# Mutually unbiased frames

Fabian Caro Pérez  Departamento de Física, Facultad de Ciencias Básicas,
Universidad de Antofagasta, Casilla 170, Antofagasta, Chile
<EMAIL_ADDRESS>Victor González Avella  Departamento de Física,
Facultad de Ciencias Básicas, Universidad de Antofagasta, Casilla 170,
Antofagasta, Chile<EMAIL_ADDRESS>Dardo Goyeneche  Departamento
de Física, Facultad de Ciencias Básicas, Universidad de Antofagasta, Casilla
170, Antofagasta, Chile<EMAIL_ADDRESS>https://www.quiga.cl/

(May 11, 2022)

###### Abstract

In this work, the concept of mutually unbiased frames is introduced as the
most general notion of unbiasedness for sets composed by linearly independent
and normalized vectors. It encompasses the already existing notions of
unbiasedness for orthonormal bases, regular simplices, equiangular tight
frames, positive operator valued measure, and also includes symmetric
informationally complete quantum measurements. After introducing the tool, its
power is shown by finding the following results about the last mentioned class
of constellations: (i) real fiducial states do not exist in any even
dimension, and (ii) unknown $d$-dimensional fiducial states are parameterized,
a priori, with roughly $3d/2$ real variables only, without loss of generality.
Furthermore, multi-parametric families of pure quantum states having minimum
uncertainty with regard to several choices of $d+1$ orthonormal bases are
shown, in every dimension $d$. These last families contain all existing
fiducial states in every finite dimension, and the bases include maximal sets
of $d+1$ mutually unbiased bases, when $d$ is a prime number.

##  1 Introduction

Finite tight frames [1, 2] are a natural generalization of orthonormal bases
[3]. These frames play a central role in quantum physics, as they are one-to-
one related to rank-one Positive Operator Valued Measures (POVM), a
generalized notion of measurements in quantum mechanics. Tight frames find
applications in signal theory [4, 5], coding and communication [6], design of
polarimeters that maximize the signal-to-noise ratio, and reduction of error
propagation [7, 8]. Optimal ways to implement quantum mechanical protocols
typically require to consider sets of quantum measurements induced by vectors
with special geometrical properties, e.g. mutually unbiased bases (MUB) [9,
10] and symmetric informationally complete (SIC)-POVM [11]. These
constellations are relevant to quantum state estimation [10, 12], entanglement
detection [13], quantum key distribution [14], and quantum-nonlocality [15,
16], among others. On the other hand, sets of frames are relevant in quantum
state tomography [17], quantum cloning [18] and quantum state discrimination
[19].

Along the last years, the notion of mutually unbiasedness has be extended to
mutually unbiased simplices [20, 21], mutually unbiased equiangular tight
frames [22], and mutually unbiased POVM [23]. Motivated by trying to
understand the tree from the exploration of the forest, the most general
notion of mutually unbiasedness for constellations composed by linearly
independent vectors is introduced, namely _mutually unbiased frames_ (MUF).
This notion allows to study some geometrical problems from a general
perspective, revealing interesting features that are not simple to see
otherwise. For instance, a series of new results about SIC-POVM, holding in
every finite dimension where they exist, can be found in a simple way when
considering MUF. This tool might contribute to unlock hidden symmetries in
further classes of constellations, as well as to allow the design of potential
practical applications, see Section 5).

This work is organized as follows: in Section 2, the basic notions required to
understand the work are introduced. In Section 3, our main tool is presented
and some basic results are derived. In Section 4, a matrix approach to MUF is
shown, specially focusing on circulant matrices. Through this study, both the
mutually unbiased bases problem in prime dimension $d$ and the SIC-POVM
problem in every dimension $d$ can be seen as $d$ MUF, encoded in $d$
circulant matrices. As a further result, the most general pair of MUF for a
qubit system is displayed, emphasizing the fact that both MUB and SIC-POVM
designs arise as particular solutions. In Section 5, the MUF formalism is
applied to the SIC-POVM problem, thus deriving the most important results of
this work. Here, the Zauner’s conjecture about the existence of Weyl-
Heisenberg covariant SIC-POVM is shown to be equivalent to the simplest way to
study $d$ MUF in dimension $d$, assuming that circulant matrices define the
simplest class of commuting matrices to deal with. Also, it is proven that
real fiducial states, with respect to the standard Weyl-Heisenberg (WH) group,
do not exist in any even dimension. Moreover, WH covariant fiducial states
belong to an explicitly given $\lfloor(d-1)/2\rfloor+d-1$ dimensional set, in
every dimension $d$ where they exist. Finally, fiducial states are shown to
define minimum uncertainty states regarding a large number of subsets of $d+1$
bases, induced by Clifford unitaries.

##  2 Preliminaries

In this section, the required notions to understand the work are introduced.
Firstly, pure quantum states, denoted in kets as $|\psi\rangle$, are
normalized vectors in $\mathbb{C}^{d}$ with regard to the norm induced by the
standard scalar product. That is,
$\|\psi\|^{2}=\langle\psi|\psi\rangle=\sum_{j=1}^{d}|\langle
j|\psi\rangle|^{2}=1$, where $|j\rangle$ denotes the $j$th element of the
canonical basis, also called computational basis in quantum information
theory. Here, $|\psi\rangle$ is an element in $\mathcal{H}_{d}$,
$\langle\phi|$ an element in the dual space $\mathcal{H}_{d}^{*}$,
$\langle\phi|\psi\rangle$ and $|\phi\rangle\langle\psi|$ the inner and outer
products, respectively. Besides, the symbol $|\phi\rangle\langle\phi|$ denotes
the rank-one projector associated to the direction $|\phi\rangle$. The first
relevant ingredient is the concept of frames [24], defined as follows:

######  Definition 1.

A set of $n\geq d$ pure states $\\{|\phi_{i}\rangle\\}_{i=1,\dots,n}$ in
$\mathcal{H}_{d}$ defines a frame if constants $0<A\leq B<\infty$ exist such
that

$A\leq\sum_{i=0}^{n-1}|\langle x|\phi_{i}\rangle|^{2}\leq B,\qquad\mbox{for
all }|x\rangle\in\mathcal{H}_{d}.$ (1)

Here, $A$ and $B$ are called the lower and upper bounds of the frame, given by
the minimal and maximal eigenvalues of the _frame operator_
$S=\sum_{i=0}^{n-1}|\phi_{i}\rangle\langle\phi_{i}|$, respectively [8]. A
frame is called _tight_ if $A$ equals $B$, having in such case that $A=B=n/d$.
When $A=B=1$, a Parseval tight Frame arises, equivalent to orthonormal bases
in finite dimensions. There is a simple way to detect whether a given set of
vectors forms a frame: linearly independence of $d$ out of $n\geq d$ vectors
is a necessary and sufficient condition to have a frame in $\mathcal{H}_{d}$.
A gentle introduction to finite frames theory can be found in Ref. [25].

A special kind of tight frames occurs when its $n$ vectors are equiangular,
known as _equiangular tight frames_ (ETF) [26], meaning that there is a
constant $c>0$ such that $|\langle\phi_{i}|\phi_{j}\rangle|^{2}=c$, for every
$i\not=j$, with $i,j\in\\{0,\dots,n-1\\}$. Here, the number $\sqrt{c}$ is
typically called _coherence_ , but for convenient reasons, throughout this
work the constant $c$ is called the _overlap_ of a MUF. See [27] for a summary
of the current state of the art of the ETF problem.

Among all constellations in the Hilbert space $\mathcal{H}_{d}$, there is a
specially distinguished one: mutually unbiased bases [28].

######  Definition 2.

Two orthonormal bases $\\{|\phi_{j}\rangle\\}_{j=0,\dots,d-1}$ and
$\\{|\psi_{k}\rangle\\}_{k=0,\dots,d-1}$ defined in $\mathcal{H}_{d}$ are
unbiased if $|\langle\phi_{j}|\psi_{k}\rangle|^{2}=\frac{1}{d}$, for every
$j,k=0,\dots,d-1$. Also, a set of $m$ orthonormal bases are MUB if they are
pairwise MUB.

For instance, the eigenvectors bases of the three Pauli matrices define $m=3$
MUB in $\mathcal{H}_{2}$. More generally, let

$X=\sum_{k=0}^{d-1}|k+1\rangle\langle k|\mbox{ and
}Z=\sum_{k=0}^{d-1}\omega^{k}|k\rangle\langle k|,$ (2)

the _shift_ and _phase_ operators, respectively, where $\omega=e^{2\pi i/d}$.
Therefore, the eigenvectors bases of the operators

$X,Z,XZ,XZ^{2},\dots,XZ^{d-1},$

form $m=d+1$ MUB in $\mathcal{H}_{d}$, whenever $d$ is a prime number. Here,
and in the rest of the work, it is assumed addition modulo $d$ in kets, i.e.
$|(d-1)+1\rangle=|0\rangle$.

