset LU {i,j,n+1,...,2n}\{n+i,n+j} but the permutation σ = (ik)(jℓ) maps it into the matrix which does not lie in LU {i,j,n+1,...,2n}\{n+i,n+j} .However, the action of S n preserves the open subset LU {1,...,n} : indeed, σ By putting together (50) and (51) we conclude that the action (49) of Sp loc 2n ⋊ ϕ S n on LG F 2 (n, 2n) is equivalent to the restriction onto full-rank matrices of the action where Action on Z n .By Proposition 2.7 the Lagrangian Grassmannian LG F 2 (n, 2n) and the variety of binary symmetric principal minors Z n are in bijection, thus we can easily conclude that the action (49) of Sp loc 2n (F 2 ) ⋊ ϕ S n on LG F 2 (n, 2n) induces an action on Z n making the following diagram commutative: In the following we show that this induced action on Z n actually is a natural one.Fix (v 0 , v 1 ) a basis of F 2 2 and the induced basis |i 1 . .
By Proposition 3.8 we know that the natural action of SL(2, F 2 ) . Another natural action on P F 2 n 2 is the one of the symmetric group S n given by which permutes the tensor product entries: it is known that this action preserves Z n .
This agrees with the end of Remark 4.6.But, in general, this action does not preserve the open subsets LU I .
Moreover, the action of S n on P F 2 2 ⊗ . . .⊗ F 2 2 permuting the tensor entries induces an action of S n on SL(2, F 2 ) ×n permuting the entries of the direct product (and the latter corresponds to the action (40) via the isomorphism SL(2, F 2 ) ×n ≃ Sp loc 2n (F 2 )).We conclude that there is a natural action and that it actually corresponds, via the isomorphism SL(2, F 2 ) ×n ≃ Sp loc 2n (F 2 ), to the action on Z n in (53).
Conclusion.This section and Section 3 achieve the proof of the first part of Theorem 1 by establishing the bijection between the C loc n ⋊ S n -orbits of maximal abelian subgroups of P n , or equivalently maximal fully isotropic subspaces of W(2n − 1, 2), and the (SL(2, F 2 ) ×n ⋊ S n )-orbits of Z n ⊂ P(F 2 n 2 ): 5 Stabilizer states and their orbits under In this section we focus on a subset of the set S (P n ) of the maximal (n-fold) abelian subgroups of the Pauli group P n , and show that the action (45) restricts to an action on this subset.
Definition.A subgroup S < P n is called stabilizer group if it is abelian and −I ⊗n / ∈ S. In particular, S ∈ S (P n ) is a maximal stabilizer group if it does not contain −I ⊗n .We denote the set of maximal stabilizer state groups in P n by S + (P n ), or simply S + n .
Let S = M 1 , . . ., M n ∈ S + n : since the M i 's are mutually commuting, they admit (at least) one common eigenvector |φ ∈ (C 2 ) ⊗n .Moreover, recall that for any M ∈ P n it holds M 2 = ±I ⊗n : in particular, for any M ∈ S one gets M 2 = I ⊗n since −I ⊗n / ∈ S and M 2 ∈ S. It follows that elements in S can only have eigenvalues +1 or −1.
For any abelian subgroup S < P n not containing −I ⊗n the set of the common eigenvectors with eigenvalue +1 of S forms a subspace V S , called stabilized subspace or stabilizer code of S, and its dimension is dim Definition.A (n-qubit) stabilizer state is the (unique up to phase) common eigenvector with eigenvalue +1 of a maximal stabilizer state S ∈ S + n .We denote by Φ n 1 the set of n-qubit stabilizer states.
