Graph states and the variety of principal minors

Abstract

In Quantum Information theory, graph states are quantum states defined by graphs. In this work we exhibit a correspondence between orbits of graph states and orbits in the variety of binary symmetric principal minors, under the action of SL$(2,{\mathbb {F}}_2)^{\times n}\rtimes {\mathfrak {S}}_n$. First we study the orbits of maximal abelian subgroups of the n-fold Pauli group under the action of ${\mathcal {C}}_n^{\textrm{loc}}\rtimes {\mathfrak {S}}_n$, where ${\mathcal {C}}_n^{\textrm{loc}}$ is the n-fold local Clifford group, and 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 is in translating the action of SL$(2,{\mathbb {F}}_2)^{\times n}$ into an action of the local symplectic group Sp$_{2n}^{\textrm{loc}}({\mathbb {F}}_2)$. We conclude by showing how the former action restricts onto stabilizer groups, stabilizer states and graph states.

1 Introduction

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 ${\mathcal {S}}_{M_1,\dots ,M_n}$ of the n-fold Pauli group ${\mathcal {P}}_n$ (see Sect.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 [11]: they are in particular useful for Measured Based Quantum Computation (MBQC) [3, 11, 19], for quantum error correcting codes [8] and for secret sharing [1, 18].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 [24, 25]. We will denote by ${{\mathcal {C}}}^{\textrm{loc}} _n\subset \text {LU}$ the group of local Clifford acting on n qubit states. Under the action of ${{\mathcal {C}}}^{\textrm{loc}} _n\rtimes \mathfrak {S}_n$, graph states have been classified up to $n=12$ qubits [4, 5, 11].

The variety ${{\mathcal {Z}}}_n$ of principal minors for $n\times n$ symmetric matrices over a field ${\mathbb {K}}$ [21] is an algebraic variety of ${\mathbb {P}}({\mathbb {K}}^{2^n})$ introduced by Holtz and Sturmfels [12] 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, 14, 22]. The goal of this paper is to show another potential application of the study of this variety over the two-elements field ${\mathbb {K}}={\mathbb {F}}_2$ by establishing a bijection between classes of graph states and orbits in the variety ${{\mathcal {Z}}}_n$. The main result of this paper is the following.

Theorem 1.1

The Lagrangian mapping induces a bijection between the $\left( {{\mathcal {C}}}^{\textrm{loc}} _n\rtimes \mathfrak {S}_n\right)$-orbits of maximal abelian subgroups of ${\mathcal {P}}_n$ and the $\left( {{\,\textrm{SL}\,}}(2,{\mathbb {F}}_2)^{\times n}\rtimes \mathfrak {S}_n\right)$-orbits in ${{\mathcal {Z}}}_n\subset {\mathbb {P}}({\mathbb {F}}_2^{2^n})$. In particular, there is a one-to-one correspondence between representatives of the graph states classification, up to $\left( {{\mathcal {C}}}^{\textrm{loc}} _n\rtimes \mathfrak {S}_n\right)$-action, and representatives of the $\left( {{\,\textrm{SL}\,}}(2,{\mathbb {F}}_2)^{\times n}\rtimes \mathfrak {S}_n\right)$-orbits in ${{\mathcal {Z}}}_n$.

Regarding the cardinality of the orbits, the Lagrangian mapping (Sect.2) shows that, if ${\mathcal {O}}_i$ is a $\left( {{\,\textrm{SL}\,}}(2,{\mathbb {F}}_2)^{\times n}\rtimes \mathfrak {S}_n\right)$-orbit of ${{\mathcal {Z}}}_n$, then the number of corresponding stabilizer states is $4^n|{\mathcal {O}}_i|$.

Maximal abelian subgroups of ${\mathcal {P}}_n$ correspond to fully isotropic subspaces of maximal dimension (see Sect.2). The bijection induced by the Lagrangian mapping between the latter subspaces and points of ${{\mathcal {Z}}}_n$ was already established in [13] in order to generalize observations made in [17] 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 [26] with motivating examples from supergravity theory. It was proven [26] that, over ${\mathbb {F}}_2$, ${{\mathcal {Z}}}_n$ is the image of the spinor variety and thus a $\text {Spin}(2n+1)$-orbit. However, the correspondence of orbits as established in Theorem 1.1 was not proven in the former papers, neither was the connection with graph states classification.

The paper is organized as follows. In Sect.2 we recall the basic definitions regarding the n-qubit Pauli group, the Lagrangian Grassmannian and the Lagrangian mapping. In Sects.3 and 4 we show how the $\left( {{\mathcal {C}}}^{\textrm{loc}} _n\rtimes \mathfrak {S}_n\right)$-action on maximal abelian subgroups of the Pauli group translates into an action on the variety of principal minors, proving the first part of Theorem 1.1. In Sects.5 and 6 we recall the definitions and basic properties of stabilizer states and graph states, and we complete the proof of our main theorem. Finally Sect.7 is dedicated to applications of our correspondence.

2 Preliminaries

The Pauli group ${\mathcal {P}}_n$. The group of the elementary Pauli matrices is ${\mathcal {P}}_1=\langle i X, iZ , i Y \rangle$, where the matrices $X= { \begin{bmatrix} 0 &{} 1 \\ 1 &{} 0 \end{bmatrix}}, Z={ \begin{bmatrix} 1 &{} 0 \\ 0 &{} -1 \end{bmatrix}}, Y={ \begin{bmatrix} 0 &{} -i \\ i &{} 0 \end{bmatrix}}\in \textrm{U}(2,{\mathbb {C}})$ are such that $X^2=Z^2=Y^2=I$ and $XZ=-ZX=-iY$, $XY=-YX=iZ$, $ZY=-YZ=-iX$. The n-fold Pauli group is the subgroup of $U(2^n,{\mathbb {C}})$

$$\begin{aligned} {\mathcal {P}}_n&= \{A_1 \otimes \ldots \otimes A_n \ | \ A_i \in {\mathcal {P}}_1\} \\&= \left\{ i^\beta Z^{\mu _1}X^{\nu _1}\otimes \ldots \otimes Z^{\mu _n}X^{\nu _n} \ \big | \ \beta \in \{0,1,2,3\}, \ \mu _i,\nu _i \in \{0,1\}\right\} \ , \end{aligned}$$

having center $Z({\mathcal {P}}_n)=\{\pm I^{\otimes n}, \pm i I^{\otimes n}\}\simeq {\mathbb {Z}}/4{\mathbb {Z}}$. In particular, the quotient $V_n={\mathcal {P}}_n/Z({\mathcal {P}}_n)$ is in one-to-one correspondence with the characteristic-two vector space ${\mathbb {F}}_2^{2n}$

$$\begin{aligned} \begin{array}{ccc} V_n &{} {\mathop {\longleftrightarrow }\limits ^{1:1}} &{} {\mathbb {F}}_2^{2n}\\ {[}Z^{\mu _1}X^{\nu _1}\otimes \ldots \otimes Z^{\mu _n}X^{\nu _n}] &{} \longleftrightarrow &{} (\mu _1, \mu _2 , \ldots , \mu _n, \nu _1, \nu _2, \ldots , \nu _n) \end{array} \ . \end{aligned}$$

(2.1)

The local Clifford group ${\mathcal {C}}_n^{\textrm{loc}}$. The n-fold Clifford group is the normalizer ${\mathcal {C}}_n= N_{\textrm{U}(2^n,{\mathbb {C}})}({\mathcal {P}}_n)$, while the n-fold local Clifford group is the subgroup

$$\begin{aligned} {\mathcal {C}}_n^{\textrm{loc}}=\{U_1\otimes \ldots \otimes U_n \ | \ U_i \in {\mathcal {C}}_1\} < {\mathcal {C}}_n \ . \end{aligned}$$

It is known that ${\mathcal {C}}_n^{\textrm{loc}}=\langle H_j, \sqrt{Z}_k\rangle$, where $H= \frac{1}{\sqrt{2}}{ \begin{bmatrix} 1 &{} 1 \\ 1 &{} -1 \end{bmatrix}}$, $\sqrt{Z}={ \begin{bmatrix} 1 &{} 0 \\ 0 &{} i \end{bmatrix}}$ and $H_j=I \otimes \ldots \otimes \overbrace{H}^{j\text {-}\textrm{th}} \otimes \ldots \otimes I$ (same for $\sqrt{Z}_k$). By definition, one gets the conjugacy action

$$\begin{aligned} \begin{array}{ccc} {\mathcal {C}}_n^{\textrm{loc}}\times {\mathcal {P}}_n &{} \longrightarrow &{} {\mathcal {P}}_n \\ (U_1\otimes \ldots \otimes U_n, A_1\otimes \ldots \otimes A_n) &{} \mapsto &{} (U_1A_1U_1^\dagger )\otimes \ldots \otimes (U_nA_nU_n^\dagger ) \end{array} \ , \end{aligned}$$

(2.2)

where $U^\dagger = \ \!\! ^t\overline{U}$ is the hermitian (i.e. conjugated transposed) of U. In particular, the action of ${\mathcal {C}}_1^{\textrm{loc}}$ on ${\mathcal {P}}_1$ is given by

The sets ${\mathscr {S}}({\mathcal {P}}_n)$ and ${\mathcal {I}}^n$. Given ${\mathbb {P}}_2^{2n-1}:={\mathbb {P}} ({\mathbb {F}}_2^{2n})$ the binary projective space and the coordinates (2.1), one has

$$\begin{aligned} \begin{array}{ccccc} &{} {\mathcal {P}}_n &{} \dashrightarrow &{} {\mathbb {P}}_2^{2n-1} &{} \\ M= &{}\alpha Z^{\mu _1}X^{\nu _1}\otimes \ldots \otimes Z^{\mu _n}X^{\nu _n} &{} \mapsto &{} [\mu _1: \ldots : \mu _n : \nu _1 : \ldots : \nu _n] &{} =P_M \end{array} \end{aligned}$$

where the arrow is dashed since the above map is actually not defined in $\alpha I^{\otimes n}$. Let $M_1,M_2 \in {\mathcal {P}}_n$ and let $P_1,P_2 \in {\mathbb {P}}_2^{2n-1}$ be the corresponding points: then

$$\begin{aligned} M_1M_2 = M_2M_1 \iff \sum _{i=1}^n\left( \mu _i^{(1)}\nu _i^{(2)} -\mu _i^{(2)}\nu _i^{(1)}\right) =0 \iff \langle P_1, P_2 \rangle _J =0 \ , \end{aligned}$$

where $\langle \cdot , \cdot \rangle _J$ is the symplectic bilinear form on ${\mathbb {F}}_2^{2n}$ defined by the symplectic matrix $J={ \begin{bmatrix} &{} I_n \\ -I_n &{} \end{bmatrix}} {\mathop {=}\limits ^{{\mathbb {F}}_2}} { \begin{bmatrix} &{} I_n \\ I_n &{} \end{bmatrix}}$, that is commuting Pauli group elements correspond to isotropic points with respect to $\langle \cdot ,\cdot \rangle _J$. In particular, any $M \in {\mathcal {P}}_n$ with associated point $P \in {\mathbb {P}}_2^{2n-1}$ defines a hyperplane $H_P=\left\{ Q \in {\mathbb {P}}_2^{2n-1} \ \big | \ \langle P,Q\rangle _J=0\right\}$ and one can extend this construction to any set of Pauli group elements:

$$\begin{aligned} M_1, \ldots , M_k \ \mapsto \ H_{P_1,\ldots ,P_k} =\left\{ Q \in {\mathbb {P}}_2^{2n-1} \ \big | \ \langle P_i,Q\rangle _J=0, \ \forall i =1:k\right\} \ . \end{aligned}$$

If one assumes that the $M_i$’s are mutually commuting, then $\langle P_1,\ldots , P_k\rangle \subset H_{P_1,\ldots , P_k}=\langle P_1,\ldots , P_k\rangle ^{\perp }$ ($H_{P_1,\ldots ,P_k}$ is not fully isotropic in general). If in addition one assumes that $M_1,\ldots ,M_k \in {\mathcal {P}}_n$ are independent, i.e.

$$\begin{aligned} M_1^{c_1}\cdot \ldots \cdot M_k^{c_k}=I^{\otimes n} \iff \forall i=1:k, \ c_i=0 \ , \end{aligned}$$

then $\dim _{{\mathbb {P}}} H_{P_1,\ldots , P_k}=2n-k-1$ (hence $k\le n$). In particular, n mutually commuting and independent elements $M_1,\ldots , M_n\in {\mathcal {P}}_n$ define a subspace $H_{P_1,\ldots , P_n}=\langle P_1,\ldots , P_n\rangle ^\perp$ of dimension $n-1$, thus $H_{P_1,\ldots , P_n}=\langle P_1,\ldots , P_n\rangle$ is maximal fully isotropic.

$$ \begin{aligned} \underbrace{M_1, \ldots , M_n}_{\textrm{indep}.\ \& \ \textrm{commut}.} \quad \Leftrightarrow \quad \underbrace{P_1,\ldots , P_n}_{\textrm{lin}. \ \textrm{indep}.\ \textrm{in}\ {\mathbb {F}}_2^{2n}} \quad \Leftrightarrow \quad \underbrace{H_{P_1,\ldots , P_n}}_{\dim _{{\mathbb {P}}}=n-1} \ . \end{aligned}$$

(2.4)

We denote the set of maximal fully isotropic subspaces in ${\mathbb {P}}_2^{2n-1}$ by

$$\begin{aligned} {\mathcal {I}}^n = \left\{ W \subset {\mathbb {P}}_2^{2n-1} \ \big | \ \dim _{{\mathbb {P}}}W=n-1, \ \forall P,Q \in W, \ \langle P,Q\rangle _J=0\right\} \ . \end{aligned}$$

Moreover, we say that a subgroup of ${\mathcal {P}}_n$ generated by n independent and mutually commuting elements $M_1,\ldots , M_n$ is a maximal abelian subgroup ${\mathcal {S}}_{M_1,\ldots , M_n}<{\mathcal {P}}_n$ and we denote the set of such subgroups by ${\mathscr {S}}({\mathcal {P}}_n)$. From (2.4) we get the correspondence

$$\begin{aligned} {\mathcal {S}}_{M_1,\ldots , M_n} \in {\mathscr {S}}({\mathcal {P}}_n) \ \longleftrightarrow \ H_{P_1,\ldots , P_n} \in {\mathcal {I}}_n \ . \end{aligned}$$

