# Evidence for and against Zauner’s MUB Conjecture in $\mathbb{C}^{6}$

Gary McConnell  Controlled Quantum Dynamics Theory Group, Imperial College,
London<EMAIL_ADDRESS>,  Harry Spencer  Churchill College, University
of Cambridge<EMAIL_ADDRESS>and  Afaq Tahir  Pembroke College, University of
Oxford<EMAIL_ADDRESS>
###### Abstract.

The problem of finding provably maximal sets of mutually unbiased bases in
$\mathbb{C}^{d}$, for composite dimensions $d$ which are not prime powers,
remains completely open. In the first interesting case, $d=6$, Zauner
predicted that there can exist no more than three MUBs.

We explore possible algebraic solutions in $d=6$ by looking at their ‘shadows’
in vector spaces over finite fields. The main result is that if a counter-
example to Zauner’s conjecture were to exist, then it would leave no such
shadow upon reduction modulo several different primes, forcing its algebraic
complexity level to be much higher than that of current well-known examples.

In the case of prime powers $q\equiv 5\bmod 12$, however, we are able to show
some curious evidence which — at least formally — points in the opposite
direction. In $\mathbb{C}^{6}$, not even a single vector has ever been found
which is mutually unbiased to a set of three MUBs. Yet in these finite fields
we find sets of three ‘generalised MUBs’ together with an orthonormal set of
four vectors of a putative fourth MUB, all of which lifts naturally to a
number field.

## Introduction

The notion of _mutually unbiased bases_ or _MUBs_ arose in physics as an
optimal choice of measurement bases for quantum tomography [Sw, Iv, WF, Z];
although the concept was discovered independently in combinatorial design
theory [CS, Ca, GR]. The problem of finding provably maximal sets of MUBs in
$\mathbb{C}^{d}$ for non-prime-power dimensions $d$ remains completely open.
In the first interesting case, $d=6$, Zauner [Z] has predicted that there can
exist no more than three MUBs. Moreover there is a conjecture about orthogonal
decompositions of Lie algebras [KKU] which is equivalent to saying that for
any such non-prime-power $d$, a maximal set of $d+1$ MUBs cannot exist: see
[Be1].

A priori there is no reason to expect the entries of MUB vectors over
$\mathbb{C}$ to be algebraic. Indeed in $\mathbb{C}^{6}$ there is a catalogue
of continuously parametrised families [Jam, Sz] of sets of three MUBs which
outside a set of measure $0$ are not unitarily equivalent to any algebraic
set; showing that for arbitrary MUBs, the algebraic complexity level of the
entries of the vectors can go all the way up to the transcendental. However
since the equations are defined over $\mathbb{Z}$, one approach to exploring
the known bounds on the number of MUBs over $\mathbb{C}^{d}$ is to attack the
problem as an algebraic question over a general ring, and to see whether the
equations can be solved there.

When studying finite systems of equations with integer coefficients, a
standard technique in number theory and algebraic geometry is to look at their
reductions modulo a prime number. Nambu [N], for example, also applied the
same technique in a broad range of physical problems. Simplifying slightly for
clarity, one looks at the image modulo $p$ of their integer solutions, ‘most’
of which will survive reduction to the field $\mathbb{F}_{p}$ of integers
modulo $p$. This picture naturally extends to number fields, and consequently
the _absence_ of a solution in a finite field can under strict conditions be
used to show a corresponding failure of solvability back up in the original
number field. See [GIJM] for a recent similar approach to an affiliated
problem.

The main thrust of this paper is to add a small result arising from this
hitherto unexplored direction, to the already large body of evidence adduced
in favour of Zauner’s conjecture. Namely, in theorem 2.1 we demonstrate in a
number of finite fields that sets of MUBs of size $4$ in dimension $6$ do not
exist. Prima facie this is weak evidence that such sets of MUBs do not exist
in the sorts of small degree number fields in which solutions have previously
been studied: see for example [Be2, Go, ABD] and 4.2. Nevertheless, the
translation back up to characteristic zero is still computationally
intractable, so applying this result relies on studying each specific
situation individually.

In the case of prime powers $q\equiv 5\bmod 12$, however, proposition 2.2
points tentatively in the opposite direction. We lift a set of three
generalised MUBs plus an additional set of four vectors of a putative fourth
MUB, directly to solutions in a number field. Note however that we are
beginning with a set of three what one might deem _hyperbolic MUBs_ : ones
whose entries are all _totally real_ numbers and where the property of
unitarity is measured in terms of a real (algebraic) norm as opposed to a
complex absolute value.

It is important to note that when $F$ is finite or $p$-adic, the sesquilinear
form $\langle\ ,\ \rangle$ yields a weaker geometric structure than an inner
product [Gro]. In particular, $F^{d}$ will always contain isotropic vectors.
Moreover, searches based on local minima of analytical functions — which have
been deployed in most of the attempts to solve this problem over the complex
field, for example [D, BW, Jam, Ray, Go, GrM] — are not possible at the finite
field level. So there is no bijective geometric correspondence between our
findings in arbitrary finite or $p$-adic fields, and the solutions in Hilbert
spaces.

In the appendices we provide very brief notes on the results of the computer
searches, the equations and their reductions, and general background reference
material on the properties of these generalised MUBs.

##  1\. The MUB problem over $\mathbb{C}$ and beyond

###  1.1. State of knowledge over $\mathbb{C}$

The MUB problem over $\mathbb{C}$ has been attacked using a mixture of
geometric, combinatorial and numerical methods. To some extent these
approaches have been unified under the combinatorial umbrella of frames and
complex projective $2$-designs: see e.g. [GR, Ca, RS, MG]. We refer to the
vast body of work in the mathematical physics literature on this problem — for
example [Sw, Be1, Be2, Ch, D, Ba] — for the motivation for studying Hilbert
space MUBs and for various standard results about their geometry.

Before giving a generalised definition of MUBs in the next section, we set out
the current state of knowledge on maximal sets of MUBs in complex Hilbert
space $\mathbb{C}^{d}$. Let $\mathcal{M}_{d}{\mathbb{C}}$ denote the maximum
number of orthonormal bases of $\mathbb{C}^{d}$ which can be pairwise mutually
unbiased to one another.

######  Theorem.

[All, Iv, WF, KR] For $d\geq 2$, write $d=\prod_{i=1}^{n}p_{i}^{r_{i}}$ as a
product of its prime power factors with the $p_{i}$ ordered so that
$p_{1}^{r_{1}}<p_{2}^{r_{2}}<\ldots<p_{n}^{r_{n}}$. Then:

  1. (I)

$p_{1}^{r_{1}}+1\ \leq\ \mathcal{M}_{d}{\mathbb{C}}\ \leq\ d+1.$

  2. (II)

When $d$ is a prime power, a maximal set of $d+1$ MUBs always exists in
$\mathbb{C}^{d}$.

The lower bound, which in the general case is currently limited to the outcome
of taking tensor products [Z, Ba, ACW, KR], has in fact been marginally
improved [WB] in some non-prime-power squared dimensions like $d=676$. It
should be stressed again that no-one yet knows tight lower or upper bounds for
$\mathcal{M}_{d}{\mathbb{C}}$ for any $d$ not a power of a prime.

###  1.2. Generalised definition of MUBs

A formal definition of mutual unbiasedness may be made by analogy with the
motivating complex case. For other instances of this see for example [Ch, Boy,
vD]111Although ostensibly looking at the same question, [vD] was concerned
with infinite-dimensional Hilbert spaces. The results and techniques are
completely unrelated to those of this paper. ; and for further discussion of
our case see A.2.1 and A.2.2. Throughout, $d\in\mathbb{N}$ will denote the
dimension of our vector space. Let $F$ be any field of characteristic not
dividing $2d$, with an automorphic involution $\sigma$ whose fixed subfield is
$K$. So in the usual complex case, $F=\mathbb{C}$ and $K=\mathbb{R}$ and
$\sigma$ is complex conjugation. For any $\alpha\in F$,
${\hbox{\small{\rm{N}}}_{\scriptscriptstyle{F}/\scriptscriptstyle{K}}}{\hbox{${\alpha}$}}=\alpha\alpha^{\sigma}$
is the field norm down to $K$.