There is a one-to-one correspondence between Φ In particular, the action of C loc n on S + n is given by and it is equivalent to the action of C loc n on Φ n 1 given by for any i = 1 : n.By applying σ ∈ S n we get for any i = 1 : , where (♣) follows by definition of the action of S n on (C 2 ) ⊗n .By (59) the above chain of equalities gives . Since this holds for any i = 1 : n, then σ |φ is a common eigenvector with eigenvalue +1 of σ M 1 , . . ., σ M n = σ M σ(1) , . . ., σ M σ(n) = σ (S |φ ).But σ (S |φ ) is a maximal stabilizer group, thus σ |φ actually is its unique eigenvector with eigenvalue +1, that is σ |φ ∈ Φ n Moreover, the above action corresponds to an action on the stabilizer states By putting together the previous actions we get that the action (45) of C loc n ⋊ S n on S (P n ) restricts to the action In particular, this action corresponds to the action on the stabilizer states where, if Conclusion.This section proves an intermediate step to the second statement of Theorem 1. From Section 4 we have the one-to-one correspondence (56) between the orbits of maximal abelian subgroups under C loc n ⋊ S n and the orbits in Z n under SL(2, F 2 ) ×n ⋊ S n .But the Lagrangian mapping (Sec.2) associates maximal fully isotropic subspaces in I n to maximal abelian subgroups in S (P n ) up to phasis (since we work into the quotient V n = P n /Z(P n )): this means that the Lagrangian mapping does not distinguish between maximal abelian subgroups containing −I ⊗n and maximal stabilizer states.It follows that there is a one-to-one correspondence between the orbits of stabilizer states under C loc n ⋊ S n and the orbits in Z n under SL(2, F 2 ) ×n ⋊ S n : 6 Graph states and their orbits under In this section we investigate how the action (63) of C loc n ⋊ S n on Φ n 1 behaves with respect to a privileged subset of stabilizer states, the so-called graph states.Remark 6.1.In [26] it is pointed out that the action of C loc n does not preserve this subset, hence the action (63) does not either: in particular, one cannot talk about orbits of graph states under C loc n ⋊ S n .However, we will improperly refer to "orbit" of a graph state as to the set of all graph states belonging to the same orbit in Φ Consider a (non-oriented) graph Γ = (V, E) defined by a finite set of vertices V = {1, . . ., n} and a set of edges E ⊂ V × V .We can associate to Γ a unique symmetric matrix θ ∈ Sym 2 (F n 2 ) such that (the matrix is indeed symmetric since the graph is not oriented).The graph Γ (or equivalently its matrix θ) defines n elements in the Pauli group P n or equivalently, in a more compact notation, where δ ij is the Kronecker symbol: these elements are independent and mutually commuting, thus they generate a maximal abelian subgroup which we denote by S Γ = S θ ∈ S (P n ).Moreover, the maximal fully isotropic subspace H θ ∈ I n corresponding to S θ is described (in the coordinates (3)) by the 2n × n matrix θ In (it is already in a Plücker form) and belongs to the chart LU {n+1,...,2n} ⊂ LG F 2 (n, 2n).Remark 6.2.The association Γ → S θ is not unique: indeed, instead of defining the elements M i 's in (65), one could choose to associate to Γ the elements and J ∈ Sp loc 2n (F 2 ).On one hand, the choice of using the M ′ i allows to have an immediate transcription in the variety of symmetric principal minors Z n : via the Lagrangian mapping, the maximal abelian subgroup S ′ θ corresponds to the point [1 : θ ii : θ [i,j] : . . .: det θ] ∈ Z n .On the other hand, the choice of using the M i is more common in the literature.Thus in this section we are going to work with the association Γ → S θ , but in Section 7 we will use the one Γ → S ′ θ for exhibiting examples.Now we wonder if the maximal abelian subgroup S θ ∈ S (P n ) defined by a graph Γ θ (with adjacent matrix θ) is a maximal stabilizer group.In general, the answer is negative unless we add the assumption that the graph is loopless (a.k.a simple), i.e. θ ii = 0 for any i = 1 : n.This fact seems to be well known and implicit in the literature but we haven't be able to find a proof, thus we propose one.Proposition 6.3.Let Γ θ be a (non-oriented) graph and let S θ ∈ S (P n ) be the associated maximal abelian group S θ via (65).Then S θ is a maximal stabilizer group if and only if the graph Γ θ is loopless: n and that, by contradiction, the graph has at least one loop, that is there exists i 0 ∈ {1, . . ., n} such that θ i 0 i 0 = 1.Then the element [⇐] Assume Γ θ to be loopless.Then for any i = 1 : n it holds M 2 i = I ⊗n , that is M i can only have eigenvalues +1 and −1.Since the M i 's are mutually commuting, there exists (at least) one common eigenvector, say |γ ∈ (C 2 ) ⊗n : then for any i = 1 : n we get M i |γ = (−1) c i |γ .In particular, for any i = 1 : n it holds (−1) c i M i |γ = |γ , that is |γ is common eigenvector with eigenvalue +1 of the n elements (−1) c i M i 's.But we can always find a local Clifford element U ∈ C loc n such that U (−1) c i M i U † = M i for all i = 1 : n: if c i 1 , . . ., c i k are the only non-zero exponents, then by (5) In which describes the subgroup σ • S θ = σ (S θ ) = σ M σ(1) , . . ., σ M σ(n) : the matrix σ θ is still symmetric and uniquely defines a graph σ Γ = Γσ θ .It follows that S n preserves S Θn and the action (60) restricts to the action Remark 6.4.The action (66) reflects an action on the n-graphs described as follows: given a graph Γ and a permutation σ ∈ S n , the graph σ Γ is obtained simply by renaming the vertices from {1, . . ., n} to {σ(1), . . ., σ(n)}, without changing the edges.