The Lagrangian Grassmannian ${{\,\textrm{LG}\,}}_{{\mathbb {F}}_2}(n,2n)$. The Grassmannian ${{\,\textrm{Gr}\,}}_{{\mathbb {F}}_2}(n,2n)$ is the set of n-dimensional subspaces of ${\mathbb {F}}_2^{2n}$ gaining structure of projective variety via the Plücker embedding [10]

$$\begin{aligned} \begin{array}{ccccc} {{\,\textrm{Pl}\,}}: &{} {{\,\textrm{Gr}\,}}_{{\mathbb {F}}_2}(n,2n) &{} \hookrightarrow &{} {\mathbb {P}}\left( \bigwedge ^n {\mathbb {F}}_2^{2n}\right) &{} \simeq {\mathbb {P}}_2^{\left( {\begin{array}{c}2n\\ n\end{array}}\right) -1}\\ &{} \langle v_1, \ldots , v_n\rangle &{} \mapsto &{} [v_1\wedge \ldots \wedge v_n] \end{array} \ . \end{aligned}$$

The standard open subsets of ${{\,\textrm{Gr}\,}}_{{\mathbb {F}}_2}(n,2n)$ are

$$\begin{aligned} U_{\{i_1,\ldots ,i_n\}}:=\left\{ \langle v_1,\ldots , v_n\rangle \in {{\,\textrm{Gr}\,}}_{{\mathbb {F}}_2}(n,2n) \ \bigg | \ \det { \begin{bmatrix} v_{1,i_1} &{} \cdots &{} v_{n,i_1}\\ \vdots &{} &{} \vdots \\ v_{1,i_n} &{} \cdots &{} v_{n,i_n} \end{bmatrix}}\ne 0\right\} \end{aligned}$$

(2.5)

for any index subset $\{i_1,\ldots , i_n\}\subset \{1,\ldots , 2n\}$. Given $W \in U_{\{t_1,\ldots , t_n\}}$, one considers its Plücker basis $v_i= e_{t_i}+ \sum _{j=1}^n a_{ji}e_{s_j}$ (where $\{s_1,\ldots , s_n\}=\{1,\ldots ,2n\}\setminus \{t_1,\ldots , t_n\}$) and W is described by a $2n\times n$ matrix $M_W$ having the i-th identity row in the $t_i$-h row: then the coordinates of W in ${\mathbb {P}}_2^{\left( {\begin{array}{c}2n\\ n\end{array}}\right) -1}$ are given by the $n\times n$ minors of the matrix $M_W$. In particular, the coefficients $a_{ij}$ in the Plücker basis define the Plücker matrix $A_W\in {{\,\textrm{Mat}\,}}_n({\mathbb {F}}_2)$ of W and the open subset $U_{\{t_1,\ldots , t_n\}}$ is parameterized by all minors of matrices in ${{\,\textrm{Mat}\,}}_n({\mathbb {F}}_2)$ (see [16, page 343]).

The Lagrangian Grassmannian ${{\,\textrm{LG}\,}}_{{\mathbb {F}}_2}(n,2n):={{\,\textrm{Pl}\,}}({\mathcal {I}}^n) \subset {\mathbb {P}}\left( \bigwedge ^n{\mathbb {F}}_2^{2n}\right)$ is the image via the Plücker embedding of ${\mathcal {I}}^n$, that is the projective variety parameterizing all $(n-1)$-dimensional (fully) isotropic subspaces of ${\mathbb {P}}_2^{2n-1}$. Any standard open subset $U_I\subset {{\,\textrm{Gr}\,}}_{{\mathbb {F}}_2}(n,2n)$ as in (2.5) restricts to a standard open subset $LU_I := U_I \cap {{\,\textrm{LG}\,}}_{{\mathbb {F}}_2}(n,2n)$, giving parameterizations of ${{\,\textrm{LG}\,}}_{{\mathbb {F}}_2}(n,2n)$: given $W\in LU_I$, by isotropicity its Plücker matrix $A_W$ is symmetric and the standard open subset $LU_I\subset {{\,\textrm{LG}\,}}_{{\mathbb {F}}_2}(n,2n)$ is parameterized by all minors of matrices in ${{\,\textrm{Sym}\,}}^2({\mathbb {F}}_2^n)$.

The variety ${\mathcal {Z}}_n$. By considering only the principal minors of symmetric matrices, one gets a rational projection $\pi :{{\,\textrm{LG}\,}}_{{\mathbb {F}}_2}(n,2n) \dashrightarrow {\mathbb {P}}_2^{2^n-1}$ whose image is the variety of binary symmetric principal minors ${\mathcal {Z}}_n= \pi \left( {{\,\textrm{LG}\,}}_{{\mathbb {F}}_2}(n,2n)\right) \subset {\mathbb {P}}_2^{2^n-1}$. This definition holds over any field [20] with $\pi$ being surjective, but over ${\mathbb {F}}_2$ it is a bijection [13].

3 Orbits in ${\mathcal {Z}}_n$ induced by the action ${\mathcal {C}}_n^{\textrm{loc}} \curvearrowright {\mathcal {P}}_n$

In this section we discuss the correspondence between the action of the group ${\mathcal {C}}_n^{\textrm{loc}}$ on ${\mathscr {S}}({\mathcal {P}}_n)$ and the action of the group $\text {SL}(2,{\mathbb {F}}_2)^{\times n}$ on ${\mathcal {Z}}_n$.

The local symplectic group ${{\,\textrm{Sp}\,}}^{\textrm{loc}}_{2n}({\mathbb {F}}_2)$. The conjugacy action (2.2) of the local Clifford group ${\mathcal {C}}_n^{\textrm{loc}}$ on the Pauli group ${\mathcal {P}}_n$ induces the action

$$\begin{aligned} \begin{array}{ccc} {\mathcal {C}}_{n}^{\textrm{loc}} \ \times \ {\mathscr {S}}({\mathcal {P}}_n) &{} \longrightarrow &{} {\mathscr {S}}({\mathcal {P}}_n) \\ (\ U \ , \ \langle M_1,\ldots , M_n\rangle \ ) &{} \mapsto &{} \langle UM_1U^\dagger , \ldots , UM_nU^\dagger \rangle \end{array} \ . \end{aligned}$$

(3.1)

By definition, any element of the Clifford group ${\mathcal {C}}_n$ induces an automorphism of ${\mathcal {P}}_n$, hence of $V_n={\mathcal {P}}_n/Z({\mathcal {P}}_n)$: but automorphisms of $V_n\simeq {\mathbb {F}}_2^{2n}$ are linear maps and they preserve commutators, hence also the symplectic form J on ${\mathbb {F}}_2^{2n}$ is preserved. It follows that there exists a well-defined homomorphism

$$\begin{aligned} \begin{array}{cccc} {\mathcal {C}}_n &{} \longrightarrow &{} {{\,\textrm{Sp}\,}}({\mathbb {F}}_2^{2n}) &{} \subset {{\,\textrm{GL}\,}}({\mathbb {F}}_2^{2n})\\ g &{} \mapsto &{} \hat{g} &{} \end{array} \end{aligned}$$

such that, given $M \in V_n\simeq {\mathbb {F}}_2^{2n}$ and $\tilde{M} \in {\mathcal {P}}_n$ any lifting of M, the action of $\hat{g}$ on M is $\hat{g} \cdot M = \overline{g \tilde{M} g^{-1}} \in V_n$. The homomorphism ${\mathcal {C}}_n\rightarrow {{\,\textrm{Sp}\,}}({\mathbb {F}}_2^{2n})$ is surjective, since the symplectic group is spanned by symplectic transvections [23, Sec. II.B], but it is not injective: its kernel is exactly the Pauli group ${\mathcal {P}}_n$ [15], thus one has the isomorphism ${\mathcal {C}}_n/{\mathcal {P}}_n \simeq {{\,\textrm{Sp}\,}}({\mathbb {F}}_2^{2n})$. Moreover, it restricts to a homomorphism

$$\begin{aligned} \begin{array}{ccc} {\mathcal {C}}_n^{\textrm{loc}} &{} \longrightarrow &{} {{\,\textrm{Sp}\,}}({\mathbb {F}}_2^{2n})\\ U &{} \mapsto &{} \tilde{U} \end{array} \end{aligned}$$

(3.2)

such that the elements in the image are of the form

$$\begin{aligned} \tilde{U}=\begin{bmatrix} a_1 &{} &{} &{} b_1 &{} &{} \\ &{} \ddots &{} &{} &{} \ddots &{} \\ &{} &{} a_n &{} &{} &{} b_n\\ c_1 &{} &{} &{} d_1 &{} &{} \\ &{} \ddots &{} &{} &{} \ddots &{} \\ &{} &{} c_n &{} &{} &{} d_n\\ \end{bmatrix} \in {{\,\textrm{Sp}\,}}({\mathbb {F}}_2^{2n}) \ , \end{aligned}$$

(3.3)

as one can check by looking at the action of ${\mathcal {C}}_n^{\textrm{loc}}$ on the Pauli elements

$$\begin{aligned} U\left( I\otimes \ldots \otimes \underbrace{Z^{\mu _k}X^{\nu _k}}_{k-th} \otimes \ldots \otimes I \right) U^\dagger&= U_1U_1^\dagger \otimes \ldots \otimes U_k\left( Z^{\mu _k}X^{\nu _k}\right) U_k^\dagger \otimes \ldots \otimes U_nU_n^\dagger \end{aligned}$$

(3.4)

$$\begin{aligned}&= I \otimes \ldots \otimes \underbrace{Z^{a_k\mu _k+b_k\nu _k} X^{c_k\mu _k+d_k\nu _k}}_{k-th}\otimes \ldots \otimes I \ , \end{aligned}$$

(3.5)

which in coordinates corresponds to

$$\begin{aligned} \tilde{U} (0,\ldots , \mu _k ,\ldots , \nu _k, \ldots , 0) =(0,\ldots , \ a_k\mu _k+b_k\nu _k \ ,\ldots , \ c_k\mu _k +d_k\nu _k \ , \ldots , 0) \ . \end{aligned}$$

The local symplectic group ${{\,\textrm{Sp}\,}}_{2n}^{\textrm{loc}}({\mathbb {F}}_2)$ is the image of the homomorphism (3.2), i.e.

$$\begin{aligned} {{\,\textrm{Sp}\,}}_{2n}^{\textrm{loc}}({\mathbb {F}}_2) :=\left\{ S \in {{\,\textrm{Sp}\,}}({\mathbb {F}}_2^{2n}) \ \big | \ S \ \text {of the form } (3.3) \right\} \ . \end{aligned}$$

Remark 3.1

The local symplectic group ${{\,\textrm{Sp}\,}}_{2n}^{\textrm{loc}}({\mathbb {F}}_2)$ is isomorphic to ${{\,\textrm{SL}\,}}(2,{\mathbb {F}}_2)^{\times n}$:

$$\begin{aligned} \begin{array}{ccccc} {\mathcal {C}}_n^{\textrm{loc}} &{} \longrightarrow &{} {{\,\textrm{Sp}\,}}_{2n}^{\textrm{loc}}({\mathbb {F}}_2) &{} {\mathop {\longrightarrow }\limits ^{\simeq }} &{} {{\,\textrm{SL}\,}}(2,{\mathbb {F}}_2)^{\times n}\\ U_1 \otimes \ldots \otimes U_n &{} \mapsto &{} \tilde{U} \ \text {as in}\ (3.3) &{} \mapsto &{} (\tilde{U}_1, \ldots , \tilde{U}_n) \end{array} \ , \end{aligned}$$

where $\tilde{U}_i={ \begin{bmatrix} a_i &{} b_i \\ c_i &{} d_i \end{bmatrix}}$.

From now on, we will denote by $\tilde{U} \in {{\,\textrm{Sp}\,}}_{2n}^{\textrm{loc}} ({\mathbb {F}}_2)$ the symplectic matrix corresponding to $U \in {\mathcal {C}}_n^{loc}$. By (3.4) we can explicit the action of $\tilde{U}\in {{\,\textrm{Sp}\,}}_{2n}^{\textrm{loc}}$ on ${\mathbb {P}}_2^{2n-1}$:

$$\begin{aligned}{} & {} \tilde{U} \cdot [\mu _1:\ldots : \mu _n :\nu _1: \ldots : \nu _n]\nonumber \\{} & {} \quad = [a_1\mu _1+b_1\nu _1: \ldots : a_n\mu _n+b_n\nu _n : c_1\mu _1 +d_1\nu _1 : \ldots : c_n\mu _n+d_n\nu _n] \ . \end{aligned}$$

(3.6)

Action on ${{\,\textrm{LG}\,}}_{{\mathbb {F}}_2}(n,2n)$. Since the local symplectic transformations preserve the dimensions and isotropicity of subspaces in ${\mathbb {P}}_2^{2n-1}$, the set ${\mathcal {I}}^n$ is ${{\,\textrm{Sp}\,}}_{2n}^{\textrm{loc}}({\mathbb {F}}_2)$-invariant and the action (3.1) induces the action

$$\begin{aligned} \begin{array}{ccc} {{\,\textrm{Sp}\,}}_{2n}^{\textrm{loc}}({\mathbb {F}}_2) \ \times \ {\mathcal {I}}^n &{} \longrightarrow &{} {\mathcal {I}}^n \\ (\ \tilde{U} \ , \ H_{P_1,\ldots , P_n} \ ) &{} \mapsto &{} H_{\tilde{U}P_1 , \ldots , \tilde{U}P_n} \end{array} \ . \end{aligned}$$

(3.7)

By applying the Plücker embedding to ${\mathcal {I}}_n$, one immediately gets the action

$$\begin{aligned} \begin{array}{ccc} {{\,\textrm{Sp}\,}}_{2n}^{\textrm{loc}}({\mathbb {F}}_2) \ \times \ {{\,\textrm{LG}\,}}_{{\mathbb {F}}_2}(n,2n) &{} \longrightarrow &{} {{\,\textrm{LG}\,}}_{{\mathbb {F}}_2}(n,2n) \\ \left( \ \tilde{U} \ , \ [v_1 \wedge \ldots \wedge v_n] \ \right) &{} \mapsto &{} \left[ \tilde{U}v_1 \wedge \ldots \wedge \tilde{U}v_n\right] \end{array} \ . \end{aligned}$$

(3.8)

Remark 3.2

In general, the action of ${{\,\textrm{Sp}\,}}^{loc}_{2n}({\mathbb {F}}_2)$ does not preserve the open subsets $LU_I\subset {{\,\textrm{LG}\,}}_{{\mathbb {F}}_2}(n,2n)$: given $H_{P_1,\ldots , P_n}\in LU_{I}$, the image $\tilde{U}\left( H_{P_1,\ldots , P_n}\right)$ may not lie in $LU_I$.