MUB have been extensively studied for the last 40 years. A maximal set of
$d+1$ MUB exists in every prime [9] and prime power [10] dimension $d$. On the
other hand, the maximal number of MUB existing in any other composite
dimension $d$ is still an open problem, even in the lowest dimensional case
$d=6$, where at most triplets of MUB are known [29, 30, 31, 32].

Another remarkable constellations are the SIC-POVM [33, 34, 35], defined as
follows:

######  Definition 3.

A set of $n=d^{2}$ pure states $\\{|\phi_{j}\rangle\\}\subset\mathcal{H}_{d}$
forms a SIC-POVM if $|\langle\phi_{j}|\phi_{k}\rangle|^{2}=\frac{1}{d+1}$, for
every $j\not=k$.

SIC-POVM are known to exist in dimensions $d=2-53$ and in several higher
dimensions [36], being $d=19603$ achieved recently [37]. Also, numerical
solutions are known in dimensions $d=2-193$ and some other dimensions up to
2208 [38, 39]. In very low dimensions, fiducial states are simple to find.
However, as the dimension increases, even the problem to find numerical
solutions becomes hard. On the one hand, the difficulty to solve the MUB and
SIC-POVM problems have motivated researchers to find solution to its
extensions to the space of full-rank density matrices, see [40] and [41],
respectively. On the other hand, the rank-one SIC-POVM problem has triggered
the search of additional symmetries in fiducial pure states. In short, every
known Weyl-Heisenberg fiducial state is eigenvector of an order 3 Clifford
operator [42]. A detailed description of symmetries in fiducial states can be
found here [33, 35, 38, 42, 43]. Among the entire set of SIC-POVM, there is a
distinguished one: the _Hoggar lines_ [44]. These solutions for 3-qubit
systems are characterized by fiducial states that are covariant with respect
to the tensor product of single qubit WH groups. The 240 fiducials existing in
this case can be divided in two classes, distinguished by the fact that they
have different amounts of three-partite entanglement [45].

The symmetries mentioned above allowed the construction of every known
fiducial state. However, there is _no single proof_ for the veracity of such
symmetries, beyond its remarkable success. In Section 5, it is provided an
important cornerstone towards this direction by revealing an elusive symmetry
that holds for any existing fiducial state. Precisely, the number of free
parameters of any fiducial state can be reduced from $2(d-1)$ to
$\lfloor(d-1)/2\rfloor+d-1$, in any dimension $d\geq 2$.

To conclude this section, it is remarked the connection existing between the
SIC-POVM problem and algebraic number theory, including a close relation to
the 12$th$ Hilbert problem [46].

##  3 Mutually unbiased frames

The notion of mutually unbiasedness for constellations of vectors has been
extensively used in quantum mechanics. One of the reasons is because two Von
Neumann observables are complementary if and only if its eigenvectors bases
are mutually unbiased, e.g. position and momentum or orthogonal directions of
spin $1/2$ observables. Mutually unbiasedness has been extended to regular
simplices [20, 21], equiangular tight frames [22] and POVM in general [23]. In
this section, a notion of unbiasedness that includes all the above notions is
introduced. The aim to present this generalization is to study the existence
and construction of some inequivalent geometrical structures under the same
framework, e.g. MUB and SIC-POVM. Despite our generalization goes beyond the
set of quantum measurements, the new point of view reveals interesting
properties when restricting the attention to POVM, as shown in Section 5. The
central concept of our work is defined as follows:

######  Definition 4.

Let $\\{|\phi_{j}\rangle\\}_{j=0,\dots,n_{1}-1}$ and
$\\{|\psi_{k}\rangle\\}_{k=0,\dots,n_{2}-1}$ be two frames in
$\mathcal{H}_{d}$. They are called unbiased if there is a constant $c>0$ such
that

$|\langle\phi_{j}|\psi_{k}\rangle|^{2}=c,$ (3)

holds for every $j=0,\dots,n_{1}-1$ and $k=0,\dots,n_{2}-1$. A set of $m$
frames is called mutually unbiased (MUF) if they are pairwise unbiased. The
constant value $c$ is called the overlap of the MUF.

Figure 1: (color online) Construction of two mutually unbiased frames composed
by two vectors each for a qubit system in the Bloch sphere. Given a frame in
$\mathbb{C}^{2}$, associated to two normalized vectors on the surface of the
Bloch sphere, red and gray vectors in the figure, one can always define
another frame that is mutually unbiased to the above one as follows: construct
two cones with axes given by the red and grey vectors, respectively. The
apertures of the cones have to be identical and large enough to have
intersection of the cones. The unbiased frame is given by vectors lying at the
intersection of the cones, i.e. the blue vectors. When the red and gray
vectors are anti-parallel then there is a unique way to define intersecting
cones, i.e. the aperture of each cone is equal to $\pi$, and both cones become
the same disk. In such case, any point in the border of the disk is unbiased
to the given basis. In this way, 3 MUB arise as 3 orthogonal lines.

Fig.1 shows an intuitive construction of two MUF for a qubit system. The
notion of MUF reduces to mutually unbiased POVM [23] when both frames are
tight, for any $n_{1}$ and $n_{2}$. Additionally, if $n_{1}=n_{2}=d$, then
they are MUB. Figure 2 illustrates the relation existing between different
notions of unbiasedness.

At first glance, a set of $m$ MUF might look as a collection of equiangular
lines. However, it is important to remark that the MUF structure has
additional constraints, as the following example shows. Suppose that $4$ MUF
exist in dimension $3$, each of them having $3$ elements, and arranged along 4
rows as follows:

$\begin{array}[]{cccc}a_{1}&a_{2}&a_{3}&\quad\mbox{$\leftarrow$ Frame 1}\\\
b_{1}&b_{2}&b_{3}&\quad\mbox{$\leftarrow$ Frame 2}\\\
c_{1}&c_{2}&c_{3}&\quad\mbox{$\leftarrow$ Frame 3}\\\
d_{1}&d_{2}&d_{3}&\quad\mbox{$\leftarrow$ Frame 4}\end{array}$

Clearly, $\\{a_{i},b_{i},c_{i},d_{i}\\}$, is a set of 4 equiangular lines for
any $i=1,2,3$, which might tempt one to claim that the above set of three
equiangular lines is equivalent to a set of four MUF. However, the MUF
constraint establishes that two elements of the above arrangement belonging to
different rows and different columns (e.g. $a_{1}$ and $b_{2}$), also have the
same overlap $c$. Indeed, the MUF constraint is equivalent to say that
$\\{a_{i},b_{j},c_{k},d_{l}\\}$ forms a set of equiangular lines, for any
$i,j,k,l=1,\dots,3$. This is an interesting combinatorial extension of the
equiangular lines problem.

Given that all sub-normalized rank-one projectors associated to a tight frame
sum up to the identity, it is simple to show that $c=1/d$ holds in (3) when
both frames are tight [23]. However, if the frames are not tight, the constant
$c$ might take another value, as shown below.

######  Proposition 1.

Let $\\{|\phi^{(\ell)}_{j}\rangle\\}$ be a set of $m$ mutually unbiased frames
in $\mathcal{H}_{d}$ composed by $n$ elements each, with overlap $c$. Also,
assume that each frame has lower and upper frame bounds $A_{\ell}$ and
$B_{\ell}$, respectively, where $\ell=0,\dots,m-1$. Therefore,
$\max_{\ell}A_{\ell}\leq cn\leq\min_{\ell}B_{\ell}$.

###### Proof.

From considering $|x\rangle=|\phi_{i}^{\ell}\rangle$, inequalities (1) imply

$A_{1}\leq\sum_{j=0}^{n-1}|\langle\phi_{i}^{(\ell)}|\phi_{j}^{(\ell^{\prime})}\rangle|^{2}\leq
B_{1},\,A_{2}\leq\sum_{i=0}^{n-1}|\langle\phi_{i}^{(\ell)}|\phi_{j}^{(\ell^{\prime})}\rangle|^{2}\leq
B_{2},\dots,\,A_{m}\leq\sum_{j=0}^{d-1}|\langle\phi_{i}^{(\ell)}|\phi_{j}^{(\ell^{\prime})}\rangle|^{2}\leq
B_{m},$ (4)

for each $\ell=0,\dots,n-1$, which reduce to $A_{1}\leq cn\leq B_{1},A_{2}\leq
cn\leq B_{2},\dotsc,A_{m}\leq cn\leq B_{m}$. Therefore, the most restrictive
constraint for $c$ is given by $\max_{\ell}A_{\ell}\leq
cn\leq\min_{\ell}B_{\ell}$. ∎

Figure 2: Mutually unbiased frames (MUF), the main concept introduced in this
work, represents the most general notion of unbiasedness for sets of linearly
independent vectors. It generalizes the existing notions of unbiasedness for
orthonormal bases (MUB) [9, 10], simplices (MUS) [20, 21], equiangular tight
frames (MU-ETF) [22] and POVM (MU-POVM) [23]. Additionally, SIC-POVM [33, 34]
can be seen as sets of $d$ MUF in dimension $d$, as shown in Section 5.

