# Discrete phase-space structure of $n$-qubit mutually unbiased bases

A. B. Klimov  Departamento de Física, Universidad de Guadalajara, 44420
Guadalajara, Jalisco, Mexico  J. L. Romero  Departamento de Física,
Universidad de Guadalajara, 44420 Guadalajara, Jalisco, Mexico  G. Björk
School of Information and Communication Technology, Royal Institute of
Technology (KTH), Electrum 229, SE-164 40 Kista, Sweden  L. L. Sánchez-Soto
Departamento de Óptica, Facultad de Física, Universidad Complutense, 28040
Madrid, Spain

###### Abstract

We work out the phase-space structure for a system of $n$ qubits. We replace
the field of real numbers that label the axes of the continuous phase space by
the finite field $\mathrm{GF}(2^{n})$ and investigate the geometrical
structures compatible with the notion of unbiasedness. These consist of
bundles of discrete curves intersecting only at the origin and satisfying
certain additional properties. We provide a simple classification of such
curves and study in detail the four- and eight-dimensional cases, analyzing
also the effect of local transformations. In this way, we provide a
comprehensive phase-space approach to the construction of mutually unbiased
bases for $n$ qubits.

###### keywords:

Mutually unbiased bases; discrete phase space; Galois fields

##  1 Introduction

Quantum mechanics describes physical systems through the density operator
$\hat{\varrho}$. For continuous systems, this operator lives in an infinite-
dimensional complex Hilbert space and its relations with the physical
properties of the system are far from obvious. To overcome these conceptual
difficulties, a number of phase-space methods have been devised, which result
in a striking formal similarity with classical mechanics [1].

Textbook examples of this subject are usually presented in terms of continuous
variables, typically position and momentum. However, there are many quantum
systems that can be appropriately described in a finite-dimensional Hilbert
space. These include, among other, spins, multilevel atoms, optical fields
with a fixed number of photons, electrons or molecules with a finite number of
sites, etc. An elegant way of approaching these systems was proposed by Weyl
[2] in his description of quantum kinematics as an Abelian group of ray
rotations. Similar results were also obtained by Schwinger [3, 4, 5], who
showed that a set of unitary operators (defined through cyclic permutations of
state vectors) can be constructed such that they are the generators of a
complete operator basis, in terms of which all possible quantities related to
the physical system can be built.

The structure of the phase space associated to a $d$-dimensional system (a
qudit, in the modern parlance of quantum information) has been addressed by a
number of authors. A possible approach was taken by Hannay and Berry [6],
considering a grid constrained to admit only periodic probability
distributions, which implies that it is effectively a $2d\times
2d$-dimensional torus. The same strategy was adopted by Leonhardt [7, 8] and
used to deal with different aspects of quantum information [9, 10, 11]. This
method offers a way for treating even-dimensional systems, since the grid has
both integer and half-odd coordinates.

However, the mainstream of research has focused on a phase space pictured as a
$d\times d$ lattice. This line was started by Buot [12], who introduced a
discrete Weyl transform that generates a Wigner function on the toroidal
lattice $\mathbb{Z}_{d}$ (with $d$ odd). More recently, these ideas have been
developed further by other authors [13, 14, 15, 16, 17, 18, 19, 20, 21, 22,
23, 24]. In particular, when the dimension is a power of a prime, one can
label the points in the $d\times d$ grid with elements of the finite Galois
field $\mathrm{GF}(d)$. At first sight, the use of elements of
$\mathrm{GF}(d)$ as coordinates could be seen as an unnecessary complication,
but it turns out to be an essential step: only by doing this we can endow the
phase space with similar geometrical properties as the ordinary plane. Note
also that though the restriction to powers of primes rules out many quantum
systems, this formulation is ideally suited for the outstanding case of $n$
qubits we deal in this paper.

In these finite descriptions, the Wigner function, being the Weyl
representative of the density operator, naturally emerges as a function that
takes values only at the points defining the discrete mesh of the phase space
(while preserving some properties that make it a special object in quantum
mechanics). A remarkable feature is that one can sum the Wigner function along
different axes (including skew ones) to obtain correct probability
distributions for observables associated with those axes. Although the axis
observables cannot be complementary in the usual sense (their commutator
cannot be proportional to the identity operator), they will have a closely
related property: every eigenstate of either one of them is a state of maximum
uncertainty with respect to the other.

This makes a deep connection with the notion of mutually unbiased bases (MUBs)
[25, 26], which were introduced as a central tool for quantum state
reconstruction [27]. They also play a relevant role in a proper understanding
of complementarity [28, 29, 30, 31], in cryptographic protocols [32, 33], and
in quantum error correction codes [34, 35]. Recently, they have also found
uses in quantum game theory, in particular to provide a convenient tool for
solving the so-called mean king problem [36, 37, 38, 39, 40, 41, 42, 43].

It has been shown [44] that the maximum number of MUBs can be at most $d+1$.
Actually, it is known that if $d$ is prime or power of a prime the maximal
number of MUBs can be achieved [45]. Different explicit constructions of MUBs
in prime power dimensions have been suggested in a number of recent papers
[46, 47, 48, 49, 50, 51, 52, 53]. Remarkably though, there is no known answer
for any other values of $d$, although there are some attempts to find a
solution in some simple cases, such as $d=6$ or when $d$ is a nonprime integer
squared [54, 55, 56, 57]. Recent work has suggested that the answer to this
question may well be related with the nonexistence of finite projective planes
of certain orders [58, 59] or with the problem of mutually orthogonal Latin
squares in combinatorics [60, 61].

The construction of MUBs is closely related to the possibility of finding
$d+1$ disjoint classes, each one having $d-1$ commuting operators [which
proves useful to arrange in a table with $(d-1)\times(d+1)$ entries], so the
corresponding eigenstates form sets of MUBs [62]. Nevertheless, these MUB
operators can be organized in several different nontrivial tables, leading to
different factorization properties of the MUB [63]. It has been recently
noticed [64] that such arrangements are related with special types of curves.
We have previously analyzed [65] those curves in the four-dimensional case
(corresponding to two qubits) and have shown that they can be obtained through
local transformations from rays (straight lines passing though the origin), so
that the six possible $3\times 5$ tables of operators lead to the same (and
unique) factorization of the MUBs.

In the present paper we go further and analyze the general situation of $n$
qubits. In particular, we classify the admissible curves in specific bundles
and show how the properties of these curves can be used to determine
nontrivial sets of MUBs.

##  2 Mutually unbiased bases and discrete phase space

###  2.1 Constructing mutually unbiased bases in prime dimensions

We start by considering a system living in a Hilbert space $\mathcal{H}_{d}$,
whose dimension $d$ is assumed for now to be a prime number $p$. The different
outcomes of a maximal test constitute an orthogonal basis of $\mathcal{H}_{d}$
[66]. One can also look for orthogonal bases that, in addition, are “as
different as possible”. This is the idea behind MUBs and can be formally
stated as follows: two bases $\\{|u_{i}\rangle\\}$ and $\\{|v_{j}\rangle\\}$
are mutually unbiased when

$|\langle u_{i}|v_{j}\rangle|^{2}=\frac{1}{d}\,.$ (2.1)

Unbiasedness also applies to measurements: two nondegenerate tests are
mutually unbiased if the bases formed by their eigenstates are MUBs.
Therefore, the measurements of the components of a spin 1/2 along $x$, $y$,
and $z$ axes are all unbiased. It is also obvious that for these finite
quantum systems unbiasedness is tantamount of complementarity.

It is useful to choose a computational basis $|n\rangle$ ($n=0,\ldots,d-1$) in
$\mathcal{H}_{d}$ and introduce the basic operators

$X|n\rangle=|n+1\rangle\,,\qquad\qquad Z|n\rangle=\omega^{n}|n\rangle,$ (2.2)

where $\omega=\exp(2\pi i/d)$ is a $d$th root of the unity and addition and
multiplication must be understood modulo $d$. These operators $X$ and $Z$,
which are generalizations of the Pauli matrices, were studied long ago by Weyl
[2] and have been used recently by many authors in a variety of applications
[67, 68]. They generate a group under multiplication known as the generalized
Pauli group and obey

$ZX=\omega\,XZ,$ (2.3)

which is the finite-dimensional version of the Weyl form of the commutation
relations.

As anticipated in the Introduction, one can construct MUBs by finding $d+1$
disjoint classes (each one having $d-1$ commuting operators), such that the
corresponding eigenstates form sets of MUBs. We follow the explicit
construction in reference [52], which starts with the following sets of
operators:

$\\{X^{k}\\},\quad\\{Z^{k}X^{mk}\\},$ (2.4)

with $k=1,\ldots,d-1$ and $m=0,\ldots,d-1$. One can easily check that

$\displaystyle\mathop{\mathrm{Tr}}\nolimits(X^{k}X^{k^{\prime}}{}^{\dagger})=d\,\delta_{k,k^{\prime}},\qquad\mathop{\mathrm{Tr}}\nolimits(Z^{k}Z^{k^{\prime}}{}^{\dagger})=d\,\delta_{k,k^{\prime}},$
(2.5)
$\displaystyle\mathop{\mathrm{Tr}}\nolimits[(Z^{k}X^{mk})(Z^{k^{\prime}}X^{m^{\prime}k^{\prime}})^{\dagger}]=d\,\delta_{k,k^{\prime}}\delta_{m,m^{\prime}}.$

These pairwise orthogonality relations indicate that, for every value of $m$,
we generate a maximal set of $d-1$ commuting operators and that all these
classes are disjoint. In addition, the common eigenstates of each class $m$
form different sets of MUBs.

###  2.2 Mutually unbiased bases for $n$ qubits

When the space dimension $d=p^{n}$ is a power of a prime it is natural to view
the system as composed of $n$ constituents (particles), each of dimension $p$.
We adapt the previous construction to this case, although with an eye on the
particular case of $n$ qubits (in the Appendix we summarize the basic notions
of finite fields needed for our purposes in this paper).

The main idea consists in labeling the states with elements of the finite
field $\mathrm{GF}(2^{n})$, instead of natural numbers. We denote by
$|\alpha\rangle$, with $\alpha\in\mathrm{GF}(2^{n})$, an orthonormal basis in
the Hilbert space of the system. Operationally, the elements of the basis can
be labeled by powers of a primitive element, and the basis reads

$\\{|0\rangle,\,|\sigma\rangle,\ldots,\,|\sigma^{2^{n}-1}=1\rangle\\}.$ (2.6)

These vectors are eigenvectors of the operators $Z_{\beta}$ belonging to the
generalized Pauli group, whose generators are now defined as

$Z_{\beta}=\sum_{\alpha\in\mathrm{GF}(2^{n})}\chi(\alpha\beta)\,|\alpha\rangle\langle\alpha|,\qquad\qquad
X_{\beta}=\sum_{\alpha\in\mathrm{GF}(2^{n})}|\alpha+\beta\rangle\langle\alpha|,\\\
$ (2.7)

so that

$Z_{\alpha}X_{\beta}=\chi(\alpha\beta)\,X_{\beta}Z_{\alpha},$ (2.8)

where $\chi$ is an additive character defined in (A.5).

The operators (2.7) can be factorized into tensor products of powers of
single-particle Pauli operators $\sigma_{z}$ and $\sigma_{x}$, whose
expression in the standard basis of the two-dimensional Hilbert space is

$\sigma_{z}=|1\rangle\langle 1|-|0\rangle\langle
0|,\qquad\qquad\sigma_{x}=|0\rangle\langle 1|+|1\rangle\langle 0|.$ (2.9)

This factorization can be carried out by mapping each element of
$\mathrm{GF}(2^{n})$ onto an ordered set of natural numbers as in equation
(A.7). A convenient choice for this is the selfdual basis, since the finite
Fourier transform factorizes then into a product of single-particle Fourier
operators, which leads to

$Z_{\alpha}=\sigma_{z}^{a_{1}}\otimes\ldots\otimes\sigma_{z}^{a_{n}},\qquad\qquad
X_{\beta}=\sigma_{x}^{b_{1}}\otimes\ldots\otimes\sigma_{x}^{b_{n}}$ (2.10)

where $(a_{1},\ldots,a_{n})$ and $(b_{1},\ldots,b_{n})$ are the corresponding
coefficients.