By $F^{d}$ we shall always mean a  _unitary space_ [Gro, chapter 10]: namely,
a $d$-dimensional vector space over $F$ equipped with a non-degenerate
_Hermitian_ form defined for any pair of vectors
$\mathbf{u}=(u_{j}),\mathbf{v}=(v_{j})\in F^{d}$ by:

$\langle{\mathbf{u}},{\mathbf{v}}\rangle=\mathbf{u}^{\dagger}\cdot\mathbf{v}=\sum_{j=1}^{d}u_{j}^{\sigma}v_{j},$

where for any matrix $M$ with entries in $F$, $M^{\dagger}$ denotes its
conjugate transpose with respect to the involution $\sigma$. The word _basis_
will always mean  _orthonormal basis_ , with respect to the given form; so we
may write the basis set
${\mathcal{B}}=\\{{\mathbf{b}}_{1},\ldots,{\mathbf{b}}_{d}\\}$ as the column
vectors of a _unitary_ — as defined with respect to this Hermitian form —
matrix $[{\mathbf{b}}_{1},\ldots,{\mathbf{b}}_{d}]$.

######  Definition 1.

Let $\mathbf{u},\mathbf{v}\in F^{d}$ be any choice of  _unit vectors_ : that
is, satisfying
$\langle\mathbf{u},\mathbf{u}\rangle=\langle\mathbf{v},\mathbf{v}\rangle=1$.
We say that $\mathbf{u},\mathbf{v}$ are _mutually unbiased_ or  _MU_ to one
another if

(1)
${\hbox{\small{\rm{N}}}_{\scriptscriptstyle{F}/\scriptscriptstyle{K}}}{\hbox{${\langle\mathbf{u},\mathbf{v}\rangle}$}}\
=\ 1/d.$

Let ${\mathcal{B}}$ and ${\mathcal{C}}$ be any two orthonormal bases. If
${\hbox{\small{\rm{N}}}_{\scriptscriptstyle{F}/\scriptscriptstyle{K}}}{\hbox{${\langle\mathbf{b}_{i},\mathbf{c}_{j}\rangle}$}}=1/d$
for every pair of vectors $\mathbf{b}_{i}\in\mathcal{B}$,
$\mathbf{c}_{j}\in\mathcal{C}$ then we say that ${\mathcal{B}},{\mathcal{C}}$
are _mutually unbiased bases_.

We use the same notation as for the complex case, in that $\mathcal{M}_{d}{F}$
will denote the size of a maximal set of MUBs in $F^{d}$.

The definition is motivated by the situation in $\mathbb{C}^{d}$, in that if
we require that every inner product
$\langle{\mathbf{b}}_{i},{\mathbf{c}}_{j}\rangle$ for $1\leq i,j\leq d$ have
the same absolute value, then in fact that common value is readily proven [Sw]
to be $1/\sqrt{d}$.

It is important to note that in translating _particular_ known algebraic
solutions directly from number fields embedded in $\mathbb{C}$, down to finite
and $p$-adic fields, at least half of the time we encounter a situation where
the $q$-th power Frobenius automorphic involution $\sigma$ does not act
analogously to complex conjugation on the roots of unity. This means that the
Wootters & Fields (henceforth just WF) solutions in [WF] to the equations in
$\mathbb{C}$ simply _do not exist_ modulo $p$. This is evidenced for example
in the number of primes implicitly excluded from the hypotheses of proposition
3.2, and even more prominently in the tabulation of our exhaustive search
results in A.1.1.

Conversely, any finite field solutions are by definition cyclotomic numbers;
so this constraint is inescapable. Using a set of equations derived from the
complex situation is thus rendered meaningless.

Throughout the paper, therefore, we have stuck with the _geometric_
interpretation of the MUB existence question, adjusting the equations so that
in each finite field they reflect the notion of an adjoint or Hermitian dot
product, rather than interpreting it strictly as the specialisation of a
complex algebraic variety.

The level of overdetermination in this problem is formidable. Just in the
relatively tight case in proposition 2.2, even after all sensible reductions,
we still begin with $150$ variables over $\mathbb{F}_{q}$ but $231$ equations.
Elimination-theoretic tools like Gröbner bases had to be rejected in favour of
an exhaustive search through small finite vector spaces. We have used MAGMA
software [M] throughout.

##  2\. MUBs over finite fields : implications for Zauner’s conjecture

The link between the study of complex MUBs and those over quadratic extensions
of finite fields is given explicitly by the correspondence between the
canonical complex conjugation involution whose fixed field is $\mathbb{R}$,
and the $q$-th power _Frobenius_ involution which is the unique non-trivial
automorphism of the field $F$ fixing the base field $K$. It is this structural
parallel which affords the possibility of analogous geometric results in the
two contexts. This is explained further in 4.1.

Our attempts to explore the lower bound for $\mathcal{M}_{6}{\mathbb{C}}$ via
the images of algebraic MUBs under reduction in finite fields, have led us to
the following.

######  Theorem 2.1.

$\mathcal{M}_{6}{\mathbb{F}_{q^{2}}}\leq 3$ for every prime power $q=p^{r}$ in
the set

$\\{5,25,7,49,11,13,17,19,23,29,31,37,41\\}.$

###### Proof.

The evidence from the numerical searches is tabulated in A.1.1. The
methodology is explained in A.2. ∎

Theorem 2.1 supports Zauner’s conjecture in a way which has not been explored
before. However, as mentioned in the introduction, we also have a result which
gives some slight evidence in the opposite direction.

######  Proposition 2.2.

Let $F=\mathbb{F}_{q^{2}}$ for some $q\equiv 5\bmod 12$. Then there exists in
$F^{6}$ a set
$\mathcal{S}_{F}=\\{\mathcal{B}_{0},\mathcal{B}_{1},\mathcal{B}_{2}\\}$ of
three pairwise mutually unbiased bases, together with an orthonormal set of
four vectors which are all mutually unbiased to the three bases. In each case
the solutions lift to the number field extension $K(\sqrt{3})/K$ where
$K=\mathbb{Q}(i,\sqrt{6})$.

###### Proof.

See A.1.2 where the calculations are explained. Here, as in the rest of the
paper, the initial basis set $\mathcal{B}_{0}$ is chosen to be the
_computational basis_ consisting of the $d$ vectors
$\mathbf{e}_{j}=(0,\ldots,0,1,0,\ldots,0),$ where the unique non-zero entry
occurs at the $j$-th position. $\mathcal{B}_{0}$ is represented in matrix form
by the identity matrix. An intermediate step was to lift the original
solutions from the searches over finite fields, using standard $p$-adic
techniques, up several levels in order to find the defining equations for the
entries. It then became clear that they lifted not only to unramified
extensions $F_{\wp}$ of $\mathbb{Q}_{p}$ but in fact all the way to a number
field contained in $F_{\wp}$. It is also important to note that there are
infinitely many possible number fields to which such solutions may be lifted:
we have just chosen the simplest. ∎

As far as we can determine from the published literature, the arXiv, and
personal communications with several other researchers active in this field
(thanks to Ingemar Bengtsson, Stefan Weigert, Markus Grassl, Dan McNulty,
Marcus Appleby and Dardo Goyeneche), no-one has yet been able to find a
_single vector_ in $\mathbb{C}^{6}$ which is mutually unbiased to a known set
of three MUBs [Go, WM]. Indeed, using Gröbner basis techniques for varying
values of $d$, Markus Grassl et al. have demonstrated many cases wherein the
extension of certain _non-maximal_ sets of bases in $\mathbb{C}^{d}$ _by just
a single vector_ is impossible, elucidating a bewilderingly rich variety of
possible outcomes given simple variations in the starting bases [GrM, §III],
[De].

So the existence of this extra set of four vectors in proposition 2.2, even if
it only obtains in this artificial ‘hyperbolic’ context, raises a natural
question as to whether such a phenomenon occurs as some sort of Galois image
of an actual set of MUBs, analogously for example to _ghost SICs_ [AFK] in the
cousin _SIC-POVM_ problem [Be3].

##  3\. Values of $\mathcal{M}_{d}{F}$ when $F$ is a finite field

The best bounds which may be stated in full generality are as follows.

######  Proposition 3.1.

For any quadratic extension of finite fields $F/K$ as above and any dimension
$d\geq 2$ coprime to the characteristic $p$ of $K$,

$1\leq\mathcal{M}_{d}{F}\leq d+1.$

###### Proof.

Despite there always being at least three MUBs over $\mathbb{C}^{d}$ for any
$d\geq 2$ — which [Ba] may be regarded simply as the eigenbases of the
generalised Pauli operators $X,Z$ and their product $XZ$ — the lower bound in
this finite field case cannot be improved upon in general. For example, see
table 2 in A.1.1 in which $\mathcal{M}_{3}{\mathbb{F}_{q^{2}}}$ is shown to be
$1$ for essentially half of all $q$.