For the case of orthonormal bases, Proposition 1 reduces to mutually unbiased
bases, where $c=1/d$. Indeed, the same overlap $c$ occurs when at least one of
the frames is tight, thus forming a POVM.

######  Corollary 1.

Let $\\{|\phi^{(\ell)}_{j}\rangle\\}$ be a set of $m$ mutually unbiased frames
in $\mathcal{H}_{d}$ with overlap $c$, composed by $n$ elements each, such
that at least one of them is tight. Therefore, $c=1/d$.

###### Proof.

The proof arises from Proposition 1, where
$\frac{n}{d}=A_{\ell^{\prime}}=B_{\ell^{\prime}}=cn$ holds. Thus, $c=1/d$. ∎

From Corollary 1, note that Parseval MUF are mutually unbiased bases in every
finite dimension $d$. In Section 4, examples of MUF in $\mathcal{H}_{d}$ are
shown, where the overlaps $c$ are not equal to $1/d$, in general.

##  4 Matrix approach to mutually unbiased frames

In this section, a matrix approach to the problem of mutually unbiased frames
is shown. In particular, the case of $m$ MUF composed by $d$ vectors each in
$\mathcal{H}_{d}$ is considered. Through this approach, entries of the $j$-th
element of each frame are arranged as columns belonging to a square complex
matrix of order $d$, for every $j=0,\dots,d-1$. Thus, there are $m$ square
matrices of order $d$, denoted $M_{0},\dots,M_{m-1}$, satisfying the following
relations:

$M_{j}^{{\dagger}}M_{k}=K^{(jk)},$ (5)

where $|(K^{(jk)})_{st}|^{2}=c$, for every $s,t=0,\dots,d-1$, and $j\not=k$.
Here, $(K^{(jk)})_{st}$ denotes the entry of matrix $K^{(jk)}$ located in row
$s$ and column $t$. Also, if $j=k$ the relation $(K^{(jj)})_{ss}=1$ occurs,
since the column vectors of $M_{j}$ are normalized, for every $j=0,\dots,m-1$,
and $s=0,\dots,d-1$. As a general property, it is simple to show that $M_{j}$
and $K^{(jk)}$ are full rank matrices, as their columns have to be linearly
independent in order to form a frame. However, the problem to find sets of $m$
matrices $\\{M_{j}\\}$ satisfying (5) is evidently hard, if no further
assumptions are taken. In order to simplify this problem, the following
symmetries are assumed: matrices $M_{j}$ are normal, i.e.
$[M^{{\dagger}}_{j},M_{j}]=0$, and they do commute, i.e. $[M_{j},M_{k}]=0$ and
$[M_{j},M_{k}^{{\dagger}}]=0$. These assumptions imply that
$[M_{\ell},K^{(jk)}]=0$ is satisfied, for every $j,k,\ell=0,\dots,d-1$. Thus,
the problem simplifies to finding special relations existing between the
spectrum of matrices $M_{j}$ and $K^{(jk)}$.

At this stage, the choice of the unitary transformation $U$ that diagonalizes
all matrices $M_{j}$ plays a fundamental role. For instance, if matrices
$M_{j}$ are diagonal in the computational basis, then there is no pair of MUF.
Thus, an interesting question arises: _which choices of $U$ provide a solution
to (5)_ for a given value of $m\geq 2$? This question is hard to solve, and it
is not studied through the current research. In turns, the simplest case of
matrices $M_{j}$ diagonalized by the discrete Fourier transform is considered.
That is, the matrices $M_{0},\dots,M_{m-1}$ are circulant. This choice is
justified by simplicity, as several useful properties are well-known for this
class of matrices [47]. A square matrix $M$ is called circulant if the
$j+1$-th row is a cyclic permutation of the $j$-th row, for every
$j=0,\dots-d-2$. That is,

$M=\left(\begin{array}[]{ccccc}m_{0}&m_{1}&m_{2}&\dots&m_{d-1}\\\
m_{d-1}&m_{0}&m_{1}&\dots&m_{d-2}\\\ \vdots&&&&\vdots\\\
m_{1}&m_{2}&m_{3}&\dots&m_{0}\end{array}\right),$ (6)

where $m_{0},\dots,m_{d-1}\in\mathbb{C}$. Let $\vec{\mu}(M)$ be the vector
formed by all elements of the first row of $M$, i.e.
$\vec{\mu}=(m_{0},\dots,m_{d-1})^{T}$, where $T$ denotes transposition. Also,
let $\vec{\lambda}(M)$ be a column vector containing all eigenvalues of $M$,
sorted according to the decomposition

$M=F\mathrm{diag}[\vec{\lambda}(M)]F^{{\dagger}},$ (7)

where $\mathrm{diag}[\vec{\lambda}(M)]$ is a diagonal matrix containing the
vector $\vec{\lambda}(M)$ along its main diagonal, and $F$ is the discrete
Fourier transform, having entries $F_{jk}=\frac{1}{\sqrt{d}}\omega^{jk}$,
where $\omega=e^{2\pi i/d}$. Along the work, $\vec{\lambda}(M)$ is called the
vector defined by the eigenvalues of $M$. From (7), it is simple to show that
$\vec{\lambda}(M)=\sqrt{d}F^{{\dagger}}\vec{\mu}(M)$ is satisfied. On the
other hand, the unitary operator $F^{2}$ transforms the first row of $M$, i.e.
$\vec{\mu}(M)$, to the first column of $M$, that has unit norm. Therefore,
normalized columns in $M$ translates to the condition $\|\vec{\mu}(M)\|=1$, or
equivalently $\|\vec{\lambda}(M)\|=\sqrt{d}$.

Similarly, any matrix $K$ from (5) is circulant, having the form

$K=\sqrt{c}\left(\begin{array}[]{ccccc}e^{i\alpha_{0}}&e^{i\alpha_{1}}&e^{i\alpha_{2}}&\dots&e^{i\alpha_{d-1}}\\\
e^{i\alpha_{d-1}}&e^{i\alpha_{0}}&e^{i\alpha_{1}}&\dots&e^{i\alpha_{d-2}}\\\
\vdots&&&&\vdots\\\
e^{i\alpha_{1}}&e^{i\alpha_{2}}&e^{i\alpha_{3}}&\dots&e^{i\alpha_{0}}\end{array}\right),$
(8)

where $\alpha_{0},\dots,\alpha_{d-1}\in[0,2\pi)$ and $0<c<1$. The first row of
$K$ is arranged as the vector
$\vec{\mu}(K)=\sqrt{c}(e^{i\alpha_{0}},\dots,e^{i\alpha_{d-1}})^{T}$, and the
vector $\vec{\lambda}(K)$ containing all eigenvalues of $K$ is defined
according to

$K=F\mathrm{diag}[\vec{\lambda}(K)]F^{{\dagger}}.$ (9)

As a basic property of circulant matrices, the relation
$\vec{\lambda}(K)=\sqrt{d}F^{{\dagger}}\vec{\mu}(K)$ occurs, where
$\|\vec{\mu}(K)\|=\sqrt{cd}$ and $\|\vec{\lambda}(K)\|=d\sqrt{c}$. Now, the
first result of this section is established.

######  Proposition 2.

Let $M_{0},\dots,M_{m-1}$ be $m$ circulant matrices that form $m$ MUF composed
by $d$ vectors each, in dimension $d$. Then, the normalized vectors defined by
the eigenvalues of these matrices form $m$ equiangular lines in
$\mathbb{C}^{d}$, with overlap $c$.