The simplest geometrical structures in the discrete phase space are straight
lines, i.e., collections of points
$(\alpha,\beta)\in\mathrm{GF}(2^{n})\times\mathrm{GF}(2^{n})$ satisfying the
relation

$\zeta\alpha+\eta\beta=\vartheta,$ (2.11)

where $\zeta,\eta$ and $\vartheta$ are fixed elements of $\mathrm{GF}(2^{n})$.
Two lines

$\zeta\alpha+\eta\beta=\vartheta,\qquad\qquad\zeta^{\prime}\alpha+\eta^{\prime}\beta=\vartheta^{\prime},$
(2.12)

are parallel if they have no common points, which implies that
$\eta\zeta^{\prime}=\zeta\eta^{\prime}$. If the lines (2.12) are not parallel
they cross each other. A ray is a line passing through the origin, so its
equation is

$\alpha=0,\qquad\mathrm{or}\qquad\beta=\lambda\alpha,$ (2.13)

or, in parametric form,

$\alpha(\kappa)=\mu\kappa,\qquad\qquad\beta(\kappa)=\nu\kappa,$ (2.14)

where $\kappa$ is the parameter running through the field. The rays are the
simplest nonsingular (i.e., with no selfintersection) additive structures in
phase space, in the sense that

$\displaystyle\alpha(\kappa+\kappa^{\prime})=\alpha(\kappa)+\alpha(\kappa^{\prime}),\qquad\qquad\beta(\kappa+\kappa^{\prime})=\beta(\kappa)+\beta(\kappa^{\prime}).$
(2.15)

This means that by summing the coordinates of the origin and of any point in a
ray we obtain another point on the same ray. In particular, this opens the
possibility of introducing operators that generate “translations” along these
rays [22].

The rays have a very remarkable property: the monomials $Z_{\alpha}X_{\beta}$
(labeled by phase-space points) belonging to the same ray commute

$Z_{\alpha_{1}}X_{\beta_{1}=\lambda\alpha_{1}}\,Z_{\alpha_{2}}X_{\beta_{2}=\lambda\alpha_{2}}=Z_{\alpha_{2}}X_{\beta_{2}=\lambda\alpha_{2}}\,Z_{\alpha_{1}}X_{\beta_{1}=\lambda\alpha_{1}},$
(2.16)

and thus, have a common system of eigenvectors
$\\{|\psi_{\upsilon,\lambda}\rangle\\}$, with
$\lambda,\upsilon\in\mathrm{GF}(2^{n})$:

$Z_{\alpha}X_{\lambda\alpha}|\psi_{\upsilon,\lambda}\rangle=\exp(i\xi_{\upsilon,\lambda})|\psi_{\upsilon,\lambda}\rangle,$
(2.17)

where $\lambda$ is fixed and $\exp(i\xi_{\upsilon,\lambda})$ is the
corresponding eigenvalue, so $|\psi_{\upsilon,0}\rangle=|\upsilon\rangle$ are
eigenstates of $Z_{\alpha}$ (displacement operators labeled by the ray
$\beta=0$, which we take as the horizontal axis). Indeed, we have that

$|\langle\psi_{\upsilon,\lambda}|\psi_{\upsilon^{\prime},\lambda^{\prime}}\rangle|^{2}=\delta_{\lambda,\lambda^{\prime}}\delta_{\upsilon,\upsilon^{\prime}}+\frac{1}{d}(1-\delta_{\lambda,\lambda^{\prime}}),$
(2.18)

and, in consequence, they are MUBs [27]. Since each ray defines a set of
$2^{n}-1$ commuting operators, if we introduce $2^{n}+1$ sets of commuting
operators (which from now on will be called displacement operators) as

$\\{X_{\beta}\\},\quad\\{Z_{\alpha}X_{\beta=\lambda\alpha}\\}\,,$ (2.19)

then we have a whole bundle of $2^{n}+1$ rays (which is obtained by varying
the “slope” $\lambda$) that allows us to construct a complete set of MUB
operators arranged in a $(2^{n}-1)\times(2^{n}+1)$ table.

We wish to emphasize that in our approach we do not assign a quantum state to
each line in phase space (as in reference [22]), but rather we use them to
label Pauli displacement operators.

##  3 Curves in discrete phase space

###  3.1 Additive curves and the commutativity condition

The rays are not the only additive structures that exist in the discrete phase
space. One can check that the parametric curves (passing through the origin)

$\alpha(\kappa)=\sum_{m=0}^{n-1}\alpha_{m}\,\kappa^{2^{m}},\qquad\qquad\beta(\kappa)=\sum_{m=0}^{n-1}\beta_{m}\,\kappa^{2^{m}},$
(3.1)

satisfy the condition (2.15) too. If we also require the displacement
operators labeled with the points of (3.1) to commute with each other, we must
impose

$\mathop{\mathrm{tr}}\nolimits(\alpha\beta^{\prime})=\mathop{\mathrm{tr}}\nolimits(\alpha^{\prime}\beta),$
(3.2)

where $\alpha^{\prime}=\alpha(\kappa^{\prime})$ and
$\beta^{\prime}=\beta(\kappa^{\prime})$. Then, the coefficients $\alpha_{m}$
and $\beta_{m}$ fulfill the following restrictions (the indices must be
understood modulus $n$)

$\sum_{m=0}^{n-1}\alpha_{n-m}^{2^{m}}\,\beta_{n-m+q}^{2^{m}}=\sum_{m=0}^{n-1}\beta_{n-m}^{2^{m}}\,\alpha_{n-m+q}^{2^{m}},\qquad
q=1,\ldots,n-1.$ (3.3)

This can be rewritten in an invariant form by summing up all Frobenius
automorphisms of (3.3):

$\sum_{m\neq k}\mathop{\mathrm{tr}}\nolimits(\alpha_{m}\beta_{k})=0.$ (3.4)

Whenever the condition (3.4) holds true, we can associate to each curve (3.1),
with given coefficients $\vec{\alpha}=(\alpha_{0},\ldots,\alpha_{n-1})$ and
$\vec{\beta}=(\beta_{0},\ldots,\beta_{n-1})$, a state
$|\psi_{\vec{\alpha},\vec{\beta}}\rangle$.

The curves fulfilling equation (3.4) will be called additive commutative
curves. When such a curve contains $2^{n}-1$ different points (apart from the
origin), the monomials $Z_{\alpha}X_{\beta}$ form also a set of commuting
operators (as happened for the rays).

In fact, consider a subset $Z_{\alpha(\kappa)}X_{\beta(\kappa)}$, such that

$[Z_{\alpha(\kappa)}X_{\beta(\kappa)},Z_{\alpha(\kappa^{\prime})}X_{\beta(\kappa^{\prime})}]=0,\qquad\qquad\mathop{\mathrm{Tr}}\nolimits(Z_{\alpha(\kappa)}X_{\beta(\kappa)}Z_{\alpha(\kappa^{\prime})}^{\dagger}X_{\beta(\kappa^{\prime})}^{\dagger})=0,$
(3.5)

for any $\kappa,\kappa^{\prime}\in\mathrm{GF}(p^{n})$, i.e. a disjoint set of
$d$ mutually commuting monomials, including
$Z_{\alpha(0)}X_{\beta(0)}=\leavevmode\hbox{\small
1\normalsize\kern-3.30002pt1}$. Then, the eigenstates of any two disjoint sets
of these mutually commuting monomials form MUBs.

To prove this important result, we first note that any set of commuting
monomials can be obtained by applying Clifford operations $U_{\lambda}$ to the
simplest set $\\{Z_{\kappa}\\}$:

$U_{\lambda}Z_{\kappa}U_{\lambda}^{\dagger}=\phi(\lambda,\kappa)\,D_{\lambda,\kappa},$
(3.6)

where $\phi(\lambda,\kappa)$ is an unessential phase factor
[$\phi(\lambda,0)=\phi(0,\kappa)=1$] and
$D_{\lambda,\kappa}=Z_{\alpha(\lambda,\kappa)}X_{\beta(\lambda,\kappa)}$
fulfill

$[D_{\lambda,\kappa},D_{\lambda,\kappa^{\prime}}]=0,\qquad\qquad\mathop{\mathrm{Tr}}\nolimits(D_{\lambda,\kappa}D_{\lambda,\kappa^{\prime}}^{\dagger})=d\,\delta_{\kappa,\kappa^{\prime}},$
(3.7)

since the transformations $U_{\lambda}$ are nondegenerate. For two disjoint
set of commuting monomials $\\{D_{\lambda,\kappa}\\}$ and
$\\{D_{\lambda^{\prime},\kappa^{\prime}}\\}$ we have then

$\mathop{\mathrm{Tr}}\nolimits(D_{\lambda,\kappa}D_{\lambda^{\prime},\kappa^{\prime}}^{\dagger})=\left\\{\begin{array}[]{cc}d\,\delta_{\kappa,\kappa^{\prime}},&\quad\lambda=\lambda^{\prime},\\\
d\,\delta_{\kappa,0}\delta_{\kappa^{\prime},0},&\quad\lambda\neq\lambda^{\prime}\end{array}\right..$
(3.8)

Let $\\{|\upsilon\rangle\\}$ be the basis of eigenstates of $Z_{\kappa}$. It
is worth noting the following expansion

$|\upsilon\rangle\langle\upsilon|=\frac{1}{d}\sum_{\kappa}\chi(-\upsilon\kappa)Z_{\kappa}=\frac{1}{d}\sum_{\kappa}\chi(\upsilon\kappa)Z_{\kappa}^{\dagger}.$
(3.9)

Now, if we define the states
$|\psi_{\upsilon,\lambda}\rangle=U_{\lambda}|\upsilon\rangle$ and
$|\psi_{\upsilon^{\prime},\lambda^{\prime}}\rangle=U_{\lambda^{\prime}}|\upsilon^{\prime}\rangle$,
where $U_{\lambda}$ is as in equation (3.6), then a direct calculation using
(3.9) shows that (2.18) holds true for them and so they are indeed MUBs, as
announced.

Finding sets of MUBs can be thus reduced to the problem of arranging additive
curves in $2^{n}+1$ bundles of mutually nonintersecting curves.

Due to the condition (2.15), points of a curve form a finitely generated
Abelian group, which allows us to determine all the curve from any $n$ points
and, in particular, from the “first” $n$ consecutive points. For instance,
taking the parameter $\kappa$ polynomially ordered (that is,
$\kappa=\sigma,\sigma^{2},\ldots,\sigma^{2^{n}-1}$), we have that
$\alpha(\sigma^{k})+\alpha(\sigma^{k+1})=\alpha(\sigma^{k}+\sigma^{k+1})=\alpha[\sigma^{k}(1+\sigma)]=\alpha[\sigma^{k+L(1)}],$
where $L(\lambda)$ is the Jacobi logarithm [69]. Given a primitive polynomial,
sometimes is possible to evaluate $L(1)$ in a simple form.

For $\mathrm{GF}(2^{2})$ the only irreducible polynomial is $x^{2}+x+1=0$, and
this immediately leads to $L(1)=L(\sigma^{3})=2$, in such a way that
$\alpha(\sigma^{k})+\alpha(\sigma^{k+1})=\alpha(\sigma^{k+2})$ and two points
are enough to determine any additive curve. In the case of
$\mathrm{GF}(2^{3})$, if we use the irreducible polynomial $x^{3}+x+1=0$, we
obtain that $L(1)=L(\sigma^{7})=3$, so that
$\alpha(\sigma^{k})+\alpha(\sigma^{k+1})=\alpha(\sigma^{k+3})$ and we need
three points to generate the curve.

###  3.2 Nonsingularity condition

To classify all the additive commutative curves, we first discuss the
condition of nonsingularity (i.e., nonselfintersection), which means that
there are no $\kappa^{\prime}\neq\kappa$ such that

$\alpha(\kappa)=\alpha(\kappa^{\prime}),\qquad\qquad\beta(\kappa)=\beta(\kappa^{\prime}).$
(3.10)

If one of these equations would be fulfilled, then
$\mathfrak{S}^{m}[\varepsilon(\kappa)]=0$ (with $\varepsilon=\alpha$ or
$\beta$) for $m=0,\ldots,n-1$.

Let us introduce the following matrix that will play a relevant role in our
subsequent analysis:

$\mathsf{W}_{\vec{\varepsilon}}=\left(\begin{array}[]{cccc}\varepsilon_{0}&\varepsilon_{1}&\varepsilon_{2}&\ldots\\\
\varepsilon_{n-1}^{2}&\varepsilon_{0}^{2}&\varepsilon_{1}^{2}&\ldots\\\
\varepsilon_{n-2}^{4}&\varepsilon_{n-1}^{4}&\varepsilon_{0}^{4}&\ldots\\\
\vdots&\vdots&\vdots&\ddots\end{array}\right),$ (3.11)

where the rows are determined by the coefficients in (3.1) and the
corresponding expansions of $\mathfrak{S}^{m}[\varepsilon(\kappa)]$ (with
$\varepsilon=\alpha,\beta$ in our case). If $\det\mathsf{W}_{\vec{\alpha}}$
and/or $\det\mathsf{W}_{\vec{\beta}}$ do not vanish simultaneously, the curve
(3.1) has no selfintersection.