say more: by [26, Theorem 1], each maximal stabilizer group (resp.stabilizer state) is C loc n -equivalent to a graph group (resp.graph state), thus any C loc n -orbit of stabilizer states admits a representative which is a graph states.This means that studying the orbits of stabilizer states under the action of C loc n ⋊ S n is the same as studying the orbits of graph states under the action of G n ⋊ S n : Conclusion: This section achieves the proof of Theorem 1, proving that the classification (up to C loc n ⋊ S n action) of (loopless) n-graph states is in a one-to-one correspondence with the orbits in Z n under the action of SL(2, F 2 ) ×n ⋊ S n : by putting together the correspondences (64) and (70) we get 7 Applications We propose two applications of Theorem 1. First, we show how the 4-qubit graph states classification can be deduced from the orbit stratification of Z 4 .Then, in the other direction, we show how the knowledge of the 5-qubit graph states classification can be used to obtain representatives of the orbits in Z 5 .

Classification of 4-qubit graph states from orbits of Z 4
Under the action of SL(2, F 2 ) ×4 ⋊ S 4 , the variety of principal minor Z 4 ⊂ P 15  2 splits in six orbits whose cardinalities and representatives are listed in [14,Table 3].Starting from that classification, we recover the orbits (in the sense of Remark 6.1) of 4-graphs under the action of C loc 4 ⋊ S 4 .
Remark 7.1.In order to be faithful to the notations we used in the first part of our work, we will consider representatives of the orbits O 2 , O 3 , O 6 , O 14 , O 17 , O 18 different from the ones in Table 3 [14].More precisely, we will take representatives in the chart {z 0 = 0} in order to recover more easily the corresponding graph-matrices.Moreover, since we want to obtain loopless graphs, we choose representatives whose coordinates z 1 = z 2 = z 3 = z 4 = 0 are zero: we denote by 0 those entries.By the Lagrangian mapping (Sec.2), we know that for each orbit O i of Z 4 there are 4 4 |O i | corresponding stabilizer states.Indeed for each isotropic 4-dimensional space in F 2n 2 , there are 4 different choices of the phases for the four elements that span it and therefore 4 4 different stablizer states.Thus, by the sizes in Table 3 [14] we recover the number of stabilizer states coming from each graph orbit.7.2 Orbits of Z 5 from 5-qubit graph states classification In [7] the number of orbits (in the sense of Remark 6.1) of 5-graphs was computed to be 11 (see also [12,Table IV]).In the tables (74) and (75), we exhibit 11 representatives of such orbits and by them we recover the representatives of the orbits in Z 5 ⊂ P(F 32 2 ).
Remark 7.3.The representative graphs in Table (74) and Table (75) are loopless and their vertices are not numbered because of Remark 6.4: as representative labeling, we choose the vertex 1 to be the highest one and the other vertices are clockwise ordered.
In the last column of Table (74), the observables generating the stabilizer groups are obtained by the columns of the matrix θ where θ is the matrix in the third column.In the last column of Table (75), the coordinates of the points in Z 5 ⊂ P 31 2 are given by the principal minors of the matrix θ (third column of Table ( 74)).In particular, the points are taken in the chart {z 0 = 0} and the entries z 1 = . . .= z 5 = 0 are all zero since the representative graphs are loopless (i.e.θ ii = 0 for all i = 1 : 5): for layout reasons, we compactly write such five coordinates as a zero vector 0.

