Cellular Automata-Based Methods for the Construction of Mutually Unbiased Bases

Abstract

Mutually unbiased bases (MUBs) are essential tools in quantum information science, with applications in state tomography, quantum cryptography, and quantum error correction. In this work, we introduce a constructive framework for generating MUBs using linear bipermutive cellular automata (LBCAs). By leveraging the algebraic structure of generalized Pauli operators over finite fields, we show that disjoint families of LBCAs correspond to commuting sets of such operators (CSPOs), which, in turn, generate MUBs. This correspondence enables the systematic construction of complete or incomplete sets of MUBs, depending on the number of disjoint LBCAs available in a given dimension. We also provide algebraic conditions to verify disjointness and discuss how the finite dimensionality constrains MUB completeness. Our approach reinterprets classical combinatorial structures in a quantum setting, offering new computational pathways for exploring MUBs through discrete dynamical systems.

1. Introduction

Quantum information science has seen significant advancements in recent years, particularly in the study of quantum states and their applications in computation, cryptography, and communication. Shafique et al. [1] explored number-theoretic constructions, Durr [2] proposed methods using group theory, and Kumari [3] applied graph-theoretic techniques. A fundamental aspect of this field is the ability to generate, manipulate, and classify bases in n-qudit systems, as these processes play a crucial role in quantum state reconstruction [4,5] to analyze optimal strategies for quantum-state and process tomography, focusing on resource efficiency and practical implementations in ion trap systems.

Additionally, they are closely linked to the classification and quantification of quantum entanglement, which is essential for understanding correlations in quantum systems. Wootters and Fields [6] introduced the concept of mutually unbiased measurements for optimal state determination, while Asplund and Björk [7] studied properties of the measurement bases and the reconstruction of discrete Wigner functions.

Among the various types of quantum bases, mutually unbiased bases (MUBs) hold special importance due to their unique informational properties. When a quantum measurement is performed in one MUB, it provides minimal redundancy compared to measurements taken in another MUB. Mathematically, two orthonormal bases A and B in a d-dimensional Hilbert space ${\mathcal{H}}_d$ are considered mutually unbiased if, for any basis vectors $|A,a\rangle$ and $|B,b\rangle$

$${|\langle A,a|B,b\rangle |}^2={\displaystyle \frac{1}{d}}$$

(1)

MUBs have found applications in a wide range of areas, including quantum state tomography [4,5] and quantum error correction. Gottesman [8], and Calderbank et al. [9] developed classes of quantum error-correcting codes that saturate the quantum Hamming bound, which are relevant to the design of robust MUB-based protocols, and the construction of discrete Wigner functions. References [10,11,12,13] explore the discrete Wigner function and its applications to finite-dimensional quantum systems, including its connection with mutually unbiased bases and tomographic universality. Despite their theoretical significance, generating complete sets of MUBs for n-qudit systems remains a challenging and active area of research. It is known that the maximum number of MUBs in such systems is bounded by $d+1$ and various approaches have been proposed to construct these sets efficiently. Several works [13,14,15,16,17,18,19] have proposed algebraic and combinatorial methods to construct complete or partial sets of MUBs, using techniques based on finite fields, operator sets, and graph theory.

In this paper, we introduce an alternative method for constructing MUBs based on Cellular Automata (CA), which are discrete dynamical systems defined on a regular grid of cells. Each cell evolves over time according to a local update rule that depends on its own state and the states of neighboring cells. Cellular automata have been widely studied across various disciplines due to their ability to model complex behavior through simple local interactions. The works by Wolfram [20], Toffoli [21], and Gutowitz [22] form the foundational literature on cellular automata, emphasizing their ability to model complex systems through simple local interactions.

In particular, bipermutive linear cellular automata (LBCAs) have been used in the combinatorial construction of Latin squares and their orthogonal counterparts [23]. Our work does not aim to redefine these structures, but rather to reinterpret them within the framework of quantum information theory, using LBCAs to construct disjoint sets of commuting generalized Pauli operators.

Our motivation stems from the need for scalable and constructive methods to generate MUBs in arbitrary dimensions, particularly in quantum information contexts where classical tools fall short. The motivation for employing CA in this context stems from their structured evolution, characterized by local and deterministic rules that enable efficient and scalable implementation. Leveraging this property, we propose a new framework for constructing MUBs. This approach may have significant implications for quantum information processing.

The matrix representation of CA allows for the computational verification of algebraic properties (such as disjunction), opening the door to the use of artificial intelligence tools to explore the space of possible rules. In particular, the following future research directions could be pursued: the use of genetic or optimization algorithms to discover local rules that generate disjoint CSPOs; and the training of machine learning models (e.g., neural networks) to classify local rules based on their ability to generate complete or incomplete sets of MUBs. These strategies could significantly expand the known set of MUBs and provide new computational approaches for their study, particularly in dimensions where traditional algebraic methods have been ineffective.

Structurally, the proposed method differs from most classical constructions of MUBs, which typically rely on algebraic tools such as finite fields [15], Hadamard matrices [24], or Latin squares [6]. These approaches often require strong arithmetic assumptions on the dimension d and are not always constructive in non-prime-power dimensions. In contrast, our approach is based on the local, deterministic evolution of linear bipermutive cellular automata, whose matrix representations naturally generate orthonormal sets under certain algebraic conditions. This shift enables novel algorithmic techniques and offers potential for automation and exploration through artificial intelligence.

This paper is organized as follows: Section 2 introduces the necessary background on cellular automata and MUBs. Section 3 details our construction method. Section 3.2 presents illustrative examples. Finally, Section 4 discusses the implications and potential extensions of our work.

2. Mathematical Preliminaries

In order to formalize our construction, we begin by reviewing the fundamental algebraic structures that support our approach. These include the generalized Pauli group, which underpins the algebraic formulation of quantum operations, as well as the theory of linear bipermutive cellular automata, which will serve as the core mechanism of our construction. We also recall the key definitions and properties of mutually unbiased bases (MUBs), which are the main objects of interest in this study. This section establishes the notation, assumptions, and mathematical tools used throughout the paper.

2.1. The Generalized Pauli Group

To define the generalized Pauli group in an n-qudit system, ($d=p^n$, with p being a prime number), it suffices to consider the action of its generators on an orthonormal basis in the corresponding Hilbert space ${\mathcal{H}}_{p^n}={\mathcal{H}}_p^{\otimes n}$. To achieve this objective and to ensure a consistent development of the results in this paper, we label each vector $|k_1,\dots ,k_n\rangle \in {\mathcal{H}}_p^{\otimes n}$, with $k_i\in {\mathbb{Z}}_p$, of an orthonormal basis by elements of a finite field $\kappa \in {\mathbb{F}}_{p^n}$:

$$|k_1,\dots ,k_n\rangle \iff |\kappa \rangle ,\langle {\kappa }^{\prime }|\kappa \rangle ={\delta }_{{\kappa }^{\prime },\kappa },\kappa =\sum _{i=1}^n{k}_i{\theta }_i\in {\mathbb{F}}_{p^n},$$

(2)

where $\{{\theta }_1,\dots ,{\theta }_n\}$ is a basis in ${\mathbb{F}}_{p^n}$ (considered as a vector space over ${\mathbb{F}}_p$).