Action on ${\mathcal {Z}}_n$. Via the projection $\pi : {{\,\textrm{LG}\,}}(n,2n) \longrightarrow {\mathcal {Z}}_n$, the action (3.8) translates into an action of ${{\,\textrm{Sp}\,}}_{2n}^{\textrm{loc}} ({\mathbb {F}}_2)$ on ${\mathcal {Z}}_n$

$$\begin{aligned} {{\,\textrm{Sp}\,}}_{2n}^{\textrm{loc}}({\mathbb {F}}_2) \ \times \ {\mathcal {Z}}_n \longrightarrow {\mathcal {Z}}_n \end{aligned}$$

(3.9)

By Remark 3.1 this action is equivalent, up to isomorphism, to an already known and natural action of ${{\,\textrm{SL}\,}}(2,{\mathbb {F}}_2)^{\times n}$ on ${\mathcal {Z}}_n$: the space ${\mathbb {P}}_2^{2^n-1}\simeq {\mathbb {P}} \left( {\mathbb {F}}_2^{2} \otimes \ldots \otimes {\mathbb {F}}_2^2 \right)$ is naturally acted upon by the group ${{\,\textrm{SL}\,}}(2,{\mathbb {F}}_2)^{\times n}$ and this action restricts to an action of ${{\,\textrm{SL}\,}}(2,{\mathbb {F}}_2)^{\times n}$ on ${\mathcal {Z}}_n \subset {\mathbb {P}}_2^{2^n-1}$ [20], Theorem III.14], which corresponds to the action of ${{\,\textrm{Sp}\,}}_{2n}^{\textrm{loc}}({\mathbb {F}}_2)$ via the representation

$$\begin{aligned} \begin{array}{cccc} {{\,\textrm{SL}\,}}(2,{\mathbb {F}}_2)^{\times n} &{} {\mathop {\longrightarrow }\limits ^{\rho }} &{} {{\,\textrm{Sp}\,}}\left( {\mathbb {F}}_2^{2n}\right) &{} \subset {{\,\textrm{GL}\,}}\left( {\mathbb {F}}_2^{2n} \right) \\ \left( \begin{bmatrix} a_1 &{} b_1 \\ c_1 &{} d_1 \end{bmatrix}, \ldots , \begin{bmatrix} a_n &{} b_n \\ c_n &{} d_n \end{bmatrix}\right) &{} \mapsto &{} \begin{bmatrix} a_1 &{} &{} &{} b_1 &{} &{} \\ &{} \ddots &{} &{} &{} \ddots &{} \\ &{} &{} a_n &{} &{} &{} b_n\\ c_1 &{} &{} &{} d_1 &{} &{} \\ &{} \ddots &{} &{} &{} \ddots &{} \\ &{} &{} c_n &{} &{} &{} d_n\\ \end{bmatrix} \end{array} \ . \end{aligned}$$

(3.10)

Remark 3.3

Actually, in his PhD thesis [20] L.Oeding proved the above result over ${\mathbb {C}}$, but it is straightforward that then it holds over ${\mathbb {F}}_2$ too.

We conclude that there is a correspondence between the ${\mathcal {C}}_n^{\textrm{loc}}$-orbits of maximal abelian subgroups in ${\mathcal {P}}_n$ and the ${{\,\textrm{SL}\,}}(2,{\mathbb {F}}_2)^{\times n}$-orbits in ${\mathcal {Z}}_n$:

$$\begin{aligned} {\mathscr {S}}({\mathcal {P}}_n) \big /{\mathcal {C}}_n^{\textrm{loc}} \ \longleftrightarrow \ {\mathcal {Z}}_n \big / {{\,\textrm{SL}\,}}(2,{\mathbb {F}}_2)^{\times n} \ . \end{aligned}$$

4 The orbits ${\mathscr {S}}({\mathcal {P}}_n)/({\mathcal {C}}_n^{\textrm{loc}}\rtimes {\mathfrak {S}}_n)$ and ${\mathcal {Z}}_n/({{\,\textrm{SL}\,}}(2,{\mathbb {F}}_2)^{\times n}\rtimes {\mathfrak {S}}_n)$

In this section we extend the previous group actions to the semidirect product with the symmetric group ${\mathfrak {S}}_n$ in order to prove the first part of Theorem 1.1.

4.1 The actions ${\mathcal {C}}_n^{\textrm{loc}}\rtimes _\phi \mathfrak {S}_n \curvearrowright {\mathcal {P}}_n$ and ${{\,\textrm{Sp}\,}}_{2n}^{\textrm{loc}}({\mathbb {F}}_2)\rtimes _\varphi {\mathfrak {S}}_n \curvearrowright {\mathbb {F}}_2^{2n}$

By definition of the n-fold Pauli group ${\mathcal {P}}_n =\left\{ A_1 \otimes \ldots \otimes A_n \ | \ A_i \in {\mathcal {P}}_1\right\}$, there is a natural action of the symmetric group $\mathfrak {S}_n$ on it permuting the tensor entries:

$$\begin{aligned} \forall \sigma \in {\mathfrak {S}}_n, \ \ \ \sigma \cdot (A_1 \otimes \ldots \otimes A_n) = A_{\sigma (1)} \otimes \ldots \otimes A_{\sigma (n)} \ . \end{aligned}$$

Each permutation $\sigma \in {\mathfrak {S}}_n$ induces a transformation $\tilde{\sigma }\in U(2^n,{\mathbb {C}})$ permuting the basis vectors, so that for any $\sigma \in {\mathfrak {S}}_n$ and for any $A_1\otimes \ldots \otimes A_n\in {\mathcal {P}}_n$ one gets

$$\begin{aligned} \sigma \cdot (A_1\otimes \ldots \otimes A_n) = A_{\sigma (1)} \otimes \ldots \otimes A_{\sigma (n)}=\tilde{\sigma } (A_1\otimes \ldots \otimes A_n)\tilde{\sigma }^\dagger \ . \end{aligned}$$

(4.1)

Example 4.1

For $n=2$, any observable $A_1\otimes A_2 \in {\mathcal {P}}_2$ corresponds to a $4\times 4$ matrix $A_1\boxtimes A_2 \in U(4,{\mathbb {C}})$, where $\boxtimes$ denotes the Kronecker product between matrices. Then the transposition $\sigma =(1 \ 2)\in \mathfrak {S}_2$ induces the transformation $\tilde{\sigma }= \begin{bmatrix} 1 \\ {} &{} 0 &{} 1 \\ {} &{} 1 &{} 0 \\ {} &{} &{} &{} 1 \end{bmatrix} \in U(4,{\mathbb {C}})$ and an easy computation shows that (4.1) is satisfied.

Notice that $\tilde{\sigma }^\dagger =\tilde{\sigma }^{-1}$. In particular, the above action preserves ${\mathcal {P}}_n$, thus for any $\sigma \in {\mathfrak {S}}_n$ it holds $\tilde{\sigma } \in {\mathcal {C}}_n=N_{U(2^n,{\mathbb {C}})}({\mathcal {P}}_n)$: it follows that there is an injective homomorphism

$$\begin{aligned} \begin{array}{cccc} {\mathfrak {S}}_n &{} \hookrightarrow &{} {\mathcal {C}}_n &{} \subset U(2^n,{\mathbb {C}})\\ \sigma &{} \mapsto &{} \tilde{\sigma } &{} \end{array} \end{aligned}$$

which allows to identify ${\mathfrak {S}}_n$ as a subgroup of the Clifford group ${\mathcal {C}}_n$. Moreover, the symmetric group ${\mathfrak {S}}_n$ naturally acts on the local Clifford group ${\mathcal {C}}_n^{\textrm{loc}}$ by conjugacy

$$\begin{aligned} \begin{array}{cccc} \phi : &{} {\mathfrak {S}}_n &{} \longrightarrow &{} {{\,\textrm{Aut}\,}}\left( {\mathcal {C}}_n^\mathrm{{loc}}\right) \\ &{} \sigma &{} \mapsto &{} \left( \ \phi _\sigma : U \mapsto \ ^\sigma \!U=\tilde{\sigma } U \tilde{\sigma }^{-1} \ \right) \end{array} \ , \end{aligned}$$

(4.2)

where $^\sigma \!U = \tilde{\sigma } U \tilde{\sigma }^{-1} =U_{\sigma (1)}\otimes \ldots \otimes U_{\sigma (n)}$ for any $U=U_1\otimes \ldots \otimes U_n \in {\mathcal {C}}_n^{\textrm{loc}}$.

It follows that the subgroups ${\mathcal {C}}_n^{\textrm{loc}}$ and ${\mathfrak {S}}_n$ (the second up to isomorphism) generate a subgroup in ${\mathcal {C}}_n$ which is isomorphic to the semidirect product ${\mathcal {C}}_n^{\textrm{loc}}\rtimes _\phi {\mathfrak {S}}_n$

$$\begin{aligned} \begin{array}{cccc} {\mathcal {C}}_{n}^{\textrm{loc}}\rtimes _\phi {\mathfrak {S}}_n &{} {\mathop {\longrightarrow }\limits ^{\simeq }} &{} \left\langle {\mathcal {C}}_{n}^{\textrm{loc}} \ , \ {\mathfrak {S}}_n\right\rangle &{} \subset {\mathcal {C}}_n\\ (U,\sigma ) &{} \mapsto &{} U \tilde{\sigma } &{} \end{array} \ , \end{aligned}$$

acting on ${\mathcal {P}}_n$ as follows

$$\begin{aligned} \begin{array}{ccc} \left( {\mathcal {C}}_n^{\textrm{loc}}\rtimes _\phi {\mathfrak {S}}_n\right) \times {\mathcal {P}}_n &{} \longrightarrow &{} {\mathcal {P}}_n\\ \left( (U,\sigma ), A_1\otimes \ldots \otimes A_n \right) &{} \mapsto &{} U \cdot \left( \sigma \cdot (A_1\otimes \ldots \otimes A_n) \right) \end{array} \end{aligned}$$

(4.3)

where, if $U=U_1 \otimes \ldots \otimes U_n$,

$$\begin{aligned} U \cdot \left( \sigma \cdot (A_1\otimes \ldots \otimes A_n) \right) = U_{1}A_{\sigma (1)}U_{1}^\dagger \otimes \ldots \otimes U_{n}A_{\sigma (n)}U_{n}^\dagger \ . \end{aligned}$$

Since the elements in ${\mathcal {P}}_n$ are of the form $\alpha Z^{\mu _1}X^{\nu _1}\otimes \ldots \otimes Z^{\mu _n}X^{\nu _n}$ for $\alpha \in \{\pm 1, \pm i\}$, the action (4.1) of the symmetric group ${\mathfrak {S}}_n$ on ${\mathcal {P}}_n$ can be equivalently described by

$$\begin{aligned} \sigma \cdot (\alpha Z^{\mu _1}X^{\nu _1}\otimes \ldots \otimes Z^{\mu _n}X^{\nu _n})= \alpha Z^{\mu _{\sigma (1)}} X^{\nu _{\sigma (1)}}\otimes \ldots \otimes Z^{\mu _{\sigma (n)}} X^{\nu _{\sigma (n)}} \ . \end{aligned}$$

Finally, from (3.6) we know that the action of a given $U=U_1\otimes \ldots \otimes U_n \in {\mathcal {C}}_n^{\textrm{loc}}$ on $Z^{\mu _1}X^{\nu _1}\otimes \ldots \otimes Z^{\mu _n}X^{\nu _n}$ corresponds to an action of a certain $\left( \begin{bmatrix} a_1&{} b_1\\ c_1&{} d_1 \end{bmatrix}, \ldots , \begin{bmatrix} a_n&{} b_n\\ c_n&{} d_n \end{bmatrix}\right) \in {{\,\textrm{SL}\,}}(2,{\mathbb {F}}_2)^{\times n}$, where each $\begin{bmatrix} a_i&{} b_i\\ c_i&{} d_i \end{bmatrix}$ depends on $U_i$ and it acts on the coefficients $\mu _i,\nu _i$’s. In particular, in the same notation, we have

$$\begin{aligned} U(Z^{\mu _1}X^{\nu _1}\otimes \ldots \otimes Z^{\mu _n}X^{\nu _n})U^\dagger&= U_1Z^{\mu _1}X^{\nu _1}U_1^\dagger \otimes \ldots \otimes U_nZ^{\mu _n} X^{\nu _n}U_n^\dagger \\&= Z^{a_1\mu _1+b_1\nu _1}X^{c_1\mu _1+d_1\nu _1}\otimes \ldots \otimes Z^{a_n\mu _n+b_n\nu _n}X^{c_n\mu _n+d_n\nu _n} \end{aligned}$$

and the action (4.3) of the semidirect product ${\mathcal {C}}_n^{\textrm{loc}}\rtimes _\phi {\mathfrak {S}}_n$ on ${\mathcal {P}}_n$ can be rephrased as

$$\begin{aligned}{} & {} (U,\sigma )\cdot (Z^{\mu _1}X^{\nu _1}\otimes \ldots \otimes Z^{\mu _n}X^{\nu _n}) \ = \ U(Z^{\mu _{\sigma (1)}} X^{\nu _{\sigma (1)}}\otimes \ldots \otimes Z^{\mu _{\sigma (n)}} X^{\nu _{\sigma (n)}})U^\dagger \nonumber \\{} & {} \quad = Z^{a_1\mu _{\sigma (1)}+b_1\nu _{\sigma (1)}} X^{c_1\mu _{\sigma (1)}+d_1\nu _{\sigma (1)}}\otimes \ldots \otimes Z^{a_n\mu _{\sigma (n)}+b_n\nu _{\sigma (n)}} X^{c_n\mu _{\sigma (n)}+d_n\nu _{\sigma (n)}} \ . \end{aligned}$$

(4.4)

The above formula shows that the action of ${\mathcal {C}}_n^{\textrm{loc}}\rtimes _\phi {\mathfrak {S}}_n$ on ${\mathcal {P}}_n$ induces an action on the vectors $(\mu _1, \ldots , \mu _n,\nu _1,\ldots , \nu _n) \in {\mathbb {F}}_2^{2n}$ via the bijection $V_n={\mathcal {P}}_n / Z({\mathcal {P}}_n)\simeq {\mathbb {F}}_2^{2n}$.