2. 1
The Pauli group P n and the local Clifford group C loc n

Remark 4 . 3 .
By Remark 3.2 and (17), we know that there exists a group homomorphism C n → Sp(F 2n 2 ) restricting to an homomorphism C loc n ։ Sp loc(F 2 ) 2n

n 1 :
in order to get an actual action on the graph states, one should consider only certain local Clifford operations corresponding to graph transformations (cf.[26, Sec.IV, Definition 1]).
defining the subspace H ′ θ ∈ LU {1,...,n} described by the matrix In θ .However, this ambiguity does not affect our interests and results since the two different associations Γ → M i and Γ → M ′ i give maximal abelian subgroups S θ and S ′ θ which are in the same C loc n -orbit: indeed, J

Remark 7 . 2 .
The vertices of the representative graphs in Table (72) are not numbered because of Remark 6.4: as representative labeling, we choose the vertex 1 to be the upper-left one and the other vertices are clockwise ordered.
Sp loc 2n (F 2 ) acting on LG F 2 (n, 2n) Let us keep in mind the diagrams (23) and (15).We look for the action on LG(n, 2n) induced by the action Sp loc 2n (F 2 ) I n via the Plücker embedding: for simplicity, we describe the action on the standard open subset LU {1,...,n} but all arguments apply to any standard open subset LU I .and the symplectic form •, • J .By Remark 2.6, we consider the parametrization of LU {1,...,n} ⊂ LG(n, 2n) induced by all maximal minors of 2n × n matrices of the form In S as S varies in Sym 2 (F n 2 3.2

4
Correspondence between C loc n ⋊S n S (P n ) and SL(2, F 2 ) ×n ⋊ S n Z n Action on LG F 2 (n, 2n).By definition, LG F 2 (n, 2n) = Pl(I n ) ⊂ P n F 2n 2 , where Pl is the Plücker embedding: in particular, a maximal fully isotropic subspace H P 1 ,...,Pn = P 1 , . . ., P n F 2 ∈ I n corresponds to the point [P 1 ∧ . . .∧ P n ] ∈ LG F 2 (n, 2n).Moreover, one can write the point [P 1 ∧ . . .∧ P n ] in coordinates in P n 1 and S + n : we denote by S |φ the maximal stabilizer group associated to the stabilizer state |φ .Let us study how C loc the local Clifford tranformation Û = H i 1 • ...•H i k (where H is the Hadamard matrix and H j = I ⊗ ...⊗H j−th ⊗ ...⊗I) is the one doing the work.It follows that Û |γ is common eigenvector with eigenvalue +1 of M 1 , ..., M n , hence of S θ ∈ S + n .Since we are interested in the so-called graph states (which are stabilizer states) and how C loc n ⋊ S n acts on them, from now on we restrict to considering only the maximal abelian subgroups which are defined by loopless graphs: we underline once again that graphs with loops can only define maximal abelian subgroups which are not stabilizer, thus in these cases one cannot talk about graph states as stabilizer states.Definition.A (n-qubit) graph state |Γ ∈ (C 2 ) ⊗n is the stabilizer state (i.e. common eigenvector with eigenvalue +1) of a maximal stabilizer group S θ ∈ S + n defined by a nvertex (loopless) graph Γ θ .We denote the subset of n-qubit graph states by Θ n ⊂ Φ n 1 .Moreover, we define a graph group to be a maximal stabilizer group defined by a (loopless) graph and we denote the set of such subgroups by S Θn ⊂ S + n .Θn ↔ Θ n .Next, we wonder if the actions of S n and C loc n on S + n preserve Θ n : let us investigate the two actions separately.Action of S n .Let |Γ ∈ Θ n be a graph state defined by a graph Γ with adjacent matrix θ.The graph group S θ = M 1 , . .., M n ∈ S Θn (as in (65)) is described by the 2n × n matrix θIn .Given σ ∈ S n , by (51) we know that σ • θ In