###### Proof.

The result arises from direct calculation:

$\displaystyle|\vec{\lambda}(M_{j})\cdot\vec{\lambda}(M_{k})|^{2}$
$\displaystyle=$
$\displaystyle|\mathrm{Tr}\bigl{(}\mathrm{diag}[\vec{\lambda}(M_{j})]^{{\dagger}}\mathrm{diag}[\vec{\lambda}(M_{k})]\bigr{)}|^{2}$
(10) $\displaystyle=$
$\displaystyle|\mathrm{Tr}(F^{{\dagger}}\vec{\lambda}(M_{j})]^{{\dagger}}FF^{{\dagger}}\vec{\lambda}(M_{k})]F)|^{2}$
$\displaystyle=$ $\displaystyle|\mathrm{Tr}(M_{j}^{{\dagger}}M_{k})|^{2}$
$\displaystyle=$ $\displaystyle|\mathrm{Tr}(K)|^{2}$ $\displaystyle=$
$\displaystyle cd,$

for every $j\not=k\in\\{0,\dots,m-1\\}$. Moreover, when eigenvectors
$\vec{\lambda}(M_{0}),\dots,\vec{\lambda}(M_{m-1})$ are normalized, they
define $m$ equiangular lines with overlap $c$, the same overlap of the MUF. ∎

An expert reader might recognize here the evidence of a well-known property
holding for fiducial states of SIC-POVM: the Fourier transform of a fiducial
state is a fiducial state. From combining Prop. 2 with results from
equiangular lines theory [48], the following constraints for the existence of
MUF is obtained:

######  Proposition 3.

Suppose that $M_{1},\dots,M_{m}$ are circulant matrices of order $d$, forming
$m$ MUF. Then, the inequality $m\leq d^{2}$ holds.

###### Proof.

The proof arises from the fact that there are no more than $d^{2}$ equiangular
lines in dimension $d$. ∎

Moreover, stronger bounds can be found when the eigenvalues of $M_{j}$ are
real.

######  Proposition 4.

Suppose a set MUF composed of $m$ hermitian circulant matrices, with overlap
$c$, exists. Therefore, the following properties hold:

  1. 1.

$m\leq d(d+1)/2$ (see [49]).

  2. 2.

If $m\geq 2d$, then $1/\sqrt{c}$ is an integer number (see [49]).

  3. 3.

If $c\leq 1/(d+2)$, then $m\leq d(1-c)/(1-dc)$ (see Lemma 6.1 here [50]).

  4. 4.

If $c\leq 1/d^{2}$, then $m\leq d+1$.

###### Proof.

Proofs are based on the fact that eigenvectors of all matrices forming the MUF
are real (due to hermiticity) and equiangular (due to Prop 2). Thus, the
constraints established in _1._ , _2._ and _3._ occur for real ETF, which can
be found in the described references. Item _4._ arises from assuming that
$c\leq 1/d^{2}$, and then using _3._ ∎

Further properties of eigenvalues of matrices $M_{j}$ can be derived. By using
(5), it occurs that

$\vec{\lambda}^{*}(M_{j})\circ\vec{\lambda}(M_{k})=\vec{\lambda}(K),$ (11)

where $\vec{\lambda}(M_{j})$ is the vector associated to $M_{j}$, the circle
($\circ$) denotes the entrywise (Hadamard) product and the asterisk (*)
complex conjugation in the computational basis. Note that the inequality norm
$\|\vec{v}_{1}\circ\vec{v}_{2}\|\leq\|\vec{v}_{1}\|\|\vec{v}_{2}\|$, produces
$c\leq 1$. This upper bound is tight, as two matrices $M_{1}$ and $M_{2}$ with
normalized and linearly independent columns might be arbitrarily close to a
rank one projector $P$, thus producing $c\rightarrow 1$. Also, it might occur
that $M_{1}\rightarrow P_{1}$ and $M_{2}\rightarrow P_{2}$, with orthogonal
projectors $P_{1}$ and $P_{2}$ satisfying $P_{1}P_{2}=0$, thus having
$c\rightarrow 0$.

To illustrate the above results, the most general family of 2 MUF in dimension
2, arising from circulant matrices, is derived as follows. Suppose that

$\displaystyle M_{1}=\begin{pmatrix}\cos(\theta)&\sin(\theta)e^{i\alpha}\\\
\sin(\theta)e^{i\alpha}&\cos(\theta)\end{pmatrix}\quad\mbox{and}\quad
M_{2}=\begin{pmatrix}\cos(\eta)&\sin(\eta)e^{i\beta}\\\
\sin(\eta)e^{i\beta}&\cos(\eta)\end{pmatrix},$ (12)

satisfy the relations $M^{{\dagger}}_{1}M_{2}=K$, with $|K_{ij}|^{2}=c$,
$i,j=0,\dots,d-1$. From a straight calculation, a sufficient condition to have
a pair of MUF from (12) is obtained:

$\displaystyle\sin(\alpha)\sin(\beta)\tan(2\theta)\tan(2\eta)=-1.$ (13)

Note that (13) has a solution if and only if the following properties hold:

$\alpha,\beta,\theta,\eta\not\in\\{0,\pi/2,\pi,3\pi/2\\},$
$-1\leq\sin^{-1}(\alpha)\tan^{-1}(2\theta)\tan^{-1}(2\eta)\leq 1,$

and

$-1\leq\sin^{-1}(\beta)\tan^{-1}(2\theta)\tan^{-1}(2\eta)\leq 1,$

where the negative power means inverse multiplicative. This generic solution
determines a 3-parametric nonlinear family of MUF with overlap

$c=\cos^{2}(\eta-\theta)+2\sin(\eta)\cos(\eta)\sin(\theta)\cos(\theta)(\cos(\alpha-\beta)-1),$
(14)

which includes SIC-POVM when, for instance, $\alpha=\beta$ and
$\theta-\eta=\arccos(1/\sqrt{3})$. Note that there is no real pair of MUF
within this continuous family. Nonetheless, real pairs of MUF arise from (12)
when $\theta\in\\{0,\pi\\}$ and $\eta\in\\{\pi/2,3\pi/2\\}$, also when
$\theta\in\\{\pi/2,3\pi/2\\}$ and $\eta\in\\{0,\pi\\}$, and finally when
$\alpha,\beta\in\\{\pi/2,3\pi/2\\}$. All these isolated solutions form pairs
of real MUB.

Interestingly, the above study reveals that MUF based on circulant matrices
admit solution to both the SIC-POVM and the MUB problem for a qubit system.
The same occurs for every prime dimension $d$: the eigenvectors bases of
operators $Z,XZ,\dots,XZ^{d-1}$, forming a maximal set of $d+1$ MUB, can be
arranged as $d$ circulant unitary matrices of order $d$ [51]. Moreover,
Zauner’s conjecture [33] allows to see a SIC-POVM as a set of MUF composed by
$d$ circulant matrices, as it will be shown in Section 5. Despite the
existence of $m$ MUF in dimension $d$ is a hard problem, a solution for the
case of $m=2$ is shown below, in any dimension $d$.

######  Proposition 5.

Let $M$ be a circulant matrix of order $d$ with columns forming a normalized
frame. Then, there is another matrix $\tilde{M}$ inducing a frame such that
$M$ and $\tilde{M}$ form a pair of MUF, in every dimension $d$.

###### Proof.

Matrices $M$ and $\tilde{M}$ are MUF if and only if
$M^{{\dagger}}\tilde{M}=K$, with $|K_{jk}|^{2}=c$, for every $j,k=0,\dots,d-1$
and some $0<c<1$. This is equivalent to say that their normalized vectors
defined by its eigenvalues satisfy

$\vec{\lambda}^{*}_{M}\circ\vec{\lambda}_{\tilde{M}}=\vec{\lambda}_{K}.$ (15)

By assuming the polar decompositions

$\lambda_{M}=\sum_{j=0}^{d-1}r_{j}\omega^{\alpha_{j}}|j\rangle\quad\mbox{and}\quad\lambda_{\tilde{M}}=\sum_{j=0}^{d-1}s_{j}\omega^{\beta_{j}}|j\rangle,$
(16)

where $r_{j},s_{j},\alpha_{j},\beta_{j}\in\mathbb{R}$, $j=0,\dots,d-1$, and
the fact that

$\vec{\lambda}_{K}=\sqrt{c}\,F^{{\dagger}}\sum_{j=0}^{d-1}\omega^{\gamma_{j}}|j\rangle,$
(17)

for some $\gamma_{0},\dots,\gamma_{d-1}\in[0,2\pi)$, one can re-write (15) as

$\sum_{j=0}^{d-1}r_{j}s_{j}\omega^{\beta_{j}-\alpha_{j}}|j\rangle=\sqrt{cd}\,F^{{\dagger}}\sum_{j=0}^{d-1}\omega^{\gamma_{j}}|j\rangle.$
(18)

Note that $r_{j}\not=0$ and $s_{j}\not=0$, for any $j=0,\dots,d-1$. This is so
because eigenvalues of matrices $M$ and $\tilde{M}$ cannot vanish, because
their columns form a frame. Now, if there is an -unormalized- vector
$\sum_{j=0}^{d-1}\omega^{\gamma_{j}}|j\rangle$ unbiased to the pair
$\\{\mathbb{I},F\\}$, then there exists
$\delta_{0},\dots,\delta_{d-1}\in[0,2\pi)$ such that
$F^{{\dagger}}\sum_{j=0}^{d-1}\omega^{\gamma_{j}}|j\rangle=\sum_{j=0}^{d-1}\omega^{\delta_{j}}|j\rangle$.
So, (18) reduces to