Remark 4.2

From Sect.3 we know that there exists a group homomorphism ${\mathcal {C}}_n\rightarrow {{\,\textrm{Sp}\,}}({\mathbb {F}}_2^{2n})$ restricting to an homomorphism ${\mathcal {C}}_n^{\textrm{loc}}\twoheadrightarrow {{\,\textrm{Sp}\,}}_{2n}^{\textrm{loc} ({\mathbb {F}}_2)}<{{\,\textrm{Sp}\,}}({\mathbb {F}}_2^{2n})$ with kernel ${\mathcal {P}}_n$. Actually, one can also consider the restriction to the subgroup ${\mathcal {C}}_n^{\textrm{loc}}\rtimes _\phi {\mathfrak {S}}_n$ giving

$$\begin{aligned} {\mathcal {C}}_n^{\textrm{loc}} \rtimes _\phi {\mathfrak {S}}_n \longrightarrow {{\,\textrm{Sp}\,}}({\mathbb {F}}_2^{2n}) \end{aligned}$$

which is again not injective having kernel ${\mathcal {P}}_n \rtimes _\phi {\mathfrak {S}}_n$.

On the other hand, there is a well-defined action of ${\mathfrak {S}}_n$ on ${\mathbb {F}}_2^{2n}$ given by

$$\begin{aligned} \sigma \cdot (\mu _1, \ldots , \mu _n,\nu _1,\ldots , \nu _n) = (\mu _{\sigma (1)}, \ldots , \mu _{\sigma (n)},\nu _{\sigma (1)}, \ldots , \nu _{\sigma (n)}) \ . \end{aligned}$$

In particular, any $\sigma \in {\mathfrak {S}}_n$ corresponds to a $S_\sigma \in {{\,\textrm{Sp}\,}}({\mathbb {F}}_2^{2n})$ such that

$$\begin{aligned} \sigma \cdot (\mu _1, \ldots , \mu _n,\nu _1,\ldots , \nu _n)&= (\mu _{\sigma (1)}, \ldots , \mu _{\sigma (n)},\nu _{\sigma (1)}, \ldots , \nu _{\sigma (n)}) \\&= S_\sigma (\mu _1, \ldots , \mu _n,\nu _1,\ldots , \nu _n) \ , \end{aligned}$$

and this allows to identify ${\mathfrak {S}}_n$ as a subgroup of ${{\,\textrm{Sp}\,}}({\mathbb {F}}_2^{2n})$. The matrices $S_\sigma \in {{\,\textrm{Sp}\,}}({\mathbb {F}}_2^{2n})$ are of the form $\begin{bmatrix} A_\sigma &{} 0 \\ 0 &{} A_\sigma \end{bmatrix}$ for some permutation matrix $A_\sigma \in {{\,\textrm{Sp}\,}}({\mathbb {F}}_2^n)$.

Finally, the symmetric group acts by conjugacy on ${{\,\textrm{Sp}\,}}_{2n}^{\textrm{loc}}({\mathbb {F}}_2)$ as follows

$$\begin{aligned} \begin{array}{cccc} \varphi : &{} {\mathfrak {S}}_n &{} \longrightarrow &{} {{\,\textrm{Aut}\,}}\left( {{\,\textrm{Sp}\,}}_{2n}^{\textrm{loc}}({\mathbb {F}}_2)\right) \\ &{} \sigma &{} \mapsto &{} \left( \ \varphi _\sigma : \tilde{U} \mapsto \ ^\sigma \!\tilde{U}=S_\sigma \tilde{U} S_{\sigma }^{-1} \ \right) \end{array} \ \end{aligned}$$

(4.5)

which is well-defined since

$$\begin{aligned} ^t(^\sigma \!\tilde{U})J(^\sigma \!\tilde{U}) = (^t\!S_\sigma ^{-1}) (^t\tilde{U})\underbrace{(^t\!S_\sigma ) J S_\sigma }_{=J} \tilde{U} S_\sigma ^{-1} = (^t\!S_\sigma ^{-1})\underbrace{(^t\tilde{U}) J \tilde{U}}_{=J} S_\sigma ^{-1} \ = \ J \ . \end{aligned}$$

More precisely, if $\tilde{U} \in {{\,\textrm{Sp}\,}}_{2n}^{\textrm{loc}} ({\mathbb {F}}_2)$ is as in (3.3), then

$$\begin{aligned} ^\sigma \!\tilde{U}= \begin{bmatrix} a_{\sigma (1)} &{} &{} &{} b_{\sigma (1)} &{} &{} \\ &{} \ddots &{} &{} &{} \ddots &{} \\ &{} &{} a_{\sigma (n)} &{} &{} &{} b_{\sigma (n)}\\ c_{\sigma (1)} &{} &{} &{} d_{\sigma (1)} &{} &{} \\ &{} \ddots &{} &{} &{} \ddots &{} \\ &{} &{} c_{\sigma (n)} &{} &{} &{} d_{\sigma (n)}\\ \end{bmatrix} \in {{\,\textrm{Sp}\,}}_{2n}^{\textrm{loc}}({\mathbb {F}}_2) \ . \end{aligned}$$

It follows that ${{\,\textrm{Sp}\,}}_{2n}^{\textrm{loc}}({\mathbb {F}}_2)$ and ${\mathfrak {S}}_n$ generate a subgroup of ${{\,\textrm{Sp}\,}}({\mathbb {F}}_2^{2n})$ isomorphic to the semidirect product ${{\,\textrm{Sp}\,}}_{2n}^{\textrm{loc}}({\mathbb {F}}_2)\rtimes _\varphi {\mathfrak {S}}_n$

$$\begin{aligned} \begin{array}{cccc} {{\,\textrm{Sp}\,}}_{2n}^{\textrm{loc}}({\mathbb {F}}_2)\rtimes _\varphi {\mathfrak {S}}_n &{} {\mathop {\longrightarrow }\limits ^{\simeq }} &{} \left\langle {{\,\textrm{Sp}\,}}_{2n}^{\textrm{loc}} ({\mathbb {F}}_2) \ , \ {\mathfrak {S}}_n\right\rangle &{} \subset {{\,\textrm{Sp}\,}}({\mathbb {F}}_2^{2n})\\ (\tilde{U},\sigma ) &{} \mapsto &{} \tilde{U} S_\sigma &{} \end{array} \ , \end{aligned}$$

which acts on ${\mathbb {F}}_2^{2n}$ as in the exponents in (4.4), that is

$$\begin{aligned}{} & {} (\tilde{U},\sigma )\cdot (\mu _1, \ldots , \mu _n,\nu _1,\ldots , \nu _n) \ = \ \tilde{U}S_\sigma (\mu _{1}, \ldots , \mu _{n}, \nu _{1},\ldots , \nu _{n}) \nonumber \\{} & {} \quad = \tilde{U}(\mu _{\sigma (1)}, \ldots , \mu _{\sigma (n)}, \nu _{\sigma (1)},\ldots , \nu _{\sigma (n)}) \nonumber \\{} & {} \quad = \left( a_{1}\mu _{\sigma (1)}+b_{1}\nu _{\sigma (1)}, \ \ldots , \ a_{n}\mu _{\sigma (n)}+b_{n}\nu _{\sigma (n)}, \ c_{1}\mu _{\sigma (1)}\right. \nonumber \\{} & {} \qquad \quad \left. +d_{1}\nu _{\sigma (1)}, \ \ldots , \ c_{n}\mu _{\sigma (n)}+d_{n}\nu _{\sigma (n)}\right) \ . \end{aligned}$$

(4.6)

Thus the restriction of ${\mathcal {C}}_n \rightarrow {{\,\textrm{Sp}\,}}({\mathbb {F}}_2^{2n})$ to ${\mathcal {C}}_n^{\textrm{loc}}\rtimes _\phi {\mathfrak {S}}_n$ gives the surjection

$$\begin{aligned} \begin{array}{ccc} {\mathcal {C}}_n^{\textrm{loc}} \rtimes _\phi {\mathfrak {S}}_n &{} \twoheadrightarrow &{} {{\,\textrm{Sp}\,}}_{2n}^{\textrm{loc}}({\mathbb {F}}_2) \rtimes _\varphi {\mathfrak {S}}_n \\ (U,\sigma ) &{} \mapsto &{} (\tilde{U}, \sigma )\\ U\tilde{\sigma } &{} \mapsto &{} \tilde{U} S_\sigma \end{array} \end{aligned}$$

where in the last line we formally identify $(U,\sigma )$ with the Clifford transformation $U\tilde{\sigma }\in {\mathcal {C}}_n$ and $(\tilde{U},\sigma )$ with the symplectic transformation $\tilde{U} S_\sigma \in {{\,\textrm{Sp}\,}}({\mathbb {F}}_2^{2n})$. Moreover, from (4.4) and (4.6) it follows that, although the two semidirect products are not isomorphic, the above homomorphism translates the action of ${\mathcal {C}}_n^{\textrm{loc}}\rtimes _\phi {\mathfrak {S}}_n$ on the Pauli group ${\mathcal {P}}_n$ into the action of ${{\,\textrm{Sp}\,}}_{2n}^{\textrm{loc}}({\mathbb {F}}_2)\rtimes _\varphi {\mathfrak {S}}_n$ on the symplectic space $({\mathbb {F}}_2^{2n},J)$, and viceversa. Thus we get a correspondence between orbits

$$\begin{aligned} {\mathcal {P}}_n \big / C_n^{\textrm{loc}}\rtimes _\phi {\mathfrak {S}}_n \ \longleftrightarrow \ {\mathbb {F}}_2^{2n} \big / {{\,\textrm{Sp}\,}}_{2n}^{\textrm{loc}}({\mathbb {F}}_2)\rtimes _\varphi {\mathfrak {S}}_n \ . \end{aligned}$$

(4.7)

4.2 The actions ${\mathcal {C}}_n^{\textrm{loc}}\rtimes _\phi {\mathfrak {S}}_n \curvearrowright {\mathscr {S}}({\mathcal {P}}_n)$ and ${{\,\textrm{Sp}\,}}_{2n}^{\textrm{loc}}({\mathbb {F}}_2)\rtimes _\varphi {\mathfrak {S}}_n \curvearrowright {\mathcal {I}}^n$

The next step is to extend the orbit correspondence (4.7) to the sets

$$ \begin{aligned}{} & {} {\mathscr {S}} ({\mathcal {P}}_n)=\left\{ \langle M_1,\ldots , M_n \rangle < {\mathcal {P}}_n \ | \ M_i's \ \text {independent}\ \& \ \text {mutually commuting}\right\} \ ,\\{} & {} {\mathcal {I}}^n=\left\{ W \subset {\mathbb {P}}_2^{2n-1} \ | \ W \simeq {\mathbb {P}}^{n-1}, \ \langle P,Q\rangle _J=0 \ \forall P,Q \in W \right\} \ , \end{aligned}$$

which are in bijection via (2.4). From Sect.3 we already have the orbit correspondences

$$\begin{aligned} {\mathscr {S}}({\mathcal {P}}_n) \big /{\mathcal {C}}_n^{\textrm{loc}} \ \longleftrightarrow \ {\mathcal {I}}^n \big /{{\,\textrm{Sp}\,}}_{2n}^{\textrm{loc}} ({\mathbb {F}}_2) \ \longleftrightarrow \ {\mathcal {Z}}_n \big /{{\,\textrm{SL}\,}}(2,{\mathbb {F}}_2)^{\times n}. \end{aligned}$$

Action on ${\mathscr {S}}({\mathcal {P}}_n)$. Given $M_i=A_1^{(i)}\otimes \ldots \otimes A_n^{(i)}\in {\mathcal {P}}_n$ and $\sigma \in {\mathfrak {S}}_n$, we denote

$$\begin{aligned} ^\sigma \!M_i \ = \ \sigma \cdot (A_1^{(i)}\otimes \ldots \otimes A_n^{(i)}) \ = \ A_{\sigma (1)}^{(i)}\otimes \ldots \otimes A_{\sigma (n)}^{(i)} \ . \end{aligned}$$

Let $S_{M_1,\ldots , M_n}=\langle M_1,\ldots , M_n\rangle \in {\mathscr {S}}({\mathcal {P}}_n)$ be a maximal abelian subgroup of ${\mathcal {P}}_n$ with

$$\begin{aligned} M_i= \alpha _iZ^{\mu _1^{(i)}}X^{\nu _1^{(i)}}\otimes \ldots \otimes Z^{\mu _n^{(i)}}X^{\nu _n^{(i)}} \ \ \ , \ \alpha _i \in \{\pm 1, \pm i\} \ . \end{aligned}$$