The condition

$\det\mathsf{W}_{\vec{\alpha}}=\det\mathsf{W}_{\vec{\beta}}=0,$ (3.12)

indicates that the ranks of the matrices $\mathsf{W}_{\vec{\alpha}}$ and
$\mathsf{W}_{\vec{\beta}}$ are smaller than the dimension of the system, but
it does not guarantee that there exist $\kappa^{\prime}\neq\kappa$ satisfying
(3.10), because the solutions of $\alpha(\kappa)=\alpha(\kappa^{\prime})$ and
$\beta(\kappa^{\prime\prime})=\beta(\kappa^{\prime\prime\prime})$ can form
disjoint sets. Consequently, (3.12) is necessary but not sufficient to
determine if a curve is singular. Another necessary but not sufficient
condition of singularity is
$\det(\mathsf{W}_{\vec{\alpha}}+\mathsf{W}_{\vec{\beta}})=0$.

A curve that fulfills $\det\mathsf{W}_{\vec{\alpha}}\neq 0$ and/or
$\det\mathsf{W}_{\vec{\beta}}\neq 0$ will be called a regular curve. For such
curves the coordinate $\alpha$ (when $\det\mathsf{W}_{\vec{\alpha}}\neq 0$) or
$\beta$ (when $\det\mathsf{W}_{\vec{\beta}}\neq 0$) take all the values in the
field.

Nonsingular curves satisfying (3.12) will be called exceptional curves. The
conditions (3.12) mean that $\mathfrak{S}^{m}(\alpha)$ and
$\mathfrak{S}^{m}(\beta)$ are not linearly independent (for $m=0,\ldots,n-1$),
so neither $\alpha$ or $\beta$ run through the whole field (in other words,
the values of $\alpha$ and $\beta$ are degenerate). The number of linearly
independent powers of $\alpha$ (respectively $\beta$) is equal to the rank of
the matrix $\mathsf{W}_{\vec{\alpha}}$ (respectively
$\mathsf{W}_{\vec{\beta}}$) and the quantities
$n-\mathop{\mathrm{rank}}\nolimits\mathsf{W}_{\vec{\alpha}}$ (respectively
$n-\mathop{\mathrm{rank}}\nolimits\mathsf{W}_{\vec{\beta}}$) determine the
degree of degeneration of every allowed value of $\alpha$ (respectively
$\beta$).

It is interesting to note that the determinant of the matrix (3.11) takes only
the values zero and one; i.e.,
$\det\mathsf{W}_{\vec{\varepsilon}}\in\mathbb{Z}_{2}$, which can be easily
seen by observing that
$(\det\mathsf{W}_{\vec{\varepsilon}})^{2}=\det\mathsf{W}_{\vec{\varepsilon}}$.

##  4 Regular curves

###  4.1 Explicit forms

Given a regular curve, we can invert one of the relations (3.1) and, by
substituting into the other one, we find an explicit equation of the curve.
When $\det\mathsf{W}_{\vec{\alpha}}\neq 0$ and
$\det\mathsf{W}_{\vec{\beta}}\neq 0$, the coordinates $\alpha$ and $\beta$ are
nondegenerate and the curve can be written either as

$\beta=\sum_{m=0}^{n-1}\phi_{m}\,\alpha^{2^{m}};$ (4.1)

or

$\alpha=\sum_{m=0}^{n-1}\psi_{m}\,\beta^{2^{m}}.$ (4.2)

However, when $\det\mathsf{W}_{\vec{\alpha}}\neq 0$ but
$\det\mathsf{W}_{\vec{\beta}}=0$, the coordinate $\beta$ is degenerate and the
curve cannot be expressed in the form (4.2). We will refer to the
corresponding curve as $\alpha$-curve. Similarly, if
$\det\mathsf{W}_{\vec{\beta}}\neq 0$ but $\det\mathsf{W}_{\vec{\alpha}}=0$,
the coordinate $\alpha$ is degenerate and the curve cannot be expressed in the
form (4.1): we will call it a $\beta$-curve.

###  4.2 Commutativity conditions

The commutativity condition (3.4) can be further simplified for regular
curves. When $\det\mathsf{W}_{\vec{\alpha}}\neq 0$ (or
$\det\mathsf{W}_{\vec{\beta}}\neq 0)$ we obtain, by direct substitution of the
explicit forms (4.1) or (4.2) into (3.2), the following restrictions on the
coefficients $\phi_{m}$ (or $\psi_{m})$

$\phi_{j}=\phi_{n-j}^{2^{j}},\qquad\qquad\psi_{j}=\psi_{n-j}^{2^{j}},\qquad
j=1,\ldots,[(n-1)/2],$ (4.3)

where $[\ ]$ denotes the integer part. For even values of $n$, the additional
requirements $\phi_{n/2}=\phi_{n/2}^{2^{n/2}}$
($\psi_{n/2}=\psi_{n/2}^{2^{n/2}}$) should be fulfilled, which basically
implies that $\phi_{n/2}$ ($\psi_{n/2}$) belong to the subfield
$\mathrm{GF}(2^{n/2})$.

Because the regular curves are nonsingular per definition, we do not have to
carry out the whole analysis involving the parametric forms of curves and the
properties of the corresponding $\mathsf{W}_{\vec{\alpha}}$ and
$\mathsf{W}_{\vec{\beta}}$, but just to write down explicit expressions using
directly (4.3).

Two regular curves defined explicitly as $\beta=f(\alpha,\vec{\phi})$ and
$\beta=f(\alpha,\vec{\phi}^{\prime})$, with
$\vec{\phi}=(\phi_{0},\ldots,\phi_{n-1})$ [or $\alpha=g(\beta,\vec{\psi})$ and
$\alpha=g(\beta,\vec{\psi}^{\prime})$, with
$\vec{\psi}=(\psi_{0},\ldots,\psi_{n-1})$] are not mutually intersecting
(except at the origin), if

$\det(\mathsf{W}_{\vec{\phi}}+\mathsf{W}_{\vec{\phi}^{\prime}})\neq
0,\qquad\qquad\det(\mathsf{W}_{\vec{\psi}}+\mathsf{W}_{\vec{\psi}^{\prime}})\neq
0,$ (4.4)

for $\alpha$\- and $\beta$-curves, respectively, and the matrices $\mathsf{W}$
have been defined in (3.11).

An $\alpha$-curve $\beta=f(\alpha,\vec{\phi})$ intersects with a $\beta$-curve
$\alpha=g(\beta,\vec{\psi})$ when the polynomial
$\beta=f[g(\beta,\vec{\psi}),\vec{\phi}]$ (or
$\alpha=g[f(\alpha,\vec{\phi}),\vec{\psi}]$) has at least one nonzero root.

It follows from (4.4) that the regular curves

$\beta=\phi_{0}\alpha+\sum_{m=1}^{n-1}\phi_{m}\alpha^{2^{m}},$ (4.5)

where $\phi_{m}$ ($m=1,\ldots,n-1$) are fixed and fulfill (4.3), and
$\phi_{0}$ runs through the whole field, belong to a bundle of nonintersecting
curves, since the matrices
$\mathsf{W}_{\vec{\phi}}+\mathsf{W}_{\vec{\phi}^{\prime}}$ take now the
diagonal form

$\mathsf{W}_{\vec{\phi}}+\mathsf{W}_{\vec{\phi}^{\prime}}=\mathop{\mathrm{diag}}\nolimits\left[\phi_{0}+\phi_{0}^{\prime},(\phi_{0}+\phi_{0}^{\prime})^{2},\ldots,(\phi_{0}+\phi_{0}^{\prime})^{2^{n-1}}\right],$
(4.6)

so that

$\det(\mathsf{W}_{\vec{\phi}}+\mathsf{W}_{\vec{\phi}^{\prime}})=(\phi_{0}+\phi_{0}^{\prime})^{2^{n}-1},$
(4.7)

and thus, $\det(\mathsf{W}_{\vec{\phi}}+\mathsf{W}_{\vec{\phi}^{\prime}})=1$
if $\phi_{0}\neq\phi_{0}^{\prime}$. To complete the bundle to $n+1$ curves we
add the ray $\alpha=0$, which obviously has no common points with (4.5).

Similarly, the curves

$\alpha=\psi_{0}\beta+\sum_{m=1}^{n-1}\psi_{m}\,\beta^{2^{m}}$ (4.8)

form bundles of nonintersecting curves, except that now we have to add the ray
$\beta=0$ to complete the bundle.

###  4.3 Examples

####  4.3.1 Regular curve

Let us consider the following parametric curve in $\mathrm{GF}(2^{3})$

$\displaystyle\alpha=\sigma^{2}\kappa+\kappa^{2}+\sigma^{4}\kappa^{4},\qquad\qquad\beta=\sigma^{3}\kappa+\sigma^{6}\kappa^{2}+\sigma^{6}\kappa^{4},$
(4.9)

where $\sigma$ is the primitive element. The associated matrices are

$\mathsf{W}_{\vec{\alpha}}=\left(\begin{array}[]{ccc}\sigma^{2}&1&\sigma^{4}\\\
\sigma&\sigma^{4}&1\\\
1&\sigma^{2}&\sigma\end{array}\right),\qquad\qquad\mathsf{W}_{\vec{\beta}}=\left(\begin{array}[]{ccc}\sigma^{3}&\sigma^{6}&\sigma^{6}\\\
\sigma^{5}&\sigma^{6}&\sigma^{5}\\\
\sigma^{3}&\sigma^{3}&\sigma^{5}\end{array}\right).$ (4.10)

One can check that
$\det\mathsf{W}_{\vec{\alpha}}=\det\mathsf{W}_{\vec{\beta}}=1$, and the
explicit forms of the curve are

$\beta=\sigma^{6}\alpha+\sigma^{3}\alpha^{2}+\sigma^{5}\alpha^{4},\quad\mathrm{or}\quad\alpha=\sigma^{6}\beta+\sigma^{3}\beta^{2}+\sigma^{5}\beta^{4},$
(4.11)

whose coefficients satisfy the condition (4.3). The set of commuting operators
corresponding to this curve is

$\\{Z_{\sigma^{6}}X_{\sigma^{5}},\,Z_{\sigma^{5}}X_{\sigma^{6}},\,Z_{\sigma^{4}}X_{\sigma^{2}},\,Z_{\sigma}X_{\sigma},\,Z_{\sigma^{7}}X_{\sigma^{7}},Z_{\sigma^{2}}X_{\sigma^{4}},\,Z_{\sigma^{3}}X_{\sigma^{3}}\\}.$
(4.12)

The curve belongs to a bundle of nonintersecting curves defined, for instance,
by $\beta=\phi_{0}\alpha+\sigma^{3}\alpha^{2}+\sigma^{5}\alpha^{4}$.

####  4.3.2 $\alpha$-curve

To the parametric curve

$\alpha=\sigma^{2}\kappa^{4},\qquad\qquad\beta=\sigma^{2}\kappa+\kappa^{2}+\sigma\kappa^{4},$
(4.13)

it corresponds the matrices

$\mathsf{W}_{\vec{\alpha}}=\left(\begin{array}[]{ccc}0&0&\sigma^{2}\\\
\sigma^{4}&0&0\\\
0&\sigma&0\end{array}\right),\qquad\qquad\mathsf{W}_{\vec{\beta}}=\left(\begin{array}[]{ccc}\sigma^{2}&1&\sigma\\\
\sigma^{2}&\sigma^{4}&1\\\ 1&\sigma^{4}&\sigma\end{array}\right).$ (4.14)

Now we have $\det\mathsf{W}_{\vec{\alpha}}=1$ and
$\det\mathsf{W}_{\vec{\beta}}=0$, which leads to the following explicit form
of the $\alpha$-curve

$\beta=\sigma^{6}\alpha+\sigma^{5}\alpha^{2}+\sigma^{6}\alpha^{4},$ (4.15)

whose coefficients satisfy again the condition (4.3). The corresponding
$\mathsf{W}_{\vec{\phi}}$ matrix is degenerate in this case. The set of
commuting operators is

$\\{Z_{\sigma^{6}},\,Z_{\sigma^{3}}X_{\sigma^{2}},\,Z_{\sigma^{7}}X_{\sigma^{5}},\,Z_{\sigma^{4}}X_{\sigma^{2}},\,Z_{\sigma}X_{\sigma^{3}},\,Z_{\sigma^{5}}X_{\sigma^{3}},\,Z_{\sigma^{2}}X_{\sigma^{5}}\\},$
(4.16)

and the curve belongs to the bundle
$\beta=\psi_{0}\alpha+\sigma^{5}\alpha^{2}+\sigma^{6}\alpha^{4}$.