$\sum_{j=0}^{d-1}r_{j}s_{j}\omega^{\beta_{j}-\alpha_{j}}|j\rangle=\sqrt{cd}\,\sum_{j=0}^{d-1}\omega^{\delta_{j}}|j\rangle,$
(19)

implying that $r_{j}s_{j}=\sqrt{cd}$ and $\beta_{j}-\alpha_{j}=\delta_{j}$,
for every $j=0,\dots,d-1$. Also, recall that normalization of vectors of the
frames imply the additional constraints
$\sum_{j=0}^{d-1}r^{2}_{j}=\sum_{j=0}^{d-1}s^{2}_{j}=d$. Thus, the suitable
matrix $\tilde{M}$ is characterized by $s_{j}=\sqrt{cd}/r_{j}$ and
$\beta_{j}=\alpha_{j}+\gamma_{j}$, with the constraint
$\sum_{j=0}^{d-1}s^{2}_{j}=cd\sum_{j=0}^{d-1}\frac{1}{r^{2}_{j}}=d$, implying
that $c=1/\sum_{j=0}^{d-1}\frac{1}{r^{2}_{j}}<1$. To conclude the proof, note
that there are at least $d$ vectors MU to the pair $\\{\mathbb{I},F\\}$ in
every dimension $d$, corresponding to the eigenvectors of $XZ$. ∎

Prop. 5 can be illustrated with a 3-parametric family of MUF for a qubit
system. Consider the vector $v=(1,i)^{T}$, which is MU to the pair
$\\{\mathbb{I},F\\}$ in dimension $d=2$. Thus, taking
$r_{0}=\sqrt{2}\cos(\nu)$ and $r_{1}=\sqrt{2}\sin(\nu)$, it holds that

$\displaystyle M$ $\displaystyle=$ $\displaystyle
Fdiag[\vec{\lambda}(M)]F^{{\dagger}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\begin{array}[]{cc}1&1\\\
1&-1\end{array}\right)\left(\begin{array}[]{cc}\sqrt{2}\cos(\nu)\omega^{\alpha_{0}}&0\\\
0&\sqrt{2}\sin(\nu)\omega^{\alpha_{1}}\end{array}\right)\left(\begin{array}[]{cc}1&1\\\
1&-1\end{array}\right)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(\begin{array}[]{cc}\cos(\nu)\omega^{\alpha_{0}}+\sin(\nu)\omega^{\alpha_{1}}&\cos(\nu)\omega^{\alpha_{0}}-\sin(\nu)\omega^{\alpha_{1}}\\\
\cos(\nu)\omega^{\alpha_{0}}-\sin(\nu)\omega^{\alpha_{1}}&\cos(\nu)\omega^{\alpha_{0}}+\sin(\nu)\omega^{\alpha_{1}}\end{array}\right),$

and

$\displaystyle\tilde{M}$ $\displaystyle=$ $\displaystyle
Fdiag[\vec{\lambda}(\tilde{M})]F^{{\dagger}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\begin{array}[]{cc}1&1\\\
1&-1\end{array}\right)\left(\begin{array}[]{cc}\frac{\sqrt{c}}{\cos(\nu)}\omega^{\alpha_{0}+1/4}&0\\\
0&\frac{\sqrt{c}}{\sin(\nu)}\omega^{\alpha_{1}-1/4}\end{array}\right)\left(\begin{array}[]{cc}1&1\\\
1&-1\end{array}\right)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sqrt{c}\left(\begin{array}[]{cc}\sec(\nu)w^{\alpha_{0}+\frac{1}{4}}+\csc(\nu)w^{\alpha_{1}-\frac{1}{4}}&w^{\alpha_{0}+\frac{1}{4}}\sec(\nu)-w^{\alpha_{1}-\frac{1}{4}}\csc(\nu)\\\
w^{\alpha_{0}+\frac{1}{4}}\sec(\nu)-w^{\alpha_{1}-\frac{1}{4}}\csc(\nu)&\sec(\nu)w^{\alpha_{0}+\frac{1}{4}}+\csc(\nu)w^{\alpha_{1}-\frac{1}{4}}\end{array}\right),$

with overlap $c=2\cos^{2}(\nu)\sin^{2}(\nu)$. In particular, for $\nu=\pi/4$
and $\nu=-\arctan\left(\sqrt{\frac{3-\sqrt{3}}{3+\sqrt{3}}}\right)$, two MUB
and a SIC-POVM occur, respectively.

##  5 Application to the SIC-POVM problem

Along this section, SIC-POVM are organized as $d$ sets of mutually unbiased
frames, composed by $d$ vectors each. This slightly different way of seeing
these constellations carries out remarkably interesting consequences, as shown
below. Based on the matrix approach to MUF, introduced in Section 4, one can
assume that the $d^{2}$ vectors of a SIC-POVM can be arranged as column
vectors in $m=d$ circulant matrices of order $d$, $M_{0},\dots,M_{d-1}$, each
of them defining a frame. This assumption reduces the problem to finding a
single suitable matrix of order $d$, called $\mathcal{M}$, that contains the
normalized vectors
$\vec{\lambda}(M_{0})/\sqrt{d},\dots,\vec{\lambda}(M_{d-1})/\sqrt{d}$ along
its columns. The problem of finding a solution $\mathcal{M}$ is hard, since it
has a quadratic number of free parameters as a function of the dimension $d$.
In order to make the problem simpler, a further reasonable assumption is
adopted: the matrix $\mathcal{M}$ is circulant. Thus, all the problem reduces
to find a suitable first row of $\mathcal{M}$, that has $2(d-1)$ free
parameters without loss of generality. After these intuitive, but drastic
assumptions, one might wonder whether there is a SIC-POVM satisfying the above
symmetries. Below, a positive answer is provided in every dimension $d$ where
a SIC-POVM is known.

######  Proposition 6.

The assumption that matrices $M_{0},\dots,M_{d-1}$ and $\mathcal{M}$ are
circulant is equivalent to the assumption that a SIC-POVM can be defined as an
orbit of a fiducial state under the Weyl-Heisenberg group.

###### Proof.

The fact that the $d^{2}$ vectors of a SIC-POVM are arranged as columns in
circulant matrices $M_{0},\dots,M_{d-1}$ imply that the $d$ vectors within
each of these matrices are connected by a shift of its entries. Evenmore, the
fact that $\mathcal{M}$ is circulant means that
$\vec{\lambda}(M_{j})=X^{j}\vec{\lambda}(M_{0})$, implying that
$M_{j}=Z^{j}M_{0}(Z^{j})^{{\dagger}}$. Thus, the $d^{2}$ vectors of the SIC-
POVM are given by $|\phi_{jk}\rangle=X^{j}Z^{k}|\phi_{00}\rangle$, where
$|\phi_{00}\rangle$ is an unknown fiducial state. ∎

Note that normalized vectors $\vec{\lambda}(M_{j})/\sqrt{d}$, $j=0,\dots,d-1$,
are also fiducial states. This is simple to see, as the first row of $M_{j}$
is connected with the normalized vector $\vec{\lambda}(M_{j})/\sqrt{d}$ via
the Fourier transform $F$, and the first row of $M_{j}$ is connected with its
first column via $F^{2}$. Given that $F$ belongs to the Clifford group, it
transforms fiducial states into fiducial states. Note that this result is in
agreement with Prop. 2.

A fundamental ingredient to have a frame is the linearly independence of its
elements. In the matrix approach, this is equivalent to say that matrices
$M_{0},\dots,M_{d-1}$ do not have vanishing determinant, i.e. the vector
$\lambda(M_{0})$ has non-zero entries. As this vector is also a fiducial
state, up to a rescaling factor, it is interesting to wonder whether a
fiducial state can have a vanishing entry. Zauner already noted the existence
of such fiducial states in dimension 3 [33]. Some recent explorations revealed
the existence of fiducials with zero entries in dimensions $d=26,28,62,98$ and
$228$ [52]. For an extended study about linearly dependencies of elements of a
SIC-POVM see here [53]. When having a fiducial state $|\phi\rangle$ with zero
entries one might consider another fiducial $U|\phi\rangle$, where $U$ is an
element of the Clifford group. Most likely, many of these transformations
produce a fiducial state having non-zero entries but, unfortunately, it was
not possible to demonstrate it. This property is left as a conjecture.