Then, for any $\sigma \in {\mathfrak {S}}_n$, the observables $^\sigma M_i$’s are such that

$$\begin{aligned} \sum _{j=1}^n\left( \mu _{\sigma (j)}^{(h)}\nu _{\sigma (j)}^{(k)} -\mu _{\sigma (j)}^{(k)}\nu _{\sigma (j)}^{(h)}\right) {\mathop {=}\limits ^{l=\sigma ^{-1}(j)}}\sum _{l=1}^n \left( \mu _l^{(h)}\nu _l^{(k)}-\mu _l^{(k)}\nu _l^{(h)}\right) {\mathop {=}\limits ^{(\clubsuit )}}0 \end{aligned}$$

and

$$\begin{aligned} (^\sigma M_1)^{c_1}\cdot \ldots \cdot (^\sigma M_n)^{c_n}&= (A_{\sigma (1)}^{(1)})^{c_1}\cdots (A_{\sigma (1)}^{(n)})^{c_n} \otimes \ldots \otimes (A_{\sigma (n)}^{(1)})^{c_1}\cdots (A_{\sigma (n)}^{(n)})^{c_n}\\&{\mathop {=}\limits ^{(\spadesuit )}} I_2\otimes \ldots \otimes I_2 = I^{\otimes n} \ , \end{aligned}$$

where $I_2$ is the $2\times 2$ identity matrix and the equalities $(\clubsuit )$ and $(\spadesuit )$ respectively follow from the commutation and the independence of the $M_i$’s. It follows that $^\sigma \!M_1, \ldots , ^\sigma \!M_n$ are independent and mutually commuting too: thus we get the following well-defined action

$$\begin{aligned} \begin{array}{ccc} {\mathfrak {S}}_n \times {\mathscr {S}}({\mathcal {P}}_n) &{} \longrightarrow &{} {\mathscr {S}}({\mathcal {P}}_n)\\ \left( \sigma \ , \ S_{M_1,\ldots , M_n} \right) &{} \mapsto &{} S_{^\sigma \!M_1, \ldots , ^\sigma \!M_n} \end{array} \ . \end{aligned}$$

Actually, since the symmetric group $\mathfrak {S}_n$ 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

$$\begin{aligned} \begin{array}{ccc} {\mathfrak {S}}_n \times {\mathscr {S}}({\mathcal {P}}_n) &{} \longrightarrow &{} {\mathscr {S}}({\mathcal {P}}_n)\\ \left( \sigma \ , \ \langle M_1,\ldots , M_n\rangle \right) &{} \mapsto &{} \langle ^\sigma \!M_{\sigma (1)}, \ldots , ^\sigma \!M_{\sigma (n)}\rangle \end{array} \end{aligned}$$

(4.8)

where $^\sigma M_{\sigma (i)}=Z^{\mu _{\sigma (1)}^{(\sigma (i))}} X^{\nu _{\sigma (1)}^{(\sigma (i))}}\otimes \ldots \otimes Z^{\mu _{\sigma (n)}^{(\sigma (i))}}X^{\nu _{\sigma (n)}^{(\sigma (i))}}$. We conclude that the action of the group ${\mathcal {C}}_n^{\textrm{loc}}\rtimes _\phi {\mathfrak {S}}_n$ on ${\mathcal {P}}_n$ extends to the action

$$\begin{aligned} \begin{array}{ccc} \left( {\mathcal {C}}_n^{\textrm{loc}}\rtimes _\phi {\mathfrak {S}}_n \right) \times {\mathscr {S}}({\mathcal {P}}_n) &{} \longrightarrow &{} {\mathscr {S}}({\mathcal {P}}_n)\\ \left( (U,\sigma ) \ , \ \langle M_1,\ldots , M_n\rangle \right) &{} \mapsto &{} \langle \ U(^\sigma \!M_{\sigma (1)})U^\dagger , \ldots , U(^\sigma \!M_{\sigma (n)})U^\dagger \ \rangle \end{array} \end{aligned}$$

(4.9)

where, for $U=U_1\otimes \ldots \otimes U_n\in {\mathcal {C}}_n^{\textrm{loc}}$ and $M_i=Z^{\mu _1^{(i)}}X^{\nu _1^{(i)}}\otimes \ldots \otimes Z^{\mu _n^{(i)}}X^{\nu _n^{(i)}}$,

$$\begin{aligned} U(^\sigma \!M_{\sigma (i)})U^\dagger =U_1Z^{\mu _{\sigma (1)}^{(\sigma (i))}}X^{\nu _{\sigma (1)}^{(\sigma (i))}} U_1^\dagger \otimes \ldots \otimes U_nZ^{\mu _{\sigma (n)}^{(\sigma (i))}} X^{\nu _{\sigma (n)}^{(\sigma (i))}}U_n^\dagger \ . \end{aligned}$$

Action on ${\mathcal {I}}^n$. From the correspondence (2.4), any maximal abelian subgroup $S_{M_1,\ldots , M_n} \in {\mathscr {S}}(P_n)$ corresponds to the maximal isotropic subspace $H_{P_1,\ldots , P_n}=\langle P_1,\ldots , P_n\rangle \in {\mathcal {I}}^n$. By putting together (4.6) and (4.9), one gets the action

$$\begin{aligned} \begin{array}{ccc} \left( {{\,\textrm{Sp}\,}}_{2n}^{\textrm{loc}}({\mathbb {F}}_2)\rtimes _\varphi {\mathfrak {S}}_n \right) \times {\mathcal {I}}^n &{} \longrightarrow &{} {\mathcal {I}}^n\\ \left( (\tilde{U},\sigma ) \ , \ H_{P_1,\ldots , P_n} \right) &{} \mapsto &{} H_{\tilde{U}S_\sigma P_{\sigma (1)}, \ldots , \tilde{U}S_\sigma P_{\sigma (n)}} \end{array} \end{aligned}$$

(4.10)

where, for $P_i=(\mu _{1}^{(i)}, \ldots , \mu _{n}^{(i)},\nu _{1}^{(i)},\ldots , \nu _{n}^{(i)})$,

$$\begin{aligned} \tilde{U}S_\sigma P_{\sigma (i)}= & {} \tilde{U}S_\sigma \left( \mu _{1}^{(\sigma (i))}, \ldots , \mu _{n}^{(\sigma (i))}, \nu _{1}^{(\sigma (i))},\ldots , \nu _{n}^{(\sigma (i))}\right) \nonumber \\= & {} \tilde{U}\left( \mu _{\sigma (1)}^{(\sigma (i))}, \ldots , \mu _{\sigma (n)}^{(\sigma (i))},\nu _{\sigma (1)}^{(\sigma (i))}, \ldots , \nu _{\sigma (n)}^{(\sigma (i))}\right) \nonumber \\= & {} \left( a_{1}\mu _{\sigma (1)}^{(\sigma (i))}+b_{1} \nu _{\sigma (1)}^{(\sigma (i))}, \ \ldots , \ a_{n} \mu _{\sigma (n)}^{(\sigma (i))}+b_{n}\nu _{\sigma (n)}^{(\sigma (i))}, \ c_{1}\mu _{\sigma (1)}^{(\sigma (i))}\right. \nonumber \\{} & {} \left. +d_{1} \nu _{\sigma (1)}^{(\sigma (i))}, \ \ldots , \ c_{n} \mu _{\sigma (n)}^{(\sigma (i))}+d_{n} \nu _{\sigma (n)}^{(\sigma (i))}\right) \ . \end{aligned}$$

(4.11)

Notice that each point $\tilde{U}S_\sigma P_{\sigma (i)}\in {\mathbb {F}}_2^{2n}$ corresponds to the element $U(^\sigma \!M_{\sigma (i)})U^\dagger \in {\mathcal {P}}_n$. We conclude that the correspondence (4.7) extends to an orbit correspondence

$$\begin{aligned} {\mathscr {S}}({\mathcal {P}}_n)\big / C_n^{\textrm{loc}} \rtimes _\phi {\mathfrak {S}}_n \ \longleftrightarrow \ {\mathcal {I}}^n \big /{{\,\textrm{Sp}\,}}_{2n}^{\textrm{loc}} ({\mathbb {F}}_2)\rtimes _\varphi {\mathfrak {S}}_n \ . \end{aligned}$$

(4.12)

4.3 The actions of ${{\,\textrm{Sp}\,}}_{2n}^{\textrm{loc}}({\mathbb {F}}_2)\rtimes _\varphi {\mathfrak {S}}_n$ on ${{\,\textrm{LG}\,}}_{{\mathbb {F}}_2}(n,2n)$ and ${\mathcal {Z}}_n$

In order to prove the first statement of Theorem 1.1, it only remains to translate the action (4.10) into actions on the Lagrangian Grassmannian ${{\,\textrm{LG}\,}}_{{\mathbb {F}}_2}(n,2n)$ and on ${\mathcal {Z}}_n$.

Action on ${{\,\textrm{LG}\,}}_{{\mathbb {F}}_2}(n,2n)$. Via the Plücker embedding, a maximal fully isotropic subspace $H_{P_1,\ldots , P_n}=\langle P_1,\ldots , P_n\rangle \in {\mathcal {I}}^n$ corresponds to the point $[P_1\wedge \ldots \wedge P_n]\in {{\,\textrm{LG}\,}}_{{\mathbb {F}}_2}(n,2n)$. It is straightforward that the action (4.10) of ${{\,\textrm{Sp}\,}}_{2n}^{\textrm{loc}} ({\mathbb {F}}_2)\rtimes _\varphi {\mathfrak {S}}_n$ on ${\mathcal {I}}^n$ is equivalent to the following action on the Lagrangian Grassmannian (extending (3.8)):

$$\begin{aligned} \begin{array}{ccc} \left( {{\,\textrm{Sp}\,}}_{2n}^{\textrm{loc}}({\mathbb {F}}_2)\rtimes _\varphi {\mathfrak {S}}_n \right) \times {{\,\textrm{LG}\,}}_{{\mathbb {F}}_2}(n,2n) &{} \longrightarrow &{} {{\,\textrm{LG}\,}}_{{\mathbb {F}}_2}(n,2n)\\ \left( (\tilde{U},\sigma ) \ , \ [P_1\wedge \ldots \wedge P_n] \right) &{} \mapsto &{} \left[ \tilde{U}S_\sigma P_{\sigma (1)} \wedge \ldots \wedge \tilde{U}S_\sigma P_{\sigma (n)}\right] \end{array} \end{aligned}$$

(4.13)

where $\tilde{U}S_\sigma P_{\sigma (i)}$ are as in (4.11). One can write the point $[P_1\wedge \ldots \wedge P_n]$ in coordinates in ${\mathbb {P}}_2^{\left( {\begin{array}{c}2n\\ n\end{array}}\right) -1}$: given $N=\left[ P_1 | \ldots | P_n\right]$ the $2n \times n$ matrix representing the subspace $H_{P_1,\ldots , P_n}$, the coordinates of $[P_1\wedge \ldots \wedge P_n]$ are given by the $n\times n$ minors of N, that is

$$\begin{aligned} \begin{array}{ccc} {\mathbb {P}}\left( \bigwedge ^n{\mathbb {F}}_2^{2n}\right) &{} \longleftrightarrow &{} {\mathbb {P}}_2^{\left( {\begin{array}{c}2n\\ n\end{array}}\right) -1}\\ {[}P_1\wedge \ldots \wedge P_n] &{} \longleftrightarrow &{} \left[ N_{\{1,\ldots ,n\}}: N_{\{1,\ldots , n-1,n+1\}}: \ldots : N_{\{n+1,\ldots , 2n\}}\right] \end{array} \end{aligned}$$

where $N_I$ is the minor of N given by the I-indexed rows and all the n columns. The action (3.7) leads to an action of ${{\,\textrm{Sp}\,}}_{2n}^{\textrm{loc}}({\mathbb {F}}_2)$ on the full-rank $2n\times n$ matrices by left-multiplication, that is a transformation $\tilde{U}={ \begin{bmatrix} A&{} B \\ C &{} D \end{bmatrix}}\in {{\,\textrm{Sp}\,}}_{2n}^{\textrm{loc}}({\mathbb {F}}_2)$ maps a certain full-rank $2n\times n$ matrix $N={ \begin{bmatrix} F \\ G \end{bmatrix}}$ into the full-rank $2n\times n$ matrix

$$\begin{aligned} \tilde{U} \cdot N \ = \ \begin{bmatrix} AF+BG \\ CF+DG \end{bmatrix} \ . \end{aligned}$$

(4.14)

By substituting the identity matrix $\tilde{U}=I$ in (4.11) we deduce that a permutation $\sigma \in {\mathfrak {S}}_n$ acts on a full-rank $2n\times n$ matrix ${\begin{bmatrix} F\\ G\end{bmatrix}}=[P_1| \ldots |P_n]$ as

$$\begin{aligned} \sigma \cdot \begin{bmatrix} F\\ G \end{bmatrix}= \left[ S_\sigma P_{\sigma (1)}| \ldots | S_\sigma P_{\sigma (n)}\right] =\begin{bmatrix} \begin{matrix} \mu _{\sigma (1)}^{(\sigma (1))} &{} \mu _{\sigma (1)}^{(\sigma (2))} &{} \cdots &{} \mu _{\sigma (1)}^{(\sigma (n))}\\ \mu _{\sigma (2)}^{(\sigma (1))} &{} \mu _{\sigma (2)}^{(\sigma (2))} &{} \cdots &{} \mu _{\sigma (2)}^{(\sigma (n))}\\ \vdots &{} \vdots &{} &{} \vdots \\ \mu _{\sigma (n)}^{(\sigma (1))} &{} \mu _{\sigma (n)}^{(\sigma (2))} &{} \cdots &{} \mu _{\sigma (n)}^{(\sigma (n))} \end{matrix}\\ ---------------\\ \begin{matrix} \nu _{\sigma (1)}^{(\sigma (1))} &{} \nu _{\sigma (1)}^{(\sigma (2))} &{} \cdots &{} \nu _{\sigma (1)}^{(\sigma (n))}\\ \nu _{\sigma (2)}^{(\sigma (1))} &{} \nu _{\sigma (2)}^{(\sigma (2))} &{} \cdots &{} \nu _{\sigma (2)}^{(\sigma (n))}\\ \vdots &{} \vdots &{} &{} \vdots \\ \nu _{\sigma (n)}^{(\sigma (1))} &{} \nu _{\sigma (n)}^{(\sigma (2))} &{} \cdots &{} \nu _{\sigma (n)}^{(\sigma (n))} \end{matrix} \end{bmatrix} = \begin{bmatrix} ^\sigma \!F \\ ^\sigma \!G \end{bmatrix} \end{aligned}$$

(4.15)

where the $n\times n$ matrix $^\sigma \!F$ (resp. $^\sigma \!G$) is obtained by the $n \times n$ matrix F (resp. G) by permuting both columns and rows by $\sigma \in {\mathfrak {S}}_n$: more precisely, if $S_\sigma = { \begin{bmatrix} A_\sigma \\ &{} A_\sigma \end{bmatrix}}$ where $A_\sigma$ is the $n\times n$ permutation matrix defined by $\sigma$, then the action of $\sigma$ onto ${ \begin{bmatrix} F\\ G\end{bmatrix}}$ corresponds to the conjugacy action by $A_\sigma$ onto F and G separately, that is

$$\begin{aligned} \sigma \cdot \begin{bmatrix} F\\ G \end{bmatrix}= \begin{bmatrix} ^\sigma \!F\\ ^\sigma \!G \end{bmatrix}=\begin{bmatrix} A_\sigma F A_\sigma ^{-1}\\ A_\sigma G A_\sigma ^{-1} \end{bmatrix} \ . \end{aligned}$$

By putting together (4.14) and (4.15) we conclude that the action (4.13) of ${{\,\textrm{Sp}\,}}_{2n}^{\textrm{loc}}\rtimes _\varphi {\mathfrak {S}}_n$ on ${{\,\textrm{LG}\,}}_{{\mathbb {F}}_2}(n,2n)$ is equivalent to the restriction onto full-rank matrices of the action

$$\begin{aligned} \begin{array}{ccc} \left( {{\,\textrm{Sp}\,}}_{2n}^{\textrm{loc}}({\mathbb {F}}_2)\rtimes _\varphi {\mathfrak {S}}_n \right) \times {{\,\textrm{Mat}\,}}_{2n\times n}({\mathbb {F}}_2) &{} \longrightarrow &{} {{\,\textrm{Mat}\,}}_{2n\times n}({\mathbb {F}}_2)\\ \left( (\tilde{U},\sigma ) \ , \ {\begin{bmatrix} F \\ G \end{bmatrix}} \right) &{} \mapsto &{} \begin{bmatrix} A(^\sigma \!F)+B(^\sigma \!G)\\ C(^\sigma \!F)+D(^\sigma \!G) \end{bmatrix} \end{array} \end{aligned}$$

(4.16)

where $\tilde{U}={ \begin{bmatrix} A&{} B \\ C &{} D \end{bmatrix}}$.