For the upper bound, which again is saturated in many cases, we adapt the
argument from the complex case in [Iv], following the exposition in §4 of
[Be1].

Define a non-degenerate Hermitian trace form $\hbox{\rm
Tr}\left(A^{\dagger}B\right)$ on the matrix algebra ${\hbox{\rm M}_{d\times
d}(F)}$ of degree $d$ over $F$. Consider any orthonormal basis
$\mathcal{B}=\\{\mathbf{v}_{1},\ldots,\mathbf{v}_{d}\\}$ of $F^{d}$. The
corresponding rank $1$ trace $1$ projectors
$\pi_{\mathbf{v_{k}}}=\mathbf{v}_{k}\otimes\mathbf{v}_{k}^{\dagger}$ are
orthogonal unit vectors in ${\hbox{\rm M}_{d\times d}(F)}$ under this trace
form. Their endpoints span a $d$-simplex whose barycentre is
$\frac{1}{d}\mathbb{I}$, where $\mathbb{I}$ denotes the identity matrix in
${\hbox{\rm M}_{d\times d}(F)}$. The set resulting from subtracting
$\frac{1}{d}\mathbb{I}$ from each $\pi_{\mathbf{v_{k}}}$ spans a $K^{d-1}$
subspace of the space $\mathcal{H}_{0}\cong K^{d^{2}-1}$ of trace $0$
Hermitian operators in ${\hbox{\rm M}_{d\times d}(F)}$. The pairwise Hermitian
products between distinct shifted projectors arising from $\mathcal{B}$ are
now $\frac{-1}{d}$ rather than zero.

However, given any $\mathbf{b}\in\mathcal{B}$ and $\mathbf{c}\in\mathcal{C}$
from mutually unbiased bases $\mathcal{B}$ and $\mathcal{C}$, the shifted
trace product is zero:

$\displaystyle\hbox{\rm
Tr}\big{(}(\mathbf{b}\otimes\mathbf{b}^{\dagger}-\frac{1}{d}\mathbb{I})^{\dagger}(\mathbf{c}\otimes\mathbf{c}^{\dagger}-\frac{1}{d}\mathbb{I})\big{)}$
$\displaystyle=$ $\displaystyle\hbox{\rm
Tr}\big{(}(\mathbf{b}\otimes\mathbf{b}^{\dagger})(\mathbf{c}\otimes\mathbf{c}^{\dagger})\big{)}-\frac{1}{d}\hbox{\rm
Tr}\big{(}(\mathbf{b}\otimes\mathbf{b}^{\dagger}+\mathbf{c}\otimes\mathbf{c}^{\dagger})\big{)}+\hbox{\rm
Tr}\frac{1}{d^{2}}\mathbb{I}$ $\displaystyle=$
$\displaystyle\langle\mathbf{b},\mathbf{c}\rangle\langle\mathbf{c},\mathbf{b}\rangle-\frac{2}{d}+\frac{1}{d}$
$\displaystyle=$ $\displaystyle 0.$

That is to say, mutual unbiasedness between the bases down in $F^{d}$ lifts to
orthogonality between the corresponding translated operator subspaces in
$\mathcal{H}_{0}$.

Hence $n$ MUBs of $F^{d}$ produce $n$ mutually orthogonal $(d-1)$-dimensional
$K$-subspaces inside $K^{d^{2}-1}$. If $p\mid{\rm char}(K)$ then for example
the identity matrix lies in $\mathcal{H}_{0}$ and its trace product with every
other matrix in $\mathcal{H}_{0}$ is zero, rendering degenerate the $K$-valued
restriction to $\mathcal{H}_{0}$ of this trace form. However when $p\nmid{\rm
char}(K)$, by considering a $K$-basis for $\mathcal{H}_{0}$ formed from simple
$F$-linear combinations of the elementary matrices $E_{ij}$ we see that the
trace form remains non-degenerate; and the result follows. ∎

######  Remark.

The diagonalisation-based argument of [Ba] does not work here, because no
Hermitian structure is possible. Indeed, the Galois group
$\hbox{\small{\rm{Gal}}}_{\overline{F}/F}\cong\widehat{\mathbb{Z}}$ of the
algebraic closure $\overline{F}$ of a finite field $F$ is torsion-free. So in
particular there is no surrogate operator for complex conjugation to match
that of $\hbox{\small{\rm{Gal}}}_{\mathbb{C}/\mathbb{R}}$.

###  3.1. Values of $\mathcal{M}_{d}{\mathbb{F}_{q^{2}}}$ when $d$ is a prime
power

Given any prime power dimension $d=l^{k}$, the next result yields a class of
primes $p$ of positive density, by Dirichlet’s theorem on primes in arithmetic
progressions, for which the complex WF solutions have _good reduction_ over
fields of characteristic $p$.

######  Proposition 3.2.

Let $l$ be a prime and $k$ a positive integer. Let the prime $p\neq l$ and
$r\geq 1$ satisfy $p^{r}\equiv-1\bmod l^{k}$ when $l$ is odd, or simply
$p^{r}\equiv-1\bmod 4$ when $l=2$. Then writing $q=p^{r}$,

$\mathcal{M}_{l^{k}}{\mathbb{F}_{q^{2}}}={l^{k}+1}.$

Notice that if $p^{r}\equiv-1\bmod l^{k}$ then the same is true of $p^{ar}$
for every odd integer $a$. In other words, for each particular dimension
$d=l^{k}$, every such prime $p$ additionally furnishes us with an infinite set
of finite fields for which the statement holds.

The same techniques can be applied to more complicated sets of MUBs: we
illustrate this in 4.1. Moreover, as illustrated in A.1.1, whenever we have
done an exhaustive search in a prime-power dimension $d=2,3,4,5,7$ over some
finite field, we have found that all _maximal_ solutions of $d+1$ MUBs are
_equivalent_ — in the sense of [BWB] as explained in A.2.2 — to the reduced WF
solutions, possibly modified by $\sqrt{-1}$ as in the proof below.

On the other hand, for prime powers where the structure in proposition 3.2
does not obtain, we have shown that for $d\leq 7$, $d+1$ MUBs do _not_ exist
for the small primes in our searches222Other than in the strange anomalous
case $d=7$, $p=3$ in table 3 of A.1.1, where _a priori_ we only know that
solutions exist for powers of $q=p^{3}=27$, by proposition 3.2. . So on the
basis of this tiny body of evidence, the WF construction — with the slight
modifications needed in the proof below — would seem to be universal for
creating maximal sets of MUBs over finite fields. See [Ba] for a comparison
with the complex case.

The sub-maximal sets in these defective dimension-prime combinations display
behaviour akin to _bad reduction_ of abelian varieties; although it is not
because of the introduction of mod-$p$ singularities but rather because the
Hermitian form collapses. In other words, as we observed in 1.2, because the
Hermitian form is encapsulated (via the Galois action) within the polynomial
system, we are forced to use a different set of ‘real’ equations in these
finite fields from that in the complex case or indeed in the finite fields in
proposition 3.2. It is not clear what the ‘geometric’ meaning of the solutions
is when there is a square root of $-1$ in the ground field, as is the case for
example in proposition 2.2.

Given any field $F$ and any $N\in\mathbb{N}$, let $\zeta_{N}\in\overline{F}$
denote a fixed primitive $N$-th root of unity in an algebraic closure
$\overline{F}$ of $F$ and let $\mu_{N}\ =\ <\zeta_{N}>$ be the group it
generates. Similarly the symbol $\sqrt{T}$ will denote a fixed square root of
a field element $T$ inside an algebraic closure. A reference for the facts we
use on finite fields is [Se].

###### Proof of proposition 3.2.

We show by construction that the upper bound is attained; that it cannot be
breached is proposition 3.1. In [WF], complete sets of $l^{k}+1$ MUBs
$\\{\mathcal{B}_{0},\mathcal{B}_{1},\ldots,,\mathcal{B}_{l^{k}}\\}$ are
constructed in $\mathbb{C}^{l^{k}}$, where $\mathcal{B}_{0}$ is the
computational basis represented by the identity matrix. All vector entries in
the other $\mathcal{B}_{k}$ are of the form
$\frac{\zeta_{l}^{t}}{\sqrt{l^{k}}}$ for some $t\in\mathbb{Z}$ when $l$ is
odd; and $\frac{i^{t}}{\sqrt{2^{k}}}$ when $l=2$, where $i=\sqrt{-1}$.