Example 1.

Let $p=2$ and $n=2$. Then, ${\mathbb{F}}_{2^2}$ is a 4-element field, and ${\mathcal{H}}_2^{\otimes 2}$ has basis states $|00\rangle ,|01\rangle ,|10\rangle ,|11\rangle$. Let $\{{\theta }_1=1,{\theta }_2=\alpha \}$ be a basis of ${\mathbb{F}}_{2^2}$ over ${\mathbb{F}}_2$, where α is a root of the irreducible polynomial $x^2+x+1$.

We can associate:

$$|00\rangle \leftrightarrow |0\rangle ,|01\rangle \leftrightarrow |\alpha \rangle ,|10\rangle \leftrightarrow |1\rangle ,|11\rangle \leftrightarrow |1+\alpha \rangle .$$

This identification allows us to write vectors in ${\mathcal{H}}_2^{\otimes 2}$ using field elements of ${\mathbb{F}}_4$ and to define operators acting on these states using the field structure.

The components $k_i$, $i=1,\dots ,n$ are obtained through the trace operation with a dual basis, $\{{\tilde{\theta }}_1,\dots ,{\tilde{\theta }}_n\}$, (i.e., $tr\left({\theta }_i{\tilde{\theta }}_j\right)={\delta }_{ij}$, where $\delta$ denotes the Kronecker delta, i.e., ${\delta }_{ij}=1$ if $i=j$, and 0 otherwise, that is, in the language of finite fields, the equivalent of an orthonormal basis, with $tr\left(\alpha \right)=\alpha +{\alpha }^2+\dots +{\alpha }^{p^{n-1}}$, $\alpha \in {\mathbb{F}}_{p^n}$ [25].

In this way, the ${\widehat{Z}}_{\alpha }$ position and generalized moment ${\widehat{X}}_{\beta }$ operators, which generate the generalized Pauli group ${\mathcal{P}}_n$ can be defined as follows:

$$\begin{array}{ccc}\hfill {\widehat{Z}}_{\alpha }& =& \sum _{\kappa }{\omega }^{tr\left(\alpha \kappa \right)}|\kappa \rangle \langle \kappa |,{\widehat{X}}_{\beta }=\sum _{\kappa }|\kappa +\beta \rangle \langle \kappa |,\hfill \end{array}$$

(3)

$$\begin{array}{ccc}\hfill {\widehat{Z}}_{\alpha }^{†}& =& {\widehat{Z}}_{-\alpha },{\widehat{X}}_{\beta }^{†}={\widehat{X}}_{-\beta },{\widehat{Z}}_{\alpha }^p={\widehat{X}}_{\beta }^p=\widehat{I},\hfill \end{array}$$

(4)

where $\omega =e^{2\pi i/p}$ and satisfy the commutation relation:

$${\widehat{Z}}_{\alpha }{\widehat{X}}_{\beta }={\omega }^{tr\left(\alpha \kappa \right)}{\widehat{X}}_{\beta }{\widehat{Z}}_{\alpha }.$$

(5)

The set $\{{\widehat{Z}}_{\alpha }{\widehat{X}}_{\beta }:\alpha ,\beta \in {\mathbb{F}}_{p^n}\}$ of $p^{2n-1}$ operators (excluding the identity operator, as it commutes with all) can be partitioned into $p^n+1$ commutative sets $Y^{\lambda }$, $\lambda =0,1,...,p^n$, each containing $p^{n-1}$ elements, ${\widehat{Z}}_{{\alpha }_{\lambda }\left(\tau \right)}{\widehat{X}}_{{\beta }_{\lambda }\left(\tau \right)},\tau \in {\mathbb{F}}_{p^n}$ [17,18,19]:

$$Y^{\lambda }=\left\{{\widehat{Z}}_{{\alpha }_{\lambda }\left(\tau \right)}{\widehat{X}}_{{\beta }_{\lambda }\left(\tau \right)}:\left[{\widehat{Z}}_{{\alpha }_{\lambda }\left(\tau \right)}{\widehat{X}}_{{\beta }_{\lambda }\left(\tau \right)},{\widehat{Z}}_{{\alpha }_{\lambda }\left({\tau }^{\prime }\right)}{\widehat{X}}_{{\beta }_{\lambda }\left({\tau }^{\prime }\right)}\right]=0\forall \tau ,{\tau }^{\prime }\in {\mathbb{F}}_{p^n}\right\},$$

(6)

where, from Equation (5), it follows that:

$$\left[{\widehat{Z}}_{{\alpha }_{\lambda }\left(\tau \right)}{\widehat{X}}_{{\beta }_{\lambda }\left(\tau \right)},{\widehat{Z}}_{{\alpha }_{\lambda }\left({\tau }^{\prime }\right)}{\widehat{X}}_{{\beta }_{\lambda }\left({\tau }^{\prime }\right)}\right]=0\iff tr\left({\alpha }_{\lambda }\left({\tau }^{\prime }\right){\beta }_{\lambda }\left(\tau \right)-{\beta }_{\lambda }\left({\tau }^{\prime }\right){\alpha }_{\lambda }\left(\tau \right)\right)=0.$$

(7)

The operators corresponding to each CSPO $Y^{\lambda }$ can be diagonalized simultaneously and their common eigenvectors form an orthonormal basis

$${\widehat{Z}}_{{\alpha }_{\lambda }\left(\tau \right)}{\widehat{X}}_{{\beta }_{\lambda }\left(\tau \right)}|{\psi }_{\kappa }^{\lambda }\rangle ={\omega }^{tr\left(\kappa \tau \right)}|{\psi }_{\kappa }^{\lambda }\rangle ,\kappa ,\tau \in {\mathbb{F}}_{p^n}.$$

(8)