Action on ${\mathcal {Z}}_n$. Via the bijection $\pi :{{\,\textrm{LG}\,}}_{{\mathbb {F}}_2}(n,2n)\rightarrow {\mathcal {Z}}_n$, the action (4.13) of ${{\,\textrm{Sp}\,}}_{2n}^{\textrm{loc}}({\mathbb {F}}_2)\rtimes _\varphi {\mathfrak {S}}_n$ on ${{\,\textrm{LG}\,}}_{{\mathbb {F}}_2}(n,2n)$ induces an action on ${\mathcal {Z}}_n$ making the following diagram commutative:

In the following we show that this induced action on ${\mathcal {Z}}_n$ actually is a natural one.

Fix $(v_0,v_1)$ a basis of ${\mathbb {F}}_2^2$ and the induced basis $\left( |i_1\ldots i_n\rangle = v_{i_1}\otimes \ldots \otimes v_{i_n} \ | \ i_k\in \{0,1\}\right)$ (in Dirac notation) of ${\mathbb {F}}_2^2 \otimes \ldots \otimes {\mathbb {F}}_2^2$. Consider the isomorphism

$$\begin{aligned} \begin{array}{ccc} {\mathbb {F}}_2^2 \otimes \ldots \otimes {\mathbb {F}}_2^2 &{} {\mathop {\longrightarrow }\limits ^{\simeq }} &{} {\mathbb {F}}_2^{2^n}\\ |0\ldots 0\rangle &{} \mapsto &{} (1, 0 , \ldots ,0)\\ |0\ldots \underbrace{1}_{n-i}\ldots 0 \rangle &{} \mapsto &{} (0,\ldots , \underbrace{1}_{i+2}, \ldots , 0)\\ |1\ldots 1\rangle &{}\mapsto &{} (0,\ldots ,0,1) \end{array} \end{aligned}$$

(4.18)

From [20], Theorem III.14], the natural action of ${{\,\textrm{SL}\,}}(2,{\mathbb {F}}_2)^{\times n}$ on ${\mathbb {P}}\left( {\mathbb {F}}_2^{2^n}\right)$ given by $(U_1,\ldots , U_n) \cdot [v_{1}\otimes \ldots \otimes v_n] = \left[ U_1v_1 \otimes \ldots \otimes U_nv_n\right]$ induces an action of ${{\,\textrm{SL}\,}}(2,{\mathbb {F}}_2)^{\times n}$ on ${\mathcal {Z}}_n\subset {\mathbb {P}}_2^{2^n-1}$. Also the symmetric group ${\mathfrak {S}}_n$ has a natural action on ${\mathbb {P}}\left( {\mathbb {F}}_2^{2^n}\right)$ which permutes the tensor product entries, that is $\sigma \cdot [v_1\otimes \ldots \otimes v_n]= [v_{\sigma (1)}\otimes \ldots \otimes v_{\sigma (n)}]$: it is known that this action preserves ${\mathcal {Z}}_n$. Moreover, the action of ${\mathfrak {S}}_n$ on ${\mathbb {P}}\left( {\mathbb {F}}_2^2 \otimes \ldots \otimes {\mathbb {F}}_2^2\right)$ permuting the tensor entries induces an action of ${\mathfrak {S}}_n$ on ${{\,\textrm{SL}\,}}(2,{\mathbb {F}}_2)^{\times n}$ permuting the entries of the direct product (and the latter corresponds to the action (4.5) via the isomorphism ${{\,\textrm{SL}\,}}(2,{\mathbb {F}}_2)^{\times n}\simeq {{\,\textrm{Sp}\,}}_{2n}^{\textrm{loc}}({\mathbb {F}}_2)$). We conclude that there is a natural action

$$\begin{aligned} \left( {{\,\textrm{SL}\,}}(2,{\mathbb {F}}_2)^{\times n}\rtimes {\mathfrak {S}}_n \right) \times {\mathcal {Z}}_n \ \longrightarrow \ {\mathcal {Z}}_n \end{aligned}$$

and that it actually corresponds, via the isomorphism ${{\,\textrm{SL}\,}}(2,{\mathbb {F}}_2)^{\times n}\simeq {{\,\textrm{Sp}\,}}_{2n}^{\textrm{loc}}({\mathbb {F}}_2)$, to the action on ${\mathcal {Z}}_n$ in (4.17). This achieves the proof of the first part of Theorem 1.1 by establishing the bijection between the $\left( {{\mathcal {C}}}^{\textrm{loc}} _n\rtimes \mathfrak {S}_n\right)$-orbits of maximal abelian subgroups of ${\mathcal {P}}_n$ (or equivalently, maximal fully isotropic subspaces of ${\mathbb {P}}_2^{2n-1}$) and the $\left( {{\,\textrm{SL}\,}}(2,{\mathbb {F}}_2)^{\times n}\rtimes \mathfrak {S}_n\right)$-orbits in ${{\mathcal {Z}}}_n\subset {\mathbb {P}}({\mathbb {F}}_2^{2^n})$:

$$\begin{aligned} {\mathscr {S}}({\mathcal {P}}_n) \big / C_n^{\textrm{loc}}\rtimes _\phi {\mathfrak {S}}_n \ \longleftrightarrow \ {\mathcal {Z}}_n \big /{{\,\textrm{Sp}\,}}_{2n}^{\textrm{loc}}({\mathbb {F}}_2) \rtimes _\varphi {\mathfrak {S}}_n \ \simeq \ {\mathcal {Z}}_n \big / {{\,\textrm{SL}\,}}(2,{\mathbb {F}}_2)^{\times n}\rtimes {\mathfrak {S}}_n \ . \end{aligned}$$

(4.19)

5 Stabilizer states and their orbits under ${\mathcal {C}}_n^{\textrm{loc}}\rtimes {\mathfrak {S}}_n$

In this section we restrict the action (4.9) onto stabilizer groups and stabilizer states.

Stabilizer states. A subgroup $S<{\mathcal {P}}_n$ is called stabilizer group if it is abelian and $-I^{\otimes n}\notin S$. In particular, $S\in {\mathscr {S}}({\mathcal {P}}_n)$ is a maximal stabilizer group if it does not contain $-I^{\otimes n}$. We denote the set of maximal stabilizer state groups in ${\mathcal {P}}_n$ by ${\mathscr {S}}^+({\mathcal {P}}_n)$, or simply ${\mathscr {S}}^+_n$.

Let $S=\langle M_1,\ldots , M_n\rangle \in {\mathscr {S}}_n^+$: since the $M_i$’s are mutually commuting, they admit (at least) one common eigenvector $|\phi \rangle \in ({\mathbb {C}}^2)^{\otimes n}$. Moreover, recall that for any $M\in {\mathcal {P}}_n$ it holds $M^2=\pm I^{\otimes n}$: in particular, for any $M \in S$ one gets $M^2=I^{\otimes n}$ since $-I^{\otimes n}\notin S$ and $M^2 \in S$. It follows that elements in S can only have eigenvalues $+ 1$ or $-1$.

For any abelian subgroup $S<{\mathcal {P}}_n$ not containing $-I^{\otimes n}$ 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 $\dim V_S=2^n/|S|$ [6, Sect. III.B, III.C]: in particular, if $S \in {\mathscr {S}}_n^+$, $V_S$ is one-dimensional.

A (n-qubit) stabilizer state is the (unique up to phase) common eigenvector with eigenvalue $+1$ of a maximal stabilizer state $S \in {\mathscr {S}}_n^+$. We denote by $\Phi _1^n$ the set of n-qubit stabilizer states. There is a one-to-one correspondence between $\Phi _1^n$ and ${\mathscr {S}}_n^+$: we denote by $S_{|\phi \rangle }$ the maximal stabilizer group associated to the stabilizer state $|\phi \rangle$. Let us study how ${\mathcal {C}}_n^{\textrm{loc}}\rtimes {\mathfrak {S}}_n$ acts on ${\mathscr {S}}_n^+$ and $\Phi _1^n$.

Action of $C_n^{\textrm{loc}}$. The action (3.1) of ${\mathcal {C}}_n^{\textrm{loc}}$ on ${\mathscr {S}}({\mathcal {P}}_n)$ preserves ${\mathscr {S}}_n^+$: indeed, given $U \in {\mathcal {C}}_n^{\textrm{loc}}$ and $S_{|\phi \rangle }=\langle M_1, \ldots , M_n\rangle \in {\mathscr {S}}_n^+$ it holds $UM_iU^\dagger (U|\phi \rangle ) = UM_i|\phi \rangle =U|\phi \rangle$ for any $i=1:n$, that is $U|\phi \rangle$ is common eigenvector with eigenvalue $+1$ of $S_{UM_1U^\dagger , \ldots , UM_nU^\dagger }$.

In particular, one gets the equivalent actions

$$\begin{aligned} \begin{array}{ccc} {\mathcal {C}}_n^{\textrm{loc}} \ \times \ {\mathscr {S}}_n^+ &{} \longrightarrow &{} {\mathscr {S}}_n^+\\ (U\ ,\ S_{|\phi \rangle }) &{} \mapsto &{} S_{U|\phi \rangle } \end{array} \ \ \ \ \ \ , \ \ \ \ \ \ \begin{array}{ccc} {\mathcal {C}}_n^{\textrm{loc}} \ \times \ \Phi _1^n &{} \longrightarrow &{} \Phi _1^n\\ (U\ ,\ |\phi \rangle ) &{} \mapsto &{} U|\phi \rangle \end{array} \end{aligned}$$

(5.1)

where, if $U=U_1\otimes \ldots \otimes U_n$ and $|\phi \rangle =\sum _{j}|\phi _j^{(1)}\rangle \otimes \ldots \otimes |\phi _j^{(n)}\rangle$ (with $|\phi _j^{(k)}\rangle \in {\mathbb {C}}^2$),

$$\begin{aligned} U|\phi \rangle = \sum _{j}U_1|\phi _j^{(1)}\rangle \otimes \ldots \otimes U_n|\phi _j^{(n)}\rangle \ . \end{aligned}$$

Action of ${\mathfrak {S}}_n$. Also the action (4.8) of ${\mathfrak {S}}_n$ on ${\mathscr {S}}({\mathcal {P}}_n)$ preserves ${\mathscr {S}}_n^+$: indeed, given $S=\langle M_1,\ldots , M_n\rangle \in {\mathscr {S}}_n^+$ (which by definition does not contain $-I^{\otimes n}$), if $^\sigma \!S=\langle ^\sigma M_{\sigma (1)},\ldots , ^\sigma \!M_{\sigma (n)}\rangle$ contained $-I^{\otimes n}$, then $\sigma ^{-1}\cdot (-I^{\otimes n}) =-I^{\otimes n}$ would be in $^{\sigma ^{-1}}(^\sigma \!S)=S$ which is a contradiction. Thus we get the induced action

$$\begin{aligned} \begin{array}{ccc} {\mathfrak {S}}_n \ \times \ {\mathscr {S}}_n^+ &{} \longrightarrow &{} {\mathscr {S}}_n^+\\ \left( \sigma \ , \ S_{|\phi \rangle }\right) &{} \mapsto &{} ^\sigma \!(S_{|\phi \rangle }) \end{array} \ . \end{aligned}$$

Actually, we can exhibit the stabilizer state corresponding to $^\sigma \!(S_{|\phi \rangle })$: given $\sigma \in {\mathfrak {S}}_n$ and $|\phi \rangle =\sum _{j}|\phi _j^{(1)}\rangle \otimes \ldots \otimes |\phi _j^{(n)}\rangle \in \Phi _1^n$, our candidate is

$$\begin{aligned} ^\sigma \!|\phi \rangle = \sum _{j}|\phi _j^{(\sigma (1))}\rangle \otimes \ldots \otimes |\phi _j^{(\sigma (n))}\rangle \in ({\mathbb {C}}^2)^{\otimes n} \end{aligned}$$

(5.2)

that is the element obtained by permuting via $\sigma$ the tensor entries of $|\phi \rangle$. This is precisely the one that we should expect: indeed, ${\mathfrak {S}}_n$ acts on a maximal stabilizer group by permuting both its generators and their tensor entries, but the eigenvector is invariant under reordering of the generators.