##  5 Exceptional curves

The analysis of exceptional curves is considerably more involved. As we have
stressed above, the points on the curve do not take all the values in the
field and their admissible values are fixed by the structural equations

$\sum_{m=0}^{r_{\alpha}}\upsilon_{m}\,\alpha^{2^{m}}=0,\qquad\qquad\sum_{m=0}^{r_{\beta}}\tau_{m}\,\beta^{2^{m}}=0,$
(5.1)

where $r_{\alpha}=\mathop{\mathrm{rank}}\nolimits\mathsf{W}_{\vec{\alpha}}\leq
n-1$ and
$r_{\beta}=\mathop{\mathrm{rank}}\nolimits\mathsf{W}_{\vec{\beta}}\leq n-1$,
which are a consequence of the linear dependence of $\alpha^{2^{m}}$ and
$\beta^{2^{m}}$. The coordinates $\alpha$ and $\beta$ of an exceptional curve
are $\deg\alpha=2^{n-r_{\alpha}}$ and $\deg\beta=2^{n-r_{\beta}}$ times
degenerate, respectively. In other words, if $(\alpha_{j},\beta_{j})$ is a
point of an exceptional curve, for each $\alpha_{j}$ there are
$2^{n-r_{\beta}}$ values of $\beta$, such that the points
$(\alpha_{j},\beta_{k})$ ($k=1,\ldots,2^{n-r_{\beta}}$) belong to the same
curve and, conversely, for each $\beta_{j}$ there are $2^{n-r_{\alpha}}$
values of $\alpha$, such that the points $(\alpha_{k},\beta_{j})$ also belong
to the same curve.

Due to the nonsingularity condition, there are $2^{n}$ different pairs of
points $(\alpha,\beta)$ belonging to the curve, so the condition
$r_{\alpha}+r_{\beta}\geq n$ is satisfied. For instance, for
$\mathrm{GF}(2^{2})$ the only possibility is $r_{\alpha}=r_{\beta}=1$, and the
only type of degeneration is $\deg\alpha=\deg\beta=2$. For
$\mathrm{GF}(2^{3})$ there are three possibilities: $r_{\alpha}=r_{\beta}=2$
($\deg\alpha=\deg\beta=2$), $r_{\alpha}=1,r_{\beta}=2$ ($\deg\alpha=4$,
$\deg\beta=2$), and $r_{\alpha}=2,r_{\beta}=1$ ($\deg\alpha=2$,
$\deg\beta=4$).

When a curve equation can be found [i.e., a relation of the type
$F(\alpha)=G(\beta)$, where $F(\alpha)$ and $G(\beta)$ are polynomials of
degrees $2^{(r_{\alpha}-1)}$ and $2^{(r_{\beta}-1)}$], it establishes a direct
correspondence between the roots of (5.1). To define uniquely the curve, the
equation $F(\alpha)=G(\beta)$ should be supplemented with the structural
equations. Nevertheless, such a relation exists only when the conditions
$r_{\alpha,\beta}\geq(n+1)/2$ hold.

There are two ways of approaching the classification of exceptional curves.
The first is a direct analysis of the parametric form (3.1) [whose
coefficients satisfy the commutativity relation (3.4) and the corresponding
determinants vanish]. We have to determine the rank of the matrices
$\mathsf{W}_{\vec{\alpha}}$ and $\mathsf{W}_{\vec{\beta}}$, find the
structural relations (5.1), and check the nonsingularity condition. The main
difficulty with this approach is the complicated form of (3.4), which is
related to that fact that there is no one-to-one correspondence between the
parametric form of a curve and points in the discrete phase space, in the
sense that the same curve can be defined by several different parametric
equations.

We shall take an alternative route and construct all the possible exceptional
curves by imposing ab initio the nonsingularity and commutativity conditions.
An important ingredient in this construction is the existence of a priori
information about the degree of degeneration in the $\alpha$ and $\beta$
directions. We shall outline the main idea and study in detail only the eight-
dimensional case.

###  5.1 Constructing exceptional curves

Let us consider a nonsingular curve with degenerations $2^{n-r_{\alpha}}$ and
$2^{n-r_{\beta}}$ along the $\alpha$ and $\beta$ axes, respectively. The
structural equations (5.1) can be represented as

$\alpha\,\prod_{j=1}^{2^{r_{\alpha}}-1}(\alpha+\alpha_{j})=0,\qquad\qquad\beta\,\prod_{j=1}^{2^{r_{\beta}}-1}(\beta+\beta_{j})=0,$
(5.2)

with all the roots $\alpha_{j}$ and $\beta_{j}$ different. Since only the
powers $\alpha^{2^{m}}$ and $\beta^{2^{m}}$ ($m=0,\ldots,r_{\alpha,\beta}$)
can appear in (5.2), we obtain the following restriction on the roots
$\alpha_{j}$ and $\beta_{j}$

$S_{k}(\alpha_{j})=0,\qquad\qquad S_{k}(\beta_{j})=0,\qquad k\neq
2^{r_{\alpha,\beta}}-2^{m}$ (5.3)

where

$S_{k}(\xi)=\sum_{i_{1}<i_{2}<\ldots<i_{k}}\xi_{i_{1}}\xi_{i_{2}}\ldots\xi_{i_{k}},$
(5.4)

are symmetrical functions of the roots. This restriction implies that only
$r_{\alpha}$ ($r_{\beta}$) roots $\alpha_{j}$ ($\beta_{j}$) are linearly
independent.

Condition (3.2) implies that, given a degree of degeneration and once one of
the structural equations is fixed, the other structural equation is not
arbitrary. In other words, having determined the admissible points along one
of the axis, all the admissible points of an additive commutative curve along
the other axis are uniquely defined.

The simplest situation corresponds to the case when degeneration along both
axes is the same: $\deg\alpha=\deg\beta=g=2^{n-r}$ ($r\geq n/2$), i.e., the
ranks of the corresponding matrices (3.11) are $r=r_{\alpha}=r_{\beta}$.
Suppose that the structural equation for $\alpha$ is fixed. Then, the
commutativity condition is equivalent to the following set of equations

$\mathop{\mathrm{tr}}\nolimits(\alpha_{k}\beta_{j})=0,\qquad\qquad\left\\{\begin{array}[]{ll}j=1,\ldots,n-r,&\quad
k=1,\ldots,r,\\\ j=1,\ldots,r,&\quad k=1,\ldots,n-r,\end{array}\right.$ (5.5)

where $(\alpha_{k},\beta_{k})$ are linearly independent roots and the points
$(\alpha_{j},0)$ and $(0,\beta_{j})$, with $j=1,\ldots,2^{n-r}$, belong to the
curve.

Equations (5.5) mean that (5.2) can be always written as
$\mathop{\mathrm{tr}}\nolimits(\alpha\,\xi)=0,\mathop{\mathrm{tr}}\nolimits(\beta\
\zeta)=0,$ with $\xi$ and $\zeta$ being fixed elements of
$\mathrm{GF}(2^{n})$.

In the doubly degenerate case, $g=2$, the value of $\beta_{1}$ is uniquely
determined from the first condition in (5.5) and the curve can be represented
as a disjoint union of two straight lines

$\beta^{(1)}=\frac{\beta_{1}}{\alpha_{1}}\,\alpha,\quad\beta^{(2)}=\frac{\beta_{1}}{\alpha_{1}}\,\alpha+\beta_{1},$
(5.6)

with $\alpha=\alpha_{1},\ldots,\alpha_{2^{r}-1}$. It is worth noting that
(5.5) in this case is just a structural equation, so that

$\beta_{1}=\frac{\upsilon_{1}}{\upsilon_{0}}=\frac{S_{2^{r}-2}}{S_{2^{r}-1}},$
(5.7)

where $\upsilon_{k}$ are the coefficients in (5.1) and $S_{r}$ are the
symmetrical functions (5.3) of arguments $\alpha_{k}$ ($k=1,\ldots,2^{r}-1$).
Each different ordered set $\\{\alpha_{1},\ldots,\alpha_{r}\\}$, with
$\alpha_{j}\neq\alpha_{k}\neq 0$ determines thus an exceptional doubly
degenerate curve.

For higher degenerations, the curve is a disjoint union of $g$ straight lines

$\beta^{(1)}=\lambda\alpha,\;\beta^{(2)}=\lambda\alpha+\beta_{1},\ldots,\;\beta^{(g)}=\lambda\alpha+\beta_{g-1},$
(5.8)

where $\beta_{m}$ ($m=r+1,\ldots,g-1$) are obtained as the possible different
linear combinations of $\beta_{j}$ ($j=1,\ldots,r$) and

$\lambda=\frac{\beta_{1}}{\alpha_{1}}=\ldots=\frac{\beta_{r}}{\alpha_{r}}.$
(5.9)

The above ordering indicates that the points $(\alpha_{j},\beta_{j})$ belong
to the same straight line. Then, $\beta_{k}=\beta_{1}\alpha_{k}/\alpha_{1}$
($k=2,\ldots,r$) and $\beta_{1}$ are uniquely expressed in terms of admissible
values of $\alpha_{j}$ from (5.5), which is convenient to rewrite in terms of
the parameter $\lambda$:
$\mathop{\mathrm{tr}}\nolimits(\lambda\alpha_{j}\,\alpha_{k})=0$, for
$j,k=1,\ldots,r$. It is clear that the exceptional curves constructed using
the equations (5.6) to (5.8) are nonsingular.

If the degeneration along $\alpha$ and $\beta$ axes are different (say,
$r_{\alpha}>r_{\beta}$), then the curves can be represented as a collection of
nonintersecting “parallel” curves

$\beta^{(1)}=f(\alpha),\;\beta^{(2)}=f(\alpha)+\delta_{1},\ldots,\;\beta^{(2^{n-r_{\alpha}})}=f(\alpha)+\delta_{2^{n-r_{\alpha}}-1},$
(5.10)

where $f(\alpha)$ is the function

$f(\alpha)=\sum_{k=0}^{n-r_{\alpha}-1}f_{k}\,\alpha^{2^{k}},$ (5.11)

and the commutativity condition leads to the restrictions
$\mathop{\mathrm{tr}}\nolimits(\delta_{i}\alpha_{j})=0$, which fix the values
of $\delta_{i}$.

The intersection problem can be studied using the same criterion as for the
regular curves, taking into account that those conditions should be satisfied
only at the admissible points of the curve.

###  5.2 Examples

####  5.2.1 Four-dimensional case

In the case of $\mathrm{GF}(2^{2})$ the only exceptional curves are doubly
degenerate, $r_{\alpha}=r_{\beta}=1$. Besides, the structural equation is of
second order: $\alpha(\alpha+\alpha_{1})=0$, so that any one of the three
possible exceptional curves can be represented as a union of straight lines:

$\beta^{(1)}=\alpha_{1}^{-2}\alpha,\qquad\qquad\beta^{(2)}=\alpha_{1}^{-2}\alpha+\alpha_{1}^{-1},$
(5.12)

with $\alpha=0,\alpha_{1}$. More explicitly, for a given value of $\alpha_{1}$
we have the following curve:

$(0,0),(\alpha_{1},\alpha_{1}^{-1}),(0,\alpha_{1}^{-1}),(\alpha_{1},0).$
(5.13)

In this case it is impossible to write down an equation that relates $\alpha$
and $\beta$.

As an example, consider the curve

$(0,0),(\sigma,\sigma^{2}),(0,\sigma^{2}),(\sigma,0),$ (5.14)

where $\alpha_{1}=\sigma$. The structural equations are then

$\alpha^{2}=\sigma\alpha,\qquad\qquad\beta^{2}=\sigma^{2}\beta.$ (5.15)

The implication of the curve type on the factorization of the basis will be
discussed in section 7.

####  5.2.2 Eight-dimensional case

Two types of exceptional curves exist in the case of $\mathrm{GF}(2^{3})$: (i)
doubly degenerate in both directions, which corresponds to
$r_{\alpha}=r_{\beta}=2$; (ii) doubly degenerate in one direction and
quadruply degenerate in the other, which corresponds to
$r_{\alpha}=2,r_{\beta}=1$ or $r_{\alpha}=1,r_{\beta}=2$.