Each pair of operators ${\widehat{Z}}_{{\alpha }_{{\lambda }^{\prime }}\left({\tau }^{\prime }\right)}{\widehat{X}}_{{\beta }_{{\lambda }^{\prime }}\left({\tau }^{\prime }\right)}$ and ${\widehat{Z}}_{{\alpha }_{{\lambda }^{\prime }}\left({\tau }^{\prime }\right)}{\widehat{X}}_{{\beta }_{{\lambda }^{\prime }}\left({\tau }^{\prime }\right)}$ belonging to disjoint sets $Y^{{\lambda }^{\prime }},Y^{\lambda }$, respectively, satisfy:

$$Tr\left[\left({\widehat{Z}}_{{\alpha }_{{\lambda }^{\prime }}\left({\tau }^{\prime }\right)}{\widehat{X}}_{{\beta }_{{\lambda }^{\prime }}\left({\tau }^{\prime }\right)}\right){\left({\widehat{Z}}_{{\alpha }_{\lambda }\left(\tau \right)}{\widehat{X}}_{{\beta }_{\lambda }\left(\tau \right)}\right)}^{†}\right]=p^n{\delta }_{\lambda ,{\lambda }^{\prime }}{\delta }_{\tau ,{\tau }^{\prime }}+p^n(1-{\delta }_{\lambda ,{\lambda }^{\prime }}){\delta }_{\tau ,0}{\delta }_{{\tau }^{\prime },0},$$

and so the corresponding bases (8) are mutually unbiased,

$$|\langle {\psi }_{{\kappa }^{\prime }}^{{\lambda }^{\prime }}|{\psi }_{\kappa }^{\lambda }\rangle {|}^2={\delta }_{\lambda ,{\lambda }^{\prime }}{\delta }_{\kappa ,{\kappa }^{\prime }}+p^{-n}(1-{\delta }_{\lambda ,{\lambda }^{\prime }}).$$

(9)

Therefore, if the complete CSPOs are determined, the MUBs sets are obtained indirectly and that methodology will be followed in this work.

2.2. Hadamard Matrices and Hamming Weight

The construction of MUBs has predominantly been achieved in dimensions that are powers of prime numbers, typically through the use of finite fields and other algebraic structures that exist exclusively in such dimensions [6,10,11,12,18,19]. In composite dimensions, it has not been possible to obtain complete sets of MUBs, and although no formal proof of this limitation currently exists, the literature reports strong numerical evidence supporting this conjecture [26]. Nevertheless, alternative constructions have been proposed to establish upper bounds on the number of such bases [27]. One of the most commonly used approaches involves Hadamard matrices [24], which we will adopt in this work; for this reason, we now present the necessary definitions:

Definition 1.

A matrix $H\in {\mathbb{C}}^{n\times n}$ is called a complex Hadamard matrix if it satisfies the following two conditions:

• All entries of H have unit modulus:

  $$|H_{ij}|=1\mathit{for}\mathit{all}i,j\in \{1,\dots ,n\}.$$
• The matrix is unitary up to a scalar factor:

  $$H{H}^{†}=n{I}_n,$$

  where $H^{†}$ denotes the conjugate transpose of H, and $I_n$ is the $n\times n$ identity matrix.

This implies that the columns of H are mutually orthogonal with norm $\sqrt{n}$ [24].

In this article, we work with the orthonormalized version of a complex Hadamard matrix, defined as $\tilde{H}={\displaystyle \frac{1}{\sqrt{n}}}H$. This normalization guarantees that $\tilde{H}$ is unitary, with columns forming an orthonormal basis in ${\mathbb{C}}^n$. This property is crucial in our construction because it guarantees that the transformation preserves inner products, a fundamental requirement for building MUBs. Moreover, using Hadamard matrices allows us to systematically generate unbiased bases, particularly in dimensions where other algebraic constructions are limited or fail. This enhances the flexibility and applicability of our method in both complete and incomplete MUB scenarios.

Definition 2.

The Hamming weight of a vector is the number of entries that are different from zero. The support of a vector is the set of indices corresponding to those non-zero entries.

Definition 3.

Let $\overrightarrow{m}\in {\{0,1\}}^d$ with Hamming weight s and support $\{r_0,r_1,\dots ,r_{s-1}\}$ and let $\overrightarrow{h}\in {\mathbb{C}}^s$. The embedding of $\overrightarrow{h}$ into ${\mathbb{C}}^d$ controlled by $\overrightarrow{m}$, denoted $\overrightarrow{h}\uparrow \overrightarrow{m}$ is given by:

$$\overrightarrow{h}\uparrow \overrightarrow{m}:=\sum _{i=0}^{s-1}{\overrightarrow{h}}_i{\overrightarrow{e}}_{r_i}$$

(10)

where $h_i$ is the $i^{th}$ entry in $\overrightarrow{h}|$ and ${\overrightarrow{e}}_{r_i}$ is the $r_i^{th}$ standard basis vector.

Example 2.

For the specific case where $\overrightarrow{m}\in {\mathbb{F}}_p^4$ and $\overrightarrow{h}\in {\mathbb{C}}^2$, we obtain:

$$\overrightarrow{m}=\left[\begin{array}{c}1\\ 0\\ 0\\ 1\end{array}\right]\overrightarrow{h}=\left[\begin{array}{c}\omega \\ 1\end{array}\right]\overrightarrow{h}\uparrow \overrightarrow{m}=\left[\begin{array}{c}\omega \\ 0\\ 0\\ 1\end{array}\right]$$

(11)

where $\omega =e^{2\pi i/p}$.

These definitions will play a key role in Section 3.2, where they are used to construct mutually unbiased bases for incomplete sets.

2.3. Cellular Automata

In this subsection, we cover only the basic concepts of cellular automata necessary for the development of the paper. For more detailed information, we refer the reader to references [28,29].

Definition 4.

A Non-Boundary Cellular Automaton (NBCA) is defined as a vector function $F:A^n\to A^{n-d+1}$, where A is a finite alphabet, $n\ge d$, $n,d\in \mathbb{N}$,

$$F\left(x_0,x_1,\dots ,x_{n-1}\right)=\left(y_0,y_1,\dots ,y_{n-d}\right)$$

(12)

and each $y_i$ is computed by a local rule $\mu :A^d\to A$:

$$y_i=\mu \left(x_i,x_{i+1},\dots ,x_{i-d}\right),\mathrm{for}0\le i\le n-d$$

(13)

That is, the function F updates the state of a one-dimensional cellular automaton of length n, using a local update rule that considers a neighborhood of size d. For each time step, the state of the automaton is updated according to a local function $\mu :A^d\to A$, which determines the new state of each cell based on the states of its d-sized neighborhood.

Unlike classical cellular automata, which often assume periodic boundary conditions (where the first and last cells are considered adjacent), an NBCA does not assume such boundary conditions. Instead, the function F is only defined for positions where a full d-cell neighborhood is available, meaning that the last $\left(d-1\right)$-cells might not be updated or require a different treatment.

Throughout the rest of the paper, the alphabet A of q symbols will be taken as the finite field ${\mathbb{F}}_q$, $q=p^n$, with p being a prime number. Consequently, the terms “multiplication” and “addition” will refer to the corresponding operations in ${\mathbb{F}}_q$.

Example 3.