There is a further symmetry that one can impose in order to reduce the number
of free parameters in SIC-POVM. One might assume that all matrices
$M_{0},\dots,M_{d-1}$ are hermitian, which holds if and only if
$\vec{\lambda}(M_{0})/\sqrt{d}$ is a real fiducial state. Along this
direction, it has been conjectured that real fiducial states exist in odd
dimensions of the form $d=4n^{2}+3$, $n\in\mathbb{Z}$ [38]. This conjecture is
supported with 18 real fiducials existing in dimensions of such form, the
highest dimensional case corresponding to $d=19603$ [37]. For instance, in
$d=3$, there is a family of real fiducial states [34]:

$|\phi(r_{0})\rangle=r_{0}|0\rangle-(r_{0}/2+\sqrt{2-3r_{0}^{2}}/2)|1\rangle-(r_{0}/2-\sqrt{2-3r_{0}^{2}}/2)|2\rangle,$
(24)

with $1/\sqrt{2}<r_{0}<\sqrt{2/3}$, inducing the following 1-parametric family
of hermitian matrices:

$\displaystyle M_{j}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}0&a_{j}&a_{j}^{*}\\\
a_{j}^{*}&0&a_{j}\\\ a_{j}&a_{j}^{*}&0\end{array}\right),\quad j=0,1,2,$

where

$\displaystyle a_{0}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\sqrt{3}r_{0}+i\sqrt{2-3r_{0}^{2}}\right)$
$\displaystyle a_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{12}\left(3-i\sqrt{3}\right)\left(\sqrt{6-9r_{0}^{2}}-3ir_{0}\right)$
$\displaystyle a_{2}$ $\displaystyle=$
$\displaystyle-\frac{1}{12}\left(3+i\sqrt{3}\right)\left(\sqrt{6-9r_{0}^{2}}-3ir_{0}\right).$

Note that all existing real fiducial states are defined in odd dimensions [38,
37, 54]. Thus, an interesting question arises: _are there real fiducial states
in even dimensions $d$?_ Below, a conclusive answer to this question is
provided.

######  Proposition 7.

Real fiducial states, covariant under the standard Weyl-Heisenberg group, do
not exist in any even dimension.

###### Proof.

Let $|\phi\rangle=\sum_{j=0}^{d-1}c_{j}|\varphi_{j}\rangle$ be a real pure
state, expanded in the Fourier basis
$|\varphi_{j}\rangle=\frac{1}{\sqrt{d}}\sum_{k=0}^{d-1}\omega^{jk}|k\rangle$,
where $\omega=e^{2\pi i/d}$. Taking in to account that
$\langle\phi|=\sum_{j,k}c_{j}\omega^{jk}\langle k|$, it occurs that

$\displaystyle|\langle\phi|X^{d/2}Z|\phi\rangle|^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{d}\sum_{j,j^{\prime},k,k^{\prime}}c_{j}\omega^{jk}\langle
k|\sum_{\ell=0}^{d-1}|\ell+d/2\rangle\langle\ell|\sum_{m=0}^{d-1}\omega^{m}|m\rangle\langle
m|c_{j^{\prime}}\omega^{j^{\prime}k^{\prime}}|k^{\prime}\rangle$
$\displaystyle=$
$\displaystyle\frac{1}{d}\sum_{j,j^{\prime},k,k^{\prime},\ell=0}^{d-1}c_{j}c_{j^{\prime}}\omega^{jk+j^{\prime}k^{\prime}+\ell}\langle
k|\ell+d/2\rangle\langle\ell|k^{\prime}\rangle$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{d-1}(-1)^{j}c_{j}c_{d-j-1}$ $\displaystyle=$
$\displaystyle(c_{0}c_{d-1}+c_{2}c_{d-3}+\dots+c_{d-4}c_{3}+c_{d-2}c_{1})-$
$\displaystyle(c_{1}c_{d-2}+c_{3}c_{d-4}+\dots+c_{d-3}c_{2}+c_{d-1}c_{0})$
$\displaystyle=$ $\displaystyle 0,$

for every even dimension $d$. Note that for SIC-POVM, the above overlap should
be equal to $1/(d+1)$. As $|\phi\rangle$ is an arbitrary real state, there is
no real fiducial state with respect to the standard WH group in any even
dimension $d$. ∎

Now, a hidden symmetry is revealed for the amplitudes of fiducial states.

######  Proposition 8.

Any fiducial state $|\phi\rangle$, with respect to a Weyl-Heisenberg covariant
SIC-POVM, satisfies

$\sum_{j=0}^{d-1}|\langle
j|\phi\rangle|^{2}|j\rangle=F^{{\dagger}}\left(\frac{1}{\sqrt{d}}|0\rangle+\frac{1}{\sqrt{d(d+1)}}\sum_{j=1}^{d-1}e^{i\alpha_{j}}|j\rangle\right),$
(26)

in every finite dimension $d$. Here, $F$ is the discrete Fourier transform of
order $d$ and $\alpha_{j}\in[0,2\pi)$ satisfies $\alpha_{d-j}=-\alpha_{j}$,
for every $j=1,\dots,d-1$. As a consequence, any fiducial state $|\phi\rangle$
depends on $\lfloor(d-1)/2\rfloor+d-1$ free parameters only, i.e.,
$\lfloor(d-1)/2\rfloor$ amplitudes given by (26) and $d-1$ complex phases.

###### Proof.

Matrix $M_{0}$ satisfies

$M^{{\dagger}}_{0}M_{0}=\tilde{K},$ (27)

where $\tilde{K}$ is a circulant matrix satisfying

$|\tilde{K}_{jk}|^{2}=\left\\{\begin{array}[]{cc}1&j=k\\\
\frac{1}{d+1}&j\not=k\end{array}\right..$ (28)

Given that $\tilde{K}=F\mathrm{diag}[\vec{\lambda}(\tilde{K})]F^{{\dagger}}$,
from (15) and (27) it occurs that
$\vec{\lambda}(M_{0})^{*}\circ\vec{\lambda}(M_{0})=\vec{\lambda}(\tilde{K})$.
Here, the circle $(\circ)$ denotes the -entrywise- Hadamard product. On the
other hand, given that $\tilde{K}$ is circulant, its eigenvalues are related
with its first row via the Fourier transform. Thus, from (28) it holds that

$\vec{\lambda}(\tilde{K})=\sqrt{\frac{d}{d+1}}F^{{\dagger}}\\{\sqrt{d+1},e^{i\alpha_{1}},\dots,e^{i\alpha_{d-1}}\\}^{T},$
(29)

where the fact that the entries of the first row of $\tilde{K}$ are given by
$\tilde{K}_{00}=1$ and $\tilde{K}_{0k}=\frac{1}{\sqrt{d+1}}e^{i\alpha_{k}}$,
$k=1,\dots,d-1$ has been used. Furthermore, the matrix $\tilde{K}$ is
hermitian from (27), so the phases $\alpha_{j}$ satisfy the symmetry relation
$\alpha_{d-j}=-\alpha_{j}$, for every $j=1,\dots,d-1$. Note that this is the
symmetry that allows to reduce the number of free parameters in fiducial
states. Combining (27) and (29), an expression for the amplitudes of
$\vec{\lambda}(M_{0})$ in the computational basis arises:

$\\{|(\vec{\lambda}(M_{0}))_{0}|^{2},\dots,|(\vec{\lambda}(M_{0}))_{d-1}|^{2}\\}=\sqrt{\frac{d}{d+1}}F^{{\dagger}}\\{\sqrt{d+1},e^{i\alpha_{1}},\dots,e^{i\alpha_{d-1}}\\}^{T}.$
(30)

Therefore, there are $\lfloor(d-1)/2\rfloor+d-1$ free parameters in
$\vec{\lambda}(M_{0})$, where $\lfloor(d-1)/2\rfloor$ of them come from the
absolute values (30) and $d-1$ from the complex phases. The proof concludes by
noting that $|\phi\rangle=\vec{\lambda}(M_{0})/\sqrt{d}$ is a fiducial state,
without loss of generality. ∎

Prop. 8 can be equivalently obtained by applying the discrete version of the
Wiener-Khintchine Theorem [55] to the fiducial state $|\phi\rangle$. This
simple observation has been surprisingly overlooked by the community, whereas
the approach to MUF played a fundamental role to straightly arrive to
Proposition 8. It is stressed that some numerical simulations made by Chris
Fuchs in low dimensions suggest that the number of free parameters in
fiducials might be $3(d-1)/2-1$ for $d$ odd and $3d/2-1$ for $d$ even, having
an almost perfect match with Proposition 8, see page 1258 at Chris Fuchs’s
_samizdat_ [56, 57], the last paragraph in [58], and Section IV in [59].