In the case (i), any exceptional curve can be represented as a union of two
lines

$\beta^{(1)}=\beta_{1}\alpha_{1}^{-1}\alpha,\quad\qquad\beta^{(2)}=\beta_{1}(\alpha_{1}^{-1}\alpha+1),$
(5.16)

where, according to equation (5.7),

$\beta_{1}=\frac{1}{\alpha_{1}+\alpha_{2}}+\frac{1}{\alpha_{1}}+\frac{1}{\alpha_{2}},$
(5.17)

and the admissible values of $\alpha$ are $0,\alpha_{1},\alpha_{2}$, and
$\alpha_{1}+\alpha_{2}$. For a fixed set $\\{\alpha_{1},\alpha_{2}\\}$ the
following exceptional curve is defined

$\displaystyle(0,0),(\alpha_{1},0),(\alpha_{1},\beta_{1}),(\alpha_{2},\beta_{1}(\alpha_{1}^{-1}\alpha_{2}+1)),(\alpha_{2},\beta_{1}\alpha_{1}^{-1}\alpha_{2}),$
$\displaystyle(\alpha_{1}+\alpha_{2},\beta_{1}\alpha_{1}^{-1}\alpha_{2}),(\alpha_{1}+\alpha_{2},\beta_{1}(\alpha_{1}^{-1}\alpha_{2}+1)),(0,\beta_{1}).$
(5.18)

From the above equation we find that there are 21 exceptional curves due to
the permutational symmetry between $\alpha_{2}$ and $\alpha_{1}+\alpha_{2}$.

As an example, consider

$(0,0),(\sigma^{4},0),(\sigma^{4},\sigma^{5}),(\sigma^{3},\sigma^{7}),(\sigma^{3},\sigma^{4}),(\sigma^{6},\sigma^{4}),(\sigma^{6},\sigma^{7}),(0,\sigma^{5}),$
(5.19)

where $\alpha_{1}=\sigma^{4},\alpha_{2}=\sigma^{3}$. The structural equations
are

$\sigma^{6}\alpha+\sigma^{4}\alpha^{2}+\alpha^{4}=0,\qquad\qquad\sigma^{2}\beta+\sigma^{6}\beta^{2}+\beta^{4}=0,$
(5.20)

and the curve has the form

$\beta^{2}+\sigma^{5}\beta=\sigma^{6}\alpha+\sigma^{2}\alpha^{2}.$ (5.21)

In the case (ii), one of coordinates (say, for instance, $\alpha$) is still
doubly degenerate, while the other one is quadruply degenerate. Then, the
coordinate $\beta$ takes only two values: $0$ and $\delta$, while the allowed
values of $\alpha$ are $0,\alpha_{1},\alpha_{2}$, and $\alpha_{1}+\alpha_{2}$,
so such a curve has the form

$\beta^{(1)}=f(\alpha),\qquad\qquad\beta^{(2)}=f(\alpha)+\delta,$ (5.22)

where

$f(\alpha)=\frac{\delta}{\alpha_{2}(\alpha_{1}+\alpha_{2})}(\alpha_{1}\alpha+\alpha^{2}),$
(5.23)

and $\delta$ satisfies
$\mathop{\mathrm{tr}}\nolimits(\delta\,\alpha_{1,2})=0$, which leads to
$\delta=\beta_{1}$.

Explicitly, the points of such a curve are

$(0,0),(\alpha_{1},0),(\alpha_{2},0),(\alpha_{1}+\alpha_{2},0),(\alpha_{1},\delta),(\alpha_{2},\delta),(\alpha_{1}+\alpha_{2},\delta),(0,\delta),$
(5.24)

so there are seven different curves of this type due to permutational symmetry
between $\alpha_{1},\alpha_{2}$, and $\alpha_{1}+\alpha_{2}$.

As an example of such a degenerate curve, consider the points

$(0,0),(\sigma^{3},0),(\sigma^{5},0),(\sigma^{2},0),(\sigma^{3},\sigma^{6}),(\sigma^{5},\sigma^{6}),(\sigma^{2},\sigma^{6}),(0,\sigma^{6}),$
(5.25)

where $\alpha_{1}=\sigma^{3},\alpha_{2}=\sigma^{5}$. The corresponding
structural equations are

$\sigma^{3}\alpha+\sigma^{2}\alpha^{2}+\alpha^{4}=0,\qquad\qquad\sigma^{6}\beta+\beta^{2}=0.$
(5.26)

##  6 Local transformations

Local transformations induce nontrivial transformations in the curve, although
they preserve the factorization properties (in a given basis). We recall that
in any selfdual basis we can represent the monomial $Z_{\alpha}X_{\beta}$ in
the following way

$\displaystyle
Z_{\alpha}X_{\beta}=\otimes\prod_{j=1}^{n}\sigma_{z}^{a_{j}}\sigma_{x}^{b_{j}}\equiv\otimes\prod_{j=1}^{n}(a_{j},b_{j}),$
(6.1)

where $\otimes\prod_{j}$ denotes the tensor product over the index $j$. Under
local transformations (rotations of angle $\pi/2$ around $z,x$, and $y$ axes,
which we call $z$-, $x$\- and $y$-rotations) applied to the $j$th particle, it
transforms as

$\displaystyle z$ $\displaystyle:$
$\displaystyle(a_{j},b_{j})\mapsto(a_{j}+b_{j},b_{j}),$ $\displaystyle x$
$\displaystyle:$ $\displaystyle(a_{j},b_{j})\mapsto(a_{j},b_{j}+a_{j}),$ (6.2)
$\displaystyle y$ $\displaystyle:$
$\displaystyle(a_{j},b_{j})\mapsto(a_{j}+a_{j}+b_{j},b_{j}+a_{j}+b_{j})=(b_{j},a_{j}).$

To give a concrete example, suppose we consider a $z$-rotation. The operator
$\sigma_{z}$, corresponding to $(a_{j}=1,b_{j}=0)$, is transformed into
$(a_{j}=1+0=1,b_{j}=0)$; i.e., into itself, while the operator $\sigma_{x}$,
corresponding to $(a_{j}=0,b_{j}=1)$, is mapped onto $(a_{j}=0+1=1,b_{j}=1)$,
which coincides with $\sigma_{y}$. In the same way $\sigma_{y}$ is mapped onto
$\sigma_{x}$, while the identity ($a_{j}=0,b_{j}=0$) is mapped onto itself.

In terms of the field elements it is equivalent to

$\displaystyle z$ $\displaystyle:$
$\displaystyle\left\\{\begin{array}[]{l}\displaystyle\alpha\mapsto\alpha^{\prime}=\alpha+\sum_{k}\theta_{k}\mathop{\mathrm{tr}}\nolimits(\beta\theta_{k}),\\\
\beta\mapsto\beta^{\prime}=\beta,\end{array}\right.$ (6.5) $\displaystyle x$
$\displaystyle:$
$\displaystyle\left\\{\begin{array}[]{l}\alpha\mapsto\alpha^{\prime}=\alpha,\\\
\displaystyle\beta\mapsto\beta^{\prime}=\beta+\sum_{k}\theta_{k}\mathop{\mathrm{tr}}\nolimits(\alpha\theta_{k}),\end{array}\right.$
(6.8) $\displaystyle y$ $\displaystyle:$
$\displaystyle\left\\{\begin{array}[]{l}\displaystyle\alpha\mapsto\alpha^{\prime}=\alpha+\sum_{k}\theta_{k}\mathop{\mathrm{tr}}\nolimits[(\alpha+\beta)\theta_{k}],\\\
\displaystyle\beta\mapsto\beta^{\prime}=\beta+\sum_{k}\theta_{k}\mathop{\mathrm{tr}}\nolimits[(\alpha+\beta)\theta_{k}],\end{array}\right.$
(6.11)

where ${\theta}$ is the selfdual basis.

These transformations are nonlinear in the field elements: starting with a
standard set of MUB operators related to rays, we obtain another set of MUB
operators parametrized with points of curves, but leading to the same
factorization structure. Indeed, consider a ray as in equation (2.14). Then,
under $z$-, $x$-, and $y$-rotations we have

$\displaystyle z$ $\displaystyle:$
$\displaystyle\left\\{\begin{array}[]{l}\displaystyle\alpha\mapsto\alpha^{\prime}=\mu\kappa+\sum_{m=0}^{n-1}\kappa^{2^{m}}\nu^{2^{m}}\sum_{k}\theta_{k}^{2^{m}+1},\\\
\displaystyle\beta\mapsto\beta^{\prime}=\nu\kappa,\end{array}\right.$ (6.14)
$\displaystyle x$ $\displaystyle:$
$\displaystyle\left\\{\begin{array}[]{l}\alpha\mapsto\alpha^{\prime}=\mu\kappa,\\\
\displaystyle\beta\mapsto\beta^{\prime}=\nu\kappa+\sum_{m=0}^{n-1}\kappa^{2^{m}}\mu^{2^{m}}\sum_{k}\theta_{k}^{2^{m}+1},\end{array}\right.$
(6.17) $\displaystyle y$ $\displaystyle:$
$\displaystyle\left\\{\begin{array}[]{c}\displaystyle\alpha\mapsto\alpha^{\prime}=\mu\kappa+\sum_{m=0}^{n-1}\kappa^{2^{m}}(\nu+\mu)^{2^{m}}\sum_{k}\theta_{k}^{2^{m}+1}\\\
\displaystyle\beta\mapsto\beta^{\prime}=\nu\kappa+\sum_{m=0}^{n-1}\kappa^{2^{m}}(\nu+\mu)^{2^{m}}\sum_{k}\theta_{k}^{2^{m}+1}\end{array}\right.$
(6.20)

Note that the $z$\- and $x$-rotations produce regular curves

$\displaystyle z$ $\displaystyle:$
$\displaystyle\alpha=\mu\nu^{-1}\beta\mapsto\alpha^{\prime}=\mu\nu^{-1}\beta+\sum_{m=0}^{n-1}\beta^{2^{m}}\sum_{k}\theta_{k}^{2^{m}+1},$
$\displaystyle x$ $\displaystyle:$
$\displaystyle\beta=\mu^{-1}\nu\alpha\mapsto\beta^{\prime}=\mu^{-1}\nu\alpha+\sum_{m=0}^{n-1}\alpha^{2^{m}}\sum_{k}\theta_{k}^{2^{m}+1}.$

Meanwhile, the $y$-rotation may lead to exceptional curves. In this case we
always have $\kappa=(\mu+\nu)^{-1}(\alpha+\beta)$, and thus the explicit
equation of that curve is either

$\alpha=\mu(\nu+\mu)^{-1}(\alpha+\beta)+\sum_{m=0}^{n-1}(\alpha+\beta)^{2^{m}}\sum_{k}\theta_{k}^{2^{m}+1},$
(6.22)

or

$\beta=\nu(\nu+\mu)^{-1}(\alpha+\beta)+\sum_{m=0}^{n-1}(\alpha+\beta)^{2^{m}}\sum_{k}\theta_{k}^{2^{m}+1}.$
(6.23)

For instance, in the two-qubit case, starting from the ray $\beta=0$, we can
generate all the curves shown in table 1. In particular, it can be proven that
in there are only two equivalence classes of curves[65].

Table 1:  Curves generated by applying the rotations indicated in the left column to the ray $\beta=0$ in the case of two qubits. Rotation | Curves
---|---
$x\otimes\leavevmode\hbox{\small 1\normalsize\kern-3.30002pt1}$ | $\beta=\sigma\alpha+\alpha^{2},\,\beta^{2}=\sigma^{2}\beta$
$y\otimes\leavevmode\hbox{\small 1\normalsize\kern-3.30002pt1}$ | $\sigma\beta+\beta^{2}=\sigma^{2}\alpha+\alpha^{2},\,\alpha^{2}=\sigma^{2}\alpha,\,\beta^{2}=\sigma\beta$
$\leavevmode\hbox{\small 1\normalsize\kern-3.30002pt1}\otimes x$ | $\beta=\sigma^{2}\alpha+\alpha^{2},\,\beta^{2}=\sigma\beta$
$\leavevmode\hbox{\small 1\normalsize\kern-3.30002pt1}\otimes y$ | $\sigma^{2}\beta+\beta^{2}=\sigma\alpha+\alpha^{2},\,\alpha^{2}=\sigma\alpha,\,\beta^{2}=\sigma^{2}\beta$
$x\otimes x$ | $\beta=\alpha$
$y\otimes y$ | $\beta=\sigma^{2}\alpha+\alpha^{2},\,\beta^{2}=\sigma\beta$
$x\otimes y$ | $\sigma\beta+\sigma^{2}\beta^{2}=\alpha+\sigma^{2}\alpha^{2},\,\alpha^{2}=\sigma\alpha,\,\beta^{2}=\sigma\beta$
$y\otimes x$ | $\alpha=\sigma\beta+\beta^{2},\,\alpha^{2}=\sigma^{2}\alpha$

##  7 Factorization structure and curve bundles

In this section we discuss bundles leading to MUBs with different
factorization structures. As we have stated before, given a basis in the
field, any operator, labeled by a point of a curve, is factorized into a
product of one-particle Pauli operators. For qubit systems the selfdual is the
most appropriate, for the Fourier operator is factorized, and thus, the
factorization of $Z_{\alpha}$ and $X_{\alpha}$ is straightforward. Now, let us
divide each monomial $Z_{\alpha}X_{\beta}$, into two parts, so that the first
part contains $k$ Pauli operators and the second part $n-k$ operators. If any
first “block” of the set of $d-1$ commuting generalized Pauli operators
commutes with all the other “blocks”, we will say that the corresponding curve
is factorized into two sets. Obviously, the second blocks would then also
commute between them. Moreover, inside the first or second blocks may exist
some “sub-blocks” that commute with corresponding sub-blocks, etc. In the end,
we can represent any curve $\Gamma\in\mathrm{GF}(2^{n})$ in the following
factorized form:

$\Gamma=\\{m_{1},m_{2},\ldots,m_{N}\\},\qquad\qquad 0<m_{1}\leq
m_{2}\leq\ldots\leq m_{N},\;\sum_{i}m_{i}=n,$ (7.1)

where $m_{i}\in\mathbb{N}$ is the number of particles in the $i$-th block that
cannot be factorized anymore. It is clear that
$\\{m_{1},m_{2},\ldots,m_{N}\\}$ is just a partition of the integer $n$, so
the maximum number of terms is $n$, which corresponds to a completely
factorized curve, $\Gamma=\underbrace{\\{1,1,\ldots,1\\}}_{n}$, and the
minimum number of terms is one, corresponding to a completely nonfactorized
curve, $\Gamma=\\{n\\}$.