Let $F:{\mathbb{F}}_2^6\to {\mathbb{F}}_2^4$ ($n=6,d=3$) and the local rule $\mu :{\mathbb{F}}_2^3\to {\mathbb{F}}_2$ given by

$$\mu \left(x_i,x_{i+1},x_{i+2}\right)=x_i+x_{i+1}+x_{i+2},$$

(14)

where $i=1,2,3$. For example, $F\left(0,1,0,1,0,0\right)=\left(1,0,1,1\right)$ since:

$$\mu \left(0,1,0\right)=1,\mu \left(1,0,1\right)=0,\mu \left(0,1,0\right)=1,\mu \left(1,0,0\right)=1.$$

(15)

Definition 5.

An NBCA of diameter d is said to be linear (LBCA) if its local rule $\mu :{\mathbb{F}}_q^d\to {\mathbb{F}}_q$ is a linear combination of the neighboring cells, that is, if for each $x\in {\mathbb{F}}_q^d$, there exist constants $a_0,\dots ,a_{d-1}\in {\mathbb{F}}_p$ such that:

$$\mu \left(x\right)=\mu \left(x_0,\dots ,x_{d-1}\right)=a_0{x}_0+a_1{x}_1+\dots +a_{d-1}{x}_{d-1}$$

(16)

If both $a_0$ and $a_{d-1}$ are nonzero, the local rule $\mu$ is also said to be bipermutive. It can be readily verified that the automaton in Example 3 is defined by a local rule that is bipermutive.

When an LBCA is defined by a bipermutive local rule, this rule induces a permutation on the alphabet A both when restricting it by fixing the first $d-1$ elements and when fixing the last $d-1$ elements.

Definition 6.

Let $F_1,F_2:{\mathbb{F}}_q^N\to {\mathbb{F}}_q^{d-1}$ be two LBCAs, where, henceforth, we assume that a total order ≤ is defined on ${\mathbb{F}}_q^N$, with $N=2(d-1)$. The LBCAs $F_1$ and $F_2$ are said to be disjoint if the following holds:

$$\left\{\left(F_1\left(z_1\right),F_2\left(z_1\right)\right),\dots ,\left(F_1\left(z_{q^N}\right),F_2\left(z_{q^N}\right)\right):z_k\in {\mathbb{F}}_q^N\right\}={\mathbb{F}}_q^N.$$

(17)

with $z_1\le z_2\le \cdots \le z_{q^N}$.

In other words, $F_1$ and $F_2$ are disjoint if $\left(F_1\left(z_i\right),F_2\left(z_i\right)\right)\ne \left(F_1\left(z_j\right),F_2\left(z_j\right)\right)$, for all distinct pairs $z_i,z_j\in {\mathbb{F}}_q^N$.

The following result establishes a key criterion to determine whether two linear bipermutive cellular automata (LBCAs) are disjoint [23]. Although this condition has been applied in combinatorial contexts such as MOLS, its role in generating CSPOs suitable for constructing MUBs has not been explicitly formulated to the best of our knowledge.

Proposition 1.

Let $F_1,F_2:{\mathbb{F}}_q^N\to {\mathbb{F}}_q^{d-1}$ be two LBCAs, with local rules ${\mu }_1,{\mu }_2:{\mathbb{F}}_q^{d-1}\to {\mathbb{F}}_q$ defined as:

$${\mu }_1\left(x_0,\dots ,x_{d-1}\right)=a_0{x}_0+a_1{x}_1+\dots +a_{d-1}{x}_{d-1}$$

(18)

$${\mu }_2\left(x_0,\dots ,x_{d-1}\right)=b_0{x}_0+b_1{x}_1+\dots +b_{d-1}{x}_{d-1}.$$

(19)

Then, the LBCAs $F_1$ and $F_2$ are disjoint if, and only if, the matrix $N\times N$ given by:

$$M_{1,2}=\left[\begin{array}{ccccccccc}{a}_0& \dots & a_{d-1}& 0& \dots & \dots & \dots & \dots & 0\\ 0& a_0& \dots & a_{d-1}& 0& \dots & \dots & \dots & 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& \dots & \dots & \dots & \dots & 0& a_0& \dots & a_{d-1}\\ b_0& \dots & b_{d-1}& 0& \dots & \dots & \dots & \dots & 0\\ 0& b_0& \dots & b_{d-1}& 0& \dots & \dots & \dots & 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& \dots & \dots & \dots & \dots & 0& b_0& \dots & b_{d-1}\end{array}\right]$$

(20)

is not singular.

Proof. 

It is easy to see from Definition 6, that $F_1$ and $F_2$ are disjoint if, and only if, the linear transformation $H:{\mathbb{F}}_q^N\to {\mathbb{F}}_q^N$, given by

$$H\left(z_i\right)=\left(F_1\left(z_i\right),F_2\left(z_i\right)\right),$$

(21)

is bijective, which occurs if, and only if, the corresponding transformation matrix is invertible. □

This result enables a direct algebraic test for verifying disjointness, which supports our automated generation of commuting structures relevant to quantum state tomography and quantum error correction. The following corollary is a direct consequence of Proposition 1.

Corollary 1.

A set $S=\{F_1,\dots ,F_k\}$ of LBCAs, where $F_i:{\mathbb{F}}_q^N\to {\mathbb{F}}_q^{d-1}$ for $i=1,\dots ,k$, is pairwise disjoint if, for each distinct pair $F_i,F_j\in S$, the matrix $M_{i,j}$ (see (20)), constructed using the corresponding local rules ${\mu }_i$ and ${\mu }_j$, is nonsingular.

Example 4.

Let $F_1,F_2:{\mathbb{F}}_2^4\to {\mathbb{F}}_2^2$ with local rules ${\mu }_1\left(x_i,x_{i+1},x_{i+2}\right)=x_i+x_{i+1}+x_{i+2}$ and ${\mu }_2\left(x_i,x_{i+1},x_{i+2}\right)=x_i+x_{i+2}$. Then, $F_1$ and $F_2$ are disjoint since

$$det\left(M_{1,2}\right)=det\left(\left[\begin{array}{cccc}1& 1& 1& 0\\ 0& 1& 1& 1\\ 1& 0& 1& 0\\ 0& 1& 0& 1\end{array}\right]\right)=1.$$

(22)

Considering the element $z=(1,1,0,1)\in {\mathbb{F}}_2^4$, with $H\left(z\right)=\left(F_1\left(z\right),F_2\left(z\right)\right)$, we obtain:

$$\begin{array}{cc}\hfill H\left(z\right)& =\left(F_1\left((1,1,0,1)\right),F_2\left((1,1,0,1)\right)\right)\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}& =\left({\mu }_1(1,1,0),{\mu }_1(1,0,1),{\mu }_2(1,1,0),{\mu }_2(1,0,1)\right)\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill \hspace{1em}\hspace{1em} & =(1+1+0,1+0+1,1+0,1+1)\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill & =(0,0,1,0)\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hfill \end{array}$$