Suppose first that $l$ is an odd prime. The cyclic Galois group
$\hbox{\small{\rm{Gal}}}_{\mathbb{Q}(\zeta_{l})/\mathbb{Q}}$ has a unique
involution $\tau$ whose inversion action on $\mu_{l}$ is identical to that of
the restriction of complex conjugation. On the other hand $\sqrt{l}$ is real
and so it is fixed under complex conjugation. It follows that the conjugation
action on the vector entries in
$\\{\mathcal{B}_{0},\mathcal{B}_{1},\ldots,,\mathcal{B}_{l^{k}}\\}$ is
captured entirely by $\tau\colon\zeta_{l}\mapsto\zeta_{l}^{-1}$.

We have assumed that $q\equiv-1\bmod l^{k}$ and $l\neq 2$, hence
$\mathbb{F}^{\times}_{q}\cong\mu_{q-1}$ contains no $l$-th roots of unity
other than $1$; whereas
$\mathbb{F}^{\times}_{q^{2}}\supset\mu_{q+1}\supset\mu_{l^{k}}\supset\mu_{l}$.
In particular, $\mathbb{F}_{q^{2}}=\mathbb{F}_{q}(\zeta_{l})$ and the $q$-th
power Frobenius automorphism $\sigma$ in the extension
$\mathbb{F}_{q^{2}}/\mathbb{F}_{q}$ acts on the $l$-th roots of unity via
inversion, since $\zeta_{l}^{\sigma}=\zeta_{l}^{q}=\zeta_{l}^{-1}$. It follows
that any solution to the MUB equations in
$\frac{1}{\sqrt{l^{k}}}\mathbb{Z}(\zeta_{l})$ will satisfy the very same
equations down in $\mathbb{F}_{q^{2}}$, provided that $\sigma$ fixes the
square root of $l^{k}$.

If $k$ is even then $l^{k}$ is a square integer; and if $r$ is even then by
the uniqueness of the quadratic extension of $\mathbb{F}_{p}$ the square root
of $l\in\mathbb{F}_{p}$ must already be contained in $\mathbb{F}_{q}$. In such
cases the norm
${\hbox{\small{\rm{N}}}_{\scriptscriptstyle{\mathbb{F}_{q^{2}}}/\scriptscriptstyle{\mathbb{F}_{q}}}}{\hbox{${\sqrt{l^{k}}}$}}=l^{k}$
and consequently the behaviour exactly mimics that in the complex case.

Note also the trivial fact that when $p=2$, so $l$ is in fact forced to be
odd, the square root of $l^{k}$ — that is, $1$ — is automatically in the
ground field.

However if all of $p$, $l$, $k$ and $r$ are odd it is possible that $l^{k}$
will not be a square in the base field $\mathbb{F}_{q}$: hence $\sigma$ maps
$\sqrt{l^{k}}\mapsto-\sqrt{l^{k}}$. In such cases every instance of
$\sqrt{l^{k}}$ in the WF example must be twisted by some element
$\nu\in\mathbb{F}_{q^{2}}$ whose norm is $-1$. But the norm map from
$\mathbb{F}_{q^{2}}^{\times}$ to $\mathbb{F}_{q}^{\times}$ is surjective and
so there are exactly $q+1$ elements of $\mathbb{F}_{q^{2}}^{\times}$ which map
onto $-1$. Choose one such $\nu\in\mathbb{F}_{q^{2}}^{\times}$ and multiply
all of the WF vector entries by $\nu$, ignoring $\mathcal{B}_{0}$ of course;
then once again
${\hbox{\small{\rm{N}}}_{\scriptscriptstyle{\mathbb{F}_{q^{2}}}/\scriptscriptstyle{\mathbb{F}_{q}}}}{\hbox{${\nu\sqrt{l^{k}}}$}}=l^{k}$
and the set of equations derived from the geometry in $\mathbb{C}^{l^{k}}$ is
satisfied.

When $l=2$, so $p$ is odd, all of the coefficients of the WF MUBs in dimension
$2^{k}$ are of the form $\frac{i^{a}}{\sqrt{2^{k}}}$ for some
$a\in\mathbb{Z}$. In a finite field of odd characteristic, we need the
Frobenius action $\sigma$ on the fourth roots of unity and on $\sqrt{2}$,
where relevant, to mimic that in $\mathbb{C}/\mathbb{R}$: namely, as inversion
on $i=\sqrt{-1}$ and as the identity on $\sqrt{2}$. This rules out $q\equiv
1\bmod 4$, since then either $p\equiv 1\bmod 4$ or $r$ is even: either or both
of which would ensure that $-1$ is a square in $\mathbb{F}_{q}$. So we are
forced into the situation in the hypotheses of the proposition, where $p\equiv
3\bmod 4$ and $r$ is odd, which is equivalent to $q\equiv 3\bmod 4$.

Hence all remaining possibly problematic cases for $l=2$ reduce to $k$ odd,
$r$ odd, $p\equiv 3\bmod 8$ or $p\equiv 7\bmod 8$. When $p\equiv 7\bmod 8$,
$({\scriptstyle{\frac{2}{p}}})=1$ and so the WF complex solutions behave
identically down in $\mathbb{F}_{q^{2}}$.

When $p\equiv 3\bmod 8$, however, the Legendre symbols
$({\scriptstyle{\frac{-1}{p}}})=({\scriptstyle{\frac{2}{p}}})=-1$ and so both
$\sqrt{-1}$ and $\sqrt{2}$ are mapped by $\sigma$ to their negatives. The net
action of $\sigma$ is therefore to fix entries of the form $\frac{\pm
i}{\sqrt{2}}$ and to flip the signs on the ‘real’ entries $\frac{\pm
1}{\sqrt{2}}$. So we need once again to compensate by adding in a factor whose
norm is $-1$, and we follow the same procedure as above. ∎

######  Remark.

Under the  _Alltop construction_ [All, KR, Bl] the complete set of $d+1$ MUBs
may also be constructed333For odd primes $l$ or $d=2,4$; however this is
apparently still an open problem when $d=2^{k}$ for $k\geq 3$. See [Bl] for
the case $d=4$ and for the history of the problem.  as the images of a
_fiducial vector_ under the action of a _Weyl-Heisenberg_ or _generalised
Pauli_ group of degree $d$, just as in the cousin problem of SIC-POVMs [Be3].
As in the proof above, all of the matrix entries as well as those of the
fiducial vector are drawn from $\frac{1}{\sqrt{l^{k}}}\mathbb{Z}(\zeta_{l})$;
and the only group action relevant to the MUB properties is that of complex
conjugation. Hence with similar modifications, the construction carries over
faithfully to the finite field context.

###  3.2. Values of $\mathcal{M}_{d}{\mathbb{F}_{q^{2}}}$ when $d$ is not a
prime power

######  Corollary 3.3.

Write $d=\prod_{j=1}^{n}l_{j}^{k_{j}}$ as a product of its prime power
factors, ordered so that $l_{1}^{k_{1}}<l_{2}^{k_{2}}<\ldots<l_{n}^{k_{n}}$.
Then there is a set of primes $p$ of positive Dirichlet density together with
integers $r_{p}\geq 1$ such that for each such pair, writing $q=p^{r_{p}}$:

$\mathcal{M}_{d}{\mathbb{F}_{q^{2}}}\geq{l_{1}^{k_{1}}+1}.$

###### Proof.

We need to find a prime power $q=p^{r}$ which simultaneously satisfies the
hypotheses of proposition 3.2 for every $l_{j}^{k_{j}}$. By the Chinese
remainder theorem, if we choose a sequence of integers $a_{j}\bmod
l_{j}^{k_{j}}$ for each $j$ all with the same order — namely, the value $2r$
in proposition 3.2 — then this gives us a unique class
$A\in\left(\mathbb{Z}/L\mathbb{Z}\right)^{\times}$, where in principle
$L=\prod_{j}l_{j}^{k_{j}}$. However, note that if one of the $l_{j}$ is $2$
then we must adjust so that the contribution to $L$ from the $j$-th prime $2$
is $4$, $a_{j}=3$ and $r=1$ everywhere. In this case $L$ will differ from $d$.
By Dirichlet’s theorem on primes in arithmetic progressions there are
infinitely many primes $p$ congruent to $A$ modulo $L$. Such a $p$ is then by
construction congruent to $a_{j}\bmod l_{j}$ for each $j$.