###  5.1 Minimum uncertainty states

Fiducial states are minimum uncertainty states for a complete set of MUB in
prime dimensions [58]. To show this property, let
$\\{|\varphi^{j}_{k}\rangle\\}$ be a set of $d+1$ MUB in prime dimension $d$
and $H_{j}=-\log_{2}[\sum_{k=0}^{d-1}(p^{j}_{k})^{2}]$ be the quadratic Rényi
entropy [60] associated to the $j$th MUB, where
$p^{j}_{k}=|\langle\phi|\varphi^{j}_{k}\rangle|^{2}$. Thus, it holds that
[61]:

$\sum_{m=0}^{d}H_{m}\geq(d+1)\log_{2}\left[\frac{d+1}{2}\right],$ (31)

for any pure quantum state $|\phi\rangle$. The saturation of inequality (31)
defines a minimum uncertainty states if and only if
$\sum_{k=0}^{d-1}(p^{j}_{k})^{2}=\frac{2}{d+1}$, for every $j=0,\dots,d$. It
has been shown that this last property is satisfied by any fiducial state
$|\phi\rangle$ in every prime dimension $d$ [58]. However, there are further
quantum states minimizing (31). An interesting question is the following: are
there further sets of orthonormal bases such that fiducial states are minimum
uncertainty states? The following result, based on Prop. 8, allows to provide
a positive answer to the question, in every dimension where a fiducial state
exists.

######  Proposition 9.

Let $|\phi\rangle$ be a $d$-dimensional quantum state satisfying (26),
$|\xi_{0}\rangle,\dots,|\xi_{d-1}\rangle$ the columns of any unitary matrix
belonging to the Clifford group, and $p_{k}=|\langle\phi|\xi_{k}\rangle|^{2}$.
Therefore,

$\sum_{j=0}^{d-1}p_{j}p_{j+k}=\frac{1+\delta_{k,0}}{d+1}.$ (32)

In particular, (32) holds for fiducial states $|\phi\rangle$.

###### Proof.

For $\\{|\xi_{j}\rangle\\}$ equal to the computational basis, (32) arises by
taking the inner product between the left hand side of (26) and the same
expression after applying operator $X^{k}$. For the right hand side, it has to
be used the fact that $X^{k}F^{{\dagger}}=F^{{\dagger}}Z^{k}$. Next, assume
that $\\{\xi_{j}\\}$ are the columns of any unitary $U$ belonging to the
Clifford group, and consider that $|\phi\rangle=U|\phi^{\prime}\rangle$, where
$|\phi^{\prime}\rangle$ is another fiducial state. Thus,
$p_{k}=|\langle\phi|k\rangle|^{2}=|\langle\phi|U^{{\dagger}}U|k\rangle|^{2}=|\langle\phi^{\prime}|\xi_{k}\rangle|^{2}$.
The generality of the fiducial state $|\phi^{\prime}\rangle$ allows to
conclude that (32) holds for any basis $|\xi_{k}\rangle=U|k\rangle$,
$k=0\dots,d-1$, given by the columns of $U$. ∎

When considering the identity matrix as the chosen element of the Clifford
group, Proposition 9 reduces to Corollary 2.3 in [54], also shown in eq.(10)
in [58] and as a special case of Theorem 1 in [62]. Proposition 9 has the
following immediate consequence.

######  Corollary 2.

Any $d$-dimensional quantum state $|\phi\rangle$ that satisfies (26) is a
minimum uncertainty state with respect to every subset of $d+1$ orthonormal
bases formed by the columns of $d+1$ elements of the Clifford group. In
particular, the result holds for any fiducial state in every dimension $d$,
and for $d+1$ MUB in every prime dimension $d$.

###### Proof.

From Prop. 9, inequality (31) is saturated for any subset of $d+1$ bases,
induced by the columns of Clifford unitaries. In particular, a maximal set of
MUB arises from the columns of a subset of elements of the Clifford group in
every prime dimension $d$. This is so, because there exists a subset of $d+1$
unitaries $U_{0},\dots,U_{d}$ within the Clifford group that diagonalize
operators $Z,X,XZ,XZ^{2},\dots,XZ^{d-1}$. Indeed, $Z$ belongs to the Clifford
group, $F$ diagonalizes $X$, and the Clifford unitaries
$U_{H_{1}},\dots,U_{H_{d-1}}$, where $H_{k}=\\{\\{k,-1\\},\\{-1,0\\}\\}$,
produce $U_{H_{k}}XZ^{k}U^{{\dagger}}_{H_{k}}=\tau^{-k}Z$, with
$\tau=-e^{i\pi/d}$. In particular, the unique eigenvectors bases of operators
$Z,X,XZ,XZ^{2},\dots,XZ^{d-1}$ form a maximal set of $d+1$ MUB in every prime
dimension $d$ [51]. ∎

A relation between some classes of fiducial states and mutually unbiased bases
can be shown. Assume that $|\phi\rangle$ is a fiducial state, eigenvector of
the Zauner’s operator [33]

$\mathcal{Z}=e^{i\pi(d-1)/12}FG,$ (33)

where $G$ is a diagonal unitary matrix having diagonal entries
$G_{jj}=e^{i\pi(d+1)j^{2}/d}$, $j=0,\dots,d-1$. Thus, the following result
arises.

######  Proposition 10.

Any eigenvector of the Zauner operator (33) has identical probability
distributions with respect to the triplet of MUB given by the columns of
$\\{\mathbb{I},\mathcal{Z},\mathcal{Z}^{2}\\}$, in every dimension $d$. In
particular, the result holds for fiducial states that are eigenvectors of
$\mathcal{Z}$.

###### Proof.

Zauner’s operator $\mathcal{Z}$ is unitary, thus having $|\langle
k|\phi\rangle|=|\langle
k|\mathcal{Z}^{{\dagger}}\mathcal{Z}|\phi\rangle|=|\langle
k|(\mathcal{Z}^{2})^{{\dagger}}\mathcal{Z}^{2}|\phi\rangle|$, for every
$k=0,\dots,d-1$. Furthermore, if $|\phi\rangle$ is eigenvector of
$\mathcal{Z}$ then $|\langle k|\phi\rangle|=|\langle
k|\mathcal{Z}^{{\dagger}}|\phi\rangle|=|\langle
k|(\mathcal{Z}^{2})^{{\dagger}}|\phi\rangle|$, for every $k=0,\dots,d-1$. This
implies that $|\phi\rangle$ has identical probability distributions in the
three orthonormal bases $\\{k\\}$, $\\{\mathcal{Z}|k\rangle\\}$ and
$\\{\mathcal{Z}^{2}|k\rangle\\}$. On the other hand, from (33) it is simple to
check that $|\langle j|\mathcal{Z}|k\rangle|^{2}=|\langle
j|\mathcal{Z}^{{\dagger}}\mathcal{Z}^{2}|k\rangle|^{2}=1/d$ and $|\langle
j|\mathcal{Z}^{2}|k\rangle|^{2}=1/d$, for every $j,k=0,\dots,d-1$. Therefore,
the three bases are mutually unbiased. ∎

For $d=2$, any eigenvector of $\mathcal{Z}$ is a _MUB-balanced_ state [63], in
particular when it is a fiducial state. The MUB-balanced property means that a
$d$-dimensional quantum state has identical probability distributions with
respect to a maximal set of $d+1$ MUB.

After completing the work, it was noted that Falk Unger obtained (26) in a
slightly different way (unpublished notes [64]). Namely, if
$|\phi\rangle=\sum_{j}c_{j}|j\rangle$ is a fiducial state then
$e^{i\alpha_{k}}/\sqrt{d+1}=\langle\phi|Z^{k}|\phi\rangle=\sum_{j}|c_{j}|^{2}\omega^{jk}=\sqrt{d}[F\sum_{j}|c_{j}|^{2}|j\rangle]_{k}$,
for every $k=1,\dots,d-1$.

##  6 Conclusions

This work introduced the ultimate notion of unbiasedness for sets of linearly
independent and normalized vectors in $d$-dimensional Hilbert spaces:
_mutually unbiased frames_ (MUF). These sets are characterized by a constant
overlapping that can take any value between the absolute bounds 0 and 1. The
overlap reduces to $1/d$ when at least one of the frames is tight, where
tightness is equivalent to impose that the frame defines a rank-one positive
operator valued measure (POVM). The introduced notion was illustrated by
finding the most general pair of MUF composed by two vectors each for a qubit
system. The concept of MUF allowed to think a Symmetric Informationally
Complete (SIC)-POVM in dimension $d$ as a set of $d$ MUF composed by $d$
vectors each, thus having a common root with the mutually unbiased bases
problem. Interestingly, Zauner’s conjecture about the existence of fiducial
states arose in a natural way when studying sets of $d$ MUF in dimension $d$
(see Prop. 6). The approach also allowed to unlock two fundamental properties
of SIC-POVM: real fiducial states do not exist in any even dimension (see
Prop. 7) and, any unknown $d$-dimensional fiducial state can be parameterized,
a priori, with roughly $3d/2$ real variables (see Prop. 8). Additionally, some
classes of quantum pure states, those satisfying (26), are minimum uncertainty
states with respect to $d+1$ orthonormal bases given by the columns of any
subset of $d+1$ Clifford operations. This result holds for any fiducial state
in every dimension $d$ where a Weyl-Heisenberg covariant SIC-POVM exists (see
Prop. 9 and Corol. 2). Finally, any eigenvector of the Zauner’s operator
‘looks the same’ for three MUB, in every dimension $d$ (see Prop. 10).