(26)

The same result can also be obtained using the matrix in Equation (22). By performing the matrix product with any $z_i\in {\mathbb{F}}_2^4$, we obtain ${\left(H\left(z_i\right)\right)}^T$, where the first two entries correspond to $F_1$ and the last two to $F_2$. For example:

$${\left(H(1,1,0,1)\right)}^T=\left[\begin{array}{cccc}1& 1& 1& 0\\ 0& 1& 1& 1\\ 1& 0& 1& 0\\ 0& 1& 0& 1\end{array}\right]\left[\begin{array}{c}1\\ 1\\ 0\\ 1\end{array}\right]=\left[\begin{array}{c}0\\ 0\\ 1\\ 0\end{array}\right]$$

(27)

Table 1. The range of the linear transformation H, with the domain ordered lexicographically.

$\mathit{H}\left((0,0,0,0)\right)=(0,0,0,0)$

$H\left((0,0,0,1)\right)=(0,1,0,1)$

$H\left((0,0,1,0)\right)=(1,1,1,0)$

$H\left((0,0,1,1)\right)=(1,0,1,1)$

$H\left((0,1,0,0)\right)=(1,1,0,1)$

$H\left((0,1,0,1)\right)=(1,0,0,0)$

$H\left((0,1,1,0)\right)=(0,0,1,1)$

$H\left((0,1,1,1)\right)=(0,1,1,0)$

$H\left((1,0,0,0)\right)=(1,0,1,0)$

$H\left((1,0,0,1)\right)=(1,1,1,1)$

$H\left((1,0,1,0)\right)=(0,1,0,0)$

$H\left((1,0,1,1)\right)=(0,0,0,1)$

$H\left((1,1,0,0)\right)=(0,1,1,1)$

$H\left((1,1,0,1)\right)=(0,0,1,0)$

$H\left((1,1,1,0)\right)=(1,0,0,1)$

$H\left((1,1,1,1)\right)=(1,1,0,0)$

shows the range of the linear transformation H, evaluated over a lexicographically ordered domain. This confirms that H is a bijection, as its image is ${\mathbb{F}}_2^4$.

Having established the necessary algebraic and structural background, we now proceed to describe our proposed construction method in detail.

3. LBCA-Based Construction of Mutually Unbiased Bases

Building on the mathematical preliminaries introduced in Section 2, we now turn to the central constructive part of the paper. In this section, we develop the LBCA-based machinery that produces commuting sets of generalized Pauli operators (CSPOs) and, from these operator families, mutually unbiased bases (MUBs).

3.1. Complete Sets of MUBs

This subsection describes the construction of complete sets of MUBs via disjoint CSPOs induced by families of LBCAs. Rather than building bases directly, our approach proceeds indirectly: We construct $q+1$ pairwise disjoint CSPOs (each containing $q-1$ nontrivial operators) and obtain their common eigenvectors as the desired bases. Concretely, the argument is organized in two main steps: (i) present families of linear bipermutive cellular automata and the algebraic condition for pairwise disjointness; (ii) map each disjoint LBCA family to a commutative set of generalized Pauli operators and verify the required commutation relations The main intermediate results to follow are Lemma 1, Proposition 1 (algebraic test of disjointness), and Theorem 1 (existence of $q+1$ disjoint CSPOs).

As mentioned in Section 2.1, an indirect method for generating complete sets of mutually unbiased bases is to construct $q+1$ disjoint of $q-1$ CSPOs $Y^{\lambda }$ (excluding the identity operator; see Equation (6)). In this section, we establish the connection between two LBCAs, $F_1$ and $F_2$, and their corresponding disjoint CSPOs, $Y^{{\lambda }_1}$ and $Y^{{\lambda }_2}$. Based on this connection, we derive the necessary conditions for generating MUBs from cellular automata.

To construct complete sets of MUBs, we consider the family of LBCAs given in Lemma 1, which naturally correspond bijectively to disjoint sets of Pauli operators. We then extend this idea to other disjoint families of LBCAs.

Lemma 1.

Let $F_{\lambda }:{\mathbb{F}}_q^2\to {\mathbb{F}}_q$ be a LBCA defined by the local rule ${\mu }_{\lambda }:{\mathbb{F}}_q^2\to {\mathbb{F}}_q$ given by:

$${\mu }_{\lambda }\left(x_0,x_1\right)=x_0+\lambda x_1.$$

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Then, the set $S=\{F_{\lambda }:\lambda \in {\mathbb{F}}_q^{*}\}$ is pairwise disjoint.

Proof. 

Given $F_{{\lambda }_1},F_{{\lambda }_2}\in S$ with ${\lambda }_1\ne {\lambda }_2$, they are disjoint since for Proposition 1

$$det\left(M_{{\lambda }_1,{\lambda }_2}\right)=det\left(\left[\begin{array}{cc}1& {\lambda }_1\\ 1& {\lambda }_2\end{array}\right]\right)={\lambda }_2-{\lambda }_1\ne 0.$$

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□

In summary, Lemma 1 ensures that the disjointness condition can be verified purely through the algebraic representation of the LBCA rules. This step is essential, as it transforms the problem of constructing MUBs into a tractable algebraic verification process.

Before stating Lemma 2, we emphasize the provenance of the ideas involved to avoid ambiguity. The general algebraic framework for analyzing disjointness conditions among cellular automata over finite fields has been considered in related contexts, for example, in [23,27]. However, the specific formulation presented here, which connects these disjointness conditions directly to the generation of commuting sets of Pauli operators (CSPOs) suitable for MUB construction, is original to this work. The lemma below encapsulates this new formulation while building upon certain structural observations from the cited literature.

Lemma 2.

Let the set of local rules be defined as $\{{\mu }_{\lambda }=x_0+\lambda x_1:\lambda \in {\mathbb{F}}_q^{*}\}$, associated with the disjoint LBCAs $F_{\lambda }:{\mathbb{F}}_q^2\to {\mathbb{F}}_q$. Then, for each $\lambda \in {\mathbb{F}}_q^{*}$, the Pauli operators in the set $Y^{\lambda }=\{{\widehat{Z}}_{x_1}{\widehat{X}}_{\lambda x_1}:x_1\in {\mathbb{F}}_q^{*}\}$ commute.

Proof. 