Proposition 5.1

For any stabilizer state $|\phi \rangle \in \Phi _1^n$ and any permutation $\sigma \in {\mathfrak {S}}_n$, the tensor element $^\sigma \!|\phi \rangle$ in (5.2) is also a stabilizer state, i.e. $^\sigma \!|\phi \rangle \in \Phi _1^n$. In particular,

$$\begin{aligned} ^\sigma \!(S_{|\phi \rangle })=S_{^\sigma \!|\phi \rangle } \in {\mathscr {S}}_n^+ \ . \end{aligned}$$

Proof

Let $|\phi \rangle =\sum _j |\phi _j^{(1)}\rangle \otimes \ldots \otimes |\phi _j^{(n)}\rangle \in \Phi _1^n$ be a stabilizer state and $S_{|\phi \rangle }=\langle M_1,\ldots , M_n\rangle \in {\mathscr {S}}_n^+$ the corresponding maximal stabilizer group. If $M_i=A_1^{(i)}\otimes \ldots \otimes A_n^{(n)}$ for any $i=1:n$, then by definition

$$\begin{aligned} \sum _j A_1^{(i)}|\phi _j^{(1)}\rangle \otimes \ldots \otimes A_n^{(i)}|\phi _j^{(n)}\rangle = M_i|\phi \rangle = |\phi \rangle = \sum _j |\phi _j^{(1)}\rangle \otimes \ldots \otimes |\phi _j^{(n)}\rangle \end{aligned}$$

for any $i=1:n$. By applying $\sigma \in {\mathfrak {S}}_n$ we get for any $i=1:n$

$$\begin{aligned} \sum _j A_{\sigma (1)}^{(i)}|\phi _j^{(\sigma (1))}\rangle \otimes \ldots \otimes A_{\sigma (n)}^{(i)}|\phi _j^{(\sigma (n))}\rangle&{\mathop {=}\limits ^{(\clubsuit )}} \sum _j \sigma \cdot \left( A_1^{(i)}|\phi _j^{(1)}\rangle \otimes \ldots \otimes A_n^{(i)}|\phi _j^{(n)}\rangle \right) \\&= \sigma \cdot \left( \sum _j A_1^{(i)}|\phi _j^{(1)}\rangle \otimes \ldots \otimes A_n^{(i)}|\phi _j^{(n)}\rangle \right) \\&= \sigma \cdot \left( \sum _j |\phi _j^{(1)}\rangle \otimes \ldots \otimes |\phi _j^{(n)}\rangle \right) \\&= \sum _j |\phi _j^{(\sigma (1))}\rangle \otimes \ldots \otimes |\phi _j^{(\sigma (n))}\rangle \ , \end{aligned}$$

where $(\clubsuit )$ follows by definition of the action of ${\mathfrak {S}}_n$ on $({\mathbb {C}}^2)^{\otimes n}$. By (5.2) one gets $^\sigma \!M_i( ^\sigma \!|\phi \rangle ) =\left( A_{\sigma (1)}^{(i)}\otimes \ldots \otimes A_{\sigma (n)}^{(i)}\right) (^\sigma \!|\phi \rangle ) \ = \ ^\sigma \!|\phi \rangle$, that is $^\sigma \!|\phi \rangle$ is eigenvector with eigenvalue $+1$ for $^\sigma \!M_i=A_{\sigma (1)}^{(i)}\otimes \ldots \otimes A_{\sigma (n)}^{(i)}$. Since this holds for any $i=1:n$, then $^\sigma \!|\phi \rangle$ is a common eigenvector with eigenvalue $+1$ of $\langle ^\sigma \!M_1,\ldots , ^\sigma \!M_n\rangle = \langle ^\sigma \!M_{\sigma (1)},\ldots , ^\sigma \!M_{\sigma (n)}\rangle = \ ^\sigma \!(S_{|\phi \rangle })$. But $^\sigma \!(S_{|\phi \rangle })$ is a maximal stabilizer group, thus $^\sigma |\phi \rangle$ actually is its unique eigenvector with eigenvalue $+1$, that is $^\sigma \!|\phi \rangle \in \Phi _1^n$ and $^\sigma \!(S_{|\phi \rangle })=S_{^\sigma \!|\phi \rangle }$. $\square$

Corollary 5.2

The action (4.8) of ${\mathfrak {S}}_n$ on ${\mathscr {S}}({\mathcal {P}}_n)$ preserves ${\mathscr {S}}_n^+$ and induces the actions

$$\begin{aligned} \begin{array}{ccc} {\mathfrak {S}}_n \ \times \ {\mathscr {S}}_n^+ &{} \longrightarrow &{} {\mathscr {S}}_n^+\\ \left( \sigma \ , \ S_{|\phi \rangle }\right) &{} \mapsto &{} ^\sigma \!(S_{|\phi \rangle })=S_{^\sigma \!|\phi \rangle } \end{array} \ \ \ \ \ , \ \ \ \ \ \begin{array}{ccc} {\mathfrak {S}}_n \ \times \ \Phi _1^n &{} \longrightarrow &{} \Phi _1^n\\ \left( \sigma \ , \ |\phi \rangle \right) &{} \mapsto &{} ^\sigma \!|\phi \rangle \end{array} \ . \end{aligned}$$

(5.3)

Conclusion. The action (4.9) of ${\mathcal {C}}_n^{\textrm{loc}}\rtimes {\mathfrak {S}}_n$ on ${\mathscr {S}}({\mathcal {P}}_n)$ restricts to the actions

$$\begin{aligned} \begin{array}{ccc} \left( {\mathcal {C}}_n^{\textrm{loc}}\rtimes {\mathfrak {S}}_n\right) \ \times {\mathscr {S}}_n^+ &{} \longrightarrow &{} {\mathscr {S}}_n^+\\ \left( (U,\sigma ) \ , \ S_{|\phi \rangle }\right) &{} \mapsto &{} U(^\sigma \!S_{|\phi \rangle })U^\dagger \end{array} \ \ \ \ \ , \ \ \ \ \begin{array}{ccc} \left( {\mathcal {C}}_n^{\textrm{loc}}\rtimes {\mathfrak {S}}_n\right) \times \ \Phi _1^n &{} \longrightarrow &{} \Phi _1^n\\ \left( (U,\sigma ) \ ,\ |\phi \rangle \right) &{} \mapsto &{} U(^\sigma \!|\phi \rangle ) \end{array} \end{aligned}$$

(5.4)

where, if $U=U_1\otimes \ldots \otimes U_n$ and $|\phi \rangle =\sum _{j}|\phi _j^{(1)}\rangle \otimes \ldots \otimes |\phi _j^{(n)}\rangle$, then

$$\begin{aligned} U(^\sigma \!|\phi \rangle ) = \sum _{j}U_{1}|\phi _j^{(\sigma (1))}\rangle \otimes \ldots \otimes U_{n}|\phi _j^{(\sigma (n))}\rangle \ . \end{aligned}$$

This proves an intermediate step to the second statement of Theorem 1.1. From Sect.4 we have the one-to-one correspondence (4.19) between the orbits in ${\mathscr {S}}({\mathcal {P}}_n)$ under ${\mathcal {C}}_n^{\textrm{loc}} \rtimes {\mathfrak {S}}_n$ and the orbits in ${\mathcal {Z}}_n$ under ${{\,\textrm{SL}\,}}(2,{\mathbb {F}}_2)^{\times n}\rtimes {\mathfrak {S}}_n$. But the Lagrangian mapping (Sect.2) associates maximal fully isotropic subspaces in ${\mathcal {I}}^n$ to maximal abelian subgroups in ${\mathscr {S}}({\mathcal {P}}_n)$ up to phasis (since we work into the quotient $V_n={\mathcal {P}}_n/Z ({\mathcal {P}}_n)$): this means that it does not distinguish between maximal abelian subgroups containing $-I^{\otimes n}$ and maximal stabilizer states. Hence there is a one-to-one correspondence between the orbits of stabilizer states under ${\mathcal {C}}_n^{\textrm{loc}}\rtimes {\mathfrak {S}}_n$ and the orbits in ${\mathcal {Z}}_n$ under ${{\,\textrm{SL}\,}}(2,{\mathbb {F}}_2)^{\times n}\rtimes {\mathfrak {S}}_n$:

$$\begin{aligned} \Phi _1^n\big / C_n^{\textrm{loc}}\rtimes _\phi {\mathfrak {S}}_n \ \longleftrightarrow \ {\mathscr {S}}_n^+ \big / C_n^{\textrm{loc}} \rtimes _\phi {\mathfrak {S}}_n \ \longleftrightarrow \ {\mathcal {Z}}_n \big /{{\,\textrm{SL}\,}}(2,{\mathbb {F}}_2)^{\times n} \rtimes {\mathfrak {S}}_n \ . \end{aligned}$$

(5.5)

6 Graph states and their orbits under ${\mathcal {C}}_n^{\textrm{loc}}\rtimes {\mathfrak {S}}_n$

In this section we investigate how the action of ${\mathcal {C}}_n^{\textrm{loc}}\rtimes {\mathfrak {S}}_n$ (5.4) behaves on the graph states, which are a privileged class of stabilizer states .

Remark 6.1

The action of ${\mathcal {C}}_n^{\textrm{loc}}$ does not preserve graph states [24]. Thus we cannot properly talk about orbits of graph states under ${\mathcal {C}}_n^{\textrm{loc}}\rtimes {\mathfrak {S}}_n$, but we refer to “orbit” of a graph state as to the set of all graph states belonging to the same orbit in $\Phi _1^n$. In order to get an actual action on the graph states, one has to consider only local Clifford operations corresponding to graph transformations ([24, Sect. IV, Definition 1]).

Consider a (non-oriented) graph $\Gamma =(V,E)$ defined by a finite set of vertices $V=\{1,\ldots , n\}$ and a set of edges $E\subset V\times V$. We can associate to $\Gamma$ a unique symmetric matrix $\theta \in {{\,\textrm{Sym}\,}}^2({\mathbb {F}}_2^n)$ such that $\theta _{ij}=1 \iff (i,j) \in E$ (the matrix is indeed symmetric since the graph is not oriented), called the adjacency matrix of the graph. The graph $\Gamma _\theta$ defines n Pauli group elements

$$\begin{aligned} M_i=Z^{\theta _{i1}}X^{\delta _{1i}}\otimes \ldots \otimes Z^{\theta _{in}}X^{\delta _{ni}} \in {\mathcal {P}}_n \ \ \ , \ \forall i=1:n \ , \end{aligned}$$

(6.1)

where $\delta _{ij}$ is the Kronecker symbol: these elements are independent and mutually commuting, thus they generate a maximal abelian subgroup denoted by $S_\Gamma =S_\theta \in {\mathscr {S}}({\mathcal {P}}_n)$. Moreover, the maximal fully isotropic subspace $H_\theta \in {\mathcal {I}}^n$ corresponding to $S_\theta$ is described by the $2n \times n$ matrix ${\begin{bmatrix} \theta \\ I_n \end{bmatrix}}$ and belongs to the chart $LU_{\{n+1,\ldots , 2n\}}\subset {{\,\textrm{LG}\,}}_{{\mathbb {F}}_2}(n,2n)$.

Remark 6.2

The association $\Gamma \mapsto S_\theta$ is not unique: indeed, one could choose to associate to $\Gamma$ the elements $M_i':=Z^{\delta _{1i}}X^{\theta _{i1}}\otimes \ldots \otimes Z^{\delta _{ni}}X^{\theta _{in}}$ defining the subspace $H_\theta '\in LU_{\{1,\ldots ,n\}}$ with associated matrix ${ \begin{bmatrix} I_n \\ \theta \end{bmatrix}}$. However, the two different associations $\Gamma \mapsto M_i$ and $\Gamma \mapsto M_i'$ give maximal abelian subgroups $S_\theta$ and $S_\theta '$ which are in the same ${\mathcal {C}}_n^{\textrm{loc}}$-orbit, since $J{ \begin{bmatrix} I_n \\ \theta \end{bmatrix}} \ = \ { \begin{bmatrix} 0 &{} I_n \\ I_n &{} 0 \end{bmatrix}\begin{bmatrix} I_n \\ \theta \end{bmatrix}} \ = \ { \begin{bmatrix} \theta \\ I_n \end{bmatrix}}$ and $J \in {{\,\textrm{Sp}\,}}_{2n}^{\textrm{loc}}({\mathbb {F}}_2)$. On one hand, the choice of using the $M_i'$ allows to have an immediate transcription in ${\mathcal {Z}}_n$: via the Lagrangian mapping, the maximal abelian subgroup $S_\theta '$ corresponds to the point $[1: \theta _{ii}: \theta _{[i,j]}: \ldots : \det \theta ]\in {\mathcal {Z}}_n$. On the other hand, the choice of the $M_i$’s is more common in the literature. In this section we work with the association $\Gamma \mapsto S_\theta$, but in Sect.7 we will use the one $\Gamma \mapsto S_\theta '$ for exhibiting examples.

In general, the maximal abelian subgroup $S_\theta \in {\mathscr {S}}({\mathcal {P}}_n)$ defined by a graph $\Gamma _\theta$ is not a maximal stabilizer group, unless we add the assumption that the graph is loopless (or simple), i.e. $\theta _{ii}=0$ 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 $\Gamma _\theta$ be a (non-oriented) graph and let $S_\theta \in {\mathscr {S}}({\mathcal {P}}_n)$ be the associated maximal abelian group via (6.1). Then $S_\theta \in {\mathscr {S}}_n^+$ if and only if $\Gamma _\theta$ is loopless.

Proof

Recall that $S_\theta \in {\mathscr {S}}_n^+ \ \iff \ \left( \ -I^{\otimes n}\notin S_\theta \ \ \wedge \ \ S_\theta \in {\mathscr {S}}({\mathcal {P}}_n) \ \right)$.

$[\Rightarrow ]$ Assume that $S_\theta \in {\mathscr {S}}_n^+$ and that, by contradiction, the graph has at least one loop, i.e. there exists $i_0 \in \{1,\ldots , n\}$ such that $\theta _{i_0i_0}=1$. Then the element $M_{i_0}=Z^{\theta _{i_01}}\otimes \ldots \otimes ZX \otimes \ldots \otimes Z^{\theta _{i_0n}}$ squares to $M_{i_0}^2= -I^{\otimes n}$, hence $-I^{\otimes n}\in S_\theta$ leading to a contradiction.