One can construct bundles that contain only regular curves, as it was shown in
section 4. A systematic construction of bundles containing both regular and
exceptional curves is a more involved task, which can be carried out
numerically for low dimensions.

As an example, consider the ray $\beta=0$ over $\mathrm{GF}(2^{2})$, so the
corresponding set of operators is
$\\{Z_{\sigma},Z_{\sigma^{2}},Z_{\sigma^{3}}\\}$. In the selfdual basis
$(\sigma,\sigma^{2})$ this set is factorized into
$(\sigma_{z}\leavevmode\hbox{\small
1\normalsize\kern-3.30002pt1},\leavevmode\hbox{\small
1\normalsize\kern-3.30002pt1}\sigma_{z},\sigma_{z}\sigma_{z})$. Then, the
curve $\beta=0$ is represented as $\Gamma=\\{1,1\\}$, i.e., both particles are
factorized. The ray $\beta=\sigma\alpha$, whose points label the set
$\\{Z_{\sigma}X_{\sigma^{2}},Z_{\sigma^{2}}X_{\sigma^{3}},Z_{\sigma^{3}}X_{\sigma}\\}$,
has the following factorization
$(\sigma_{z}\sigma_{x},\sigma_{x}\sigma_{y},\sigma_{y}\sigma_{z})$, so that it
can be represented as $\Gamma=\\{2\\}$, which means that there are no
factorized blocks. In the case of three qubits the possible partitions are
$\\{1,1,1\\}$ (e.g., the ray $\beta=0$), $\\{1,2\\}$ (e.g., the regular curve
$\beta=\sigma^{6}\alpha^{2}+\sigma^{3}\alpha^{4}$), and $\\{3\\}$ (e.g., the
ray $\beta=\sigma^{3}\alpha$).

The representation (7.1) is invariant under local transformations. The
corresponding basis preserves the factorization of the operator set, and local
transformations preserve the factorization structure of the curve. This means
that all the completely factorized curves can be obtained from a single
factorized ray, say $\beta=0$. Nevertheless, the curves with the same
factorization structure are not necessarily equivalent under local
transformations (except in the trivial two-qubit case [65]).

A bundle may contain curves with different factorizations. We can characterize
different bundles with a set of numbers that indicate the number of completely
factorized curves ($\underbrace{\\{1,1,\ldots,1\\}}_{n}$ structure),
completely factorized except a single two-particle block (curve of the type
$\\{\underbrace{1,1,\ldots,1}_{n-2},2\\}$), etc., until completely
nonfactorized curves $\\{n\\}$. In other words, we assign to the bundle the
set of numbers

$(k_{1},k_{2},\ldots,k_{p(n)}),\qquad\qquad\sum_{j}k_{j}=2^{n}+1,$ (7.2)

which indicate the number of curves factorized in $n$ one-dimensional blocks,
$k_{1}$; the number of curves factorized in $n-2$ one-dimensional blocks and
one two-dimensional block, $k_{2}$, etc, and $p(n)$ is the number of
partitions of an integer $n$.

###  7.1 Curves over $\mathrm{GF}(2^{2})$

As we have discussed, an additive commutative curve over $\mathrm{GF}(2^{2})$
can be expressed as

$\alpha(\kappa)=\alpha_{0}\kappa+\alpha_{1}\kappa^{2},\qquad\qquad\beta(\kappa)=\beta_{0}\kappa+\beta_{1}\kappa^{2},$
(7.3)

where the commutativity condition (3.3) impose the following restrictions on
the coefficients $\alpha_{j}$ and $\beta_{j}$

$\alpha_{1}\beta_{0}+(\alpha_{1}\beta_{0})^{2}=\alpha_{0}\beta_{1}+(\alpha_{0}\beta_{1})^{2}.$
(7.4)

In this simple case, the whole analysis can be carried out from the parametric
form. Nevertheless, it is more convenient to separate types of curves on
regular and exceptional, according to our discussion in sections 4 and 5.

All the possible additive commutative structures can be divided into two
types:

a) 12 regular curves, which can be constructed according to the general rule
(4.3), among which there are four rays

$\beta=\lambda\alpha,\qquad\qquad\alpha=0,$ (7.5)

and 8 curves

$\alpha\mathrm{-curves}:\;\beta=\eta\alpha+\alpha^{2},\qquad\qquad\beta\mathrm{-curves}:\;\alpha=\eta\beta+\beta^{2}.$
(7.6)

b) 3 exceptional curves that can be represented as a union of two parallel
lines (5.12) or in the parametric form

$\alpha=\mu(\kappa+\kappa^{2}),\qquad\beta=\mu^{2}(\sigma\kappa+\sigma^{2}\kappa^{2}).$
(7.7)

Every point of these exceptional curves is doubly degenerate and the
admissible values of $\alpha$ and $\beta$ are $\\{0,\mu\\}$ and
$\\{0,\mu^{2}\\}$, respectively.

It is important to stress that it is possible to obtain all the curves of form
(7.5), (7.6), and (7.7) from the rays after some (nonlinear) operations,
corresponding to local transformations of operators [65]. The families of such
transformations are the following: 8 curves (the rays $\beta=\alpha$ and
$\beta=0$ among them) can be obtained from the single ray $\alpha=0$
(corresponding to the vertical axis) and the other 5 curves (the ray
$\beta=\sigma^{2}\alpha$ among them) from the ray $\beta=\sigma\alpha$.

The simplest curve bundle contains just rays. There are three rays
($\beta=\alpha$, $\beta=0$, and $\alpha=0$) with completely factorizable
structure $\\{1,1\\}$ and two rays ($\beta=\sigma\alpha$ and
$\beta=\sigma^{2}\alpha$ with nonfactorizable structure $\\{2\\}$. Since any
other bundle can be obtained from the ray bundle by applying some local
transformations, the only bundle structure is $(3,2)$, i.e., in any bundle
there are three factorizable curves and two nonfactorizable (having EPR-states
as basis states).

###  7.2 Curves over $\mathrm{GF}(2^{3})$

A generic additive commutative curve over $\mathrm{GF}(2^{3})$ is given by

$\alpha=\alpha_{0}\kappa+\alpha_{1}\kappa^{2}+\alpha_{2}\kappa^{4},\qquad\qquad\beta=\beta_{0}\kappa+\beta_{1}\kappa^{2}+\beta_{2}\kappa^{4}.$
(7.8)

The commutative condition in this case is much more complicated than for
$\mathrm{GF}(2^{2})$, so a full analysis of all the possible curves becomes
cumbersome if we start with (7.8). Instead, we can follow the procedure of
sections 4 and 5.

A generic regular curves has always one of the following forms

$\displaystyle\beta=\phi_{0}\alpha+\phi^{2}\alpha^{2}+\phi\alpha^{4},\qquad\qquad\alpha$
$\displaystyle=$ $\displaystyle\psi_{0}\beta+\psi^{2}\beta^{2}+\psi\beta^{4}.$
(7.9)

for $\alpha$\- and $\beta$-curves, respectively. All in all, we get 100
different regular curves. There are 21 doubly degenerate exceptional curves of
the form (5.2.2) and 14 exceptional curves (5.24), which are quadruply
degenerate in one direction and doubly degenerate in the other.

The simplest way of forming bundles of commutative curves is given in (4.5).
Then, four bundles of nine curves each

$\beta=\phi_{0}\alpha+\phi^{2}\alpha^{2}+\phi\alpha^{4},\qquad\qquad\alpha=0,$
(7.10)

where $\phi_{0}\in\mathrm{GF}(2^{3})$ and
$\mathop{\mathrm{tr}}\nolimits(\phi)=0$, have the factorization structure
$(3,0,6)$. The choice $\phi=0$ leads to the ray structure (2.13). Among them,
only three rays ($\beta=0,\alpha=0,\beta=\alpha$) have the structure
$\\{1,1,1\\}$, while the other six rays have the structure $\\{3\\}$.

All the other bundles with $\mathop{\mathrm{tr}}\nolimits(\phi)=0$ (i.e.,
$\phi$ takes the values $\sigma$, $\sigma^{2}$, and $\sigma^{4}$) can be
generated from the bundle with $\phi=0$ by applying local transformations. In
particular, $\phi=\sigma$ is generated by an $x$-rotation of the first qubit,
$\phi=\sigma^{2}$ by an $x$-rotation of the second qubit, and
$\phi=\sigma^{4}$ by an $x$-rotation of the first and second qubits.

Another four bundles of nine curves

$\beta=\phi_{0}\alpha+\phi^{2}\alpha^{2}+\phi\alpha^{4},\qquad\qquad\alpha=0,$
(7.11)

where $\phi_{0}\in\mathrm{GF}(2^{3})$ and
$\mathop{\mathrm{tr}}\nolimits(\phi)=1$ generate all the structures $(1,6,2)$.
For instance, in the bundle with $\phi=1$ the curves with
$\phi_{0}=0,\sigma^{7}=1$ have the factorization $\\{3\\}$ and all the other
values of $\phi_{0}\in\mathrm{GF}(2^{3})$ generate the curves with the
factorization $\\{1,2\\}$. The ray $\alpha=0$ has the factorization
$\\{1,1,1\\}$.

As in the previous case, all the bundles with
$\mathop{\mathrm{tr}}\nolimits(\phi)=1$
($\phi=\sigma^{3},\sigma^{5},\sigma^{6}$) can be obtained form the bundle with
$\phi=1$ by some local transformations: $\phi=\sigma^{3}$ by an $x$-rotation
of the first qubit, $\phi=\sigma^{6}$ by an $x$-rotation of the second qubit
and $\phi=\sigma^{5}$ by an $x$-rotation of the third qubits.

The aforementioned local rotations can be also viewed as a relabeling of the
points of the curves $\beta=\phi_{0}\alpha+\alpha^{2}+\alpha^{4}$ according to
$\alpha\mapsto\sigma^{k}\alpha$ and $\beta\mapsto\sigma^{-k}\beta$, with
$k=2,4,1$, respectively. On the other hand, $k=3,5,6$ correspond to nonlocal
transformations and lead to the bundles of curves corresponding to the
$(3,0,6)$ structure.

