## 1. Introduction

**Theorem**

**1**

**.**Given three Pauli classes ${\mathcal{C}}_{1},{\mathcal{C}}_{2},{\mathcal{C}}_{3}$ belonging to a complete set $\mathcal{S}$ of classes in dimension $=4$, there exists exactly one more maximal commuting class $\mathcal{C}$ of Pauli operators in ${\mathcal{C}}_{1}\cup {\mathcal{C}}_{2}\cup {\mathcal{C}}_{3}$. The class $\mathcal{C}$ together with the remaining two classes ${\mathcal{C}}_{4}$ and ${\mathcal{C}}_{5}$ of $\mathcal{S}$ forms an unextendible set of Pauli classes, whose common eigenbases form a weakly UMUB of order 3.

^{2}(in fact, we present a construction of a new class of maximal partial spreads of the symplectic polar space ${\mathcal{W}}_{3}(d)$ for any odd prime power d, which translates to UMUBs in the case d is a prime.) In dimension $\ell =8$, a similar result is obtained in [9].

**Conjecture**

**1**

**.**Given $\ell /2+1$ maximal commuting Pauli classes ${\mathcal{C}}_{1},{\mathcal{C}}_{2},\dots ,{\mathcal{C}}_{\ell /2+1}$ belonging to a complete set $\mathcal{S}$ of classes in dimension $\ell ={2}^{N}=:{d}^{N}$, there exists exactly one more maximal commuting class $\mathcal{C}$ of Pauli operators in ${\cup}_{1\le i\le \ell /2+1}{\mathcal{C}}_{i}$. The class $\mathcal{C}$ together with the remaining classes of $\mathcal{S}$ forms an unextendible set of Pauli classes of size $\ell /2+1$, whose common eigenbases form a weakly UMUB of order $\ell /2+1$.

## 2. The General Pauli Group

## 3. Unextendible Sets of MUBs and Operator Classes

#### 3.1. Maximal Commuting Operator Classes

- (a)
- the elements of ${\mathcal{C}}_{j}$ commute for all $1\le j\le \ell $, and
- (b)
- ${\mathcal{C}}_{j}\cap {\mathcal{C}}_{k}=\varnothing $ for all $j\ne k$.

**Lemma**

**1**

**.**The common eigenbases of ℓ mutually disjoint maximal commuting operator classes form a set of ℓ mutually unbiased bases.

#### 3.2. Unextendibility of MUBs and Operator Classes

**Definition**

**1**

**.**A set of mutually disjoint maximal commuting classes $\{{\mathcal{C}}_{1},{\mathcal{C}}_{2},\dots ,{\mathcal{C}}_{\ell}\}$ of operators drawn from a unitary basis $\mathcal{U}$ is said to be unextendibleif no other maximal class can be formed out of the remaining operators in $\mathcal{U}\setminus (\{\mathbf{I}\}\cup {\bigcup}_{i=1}^{\ell}{\mathcal{C}}_{i})$.

**Definition**

**2**

**.**Given a set of MUBs $\{{B}_{1},{B}_{2},\dots ,{B}_{\ell}\}$ that are realized as common eigenbases of a set of ℓ operator classes comprising operators from $\mathcal{U}$, the set $\{{B}_{1},{B}_{2},\dots ,{B}_{\ell}\}$ is weakly unextendible if there does not exist another unbiased basis that can be realized as the common eigenbasis of a maximal commuting class of operators in $\mathcal{U}$.

## 4. Symplectic Polar Spaces and the Pauli Group

**Remark**

**1**(Number of points)

**.**

**Proposition**

**1**

**.**

- (i)
- The derived group ${\mathbf{P}}^{\prime}$ equals the center $Z(\mathbf{P})$ of $\mathbf{P}$.
- (ii)
- We have $Z(\mathbf{P})=\langle \omega \mathbf{I}\rangle $, so that $|Z(\mathbf{P})|=d$.
- (iii)
- $\mathbf{P}$ is nonabelian of exponent d if d is odd.
- (iv)
- $\mathbf{P}$ is nonabelian of exponent 4 if d is 2.
- (v)
- We have the following short exact sequence of groups:$$1\mapsto Z(\mathbf{P})\mapsto \mathbf{P}\mapsto V(2N,d)\mapsto 1.$$

**Theorem**

**2**

**.**Two elements of ${\mathbb{P}}^{\times}$ commute if and only if the corresponding points of ${\mathcal{W}}_{2N-1}(d)$ are collinear. In other words, the commuting structure of $\mathbf{P}$ (and $\mathbb{P}$) is governed by that of the symplectic polar space ${\mathcal{W}}_{2N-1}(d)$.

## 5. Unextendible Mutually Unbiased Bases and Pauli Classes

**Theorem**

**3**(General connecting theorem)

**.**

- Step 1
- To $\mathcal{S}$ corresponds a set of $M+1$ subgroups ${H}_{i}$, $i\in \{0,1,\dots ,M\}$, of $\mathbf{P}$ of size ${d}^{N+1}$ which mutually (pairwise) intersect (precisely) in $Z(\mathbf{P})$.
- Step 2
- In each ${H}_{j}$, one chooses ${d}^{N}-1$ elements ${H}_{j}^{k}$ ($k=1,2,\dots ,{d}^{N}-1$) which are not contained in $Z(\mathbf{P})$, so that no two such elements are in the same $Z(\mathbf{P})$-coset.
- Step 3
- Then, $\mathcal{U}(\mathcal{S}):=\left\{\{{H}_{\alpha}^{\beta}|\beta \in \{1,2,\dots ,{d}^{N}-1\}\}\phantom{\rule{4pt}{0ex}}|\phantom{\rule{4pt}{0ex}}\alpha \in \{0,1,\dots ,M\}\right\}$ is a set of commuting unitary classes.
- Step 4
- If $\mathcal{S}$ is a complete partial spread of ${\mathcal{W}}_{2N-1}(d)$, that is, if $\mathcal{S}$ is not strictly contained in another partial spread, then $\mathcal{U}(\mathcal{S})$ is unextendible, and the corresponding set of MUBs is weakly unextendible of size $M+1$.