$[\Leftarrow ]$ Assume $\Gamma _\theta$ to be loopless. Then for any $i=1:n$ the observable $M_i$ does not have any tensor entry equal to $ZX=iY$. In particular, it holds $M_i^2=I^{\otimes n}$, 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 $|\gamma \rangle \in ({\mathbb {C}}^2)^{\otimes n}$: then for any $i=1:n$ we get $M_i|\gamma \rangle = (-1)^{c_i}|\gamma \rangle$. In particular, for any $i=1:n$ it holds $(-1)^{c_i}M_i|\gamma \rangle = |\gamma \rangle$, that is $|\gamma \rangle$ 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 $U\in {\mathcal {C}}_n^{\textrm{loc}}$ such that $U(-1)^{c_i}M_iU^\dagger =M_i$ for all $i=1:n$. Indeed, if $c_{i_1}, \ldots , c_{i_k}$ are the only non-zero exponents, then by (2.3) the local Clifford tranformation $\hat{U}=H_{i_1}\cdot \ldots \cdot H_{i_k}$ works. It follows that $\hat{U}|\gamma \rangle$ is common eigenvector with eigenvalue $+1$ of $M_1,\ldots , M_n$, hence of $S_{\theta }\in {\mathscr {S}}_n^+$. $\square$

From now on we consider only loopless graphs. A (n-qubit) graph state $|\Gamma \rangle \in ({\mathbb {C}}^2)^{\otimes n}$ is the stabilizer state of a maximal stabilizer group $S_\theta \in {\mathscr {S}}_n^+$ defined by a n-vertex graph $\Gamma _\theta$. We denote the subset of n-qubit graph states by $\Theta _n\subset \Phi _1^n$. Moreover, we define a graph group to be a maximal stabilizer group defined by a graph and we denote the set of such subgroups by ${\mathscr {S}}_{\Theta _n} \subset {\mathscr {S}}_n^+$. Clearly, the one-to-one correspondence ${\mathscr {S}}_n^+ \leftrightarrow \Phi _1^n$ restricts to a one-to-one correspondence ${\mathscr {S}}_{\Theta _n} \leftrightarrow \Theta _n$.

Action of ${\mathfrak {S}}_n$. Let $|\Gamma \rangle \in \Theta _n$ be a graph state defined by a graph $\Gamma$ with adjacent matrix $\theta$. The graph group $S_\theta =\langle M_1,\ldots , M_n\rangle \in {\mathscr {S}}_{\Theta _n}$ (as in (6.1)) is described by the $2n\times n$ matrix $\begin{bmatrix} \theta \\ I_n\end{bmatrix}$. Given $\sigma \in {\mathfrak {S}}_n$, by (4.15) we know that $\sigma \cdot { \begin{bmatrix} \theta \\ I_n\end{bmatrix}}={ \begin{bmatrix} ^\sigma \!\theta \\ I_n\end{bmatrix}}$ which describes the subgroup $\sigma \cdot S_\theta = \ ^\sigma \!(S_\theta )=\langle ^\sigma \!M_{\sigma (1)}, \ldots , \ ^\sigma \!M_{\sigma (n)}\rangle$: the matrix $^\sigma \!\theta$ is still symmetric and uniquely defines a graph $^\sigma \!\Gamma =\Gamma _{^\sigma \!\theta }$. It follows that ${\mathfrak {S}}_n$ preserves ${\mathscr {S}}_{\Theta _n}$ and the actions (5.3) restrict to the actions

$$\begin{aligned} \begin{array}{ccc} {\mathfrak {S}}_n \ \times \ {\mathscr {S}}_{\Theta _n} &{} \longrightarrow &{} {\mathscr {S}}_{\Theta _n}\\ (\sigma \ , \ S_\theta ) &{} \mapsto &{} ^\sigma \!(S_\theta )=S_{^\sigma \!\theta } \end{array} \ \ \ \ \ , \ \ \ \ \ \begin{array}{ccc} {\mathfrak {S}}_n \ \times \ \Theta _n &{} \longrightarrow &{} \Theta _n\\ (\sigma \ , \ |\Gamma \rangle ) &{} \mapsto &{} ^\sigma \!|\Gamma \rangle =|^\sigma \!\Gamma \rangle \end{array} \end{aligned}$$

(6.2)

where $|^\sigma \Gamma \rangle$ is obtained simply by permuting via $\sigma$ the tensor entries of $|\Gamma \rangle$ as in (5.2).

Remark 6.4

(Graphical description) Given a graph $\Gamma$ and a permutation $\sigma \in {\mathfrak {S}}_n$, the graph $^\sigma \!\Gamma$ is obtained simply by renaming the vertices from $\{1,\ldots ,n\}$ to $\{\sigma (1),\ldots , \sigma (n)\}$, without changing the edges. Roughly speaking, two graphs are in the same ${\mathfrak {S}}_n$-orbit if and only if they have the same drawing representation as “dots-edges”.

Action of ${\mathcal {C}}_n^{\textrm{loc}}$. Given a local Clifford transformation $U \in {\mathcal {C}}_n^{\textrm{loc}}$ and a graph group $S_\theta \in {\mathscr {S}}_{\Theta _n}$, we can consider the corresponding local symplectic transformation $\tilde{U}= \begin{bmatrix} A &{} B \\ C &{} D \end{bmatrix}\in {{\,\textrm{Sp}\,}}_{2n}^{\textrm{loc}} ({\mathbb {F}}_2)$ of type (3.3) and the matrix $\begin{bmatrix} \theta \\ I_n\end{bmatrix}$ describing $S_\theta$: the action $U \cdot S_\theta$ corresponds to the linear transformation $\tilde{U} \cdot {\begin{bmatrix} \theta \\ I_n \end{bmatrix}}= { \begin{bmatrix} A\theta + B\\ C\theta + D \end{bmatrix}}$, thus in general $U\cdot S_\theta$ is not a graph group (but a maximal stabilizer group, of course). However, the local Clifford operations preserving $\Theta _n$ are known [11, Proposition 5.3] to correspond to operations on n-graphs, called local complementations (cf. [2, Sect. 1], [11] and [24, Sect. IV, VI]). Given a graph $G=(V,E)$ one defines the local complementation with respect to a vertex v of G as the tranformation $L_v$ described as follow:

• Consider $G_v=(V_v,E_v)$ the subgraph of G consisting of the neighbourhood of v. More precisely the vertices $V_v$ are all vertices $v'\in V$ such that the edge $e_{vv'}$ belongs to E and the edges $E_v$ are all edges of E connecting vertices of $E_v'$.

• Let $G'_v$ be the complement of $G_v$, i.e. $G'_v=(V_v,E_v')$ where $E_v'$ is such that $E_v\cap E_v'=\emptyset$ and $E_v \cup E_v'$ is a complete graph for the vertices $V_v$.

• Consider $G'=(V,E\cup E_v')$.

• Then $G'=L_v(G)$ is the image of G by local complementation.

Figure (1) illustrates the local complementation of the pentagram with respect to the node marked in black.

Fig. 1

Local complementation of the pentagram (top left) with respect to one (marked) vertex (top right and bottom left). The resulting graph is the one at the bottom right

The local complementations together with Remark 6.4 give a graphical description of the action of (a subgroup of) ${\mathcal {C}}_n^{\textrm{loc}}\rtimes {\mathfrak {S}}_n$ on the graph states.

If we denote by $G_n <{\mathcal {C}}_n^{\textrm{loc}}$ the subgroup of the local complementations, and by u the graph transformation corresponding to $U \in G_n$, then one gets the actions

$$\begin{aligned} \begin{array}{ccc} G_n \ \times \ {\mathscr {S}}_{\Theta _n} &{} \longrightarrow &{} {\mathscr {S}}_{\Theta _n}\\ (U \ , \ S_\theta ) &{} \mapsto &{} S_{u\cdot \theta } \end{array} \ \ \ \ \ , \ \ \ \ \ \begin{array}{ccc} G_n \ \times \ \Theta _n &{} \longrightarrow &{} \Theta _n\\ (U \ , \ |\Gamma \rangle ) &{} \mapsto &{} U|\Gamma \rangle \end{array} \end{aligned}$$

(6.3)

which, together with (6.2), give the actions of the semidirect product $G_n\rtimes {\mathfrak {S}}_n$

$$\begin{aligned} \begin{array}{ccc} \left( G_n\rtimes {\mathfrak {S}}_n\right) \ \times {\mathscr {S}}_{\Theta _n} &{} \longrightarrow &{} {\mathscr {S}}_{\Theta _n}\\ \left( (U,\sigma ) \ , \ S_{\theta }\right) &{} \mapsto &{} S_{u\cdot (^\sigma \!\theta )} \end{array} \ \ \ \ , \ \ \ \ \begin{array}{ccc} \left( G_n\rtimes {\mathfrak {S}}_n\right) \times \ \Theta _n &{} \longrightarrow &{} \Theta _n\\ \left( (U,\sigma ) \ ,\ |\Gamma \right) &{} \mapsto &{} U|^\sigma \!\Gamma \rangle \end{array} \ . \end{aligned}$$

(6.4)

Conclusion. For any graph group $S_\theta \in {\mathscr {S}}_{\Theta _n}$ and any $U\in {\mathcal {C}}_n^{\textrm{loc}}$, the subgroup $U\cdot S_\theta$ is still a stabilizer group, thus any graph group (resp. graph state) is the representative of a ${\mathcal {C}}_n^{\textrm{loc}}$-orbit of maximal stabilizer groups (resp. stabilizer states). But there is more: by [24, Theorem 1], each maximal stabilizer group (resp. stabilizer state) is ${\mathcal {C}}_n^{\textrm{loc}}$-equivalent to a graph group (resp. graph state), thus any ${\mathcal {C}}_n^{\textrm{loc}}$-orbit of stabilizer states admits a representative which is a graph states. Thus we get the following orbit correspondence:

$$\begin{aligned} \Phi _1^n \big / C_n^{\textrm{loc}}\rtimes {\mathfrak {S}}_n \ \longleftrightarrow \ \Theta _n \big /G_n \rtimes {\mathfrak {S}}_n \ . \end{aligned}$$

This concludes the proof of Theorem 1.1, proving that the classification (up to ${\mathcal {C}}_n^{\textrm{loc}}\rtimes {\mathfrak {S}}_n$ action) of n-graph states is in one-to-one correspondence with the orbits in ${\mathcal {Z}}_n$ under the action of ${{\,\textrm{SL}\,}}(2,{\mathbb {F}}_2)^{\times n}\rtimes {\mathfrak {S}}_n$:

$$\begin{aligned} \Theta _n\big / G_n\rtimes _\phi {\mathfrak {S}}_n \ \longleftrightarrow \ {\mathcal {Z}}_n \big /{{\,\textrm{SL}\,}}(2,{\mathbb {F}}_2)^{\times n}\rtimes {\mathfrak {S}}_n \ . \end{aligned}$$

(6.5)

7 Applications

7.1 Classification of 4-qubit graph states from orbits of ${\mathcal {Z}}_4$

Under the action of ${{\,\textrm{SL}\,}}(2,{\mathbb {F}}_2)^{\times 4}\rtimes {\mathfrak {S}}_4$, the variety ${\mathcal {Z}}_4 \subset {\mathbb {P}}_2^{15}$ splits in six orbits [13, Table 3]: in Table 1 we recover the representatives of 4-graphs up to $({\mathcal {C}}_4^{\textrm{loc}}\rtimes {\mathfrak {S}}_4)$-action.

Remark 7.1

We consider representatives of the orbits ${\mathcal {O}}_2, {\mathcal {O}}_3, {\mathcal {O}}_6 , {\mathcal {O}}_{14}, {\mathcal {O}}_{17}, {\mathcal {O}}_{18}$ different from the ones in [13, Table 3]: we take them in the chart $\{z_0\ne 0\}$ in order to recover more easily the graph-matrices. Since we want to obtain loopless graphs, we choose representatives with coordinates $z_1=z_2=z_3=z_4=0$, denoted by ${{{\underline{\varvec{0}}}}}$.

Table 1 From orbits in ${\mathcal {Z}}_4$ to classification of 4-graphs

Remark 7.2

The vertices of the graphs in Table 1 are not numbered because of Remark 6.4: we label the upper-left vertex as vertex 1 and order the others clockwisely.

By the Lagrangian mapping, we know that for each orbit ${\mathcal {O}}_i\subset {{\mathcal {Z}}}_4$ there are $4^4|{\mathcal {O}}_i|$ corresponding stabilizer states, since for subspace in ${\mathcal {I}}^4$ there are 4 different phases for the 4 elements spanning it, that is $4^4$ different stabilizer states. From [13, Table 3] we recover the number of stabilizer states for each graph orbit and list them in Table 2.

Table 2 Cardinality of orbits for $n=4$

Table 3 From representatives of 5-graphs to orbits in ${\mathscr {S}}({\mathcal {P}}_5)$

Table 4 From representatives of 5-graphs to orbits in ${\mathcal {Z}}_5$

7.2 Orbits of ${\mathcal {Z}}_5$ from 5-qubit graph states classification

In [7] the number of representatives of 5-graphs was computed to be 11 (see also [11, Table IV]). In the Tables 3 and 4 we exhibit 11 representatives of such orbits and by them we recover the representatives of the orbits in ${\mathcal {Z}}_5 \subset {\mathbb {P}}({\mathbb {F}}_2^{32})$.

Remark 7.3

The representative graphs in Tables 3 and 4 are loopless and their vertices are not numbered because of Remark 6.4: we choose the vertex 1 to be the highest one and the other vertices are clockwise ordered. In the last column of Table 3, the observables generating the stabilizer groups are obtained by the columns of the matrix ${\begin{bmatrix} I_5\\ \theta \end{bmatrix}}$. In the last column of Table 4, the coordinates of the points in ${\mathcal {Z}}_5\subset {\mathbb {P}}_2^{31}$ are given by the principal minors of the matrix $\theta$. In particular, the points are taken in the chart $\{z_0\ne 0\}$ and they have entries $z_{1}=\ldots =z_{5}=0$ since the representative graphs are loopless (we compact such coordinates as a zero vector ${{{\underline{\varvec{0}}}}}$).