It is possible to obtain one more type of bundles with different factorization
structure and constituted only by regular curves. To this end we recall that
the nonintersection condition between two regular curves has the form

$(\phi_{0}+\phi_{0}^{\prime})^{7}+\mathop{\mathrm{tr}}\nolimits[(\phi_{0}+\phi_{0}^{\prime})(\phi+\phi^{\prime})]=1,$
(7.12)

so $\phi_{0}$ and $\phi_{0}^{\prime}$ never coincide. Now, let us take three
nonintersecting regular curves satisfying (7.12)

$\displaystyle\mathop{\mathrm{tr}}\nolimits[(\phi_{0}+\phi_{0}^{\prime})(\phi+\phi^{\prime})]$
$\displaystyle=$ $\displaystyle 0,$
$\displaystyle\mathop{\mathrm{tr}}\nolimits[(\phi_{0}^{\prime}+\phi_{0}^{\prime\prime})(\phi^{\prime}+\phi^{\prime\prime})]$
$\displaystyle=$ $\displaystyle 0,$ (7.13)
$\displaystyle\mathop{\mathrm{tr}}\nolimits[(\phi_{0}+\phi_{0}^{\prime\prime})(\phi+\phi^{\prime\prime})]$
$\displaystyle=$ $\displaystyle 0,$

and construct a set of curves according to

$\beta=(\phi_{0}^{a}+\phi_{0}^{b})\alpha+(\phi^{a}+\phi^{b})^{2}\alpha^{2}+(\phi^{a}+\phi^{b})\alpha^{4},$
(7.14)

where $\phi_{0}^{a}$ , $\phi_{0}^{b}$ and $\phi^{a}$, $\phi^{b}$ are
coefficients of the previously defined curves. In this way we generate five
additional new curves:

$\displaystyle\beta$ $\displaystyle=$
$\displaystyle(\phi_{0}+\phi_{0}^{\prime})\alpha+(\phi+\phi^{\prime})^{2}\alpha^{2}+(\phi+\phi^{\prime})\alpha^{4},$
$\displaystyle\beta$ $\displaystyle=$
$\displaystyle(\phi_{0}+\phi_{0}^{\prime\prime})\alpha+(\phi+\phi^{\prime\prime})^{2}\alpha^{2}+(\phi+\phi^{\prime\prime})\alpha^{4},$
$\displaystyle\beta$ $\displaystyle=$
$\displaystyle(\phi_{0}^{\prime}+\phi_{0}^{\prime\prime})\alpha+(\phi^{\prime}+\phi^{\prime\prime})^{2}\alpha^{2}+(\phi^{\prime}+\phi^{\prime\prime})\alpha^{4},$
(7.15) $\displaystyle\beta$ $\displaystyle=$
$\displaystyle(\phi_{0}+\phi_{0}^{\prime}+\phi_{0}^{\prime\prime})\alpha+(\phi+\phi^{\prime}+\phi^{\prime\prime})^{2}\alpha^{2}+(\phi+\phi^{\prime}+\phi^{\prime\prime})\alpha^{4},$
$\displaystyle\beta$ $\displaystyle=$ $\displaystyle 0,$

and one has to add the last curve $\alpha=0$ to complete the set of nine
curves.

We observe that the three “initial” curves can be obtained in the same way
using some of the curves in (7.2). This implies that any curve constructed in
this way should satisfy the condition

$\mathop{\mathrm{tr}}\nolimits(\phi_{0}\,\phi)=0.$ (7.16)

All these sets of curves lead to the factorization structure $(2,3,4)$. One
example of such type of bundle is

$\begin{array}[]{lcl}\\{1,1,1\\}&\mapsto&\alpha=0,\beta=0\\\
\\{1,2\\}&\mapsto&\left\\{\begin{array}[]{l}\beta=\sigma^{6}\alpha+\sigma^{3}\alpha^{2}+\sigma^{5}\alpha^{4},\\\
\beta=\sigma^{2}\alpha+\sigma^{5}\alpha^{2}+\sigma^{6}\alpha^{4},\\\
\beta=\sigma^{4}\alpha+\sigma^{3}\alpha^{2}+\sigma^{5}\alpha^{4}\end{array}\right.\\\
\\{3\\}&\mapsto&\left\\{\begin{array}[]{l}\beta=\sigma^{3}\alpha,\\\
\beta=\sigma^{5}\alpha+\sigma^{5}\alpha^{2}+\sigma^{6}\alpha^{4},\\\
\beta=\sigma\alpha+\sigma^{2}\alpha^{2}+\sigma\alpha^{4},\\\
\beta=\alpha+\sigma^{2}\alpha^{2}+\sigma\alpha^{4}.\end{array}\right.\end{array}$
(7.17)

There is one more type of bundles with the structure $(0,9,0)$, i.e., which
contains only curves with the factorization $\\{1,2\\}$. Such bundles always
contain exceptional curves. One example of those bundles is

a) Regular curves

$\begin{array}[]{ll}\alpha=\sigma^{2}\beta+\sigma^{3}\beta^{2}+\sigma^{5}\beta^{4},&\alpha=\sigma^{6}\beta+\sigma^{3}\beta^{2}+\sigma^{5}\beta^{4},\\\
\beta=\sigma^{2}\alpha+\sigma^{3}\alpha^{2}+\sigma^{5}\alpha^{4},&\beta=\sigma^{6}\alpha^{2}+\sigma^{3}\alpha^{4},\\\
\alpha=\beta+\sigma^{6}\beta^{2}+\sigma^{3}\beta^{4},&\beta=\alpha+\sigma^{3}\alpha^{2}+\sigma^{5}\alpha^{4},\\\
\alpha=\sigma^{3}\beta^{2}+\sigma^{5}\beta^{4},&\end{array}$ (7.18)

b) Exceptional curves

$\displaystyle\beta^{2}+\sigma^{5}\beta$ $\displaystyle=$
$\displaystyle\sigma^{2}\alpha^{2}+\sigma^{6}\alpha,\qquad\qquad\mathop{\mathrm{tr}}\nolimits(\sigma^{4}\beta)=0,\quad\mathop{\mathrm{tr}}\nolimits(\sigma^{5}\alpha)=0;$
$\displaystyle\beta^{2}+\sigma^{2}\beta$ $\displaystyle=$
$\displaystyle\sigma^{6}\alpha^{2}+\sigma^{5}\alpha,\qquad\qquad\mathop{\mathrm{tr}}\nolimits(\sigma^{6}\beta)=0,\quad\mathop{\mathrm{tr}}\nolimits(\sigma^{2}\alpha)=0,$

where the structural equations are written in the trace form.

The rays with $\\{1,2\\}$ factorization structure cannot be obtained from rays
by local transformations. Moreover, not all the curves with factorization
$\\{3\\}$ can be obtained from the rays of the same type.

All these structures can be obtained form each other by nonlocal
transformations, which can be always reduced to a combination of CNOT gates
and local transformations. To each bundle with a given factorization structure
corresponds a set of nonlocal transformations preserving such structure. Such
a symmetry can be used to determine the optimum tomographic procedure and will
be discussed elsewhere.

The phase-space approach presented here also provides an alternative to the
graph-state classification of all the possible stabilizers states for
$n$-qubit systems. In fact, each additive curve represents a basis in the
$2^{n}$ dimensional Hilbert space, so that the stabilizer state is one element
of such basis. Not each curve can be directly associated with a graph state
[70], but it can be reduced to an appropriate graph state through local
Clifford transformations [71]. While the classification of the graph states
represents a formidable task for large numbers of qubits, the phase-space
approach allows working with algebraic structures. Although the local
equivalence is still an open problem, at least we can determine some elements
of equivalence classes under local Clifford transformations in a relatively
simple form. Besides, several nonlocal qubit operations as SWAP or CNOT gates
can be nicely represented in terms of curves transformations curves. The
above-mentioned problems will be analyzed in future work.

##  8 Conclusions

It has relatively recently been realized that several different types of MUBs,
with respect to their factorization properties, exist for a system of $n$
qubits. Such bases are related to different arrangements of generalized Pauli
monomials into sets of commuting operators. The construction of a whole set of
MUBs is an involved problem, especially for large number of qubits. The
simplest MUB construction was discovered by Wootters and it is related to
straight lines in the discrete phase space. We have shown that all the other
MUBs are connected with a special type of discrete curves. Although, in
principle, we can classify all the possible additive commutative curves and
even determine which are related through local transformations, arranging them
in bundles of nonintersecting curves is still an involved problem.
Nevertheless, we can find some of such bundles according to a “recipe”
[specifically equations (4.5) and (4.8)], which represents an essential
progress in this field.

The authors would like to acknowledge fruitful discussions with Dr. Iulia
Ghiu, from University of Bucharest. This work was supported by the Grant No
45704 of Consejo Nacional de Ciencia y Tecnología (CONACyT), the Swedish
Foundation for International Cooperation in Research and Higher Education
(STINT), the Swedish Research Council (VR), the Swedish Foundation for
Strategic Research (SSF), and the Spanish Research Directorate (Grant
FIS2005-06714). A. B. K. was also supported by the Spanish Sabbatical Program
(Grant SAB2006-0064).

##  Appendix A Galois fields

In this appendix we briefly recall the minimum background needed in this
paper. The reader interested in more mathematical details is referred, e.g.,
to the excellent monograph by Lidl and Niederreiter [69].

A commutative ring is a set $R$ equipped with two binary operations, called
addition and multiplication, such that it is an Abelian group with respect the
addition, and the multiplication is associative. Perhaps, the motivating
example is the ring of integers $\mathbb{Z}$ with the standard sum and
multiplication. On the other hand, the simplest example of a finite ring is
the set $\mathbb{Z}_{n}$ of integers modulo $n$, which has exactly $n$
elements.

A field $F$ is a commutative ring with division, that is, such that 0 does not
equal 1 and all elements of $F$ except 0 have a multiplicative inverse (note
that 0 and 1 here stand for the identity elements for the addition and
multiplication, respectively, which may differ from the familiar real numbers
0 and 1). Elements of a field form Abelian groups with respect to addition and
multiplication (in this latter case, the zero element is excluded).

The characteristic of a finite field is the smallest integer $p$ such that

$p\,1=\underbrace{1+1+\ldots+1}_{\mbox{\scriptsize$p$ times}}=0$ (A.1)

and it is always a prime number. Any finite field contains a prime subfield
$\mathbb{Z}_{p}$ and has $d=p^{n}$ elements, where $n$ is a natural number.
Moreover, the finite field containing $p^{n}$ elements is unique and is called
the Galois field $\mathrm{GF}(p^{n})$.

Let us denote as $\mathbb{Z}_{p}[x]$ the ring of polynomials with coefficients
in $\mathbb{Z}_{p}$. Let $P(x)$ be an irreducible polynomial of degree $n$
(i.e., one that cannot be factorized over $\mathbb{Z}_{p}$). Then, the
quotient space $\mathbb{Z}_{p}[X]/P(x)$ provides an adequate representation of
$\mathrm{GF}(p^{n})$. Its elements can be written as polynomials that are
defined modulo the irreducible polynomial $P(x)$. The multiplicative group of
$\mathrm{GF}(p^{n})$ is cyclic and its generator is called a primitive element
of the field.

As a simple example of a nonprime field, we consider the polynomial
$x^{2}+x+1=0$, which is irreducible in $\mathbb{Z}_{2}$. If $\sigma$ is a root
of this polynomial, the elements
$\\{0,1,\sigma,\sigma^{2}=\sigma+1=\sigma^{-1}\\}$ form the finite field
$\mathrm{GF}(2^{2})$ and $\sigma$ is a primitive element.

The map $\alpha\mapsto\alpha^{p}$, where $\alpha\in\mathrm{GF}(p^{n})$ is a
linear automorphism of $\mathrm{GF}(p^{n})$:
$(\alpha+\beta)^{n}=\alpha^{n}+\beta^{n},$ and
$(\alpha\beta)^{n}=\alpha^{n}\beta^{n}$. It is called the Frobenius
automorphism and will be represented in the form

$\mathfrak{S}^{k}(\alpha)=\alpha^{p^{k}}.$ (A.2)

The elements of the prime field are invariant under action of the this
automorphism.

Another basic map is the trace

$\mathop{\mathrm{tr}}\nolimits(\alpha)=\alpha+\alpha^{2}+\ldots+\alpha^{p^{n-1}}=\sum_{k=0}^{n-1}\mathfrak{S}^{k}(\alpha),$
(A.3)

which satisfies

$\mathop{\mathrm{tr}}\nolimits(\alpha+\beta)=\mathop{\mathrm{tr}}\nolimits(\alpha)+\mathop{\mathrm{tr}}\nolimits(\beta),$
(A.4)

and also leaves the prime field invariant. In terms of it we define the
additive characters as

$\chi(\alpha)=\exp\left[\frac{2\pi
i}{p}\mathop{\mathrm{tr}}\nolimits(\alpha)\right],$ (A.5)

and posses two important properties:

$\chi(\alpha+\beta)=\chi(\alpha)\chi(\beta),\qquad\qquad\sum_{\alpha\in\mathrm{GF}(p^{n})}\chi(\alpha\beta)=p^{n}\delta_{0,\beta}.$
(A.6)

Any finite field $\mathrm{GF}(p^{n})$ can be also considered as an
$n$-dimensional linear vector space. Given a basis $\\{\theta_{k}\\}$,
($k=1,\ldots,n$) in this vector space, any field element can be represented as

$\alpha=\sum_{k=1}^{n}a_{k}\,\theta_{k},$ (A.7)

with $a_{k}\in\mathbb{Z}_{p}$. In this way, we map each element of
$\mathrm{GF}(p^{n})$ onto an ordered set of natural numbers
$\alpha\Leftrightarrow(a_{1},\ldots,a_{n})$.