Writing $q=p^{r}$, we now construct for each $j$ a MUB over
$\mathbb{F}^{l_{j}^{k_{j}}}_{q^{2}}$ as in the proof of proposition 3.2. We
then use the argument in [KR, lemma 3] or [ACW]: take the tensor product of
these component solutions, which yields a set of MUBs in dimension $d$ over
$\mathbb{F}_{q^{2}}$. Notice that in this last step we require the condition
that the $r$ be constant, in order to be able to tensor them over the same
finite field. ∎

##  4\. Applications of theorem 2.1: reduction from $\mathbb{C}$ to finite
fields

###  4.1. Using known solutions in $\mathbb{C}$ to search for solutions over
${\mathbb{F}_{q^{2}}}$

One way to approach the search for four MUBs in dimension $6$ is to begin with
one of the many known triplets of MUBs in $\mathbb{C}^{6}$ whose vector
entries lie in a complex algebraic number field $L$ — see for example [ABD] —
and to study its behaviour under reduction at certain admissible primes $\wp$
of the ring of integers $\mathbb{Z}_{L}$. We then search for more vectors over
${\mathbb{F}_{q^{2}}}/{\mathbb{F}_{q}}$ (where $q^{2}={\rm N}\wp$) which are
MU to them all, and try to lift them back up to the extension $L_{\wp}$ of
$\mathbb{Q}_{p}$. We should point out that we were unable to produce any new
evidence this way along the lines of proposition 2.2.

In [Go], [Be2], [BWB] etc., many examples are detailed of sets of complex MUBs
in low dimensions: there is also an online classification in [Br]. In most
cases the entries are given as roots of unity or algebraic numbers of low
degree [ABD]; hence they may be viewed as lying in some number field $L$ with
a fixed embedding into $\mathbb{C}$.

Choosing a prime $\wp\lvert p$ of $L$ at which the vector entries are
$\mathbb{Z}_{L_{\wp}}$-units, the key requirement once again is that the
action of the $q$-th power Frobenius map upon the vector entries reduced
modulo $\wp$, should mimic that of complex conjugation in
$\mathbb{C}/\mathbb{R}$. For example, it must act via inversion upon any roots
of unity not in the base field. This is an unavoidable clash with the
situation over $\mathbb{C}$: for a prime power $q>3$ the $(q-1)$-th roots of
unity other than $\pm 1$ are not real; whereas they do lie in
${\mathbb{F}_{q}}$ and so the $q$-power map fixes them. Indeed, the more
complicated the Galois action upon the entries, the higher we have to go in
general, to find each prime at which Frobenius precisely replicates complex
conjugation.

Locally $L_{\wp}$ should be the unramified quadratic extension
$L_{\wp}/K_{\mathfrak{p}}$ of a base field $K_{\mathfrak{p}}$, writing
$\wp\lvert\mathfrak{p}\lvert p$, such that the action of the non-trivial
(Frobenius) element
$\sigma\in\hbox{\small{\rm{Gal}}}_{L_{\wp}/K_{\mathfrak{p}}}$ replicates that
of complex conjugation on the $L$-entries of the original matrix. Let
$q=N\mathfrak{p}$ denote the absolute (field-theoretic) norm of
$\mathfrak{p}$: so
${\mathbb{F}_{q}}\cong\mathbb{Z}_{K_{\mathfrak{p}}}/\mathfrak{p}$ is the
residue field of the ring of integers $\mathbb{Z}_{K_{\mathfrak{p}}}$ of
$K_{\mathfrak{p}}$, with $\mathbb{F}_{q^{2}}\cong\mathbb{Z}_{L_{\wp}}/\wp$ the
residue field of the ring of integers $\mathbb{Z}_{L_{\wp}}$ of $L_{\wp}$, and
order 2 Galois group
$\hbox{\small{\rm{Gal}}}_{\mathbb{F}_{q^{2}}/\mathbb{F}_{q}}\cong\hbox{\small{\rm{Gal}}}_{L_{\wp}/K_{\mathfrak{p}}}$.

###  4.2. Examples of MUBs in $\mathbb{C}^{6}$ reduced modulo $p$: The
matrices $H_{1},D(0)$ from [Go, §4]

We now illustrate this methodology with a couple of simple examples. These
matrices require the $24$-th roots of unity together with an algebraic number
$b_{2}=\frac{-1+2i}{\sqrt{5}}$, where $i^{2}=-1$. Note that $b_{2}$ is a
$p$-adic unit for all $p\neq 5$. So we need a prime $p$ which allows
$\mu_{24}\subseteq\mathbb{F}_{q^{2}}\setminus\mathbb{F}_{q}$ and which has a
square root of $5$ in $\mathbb{F}_{q}$ but not a square root of $-1$. That is,
$p\equiv-1\bmod 24$, $({\scriptstyle{\frac{5}{p}}})=1$: and $p\equiv 3\bmod 4$
automatically by the first condition. Of the first few primes $23,47,71$
satisfying $p\equiv-1\bmod 24$, only $p=71$ contains $5$ as a quadratic
residue. For $q=p=71$ in the notation of [Go], the bases were as follows.
Writing $\mathbb{F}_{q^{2}}^{\times}=<\gamma>\cong C_{71^{2}-1}$ and
$u=\gamma^{70}$ for a generator of the subgroup of elements of
$\mathbb{F}_{q^{2}}$ of norm $1$, and $\delta$ for a fixed choice of element
of $\mathbb{F}_{q^{2}}$ of norm $1/d$ as explained in A.2.4:

$H_{1}=\delta\left(\begin{array}[]{cccccc}1&1&1&1&1&1\\\
u^{54}&u^{54}&u^{6}&u^{30}&u^{30}&u^{6}\\\
u^{71}&u^{35}&u^{27}&u^{63}&u^{27}&u^{63}\\\
u^{54}&u^{54}&u^{30}&u^{6}&u^{6}&u^{30}\\\
u^{35}&u^{71}&u^{39}&u^{51}&u^{15}&u^{3}\\\
u^{35}&u^{71}&u^{15}&u^{3}&u^{39}&u^{51}\end{array}\right);\ \
D(0)=\delta\left(\begin{array}[]{cccccc}1&1&1&1&1&1\\\
1&u^{36}&u^{18}&u^{54}&u^{54}&u^{18}\\\
1&u^{18}&u^{36}&u^{18}&u^{54}&u^{54}\\\
1&u^{54}&u^{18}&u^{36}&u^{18}&u^{54}\\\
1&u^{54}&u^{54}&u^{18}&u^{36}&u^{18}\\\
1&u^{18}&u^{54}&u^{54}&u^{18}&u^{36}\end{array}\right).$

## Acknowledgements

Initially phrasing it as a problem on Gröbner bases, Al Kasprzyk and GM began
studying MUBs over finite fields in 2013. It quickly became apparent that the
only computationally viable approach was an exhaustive search through the
vectors themselves. This paper is the result of the faster technology now
available [M]. In addition to Al Kasprzyk for helping to get the whole project
running, we are grateful to Kevin Buzzard, Ian Grojnowski and Stefan Weigert
for helpful conversations during the early stages, and likewise to Markus
Grassl. A particular vote of thanks is due to Mike Harrison and Ingemar
Bengtsson for many key observations, and helpful comments on an earlier draft
of this paper.

GM thanks Myungshik Kim and Terry Rudolph of the QOLS group at Imperial
College for their continuing hospitality. AT would like to express his
gratitude to the _Crankstart Internships_ Programme at the University of
Oxford for partially funding his work during the project.

##  Appendix A The searches

###  A.1. Theorem 2.1 and proposition 2.2

####  A.1.1. Proof of theorem 2.1: limits on MUBs in finite fields

For a field $F$ and dimension $d$, suppose that $\mathcal{M}_{d}{F}=n$. As a
measure of how close we can get to constructing a set of $n+1$ MUBs in
$F^{d}$, we denote by $\nu_{d}F$ the maximal size, over all sets
$\\{\mathcal{B}_{0},\ldots,\mathcal{B}_{n-1}\\}$ of $n$ MUBs in $F^{d}$, of an
orthonormal set of vectors which are mutually unbiased to each
$\mathcal{B}_{k}$. Here are results of the exhaustive searches for the case
$d=6$.