Let ${\widehat{Z}}_{x_1}{\widehat{X}}_{\lambda x_1},{\widehat{Z}}_{x_1^{\prime }}{\widehat{X}}_{\lambda x_1^{\prime }}\in Y^{\lambda }$. Then,

$$tr\left[x_1\left(\lambda x_1^{\prime }\right)-\left(\lambda x_1\right)x_1^{\prime }\right]=tr\left(0\right)=0$$

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Thus, by the given condition in Equation (7), it follows that $\left[{\widehat{Z}}_{x_1}{\widehat{X}}_{\lambda x_1},{\widehat{Z}}_{x_1^{\prime }}{\widehat{X}}_{\lambda x_1^{\prime }}\right]=0$ □

Thus, Lemma 2 formalizes the precise algebraic conditions under which disjointness is preserved when mapping LBCAs to CSPOs. This guarantees the validity of our construction in the broader context of generalized Pauli operator frameworks.

The following theorem finalizes the assignment of the disjoint family of LBCAs, defined by the local rules in Lemma 1, to disjoint CSPOs.

Theorem 1.

The sets $Y^{\lambda }=\left\{{\widehat{Z}}_{x_1}{\widehat{X}}_{\lambda x_1}:x_1\in {\mathbb{F}}_q^{*}\right\}$ for $\lambda \in {\mathbb{F}}_q^{*}$, together with the sets $\left\{{\widehat{Z}}_{x_1}:x_1\in {\mathbb{F}}_q^{*}\right\}$ and $\left\{{\widehat{X}}_{x_1}:x_1\in {\mathbb{F}}_q^{*}\right\}$, form a family of $q+1$ CSPOs that are pairwise disjoint.

Proof. 

The sets $\left\{{\widehat{Z}}_{x_1}:x_1\in {\mathbb{F}}_q^{*}\right\}$ and $\left\{{\widehat{X}}_{x_1}:x_1\in {\mathbb{F}}_q^{*}\right\}$ are clearly commuting and disjoint. The sets $Y^{\lambda }$ are commuting by Lemma 2 and disjoint as a consequence of Lemma 1. Moreover, since for every ${\widehat{Z}}_{x_1}{\widehat{X}}_{\lambda x_1}\in Y^{\lambda }$, we have $\lambda ,x_1\ne 0$, it follows that ${\widehat{Z}}_{x_1}{\widehat{X}}_{\lambda x_1}\ne {\widehat{Z}}_{x_1}$ and ${\widehat{Z}}_{x_1}{\widehat{X}}_{\lambda x_1}\ne {\widehat{X}}_{x_1}$. Therefore,

$$Y^{\lambda }\cap \left\{{\widehat{Z}}_{x_1}:x_1\in {\mathbb{F}}_q^{*}\right\}=\varnothing ,Y^{\lambda }\cap \left\{{\widehat{X}}_{x_1}:x_1\in {\mathbb{F}}_q^{*}\right\}=\varnothing .$$

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□

The operators corresponding to each CSPO mentioned in Theorem 1 can be simultaneously diagonalized, and their common eigenvectors form an orthonormal basis (see (8)). Moreover, since the corresponding sets are pairwise disjoint, the resulting bases form a complete set of MUBs (see (9)).

Example 5.

Consider the LBCAs $F_{\lambda }:{\mathbb{F}}_4^2\to {\mathbb{F}}_4$, where the finite field is given by ${\mathbb{F}}_4=\left\{0,1,\theta ,{\theta }^2\right\},\lambda \in {\mathbb{F}}_4^{*}$ and θ is a root of the irreducible polynomial $x^2+x+1$. We have:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill F_1:{\mu }_1& =x_0+x_1,\hfill \\ \hfill F_{\theta }:{\mu }_{\theta }& =x_0+\theta x_1,\hfill \\ \hfill F_{{\theta }^2}:{\mu }_{{\theta }^2}& =x_0+{\theta }^2{x}_1.\hfill \end{array}\end{array}$$

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According to Theorem 1, the family of $q+1=5$ CSPOs is given by:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill Y^1& =\left\{{\widehat{Z}}_1{\widehat{X}}_1,{\widehat{Z}}_{\theta }{\widehat{X}}_{\theta },{\widehat{Z}}_{{\theta }^2}{\widehat{X}}_{{\theta }^2}\right\},\hfill \\ \hfill Y^{\theta }& =\left\{{\widehat{Z}}_1{\widehat{X}}_{\theta },{\widehat{Z}}_{\theta }{\widehat{X}}_{{\theta }^2},{\widehat{Z}}_{{\theta }^2}{\widehat{X}}_1\right\},\hfill \\ \hfill Y^{{\theta }^2}& =\left\{{\widehat{Z}}_1{\widehat{X}}_{{\theta }^2},{\widehat{Z}}_{\theta }{\widehat{X}}_1,{\widehat{Z}}_{{\theta }^2}{\widehat{X}}_{\theta }\right\}.\hfill \end{array}\end{array}$$

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These sets correspond to the functions $F_{\lambda }$ defined in (32). The remaining two sets are:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \left\{{\widehat{Z}}_1,{\widehat{Z}}_{\theta },{\widehat{Z}}_{{\theta }^2}\right\},\\ \hfill \left\{{\widehat{X}}_1,{\widehat{X}}_{\theta },{\widehat{X}}_{{\theta }^2}\right\}.\end{array}\end{array}$$

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This example illustrates the practical application of the theoretical criteria, showing step-by-step how an LBCA pair satisfying the disjointness conditions leads to the explicit construction of mutually unbiased bases.

3.2. Incomplete Sets of MUBs

For dimensions $2^n$ with $n>2$, it is possible to construct not only complete sets of MUBs but also incomplete ones [19]. The construction we will address in this work is based on the following theorem, adapted from Theorem 3 in [30].

Theorem 2.

Let s be a positive integer and let $\tilde{H}\in {\mathbb{C}}^{s\times s}$ be a complex Hadamard matrix that is orthonormalized. Suppose there exist $k-2$ disjoint linear bipermutive cellular automata (LBCA), $F_j:{\mathbb{F}}_q^N\to {\mathbb{F}}_q^{d-1}$, $\left(j\in [1,2,\dots ,k-2]\right)$, such that $s^2=q^N$. Then, it is possible to construct k mutually unbiased bases (MUBs) in ${\mathbb{C}}^{s^2}$.