#### 5.1. The Bijection ρ

#### 5.2. Prime Dimension

## 6. The Case $\mathit{d}$ Prime, $\mathit{N}=\mathbf{2}$—Small and Large Examples

#### 6.1. Grids and Point-Line Duals

#### 6.2. Antiregularity

#### 6.3. Regularity

#### 6.4. The Case $p=2$

**Theorem**

**4**

**.**Given three Pauli classes ${\mathcal{C}}_{1},{\mathcal{C}}_{2},{\mathcal{C}}_{3}$ belonging to a complete set $\mathcal{S}$ of classes in dimension $=4$, there exists exactly one more maximal commuting class $\mathcal{C}$ of Pauli operators in ${\mathcal{C}}_{1}\cup {\mathcal{C}}_{2}\cup {\mathcal{C}}_{3}$. The class $\mathcal{C}$ together with the remaining two classes ${\mathcal{C}}_{4}$ and ${\mathcal{C}}_{5}$ of $\mathcal{S}$ forms an unextendible set of Pauli classes, whose common eigenbases form a weakly UMUB of order 3.

**Proof.**

**Theorem**

**5**

**.**Given an unextendible set of three Pauli classes ${\mathcal{C}}_{1},{\mathcal{C}}_{2},{\mathcal{C}}_{3}$ in dimension $=4$, the nine operators in ${\mathcal{C}}_{1}\cup {\mathcal{C}}_{2}\cup {\mathcal{C}}_{3}$ can be partitioned into a different set of three maximal commuting classes ${\mathcal{C}}_{1}^{\prime},{\mathcal{C}}_{2}^{\prime},{\mathcal{C}}_{3}^{\prime}$ such that each ${\mathcal{C}}_{i}^{\prime}$ has precisely one operator in common with each ${\mathcal{C}}_{j}$, $i,j\in \{1,2,3\}$.

**Proof.**

#### 6.5. General Case

**Theorem**

**6**(Existence of unextendible Pauli class sets for d prime, A)

**.**

**Proof.**

**Corollary**

**1.**

**Proof.**

**Remark**

**2.**

- (a)
- $\tilde{(\xb7)}:\{0,1,\dots ,\ell \}\u27f6\{0,1,\dots ,\ell \}$ is an involution, so that $|\{{\mathcal{S}}_{0},{\mathcal{S}}_{1},\dots ,{\mathcal{S}}_{\ell}\}|=(\ell +1)/2$;
- (b)
- for ${\mathcal{S}}_{i}\ne {\mathcal{S}}_{j}$, we have that ${\mathcal{S}}_{i}\cap {\mathcal{S}}_{j}=\{L,M\}$ (by Section 6.2).

**Theorem**

**7.**

**Proof.**

**Theorem**

**8**(Existence of unextendible Pauli class sets for d prime, B)

**.**

## 7. Solution of Conjecture 1

**Conjecture**

**2**

**.**Given $\ell /2+1$ maximal commuting Pauli classes ${\mathcal{C}}_{1},{\mathcal{C}}_{2},\dots ,{\mathcal{C}}_{\ell /2+1}$ belonging to a complete set $\mathcal{S}$ of classes in dimension $\ell ={2}^{N}=:{d}^{N}$, $N\in \mathbb{N}\setminus \{0,1\}$, there exists exactly one more maximal commuting class $\mathcal{C}$ of Pauli operators in ${\cup}_{1\le i\le \ell /2+1}{\mathcal{C}}_{i}$. The class $\mathcal{C}$ together with the remaining classes of $\mathcal{S}$ forms an unextendible set of Pauli classes of size $\ell /2+1$, whose common eigenbases form a weakly UMUB of order $\ell /2+1$.

## 8. Existence of Maximal Pauli Classes

**Proposition**

**2.**

**C.1**- each element of $\mathcal{T}$ is a subset of the point set$$\Omega (\mathcal{S},\chi ):={\cup}_{U\in {\mathcal{S}}_{\chi}}U,$$
**C.2**- the elements of $\mathcal{T}$ partition $\Omega (\mathcal{S},\chi )$,

**Proof.**

**Remark**

**3**(Back to the prime case)

**.**

#### 8.1. $\mathcal{U}$-Sets

**Proposition**

**3**(Existence of UMUBs)

**.**

**Proof.**

**Remark**

**4.**

#### 8.2. Reformulation of Conjecture 2

**Conjecture**

**3.**

## 9. “Galois MUBs”

**Theorem**

**9**

**.**Let Γ be a generalized quadrangle of finite thick order $(s,s)$, and let $\mathcal{C}$ be a complete partial spread of Γ. If Γ is not contained in a spread of Γ, then

**Corollary**

**2.**

**Remark**

**5.**

**Theorem**

**10**

**.**Up to isomorphism, there is only one complete partial spread of size 3 in ${\mathcal{W}}_{3}(2)$.

**Corollary**

**3.**

**Remark**

**6**(On isomorphisms)

**.**

## 10. Conclusions

## Acknowledgments

## Conflicts of Interest

## Appendix A. Properties of (Symplectic) Polar Spaces

#### Appendix A.1. The Map μ

#### Appendix A.2. Generators on (N − 2)-Spaces

#### Appendix A.3. Structure of Spreads

## Appendix B. Some More Properties of W_{3}(d)

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