$q$ | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 25 | 29 | 31 | 37 | 41 | 43 | 47 | 49 | 53
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$\mathcal{M}_{6}{\mathbb{F}_{q^{2}}}$ | 3 | 3 | 3 | 2 | 3 | 3 | 3 | 2 | 3 | 3 | 2 | 3 | 3 | 3 | 2 | 3
$\nu_{6}(\mathbb{F}_{q^{2}})$ | 4 | 0 | 0 | 2 | 4 | 0 | 0 | 4 | 4 | 0 | 2 | 4 | 0 | 0 | 2 | 4
Table 1. Exhaustive search results in 6 dimensions.

As stated in 1.1, $\mathcal{M}_{d}{\mathbb{C}}=d+1$ for prime powers $d$; and
it is a corollary of the upper bound proof in proposition 3.1 that
$\nu_{d}(\mathbb{C})=0$. Further, in each of the cases $d=2,3,4,5$ it is known
that there is precisely one set of $d+1$ MUBs up to equivalence [BWB]. These
facts were paralleled in the results of the finite field searches wherever
full MUB sets were found to exist.

$d$ |  Full $d+1$ MUBs | Partial Sets | Holds For
---|---|---|---
$2$ | $q\equiv 3\pmod{4}$ |  $\mathcal{M}_{2}{\mathbb{F}_{q^{2}}}=2$ for $q\equiv 1\bmod{4}$ |  all odd $q$
$3$ | $q\equiv 2\pmod{3}$ |  $\mathcal{M}_{3}{\mathbb{F}_{q^{2}}}=1$ for $q\equiv 1\bmod{3}$ | $(3,q)=1$
$4$ | $q\equiv 3\pmod{4}$ |  $\mathcal{M}_{4}{\mathbb{F}_{q^{2}}}=3$ for $q\equiv 1\bmod{4}$ |  $q<240$, $q$ odd
$5$ | $q\equiv 4\pmod{5}$ |  $\mathcal{M}_{5}{\mathbb{F}_{2^{2}}}=4$; $\mathcal{M}_{5}{\mathbb{F}_{q^{2}}}=1$ for $q\equiv 1\bmod 5$; | $q<122,(5,q)=1$
|  |  $\mathcal{M}_{5}{\mathbb{F}_{q^{2}}}=3$ for $q\equiv 2,3\bmod 5,q\neq 2$ |
Table 2. Exhaustive search results for $d\leq 5$.

Table 2 is valid for the ranges of $q$ shown. The assertions for $d=2,3$ are a
straightforward consequence of writing out the matrices with variables,
incorporating the protocol detailed in A.2.3 and using proposition 3.2. In all
cases where $\mathcal{M}_{d}{\mathbb{F}_{q^{2}}}>1$, we find that
$\nu_{d}\mathbb{F}_{q^{2}}=0$.

Finally, our limited results for $d=7$ are summarised in the following table.

$q$ | 2 | 3 | 4 | 5 | 8 | 9 | 11 | 13 | 17
---|---|---|---|---|---|---|---|---|---
$\mathcal{M}_{7}(\mathbb{F}_{q^{2}})$ | 4 | 8 | 1 | 2 | 4 | 3 | 3 | 8 | 2
$\nu_{7}(\mathbb{F}_{q^{2}})$ | 3 | 0 | 0 | 3 | 3 | 0 | 0 | 0 | 3
Table 3. Exhaustive search results in 7 dimensions.

####  A.1.2. Proof of proposition 2.2: sets of three hyperbolic MUBs with a
fourth set of four MU vectors

As may be seen in table 1, for each prime $p\equiv 5\bmod 12$, we found in
dimension $6$ a set of $3$ MUBs alongside a further set of $4$ orthonormal
vectors MU to each MUB. The first basis is always taken to be
$\mathcal{B}_{0}$.

Lifting the solutions $p$-adically it became evident that they are perfectly
general in characteristic zero: and may be regarded for example as existing in
the extension $K(\sqrt{3})/K$, where $K=\mathbb{Q}(i,\sqrt{6})$ and the
Hermitian action is via the map $\sigma\colon\sqrt{3}\mapsto-\sqrt{3}$. When
$q\equiv 5\bmod 24$, $\frac{1}{\sqrt{6}}$ is fixed under $\sigma$; whereas
when $q\equiv 17\bmod 24$ we need an extra factor of $i=\sqrt{-1}$ for each
vector as in the proof of proposition 3.2. So in particular the following
representation reduces well in characteristic $5\bmod 24$:

$\tiny\frac{1}{\sqrt{6}}\begin{pmatrix}1&1&1&1&1&1\\\
1&-2+\sqrt{3}&-1&2-\sqrt{3}&-2+\sqrt{3}&2-\sqrt{3}\\\
1&1&-2-\sqrt{3}&1&1&-2+\sqrt{3}\\\ 1&-2-\sqrt{3}&2+\sqrt{3}&-1&1&-1\\\
1&1&-2-\sqrt{3}&-2+\sqrt{3}&1&1\\\ 1&1&2+\sqrt{3}&-1&-2-\sqrt{3}&-1\\\
\end{pmatrix},$ $\tiny\frac{1}{\sqrt{6}}\begin{pmatrix}1&1&1&1&1&1\\\
1&1&-1&-2+\sqrt{3}&-1&2-\sqrt{3}\\\ 1&1&1&1&-2-\sqrt{3}&-2+\sqrt{3}\\\
1&-2-\sqrt{3}&2+\sqrt{3}&-2-\sqrt{3}&2+\sqrt{3}&-1\\\
1&1&-2-\sqrt{3}&1&1&-2+\sqrt{3}\\\
-2-\sqrt{3}&1&2+\sqrt{3}&-2-\sqrt{3}&2+\sqrt{3}&-1\\\ \end{pmatrix}.$

The four vectors, also arranged as columns in a matrix, are:

$\tiny\frac{1}{\sqrt{6}}\begin{pmatrix}1&1&1&1\\\
\omega^{2}(2-\sqrt{3})&-\omega^{2}(2-\sqrt{3})&-\omega(2-\sqrt{3})&\omega(2-\sqrt{3})\\\
\omega&\omega&\omega^{2}&\omega^{2}\\\
-\omega&\omega&\omega^{2}&-\omega^{2}\\\
\omega^{2}&\omega^{2}&\omega&\omega\\\ -1&1&1&-1\\\ \end{pmatrix},$

where $\omega$ is a primitive cube root of unity. Note that
$\sigma\colon\omega\mapsto\omega^{2}$.

###  A.2. The search algorithms

The exhaustive computer algorithm search routine is essentially self-
explanatory. We just give a few basic background facts. The starting point in
our sets of MUBs is always
$\mathcal{B}_{0}=\\{\mathbf{e}_{1},\ldots,\mathbf{e}_{d}\\}$, represented by
the identity matrix.

####  A.2.1. Formal equivalence of notions of MU in $\mathbb{C}$ and
$\mathbb{F}_{q^{2}}$

Fix an odd prime $p$. The definition in (1) is formally aligned with that in
the complex case: the square norm of the ‘absolute value’ of the Hermitian
inner product of each pair of vectors from distinct bases must equal $1/d$.
Hence our stipulation that $(d,p)=1$. It is also somewhat meaningless to have
$d$ reinterpreted modulo the characteristic $p$. For example, if $p=5$ and
$d=6$, then every unit vector is ‘MUB to itself’. So ideally, we insist
further that $d<p$.

Fix $q=p^{r}$ for some $r\geq 1$: we always work over the extension
${\mathbb{F}_{q^{2}}}/{\mathbb{F}_{q}}$ with Galois group generated by the
Frobenius element $\sigma:x\mapsto x^{q}$ of order $2$. Since the quadratic
extension of a finite field is unique up to isomorphism, we are assured of the
existence in ${\mathbb{F}_{q^{2}}}$, if not already in ${\mathbb{F}_{q}}$, of
a square root of $d$.

As an aside, however, we illustrate a major difference between our situation —
where we are in the end searching for a solution in a number field — and those
in $\mathbb{R}$ or $\mathbb{C}$ where the base field always contains
$\sqrt{d}$. Let $d>1$ be a non-square positive integer. Consider the
corresponding Hermitian structure which arises via the Galois group
$\hbox{\small{\rm{Gal}}}_{\mathbb{Q}(\sqrt{d})/\mathbb{Q}}$. Whereas in
$\mathbb{R}$ or $\mathbb{C}$ we would have
$N(1/\sqrt{d})=(1/\sqrt{d})^{2}=1/d$, here instead
$N(1/\sqrt{d})=(1/\sqrt{d})(-1/\sqrt{d})=-1/d$. This is the context in which
the _hyperbolic MUBs_ arise in proposition 2.2 and A.1.2.