Two bases $\\{\theta_{1},\ldots,\theta_{n}\\}$ and
$\\{\theta_{1}^{\prime},\ldots,\theta_{n}^{\prime}\\}$ are dual when

$\mathop{\mathrm{tr}}\nolimits(\theta_{k}\theta_{l}^{\prime})=\delta_{k,l}.$
(A.8)

A basis that is dual to itself is called selfdual.

There are several natural bases in $\mathrm{GF}(p^{n})$. One is the polynomial
basis, defined as

$\\{1,\sigma,\sigma^{2},\ldots,\sigma^{n-1}\\},$ (A.9)

where $\sigma$ is a primitive element. An alternative is the normal basis,
constituted of

$\\{\sigma,\sigma^{p},\ldots,\sigma^{p^{n-1}}\\}.$ (A.10)

The choice of the appropriate basis depends on the specific problem at hand.
For example, in $\mathrm{GF}(2^{2})$ the elements $\\{\sigma,\sigma^{2}\\}$
are both roots of the irreducible polynomial. The polynomial basis is
$\\{1,\sigma\\}$ and its dual is $\\{\sigma^{2},1\\}$, while the normal basis
$\\{\sigma,\sigma^{2}\\}$ is selfdual.

## References

  * [1] W. P. Schleich, Quantum Optics in Phase Space, Wiley-VCH, Berlin, 2001.
  * [2] H. Weyl, The Theory of Groups and Quantum Mechanics, Dover, New York, 1950.
  * [3] J. Schwinger, Unitary operator basis, Proc. Natl. Acad. Sci. USA 46 (1960) 570–576.
  * [4] J. Schwinger, Unitary transformations and the action principle, Prod. Natl. Acad. Sci. USA 46 (1960) 883–897.
  * [5] J. Schwinger, The special canonical group, Proc. Natl. Acad. Sci. USA 46 (1960) 1401–1415.
  * [6] J. H. Hannay, M. V. Berry, Quantization of linear maps on a torus-Fresnel diffraction by aperiodic grating, Physica D 1 (1980) 267–290.
  * [7] U. Leonhardt, Quantum-state tomography and discrete Wigner function, Phys. Rev. Lett. 74 (1995) 4101–4105.
  * [8] U. Leonhardt, Discrete wigner function and quantum-state tomography, Phys. Rev. A 53 (1996) 2998–3013.
  * [9] C. Miquel, J. P. Paz, M. Saraceno, E. Knill, R. Laflamme, C. Negrevergne, Interpretation of tomography and spectroscopy as dual forms of quantum computation, Nature (London) 418 (2002) 59–62.
  * [10] C. Miquel, J. P. Paz, Quantum computers in phase space, Phys. Rev. A 65 (2002) 062309.
  * [11] J. P. Paz, Discrete Wigner functions and the phase-space representation of quantum teleportation, Phys. Rev. A 65 (2002) 062311.
  * [12] F. A. Buot, Method for calculating $\mathrm{Tr}\mathcal{H}^{n}$ in solid-state theory, Phys. Rev. B 10 (1973) 3700–3705.
  * [13] W. K. Wootters, A Wigner-function formulation of finite-state quantum mechanics, Ann. Phys. (NY) 176 (1987) 1–21.
  * [14] D. Galetti, A. F. R. de Toledo Piza, An extended Weyl-Wigner transformation for special finite spaces, Physica A 149 (1988) 267–282.
  * [15] O. Cohendet, P. Combe, M. Sirugue, M. Sirugue-Collin, A stochastic treatment of the dynamics of an integer spin, J. Phys. A 21 (1988) 2875–2884.
  * [16] D. Galetti, A. F. R. de Toledo Piza, Discrete quantum phase spaces and the mod $n$ invariance, Physica A 186 (1992) 513–523.
  * [17] P. Kasperkovitz, M. Peev, Wigner-Weyl formalism for toroidal geometries, Ann. Phys. (NY) 230 (1994) 21–51.
  * [18] T. Opatrný, V. Bužek, J. Bajer, G. Drobný, Propensities in discrete phase spaces: $q$-function of a state in a finite-dimensional Hilbert space, Phys. Rev. A 52 (1995) 2419–2427.
  * [19] D. Galetti, M. A. Marchiolli, Discrete coherent states and probability distributions in finite-dimensional spaces, Ann. Phys. (NY) 249 (1996) 454–480.
  * [20] T. Opatrný, D.-G. Welsch, V. Bužek, Parametrized discrete phase-space functions, Phys. Rev. A 53 (1996) 3822–3835.
  * [21] A. M. F. Rivas, A. M. O. de Almeida, The Weyl representation on the torus, Ann. Phys. (NY) 276 (1999) 223–256.
  * [22] K. S. Gibbons, M. J. Hoffman, W. K. Wootters, Discrete phase space based on finite fields, Phys. Rev. A 70 (2004) 062101.
  * [23] A. Vourdas, Galois quantum systems, J. Phys. A 38 (2005) 8453–8472.
  * [24] A. Vourdas, Quantum systems with finite Hilbert space: Galois fields in quantum mechanics, J. Phys. A 40 (2007) R285–R331.
  * [25] P. Delsarte, J. M. Goethals, J. J. Seidel, Bounds for systems of lines and Jacobi polynomials, Philips Res. Rep. 30 (1975) 91–105.
  * [26] W. K. Wootters, Quantum mechanics without probability amplitudes, Found. Phys. 16 (1986) 391–405.
  * [27] W. K. Wootters, B. D. Fields, Optimal state-determination by mutually unbiased measurements, Ann. Phys. (NY) 191 (1989) 363–381.
  * [28] K. Kraus, Complementary observables and uncertainty relations, Phys. Rev. D 35 (1987) 3070–3075.
  * [29] J. Lawrence, Č. Brukner, A. Zeilinger, Mutually unbiased binary observable sets on $n$ qubits, Phys. Rev. A 65 (2002) 032320.
  * [30] S. Chaturvedi, Aspects of mutually unbiased bases in odd-prime-power dimensions, Phys. Rev. A 65 (2002) 032320.
  * [31] D. Petz, Complementarity in quantum systems, Rep. Math. Phys. 59 (2007) 209–224.
  * [32] H. Bechmann-Pasquinucci, A. Peres, Quantum cryptography with 3-state systems, Phys. Rev. Lett. 85 (2000) 3313–3316.
  * [33] R. Asplund, G. Björk, Reconstructing the discrete Wigner function and some properties of the measurement bases, Phys. Rev. A 64 (2001) 012106.
  * [34] D. Gottesman, Class of quantum error-correcting codes saturating the quantum Hamming bound, Phys. Rev. A 54 (1996) 1862–1868.
  * [35] A. R. Calderbank, E. M. Rains, P. W. Shor, N. J. A. Sloane, Quantum error correction and orthogonal geometry, Phys. Rev. Lett. 78 (1997) 405–408.
  * [36] Y. Aharonov, B.-G. Englert, The mean king’s problem: spin 1, Z. Naturforsch. 56a (2001) 16–19.
  * [37] B.-G. Englert, Y. Aharonov, The mean king’s problem: Prime degrees of freedom, Phys. Lett. A 284 (2001) 1–5.
  * [38] P. K. Aravind, Solution to the king’s problem in prime power dimensions, Z. Naturforsch. 58a (2003) 85–92.
  * [39] A. Hayashi, M. Horibe, T. Hashimoto, Mean king’s problem with mutually unbiased bases and orthogonal latin squares, Phys. Rev. A 71 (2005) 052331.
  * [40] A. Klappenecker, M. Rötteler, New tales of the mean king, arXiv:quant-ph/0502138.
  * [41] J. P. Paz, A. J. Roncaglia, M. Saraceno, Qubits in phase space: Wigner-function approach to quantum-error correction and the mean-king problem, Phys. Rev. A 72 (2005) 012309.
  * [42] T. Durt, About Weyl and Wigner tomography in finite-dimensional Hilbert spaces, Open Sys. Inf. Dyn. 13 (2006) 403–413.
  * [43] G. Kimura, H. Tanaka, M. Ozawa, Solution to the mean king’s problem with mutually unbiased bases for arbitrary levels, Phys. Rev. A 73 (2006) 050301(R).
  * [44] I. D. Ivanovic, Geometrical description of quantal state determination, J. Phys. A 14 (1981) 3241–3245.
  * [45] A. R. Calderbank, P. J. Cameron, W. M. Kantor, J. Seidel, $\mathbb{Z}_{4}$-kerdock codes, orthogonal spreads, and extremal Euclidean line-sets, Proc. London Math. Soc. 75 (1997) 436–480.
  * [46] A. Klappenecker, M. Rötteler, Constructions of mutually unbiased bases, in: G. Mullen, A. Poli, H. Stichtenoth (Eds.), Finite Fields and Applications, Vol. 2948 of Lecture Notes in Computer Science, Springer, Berlin, 2003, pp. 137–144.
  * [47] J. Lawrence, Mutually unbiased bases and trinary operator sets for n qutrits, Phys. Rev. A 70 (2004) 012302.
  * [48] K. R. Parthasarathy, On estimating the state of a finite level quantum system, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 7 (2004) 607–617.
  * [49] A. O. Pittenger, M. H. Rubin, Wigner functions and separability for finite systems, J. Phys. A 38 (2005) 6005–6036.
  * [50] T. Durt, About mutually unbiased bases in even and odd prime power dimensions, J. Phys. A 38 (2005) 5267–5284.
  * [51] M. Planat, H. Rosu, Mutually unbiased phase states, phase uncertainties and Gauss sums, Eur. Phys. J. D 36 (2005) 133–139.
  * [52] A. B. Klimov, L. L. Sánchez-Soto, H. de Guise, Multicomplementary operators via finite Fourier transform, J. Phys. A 38 (2005) 2747–2760.
  * [53] O. P. Boykin, M. Sitharam, P. H. Tiep, P. Wocjan, Mutually unbiased bases and orthogonal decompositions of Lie algebras, Quantum Info. Comp. 7 (2007) 371–382.
  * [54] M. Grassl, On SIC-POVMs and MUBs in dimension 6, arXiv:quant-ph/0406175 (2004).
  * [55] C. Archer, There is no generalization of known formulas for mutually unbiased bases, J. Math. Phys. 46 (2005) 022106.
  * [56] P. Wocjan, On the maximal number of real mutually unbiased bases, arXiv.org quant-ph/0502024.
  * [57] P. Butterley, W. Hall, Numerical evidence for the maximum number of mutually unbiased bases in dimension six, Phys. Lett. 369 (2007) 5–8.
  * [58] M. Saniga, M. Planat, H. Rosu, Mutually unbiased bases and finite projective planes, J. Opt. B 6 (2004) L19–L20.
  * [59] I. Bengtsson, MUBs, polytopes and finite geometries, AIP Conf. Proc. 750 (2005)
  * [60] G. Zauner, Quantendesigns: Grundzüge einer nichtkommutativen Designtheorie, Ph.D. thesis, University of Vienna (1999).
  * [61] W. K. Wootters, Quantum measurements and finite geometry, Found. Phys. 36 (2006) 112–126.
  * [62] S. Bandyopadhyay, P. O. Boykin, V. Roychowdhury, F. Vatan, A new proof for the existence of mutually unbiased bases, Algorithmica 34 (2002) 512–528.
  * [63] J. L. Romero, G. Björk, A. B. Klimov, L. L. Sánchez-Soto, Structure of the sets of mutually unbiased bases for $n$ qubits, Phys. Rev. A 72 (2005) 062310\.
  * [64] G. Björk, J. L. Romero, A. B. Klimov, L. L. Sánchez-Soto, Mutually unbiased bases and discrete Wigner functions, J. Opt. Soc. Am. B 24 (2007) 371–378.
  * [65] A. B. Klimov, J. L. Romero, G. Björk, L. L. Sánchez-Soto, Geometrical approach to mutually unbiased bases, J. Phys. A 40 (2007) 3987–3998.
  * [66] A. Peres, Quantum Theory: Concepts and Methods, Kluwer, Dordrecht, 1993.
  * [67] D. Gottesman, A. Kitaev, J. Preskill, Encoding a qubit in an oscillator, Physical Review A 64 (2001) 012310.
  * [68] S. Bartlett, H. de Guise, B. C. Sanders, Quantum encodings in spin systems and harmonic oscillators, Phys. Rev. A 65 (5) (2002) 052316.
  * [69] R. Lidl, H. Niederreiter, Introduction to finite fields and their applications, Cambridge University Press, Cambridge, 1986.
  * [70] R. Raussendorf, D. E. Brown, H. J. Briegel, Measurement-based quantum computation on cluster states, Phys. Rev. A 68 (2003) 022312.
  * [71] M. Van den Nest, J. Dehaene, B. de Moor, Graphical description of the action of local Clifford transformations on graph states, Phys. Rev. A 69 (2004) 022316\.