The proof of this theorem is carried out in several steps, as detailed in [30]. For the purposes of this article, and to aid comprehension, we outline below the essential steps of the construction:

• Two vectors $v,u\in {\mathbb{C}}^{s^2}$ are defined using repeated sequences of integers from 0 to $s-1$. Specifically,

  $$v=\left[\begin{array}{c}0\\ 1\\ ⋮\\ s-1\\ 0\\ 1\\ ⋮\\ s-1\\ ⋮\\ 0\\ 1\\ ⋮\\ s-1\end{array}\right],u=\left[\begin{array}{c}0\\ 0\\ ⋮\\ 0\\ 1\\ 1\\ ⋮\\ 1\\ ⋮\\ s-1\\ s-1\\ ⋮\\ s-1\end{array}\right]$$
  Each block of length s is repeated s times, so that both vectors have length $s^2$. These will serve as the foundation for constructing two of the mutually unbiased bases.
• Let q be a prime power, and assume there exist $k-2$ disjoint linear bipermutive cellular automata (LBCA),

  $$F_j:{\mathbb{F}}_q^N\to {\mathbb{F}}_q^{d-1},j\in \{1,\dots ,k-2\},$$

  where each automaton is defined by a local rule applied across a fixed-radius window. The domain and codomain must satisfy $q^N=s^2$ and $q^{d-1}=s$, so that the output vector of each LBCA has length $s^2$. This vector is constructed by applying the local rule to every input in the domain ${\mathbb{F}}_q^N$, ordered lexicographically.
• The Hamming weight operation, introduced in Definition 2, is applied to each of the vectors u, v, and the ones generated by the LBCAs.
• The mutually unbiased bases are constructed by associating each Hamming-weighted vector with a column of the matrix $\tilde{H}\in {\mathbb{C}}^{s\times s}$. Each basis vector is defined by the operation

  $$b_{v_i,{\tilde{h}}_j}:=v_i\uparrow {\tilde{h}}_j,$$

  where $v_i$ is a Hamming-weighted vector of length $s^2$, and ${\tilde{h}}_j$ is the j-th column of the orthonormalized Hadamard matrix. The operation ↑ is defined formally in Definition 3.

Example 6.

Let $\tilde{H}\in {\mathbb{C}}^{2\times 2}$ be the orthonormalized Hadamard matrix:

$$\tilde{H}={\displaystyle \frac{1}{\sqrt{2}}}\left[\begin{array}{cc}1& 1\\ 1& -1\end{array}\right].$$

Define the following vectors:

$$v=\left[\begin{array}{c}0\\ 1\\ 0\\ 1\end{array}\right],u=\left[\begin{array}{c}0\\ 0\\ 1\\ 1\end{array}\right].$$

Additionally, consider a linear bipermutive cellular automaton (LBCA) $F:{\mathbb{F}}_2^2\to {\mathbb{F}}_2$ defined by the local rule $\mu (x_0,x_1)=x_0+x_1$. To generate a vector from F, we evaluate the local rule on all possible input combinations $(x_0,x_1)\in {\mathbb{F}}_2^2$, ordered lexicographically. That is, the domain ${\mathbb{F}}_2^2$ is traversed in the order:

$$(0,0),(0,1),(1,0),(1,1),$$

and the corresponding outputs of μ are collected into a vector:

$$F=\left[\begin{array}{c}\mu (0,0)\\ \mu (0,1)\\ \mu (1,0)\\ \mu (1,1)\end{array}\right]=\left[\begin{array}{c}0\\ 1\\ 1\\ 0\end{array}\right].$$

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This procedure ensures that all $s^2$ configurations of the input space ${\mathbb{F}}_q^N$ (with $q=2$, $N=2$) are considered, yielding a vector of length $s^2=4$.

By applying the Hamming weight operation to each entry of the vectors v, u, and F—that is, assigning 1 to entries equal to a fixed symbol and 0 to the rest—we obtain the following Hamming-weighted vectors:

The mutually unbiased basis vectors are then constructed by associating each Hamming-weighted vector $v_j$ (

Table 2. Vectors with Hamming weight.

v	${\mathit{v}}_0$	${\mathit{v}}_1$	u	${\mathit{u}}_0$	${\mathit{u}}_1$	F	${\mathit{F}}_0$	${\mathit{F}}_1$
$\left[\begin{array}{c}0\\ 1\\ 0\\ 1\end{array}\right]$	$\left[\begin{array}{c}0\\ 1\\ 0\\ 1\end{array}\right]$	$\left[\begin{array}{c}1\\ 0\\ 1\\ 0\end{array}\right]$	$\left[\begin{array}{c}0\\ 0\\ 1\\ 1\end{array}\right]$	$\left[\begin{array}{c}0\\ 0\\ 1\\ 1\end{array}\right]$	$\left[\begin{array}{c}1\\ 1\\ 0\\ 0\end{array}\right]$	$\left[\begin{array}{c}0\\ 1\\ 1\\ 0\end{array}\right]$	$\left[\begin{array}{c}1\\ 0\\ 0\\ 1\end{array}\right]$	$\left[\begin{array}{c}0\\ 1\\ 1\\ 0\end{array}\right]$

), with a column ${\tilde{h}}_i$ of the matrix $\tilde{H}$ via the componentwise operation ↑, defined as:

$$b_{{\tilde{h}}_i,v_j}={\tilde{h}}_i\uparrow v_j.$$

For example,

$$b_{{\tilde{h}}_2,v_0}={\displaystyle \frac{1}{\sqrt{2}}}\left[\begin{array}{c}1\\ -1\end{array}\right]\uparrow \left[\begin{array}{c}1\\ 1\\ 0\\ 0\end{array}\right]={\displaystyle \frac{1}{\sqrt{2}}}\left[\begin{array}{c}1\\ -1\\ 0\\ 0\end{array}\right].$$

The resulting mutually unbiased bases are as follows:

$$B_v=\left(\begin{array}{cc}{b}_{{\tilde{h}}_1,v_0}& b_{{\tilde{h}}_2,v_0}\\ b_{{\tilde{h}}_1,v_1}& b_{{\tilde{h}}_2,v_1}\end{array}\right),B_u=\left(\begin{array}{cc}{b}_{{\tilde{h}}_1,u_0}& b_{{\tilde{h}}_2,u_0}\\ b_{{\tilde{h}}_1,u_1}& b_{{\tilde{h}}_2,u_1}\end{array}\right),B_F=\left(\begin{array}{cc}{b}_{{\tilde{h}}_1,w_0}& b_{{\tilde{h}}_2,w_0}\\ b_{{\tilde{h}}_1,w_1}& b_{{\tilde{h}}_2,w_1}\end{array}\right).$$

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Example 7.

Let the orthonormalized Hadamard matrix be of order 4,

$$\tilde{H}={\displaystyle \frac{1}{2}}\left[\begin{array}{cccc}1& 1& 1& 1\\ 1& i& -1& -i\\ 1& -1& 1& -1\\ 1& -i& -1& i\end{array}\right]$$

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and let $F_1,F_2:{\mathbb{F}}_2^4\to {\mathbb{F}}_2^2$ be given by local rules ${\mu }_1(x_i,x_{i+1},x_{i+2})=x_i+x_{i+1}+x_{i+2}$ and ${\mu }_2(x_i,x_{i+1},x_{i+2})=x_i+x_{i+2}$, The obtained vectors are of order 16 with Hamming weight.