####  A.2.2. Hadamard matrices

Now let $F/K$ be any quadratic extension of fields and let $V$ be a unitary
space. Given any matrix operator $M$ on $V$, let $M^{\dagger}$ denote the
Hermitian conjugate transpose of $M$: so in particular $\mathcal{B}$
represents an orthonormal basis if and only if its associated matrix
$\mathcal{B}$ is unitary; which in turn is true iff
$\mathcal{B}^{\dagger}\mathcal{B}=\mathcal{B}\mathcal{B}^{\dagger}=\mathcal{B}_{0}$.
In view of the fact that much of the literature on this subject is phrased in
these terms — see for example [Be2, Br, Go] — we note that a matrix
$\mathcal{B}$ which is MU to $\mathcal{B}_{0}$ is often called an  _$F$
-Hadamard matrix_ — or where the context is clear just a  _Hadamard matrix_ —
drawing upon the terminology in the complex case444Using the convention of
Bengtsson  _et al_ in [Be2], in that Hadamard matrices are  _unitary_ , as
opposed to  _unimodular_ as in say [Br]. These differ merely by a one-off
normalisation by $\tfrac{1}{\sqrt{d}}$..

A second basis ${\mathcal{C}}$ represented by another $F$-Hadamard matrix is
then in turn MU to ${\mathcal{B}}$ iff the ordinary matrix product
${{\mathcal{B}}}^{\dagger}{{\mathcal{C}}}$ is also itself an $F$-Hadamard
matrix. Indeed, the condition in definition 1 is identical to requiring that
every entry ${\langle\mathbf{b}_{i},\mathbf{c}_{j}\rangle}$ of the unitary
matrix ${\mathcal{B}}^{\dagger}{\mathcal{C}}$ of pair-wise Hermitian dot
products have that norm. Since for any unitary matrix $\mathcal{B}$ and any
matrices $\mathcal{C}$, $\mathcal{D}$ it is the case that
$(\mathcal{B}^{\dagger}\mathcal{C})^{\dagger}(\mathcal{B}^{\dagger}\mathcal{D})=\mathcal{C}^{\dagger}\mathcal{D}$,
including $\mathcal{B}_{0}$ in any set of MUBs as the first member is no
restriction. This then forces all of the entries ${{b}_{j}}\in F$ of all other
basis vectors $\mathbf{b}$ of another MUB ${\mathcal{B}}$, say, to satisfy
${\hbox{\small{\rm{N}}}_{\scriptscriptstyle{F}/\scriptscriptstyle{K}}}{\hbox{${b_{j}}$}}=1/d$.
However, although when ${\mathcal{B}}$ and ${\mathcal{C}}$ are MU to one
another, the basis associated to
${\mathcal{D}}={{\mathcal{B}}^{\dagger}{\mathcal{C}}}$ is by definition itself
MU with respect to $\mathcal{B}_{0}$, in most cases ${\mathcal{D}}$ will be MU
neither to ${\mathcal{B}}$ nor to ${\mathcal{C}}$. Parenthetically, the order
in which we take the Hermitian product is important: when
${{\mathcal{B}}^{\dagger}{\mathcal{C}}}$ satisfies our conditions, this
nevertheless reveals nothing in particular about the unitary
${{\mathcal{B}}{\mathcal{C}}}^{\dagger}$.

####  A.2.3. Transformations which preserve the MU properties: Rearrangements
of the columns and rows of the MUB matrices

[Be2, §III], [D]. Let
$\mathcal{S}=\\{\mathcal{B}_{1},\ldots,\mathcal{B}_{n}\\}$ be a set of MUBs in
$F^{d}$ which is assumed not to include $\mathcal{B}_{0}$, but to be MU to it.
Given any basis matrix $\mathcal{B}_{k}\in\mathcal{S}$ we are free to
rearrange the columns in any order without affecting the MU properties. This
amounts to right-multiplying $\mathcal{B}_{k}$ by a $d\times d$ permutation
matrix. Similarly and separately, we may rearrange the rows of every
$\mathcal{B}_{k}$ for $1\leq k\leq n$,  _provided that we do an identical
rearrangement on all $n$ matrices simultaneously_. This may be effected by
left-multiplying them all by _the same_ permutation matrix.

By analogy with the real case, two pairs of vectors
${\mathbf{u}},{\mathbf{v}}$ and ${\mathbf{u^{\prime}}},{\mathbf{v^{\prime}}}$
subtend the same ‘angle’ if and only if
${\hbox{\small{\rm{N}}}_{\scriptscriptstyle{F}/\scriptscriptstyle{K}}}{\hbox{${\langle\mathbf{u},\mathbf{v}\rangle}$}}={\hbox{\small{\rm{N}}}_{\scriptscriptstyle{F}/\scriptscriptstyle{K}}}{\hbox{${\langle\mathbf{u^{\prime}},\mathbf{v^{\prime}}\rangle}$}}$.
This angle is unchanged if we multiply ${\mathbf{u}}$ or ${\mathbf{v}}$ or
both by elements of $F$ of norm 1. Regarding each $\mathcal{B}_{k}$ separately
therefore we are free to right-multiply any or all of them by (possibly)
different diagonal matrices of norm 1 elements. So in particular, we may
assume without loss of generality that the first entry in every vector of
every one of the $\mathcal{B}_{k}$ is equal to the same value, which we choose
to be the quantity $\delta$ below, where
${\hbox{\small{\rm{N}}}_{\scriptscriptstyle{F}/\scriptscriptstyle{K}}}{\hbox{${\delta}$}}=1/d$.

Equally, we may multiply the rows of the bases by norm $1$ elements, again
provided that we use the same element on each corresponding row of each matrix
at the same time. This is achieved by left-multiplying every one of the
$\mathcal{B}_{k}$ by the same diagonal matrix with norm $1$ diagonal entries.
In this way, we are able for example to ensure that the first vector of the
first basis $\mathcal{B}_{1}$ should have all of its entries equal to
$\delta$. This was illustrated in the examples in 4.1 and A.1.2.

####  A.2.4. Transformations which preserve the MU properties: Reducing the
search space from $d$ dimensions to ${d-1}$

We make a few obvious simplifications. The norm homomorphism $N$ on
multiplicative groups of finite fields is surjective, yielding the following
short exact sequence of finite groups defining the kernel ${U}$:

$1{\ \xrightarrow{\ \ \ }\ }{U}{\ \xrightarrow{\ \ \ }\
}{\mathbb{F}^{\times}_{q^{2}}}\overset{N}{{\ \xrightarrow{\ \ \ }\
}}{\mathbb{F}^{\times}_{q}}{\ \xrightarrow{\ \ \ }\ }1.$

If $\gamma$ is any generator for the multiplicative subgroup
${\mathbb{F}^{\times}_{q^{2}}}$ of ${\mathbb{F}_{q^{2}}}$ then ${U}=\
<\gamma^{q-1}>$ is the unique subgroup of ${\mathbb{F}^{\times}_{q^{2}}}$ of
index $q-1$ (equivalently, of order $q+1$). Let
$\delta\in{\mathbb{F}^{\times}_{q^{2}}}$ lie above $d^{-1}$; then $\delta U$
is the coset containing all elements of norm $d^{-1}=N\delta=\delta^{q+1}$.

By setting the first basis to be $\mathcal{B}_{0}$, we may restrict our
attention to vectors whose entries lie in ${\delta U}$. Indeed we may view
$\left({\delta U}\right)^{d}=\delta{U}^{d}$, the cartesian product of $d$
copies of ${\delta U}$, as a sub _set_ of ${\mathbb{F}^{d}_{q^{2}}}$ and focus
our search within it. In accordance with the previous section, we are free to
choose the first vector of our first new basis $\mathcal{B}_{1}$ to consist of
all $\delta$’s; and moreover the first element of each of our
non-$\mathcal{B}_{0}$ vectors also may be chosen to be $\delta$.

Just as with the situation over the complex extension of the reals, for each
vector entry we require two variables over $\mathbb{F}_{q}$ to constitute just
one over $\mathbb{F}_{q^{2}}$, so that the Frobenius map may be invoked
separately all the way up the $p$-adic tower. So we need ten variables in
$\mathbb{F}_{q}$ for each vector in our search.

##  Appendix B The equations and reduction mod $p$

We give a very brief outline of some facts which arise when viewing this
problem through the lens of algebraic geometry. Throughout we assume we are in
the context of a unitary space over a field $F$, where $F/K$ is a quadratic
field extension as in the text.

In order for a basis $\mathcal{C}$ to be MU to an ON basis $\mathcal{B}$, it
must satisfy three separate uniform sets of highly symmetric equations: (I)
equality to $\frac{1}{d}$ of the norm of each vector entry, to prove that the
vectors are MU to the computational basis $\mathcal{B}_{0}$ — which also
forces them to be unit vectors; (II) orthogonality among themselves; and (III)
mutual unbiasedness to $\mathcal{B}$.

###  B.1. The individual defining polynomials reduced modulo a prime

Zauner’s conjecture predicts that the ideal $\mathcal{J}$ of the polynomial
ring $\mathbb{Z}[\underline{X}]$ generated by the $261$ multivariate quartic
and quadratic polynomials in $N=216$ real variables defining four MUBs in
$\mathbb{C}^{6}$, should be the whole ring $\mathbb{Z}[\underline{X}]$. This
is a Gröbner basis calculation way beyond the power of any known algorithm. In
particular, good reduction modulo a prime is unprovable, other than by
inspection for the primes $2$ and $3$. So although the following result puts
the reduction behaviour of the _individual_ MUB defining polynomials into
context, it nevertheless says nothing about the multiplicity of reduced
solutions in their intersection varieties. See B.2 for the actual polynomials
for $d=2$: the general case is entirely analogous.

######  Proposition B.1.

The defining polynomials for a set of four MUBs over the complex numbers in
dimension $6$ are each absolutely irreducible. Consequently each polynomial is
guaranteed to have good reduction outside a finite set of primes.

###### Proof.

Let $f(X)$ be one of the defining polynomials, where $X=\underline{X}$ denotes
a tuple of variables. Choosing some high enough prime $p>3$ we verified using
MAGMA the irreducibility of $f$ over $\mathbb{F}_{p}$. Examples of _simple
zeroes_ for $f$ — that is, solutions with entries in $\mathbb{F}_{p}$ where
the Jacobian does not vanish completely — are then straightforward to
construct by hand. Using [Ra, thm1] it follows therefore that $f(X)$ is
_absolutely irreducible_ : that is, it remains irreducible over the algebraic
closure $\overline{\mathbb{F}_{p}}$ of $\mathbb{F}_{p}$.

Finally by theorem 2 of the same paper, since the degree of $f$ is unchanged
under reduction modulo $p$, we may lift that absolute irreducibility to
$\mathbb{Q}$. That is to say, it is irreducible over $\overline{\mathbb{Q}}$.
The second assertion then follows from Emmy Noether’s irreducibility criteria
[Sch, §V]. ∎

The known theoretical upper bounds for these bad primes are huge [Ra]; however
in practice the set of bad primes for any known MUB solution is tiny.

###  B.2. The shape of the equations in dimension $d=2$

We give a representative example only for dimension $d=2$ for the sake of
brevity; the systems scale to higher dimensions in an entirely predictable and
uniform way. The notation for the vector entries only applies to this section.

Other than the standard assumption that the first basis of any set always be
$\mathcal{B}_{0}$, we leave them in their most general format so as to
illustrate the inherent symmetries. Representing the MUB vectors as usual by
columns of a $d\times d$ matrix, we may establish the system as follows, using
$i$ as a generic symbol for a generator of the quadratic field extension
$F/K$:

${\mathcal{B}_{0}}=\left(\begin{array}[]{cc}1&0\\\ 0&1\end{array}\right);\ \
{\mathcal{B}_{1}}=\left(\begin{array}[]{cc}e+i\epsilon&g+i\gamma\\\
f+i\phi&h+i\chi\end{array}\right),\ \
{\mathcal{B}_{2}}=\left(\begin{array}[]{cc}s+i\sigma&u+i\mu\\\
t+i\tau&v+i\nu\end{array}\right).$

We are forced to have two $K$-valued variables per $F$-valued vector entry,
per basis vector: since — with an eye to lifting eventually to characteristic
zero — we must explicitly build the Hermitian conjugate into the equations.
Prior to any adjustments along the lines of those in A.2.3, the resulting
equations are as follows.

The roman numerals refer to the introduction to this appendix. (I) First, we
must ensure that all entries in all vectors have norm $\frac{1}{d}$, to ensure
MUB-ness with the computational basis $\mathcal{B}_{0}$. This also forces them
to be unit vectors. In principle this should give $d$ inhomogeneous quadratic
equations per vector, hence $d^{2}$ per basis beyond $\mathcal{B}_{0}$, viz.:

$\displaystyle\small{e^{2}+\epsilon^{2}-1/2\hbox{\ , \ \
}f^{2}+\phi^{2}-1/2\hbox{\ , \ \ }g^{2}+\gamma^{2}-1/2\hbox{\ , \ \
}h^{2}+\chi^{2}-1/2\hbox{\ , \ \ }}$
$\displaystyle\small{s^{2}+\sigma^{2}-1/2\hbox{\ , \ \
}t^{2}+\tau^{2}-1/2\hbox{\ , \ \ }u^{2}+\mu^{2}-1/2\hbox{\ , \ \
}v^{2}+\nu^{2}-1/2}\ .$

Note that this places the candidates for MUB vector entries upon a
$2d^{2}$-torus.

(II) Secondly, each basis must be orthonormal. By (I) they are unit vectors,
so it remains to check orthogonality. There are $\binom{d}{2}$ comparisons to
be made, each with real and imaginary parts, yielding a total of $d(d-1)$
extra homogeneous quadratic equations per basis; in our case:

$\displaystyle\small{eg+\epsilon\gamma+fh+\phi\chi\hbox{\ , \ \
}-e\gamma+\epsilon g-f\chi+\phi h\hbox{\ , \ \ }}$
$\displaystyle\small{su+\sigma\mu+tv+\tau\nu\hbox{\ , \ \ }-s\mu+\sigma
u-t\nu+\tau v}.$

(III) Finally the actual MUB-ness comparisons require a recursive structure of
new equations which — in taking norms of sums of Hermitian products — gives
$d^{2}$ more inhomogeneous quartic equations per (unordered) _pair of bases_ :

$\displaystyle\small{e^{2}s^{2}+e^{2}\sigma^{2}+2efst+2ef\sigma\tau+2e\phi
s\tau-2e\phi\sigma t+\epsilon^{2}s^{2}+\epsilon^{2}\sigma^{2}-2\epsilon
fs\tau}$ $\displaystyle\small{+2\epsilon f\sigma t+2\epsilon\phi
st+2\epsilon\phi\sigma\tau+f^{2}t^{2}+f^{2}\tau^{2}+\phi^{2}t^{2}+\phi^{2}\tau^{2}-2\
,}$ $\displaystyle\small{e^{2}u^{2}+e^{2}\mu^{2}+2efuv+2ef\mu\nu+2e\phi
u\nu-2e\phi\mu v+\epsilon^{2}u^{2}+\epsilon^{2}\mu^{2}-2\epsilon
fu\nu+2\epsilon f\mu v}$ $\displaystyle\small{+2\epsilon\phi
uv+2\epsilon\phi\mu\nu+f^{2}v^{2}+f^{2}\nu^{2}+\phi^{2}v^{2}+\phi^{2}\nu^{2}-2\
,}$ $\displaystyle\small{g^{2}s^{2}+g^{2}\sigma^{2}+2ghst+2gh\sigma\tau+2g\chi
s\tau-2g\chi\sigma t+\gamma^{2}s^{2}+\gamma^{2}\sigma^{2}-2\gamma hs\tau}$
$\displaystyle\small{+2\gamma h\sigma t+2\gamma\chi
st+2\gamma\chi\sigma\tau+h^{2}t^{2}+h^{2}\tau^{2}+\chi^{2}t^{2}+\chi^{2}\tau^{2}-2\
,}$ $\displaystyle\small{g^{2}u^{2}+g^{2}\mu^{2}+2ghuv+2gh\mu\nu+2g\chi
u\nu-2g\chi\mu v+\gamma^{2}u^{2}+\gamma^{2}\mu^{2}-2\gamma hu\nu+2\gamma h\mu
v}$ $\displaystyle\small{+2\gamma\chi
uv+2\gamma\chi\mu\nu+h^{2}v^{2}+h^{2}\nu^{2}+\chi^{2}v^{2}+\chi^{2}\nu^{2}-2\
.}$

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