Therefore, the set of MUBs that is obtained is:

$$\begin{array}{c}\hfill B_{F_1}=\left(\begin{array}{cccc}{b}_{{\tilde{h}}_1,F_{1_{\left(00\right)}}}& b_{{\tilde{h}}_2,F_{1_{\left(00\right)}}}& b_{{\tilde{h}}_3,F_{1_{\left(00\right)}}}& b_{{\tilde{h}}_4,F_{1_{\left(00\right)}}}\\ b_{{\tilde{h}}_1,F_{1_{\left(01\right)}}}& b_{{\tilde{h}}_2,F_{1_{\left(01\right)}}}& b_{{\tilde{h}}_3,F_{1_{\left(01\right)}}}& b_{{\tilde{h}}_4,F_{1_{\left(01\right)}}}\\ b_{{\tilde{h}}_1,F_{1_{\left(10\right)}}}& b_{{\tilde{h}}_2,F_{1_{\left(10\right)}}}& b_{{\tilde{h}}_3,F_{1_{\left(10\right)}}}& b_{{\tilde{h}}_4,F_{1_{\left(10\right)}}}\\ b_{{\tilde{h}}_1,F_{1_{\left(11\right)}}}& b_{{\tilde{h}}_2,F_{1_{\left(11\right)}}}& b_{{\tilde{h}}_3,F_{1_{\left(11\right)}}}& b_{{\tilde{h}}_4,F_{1_{\left(11\right)}}}\end{array}\right)\end{array}$$

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$$\begin{array}{c}\hfill B_{F_2}=\left(\begin{array}{cccc}{b}_{{\tilde{h}}_1,F_{2_{\left(00\right)}}}& b_{{\tilde{h}}_2,F_{2_{\left(00\right)}}}& b_{{\tilde{h}}_3,F_{2_{\left(00\right)}}}& b_{{\tilde{h}}_4,F_{2_{\left(00\right)}}}\\ b_{{\tilde{h}}_1,F_{2_{\left(01\right)}}}& b_{{\tilde{h}}_2,F_{2_{\left(01\right)}}}& b_{{\tilde{h}}_3,F_{2_{\left(01\right)}}}& b_{{\tilde{h}}_4,F_{2_{\left(01\right)}}}\\ b_{{\tilde{h}}_1,F_{2_{\left(10\right)}}}& b_{{\tilde{h}}_2,F_{2_{\left(10\right)}}}& b_{{\tilde{h}}_3,F_{2_{\left(10\right)}}}& b_{{\tilde{h}}_4,F_{2_{\left(10\right)}}}\\ b_{{\tilde{h}}_1,F_{2_{\left(11\right)}}}& b_{{\tilde{h}}_2,F_{2_{\left(11\right)}}}& b_{{\tilde{h}}_3,F_{2_{\left(11\right)}}}& b_{{\tilde{h}}_4,F_{2_{\left(11\right)}}}\end{array}\right)\end{array}$$

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$$\begin{array}{c}\hfill B_v=\left(\begin{array}{cccc}{b}_{{\tilde{h}}_1,v_0}& b_{{\tilde{h}}_2,v_0}& b_{{\tilde{h}}_3,v_0}& b_{{\tilde{h}}_4,v_0}\\ b_{{\tilde{h}}_1,v_1}& b_{{\tilde{h}}_2,v_1}& b_{{\tilde{h}}_3,v_1}& b_{{\tilde{h}}_4,v_1}\\ b_{{\tilde{h}}_1,v_2}& b_{{\tilde{h}}_2,v_2}& b_{{\tilde{h}}_3,v_2}& b_{{\tilde{h}}_4,v_2}\\ b_{{\tilde{h}}_1,v_3}& b_{{\tilde{h}}_2,v_3}& b_{{\tilde{h}}_3,v_3}& b_{{\tilde{h}}_4,v_3}\end{array}\right)\end{array}$$

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$$\begin{array}{c}\hfill B_u=\left(\begin{array}{cccc}{b}_{{\tilde{h}}_1,u_0}& b_{{\tilde{h}}_2,u_0}& b_{{\tilde{h}}_3,u_0}& b_{{\tilde{h}}_4,u_0}\\ b_{{\tilde{h}}_1,u_1}& b_{{\tilde{h}}_2,u_1}& b_{{\tilde{h}}_3,u_1}& b_{{\tilde{h}}_4,u_1}\\ b_{{\tilde{h}}_1,u_2}& b_{{\tilde{h}}_2,u_2}& b_{{\tilde{h}}_3,u_2}& b_{{\tilde{h}}_4,u_2}\\ b_{{\tilde{h}}_1,u_3}& b_{{\tilde{h}}_2,u_3}& b_{{\tilde{h}}_3,u_3}& b_{{\tilde{h}}_4,u_3}\end{array}\right)\end{array}$$

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These examples serve to illustrate the general applicability of the proposed method and its effectiveness in producing valid MUBs in different scenarios.

4. Conclusions

The theoretical developments and examples presented in Section 3 and Section 3.2 confirm that the LBCA-based approach can systematically produce both complete and incomplete sets of MUBs. The lemmas and propositions provide precise algebraic tools to test and guarantee disjointness, while the examples demonstrate their applicability in explicit constructions. This dual perspective—rigorous theory and constructive practice—directly supports our claim that cellular automata offer an efficient, scalable, and structurally transparent framework for MUB generation.

In this paper, we introduced a framework for constructing mutually unbiased bases (MUBs) using linear bipermutive cellular automata (LBCAs) over finite alphabets. By establishing a correspondence between disjoint families of LBCAs and commuting sets of Pauli operators (CSPOs), we developed a constructive method for generating both complete and incomplete sets of MUBs. The key algebraic criterion—matrix invertibility—ensures unbiasedness among the resulting bases.

While disjoint LBCAs have been previously studied in classical combinatorial contexts, such as orthogonal Latin squares [23,31], our work translates these ideas into the quantum domain. Leveraging the structure of CSPOs, we reinterpret disjoint LBCAs as generators of operator bases directly applicable to quantum measurement theory, establishing novel links between discrete dynamical systems and quantum information processing.

Our method is applicable even in dimensions where complete sets of MUBs are conjectured not to exist, producing substantial incomplete sets that remain useful for quantum measurements. Unlike approaches based on finite fields, Latin squares, or Clifford algebra symmetries, our technique is driven by local update rules from cellular automata, allowing algorithmic rule discovery and automated verification. This makes it especially promising for high-dimensional or noisy quantum systems.

Future research could explore optimization and machine learning strategies to discover disjoint automata and investigate incomplete MUBs in complex or constrained settings. Overall, our results position cellular automata as a versatile, computationally grounded tool that complements and extends existing MUB constructions.