Acknowledgements

The authors kindly acknowledge M. Appleby, A. Araya, I. Bengtsson, C. Fuchs,
M. Grassl, F. Unger and K. Życzkowski for their valuable comments. This work
is supported by grant FONDECyT Iniciación nro. 11180474, Chile. FCP and VGA
belong to the first year of the PhD Program _Doctorado en Física, mención
Física-Matemática_ , Universidad de Antofagasta, Antofagasta, Chile.

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# Continual Learning of Visual Concepts for Robots through Limited Supervision

Ali Ayub<EMAIL_ADDRESS>The Pennsylvania State UniversityState CollegePAUSA
and  Alan R. Wagner<EMAIL_ADDRESS>The Pennsylvania State
UniversityState CollegePAUSA

(2021)

###### Abstract.

For many real-world robotics applications, robots need to continually adapt
and learn new concepts. Further, robots need to learn through limited data
because of scarcity of labeled data in the real-world environments. To this
end, my research focuses on developing robots that continually learn in
dynamic unseen environments/scenarios, learn from limited human supervision,
remember previously learned knowledge and use that knowledge to learn new
concepts. I develop machine learning models that not only produce State-of-
the-results on benchmark datasets but also allow robots to learn new objects
and scenes in unconstrained environments which lead to a variety of novel
robotics applications.

††copyright: rightsretained††journalyear: 2021††conference: Companion of the
2021 ACM/IEEE International Conference on Human-Robot Interaction; March 8–11,
2021; Boulder, CO, USA††booktitle: Companion of the 2021 ACM/IEEE
International Conference on Human-Robot Interaction (HRI ’21 Companion), March
8–11, 2021, Boulder, CO, USA††doi: 10.1145/3434074.3446357††isbn:
978-1-4503-8290-8/21/03

##  1\. Introduction

Continual adaptation and learning through limited data is the hallmark of
human intelligence. Humans continue to learn new concepts over their lifetime
without the need to relearn most previous concepts. With robots becoming an
integral part of our society, they must also continue to learn over their
lifetime to adapt to the ever-changing environments. Further, in real-world
applications, robots do not have access to a large amount of labeled data
since it is impractical for human users to provide hundreds of examples to the
robot. Thus, robots must learn using a small amount of data through limited
human supervision. The long-term goal of my research is to develop autonomous
robots for everyday environments where they can learn over their lifetime and
use the learned knowledge to assist humans in their daily lives.

Creating robots that continually learn is a challenging problem. Deep learning
is widely used to address many robot learning tasks, yet deep learning suffers
from a phenomenon called catastrophic forgetting when learning continually.
Catastrophic forgetting occurs when continually training a model (neural
network) to recognize new classes, the model forgets the previously learned
classes and the overall classification accuracy decreases. One way to address
this problem is by storing the complete data of the previously learned
classes. However, storing data of the previous classes requires a huge memory
when learning new classes continually. Robots, on the other hand, have limited
on-board memory available, hence they cannot keep storing high-dimensional
images of previous classes. In real-world scenarios, labelling a large amount
of data is costly in terms of time and effort. Hence, robots have to learn
from a small number of interactions with likely impatient human users. Deep
learning systems, however, require a large amount of labelled data for
learning.

In order to tackle these challenges, my work develops machine learning and
computer vision techniques that are inspired by concept learning models from
cognitive science. My work is informed by higher level concept learning in
children (and all humans) related to curiosity-driven, intrinsically
motivated, continual learning of visual concepts (objects and scenes).

##  2\. Related Work

Recent continual learning techniques use deep neural networks and rely on
storing a fraction of old class data when learning a set of new classes
(Rebuffi et al., 2017; Castro et al., 2018). To avoid storage of real samples,
some approaches use generative-memory and regenerate samples of old classes
using GANs or autoencoders (Ayub and Wagner, 2020d, 2021; Ostapenko et al.,
2019; Ayub and Wagner, 2021), however the performance of these approaches is
generally inferior to approaches that store real images. One major issue with
all these prior continual learning approaches is that they require a large
amount of training data. Hence using these methods for continual learning from
limited data results in poor accuracy.

Curiosity-driven learning has been explored for robotics applications in the
past to learn from limited data and supervision. In recent years, some deep
reinforcement learning approaches have been proposed that use a curiosity-
driven reward function (Burda et al., 2019; Haber et al., 2018) to train
neural networks. For object learning, many researchers have presented active
learning techniques using uncertainty sampling (Beluch et al., 2018; Gal et
al., 2017; Siddiqui et al., 2020; Yoo and Kweon, 2019; Shen et al., 2019). All
of these approaches train deep networks using specific loss terms such that
the network can predict the most uncertain samples. Although these approaches
produced good results on small, simple image datasets like MNIST (LeChun,
1998), they were not tested on a real robot.

One of the main limitations of prior curiosity-driven and active learning
approaches is that they are designed for a batch learning setting and will
thus suffer from catastrophic forgetting when attempting to learn continually.
In contrast, we present a novel approach that not only allows a robot to learn
from visual data continually but also allows it to assign curiosity scores to
unlabeled objects in a self-supervised manner.

##  3\. Methodology

In this work, I consider a general continual learning setup for learning
visual categories (object or scene classes). In each new increment $t$, the
robot gets a small set of labeled samples
$S_{t}=\\{(x_{i}^{t},y_{i}^{t})\\}_{i=1}^{n^{t}}$, where $x_{i}\in\mathcal{X}$
are the visual samples (images) and $y_{i}^{t}$ are their ground truth labels.
The samples in an increment can belong to the earlier learned classes or
completely new classes. Further, the robot has limited storage capacity, thus
it cannot store the high dimensional images of the previously learned
categories.

To learn new objects or scenes, the robot first acquires new image data
autonomously using its own cameras.The category labels for the images are
provided by the human in a textual format. I then use a neural network pre-
trained on a large dataset (e.g. ImageNet (Russakovsky et al., 2015)) to
extract feature vectors for the images. Then, I apply a novel cognitively-
inspired clustering approach (called Agg-Var clustering) on the feature
vectors of the images to learn centroids and covariance matrices for the
visual categories. In Agg-Var clustering, the model finds the Euclidean
distance of a new $i$th feature vector $x_{i}^{y}$ of a class $y$ to the
previously learned centroids of the class. If the distance is below a pre-
defined distance threshold $D$ (hyperparameter), the model performs memory
integration (Mack et al., 2018) by updating the closest centroid and the
corresponding covariance matrix using the new feature vector. If the distance
is above the distance threshold $D$, the model performs pattern separation
(Mack et al., 2018) by creating a new centroid initialized with $x_{i}^{y}$
and a new covariance matrix initialized with a zero matrix. In this way, the
model gets a set of centroids and covariance matrices for all the classes
separately. Note that even a small number of images per class are enough to
learn the centroids/covariance matrix representation for the class, hence my
model can be used to learn from limited labeled data.

For classification of test images, I use pseudorehearsal technique (Robins,
1995) in which I use the centroids and covariance matrices of the old classes
as parameters of Gaussian distributions. I then sample these Gaussian
distributions to generate pseudo-exemplars for the old classes. A shallow
neural network classifier with a single linear layer is then trained using the
pseudo-exemplars and the feature vectors of the images in the current
increment. In this way, the model mitigates catastrophic forgetting.

##  4\. Past, Current and Future Work

Towards the goal of creating continually learning robots, the first project in
my PhD was focused on the few-shot incremental learning problem (FSIL), in
which the robot learns continually from a small number of object examples
provided by a human. I developed a novel approach termed Centroid-Based
Concept Learning (CBCL) to tackle this problem (Ayub and Wagner, 2020b).
CBCL’s classification accuracy was significantly higher than the State-of-the-
art (SOTA) incremental learning approaches on benchmark datasets (Table 1). I
then applied CBCL on a real robot for a cleaning application, in which the
robot learns household objects from a few visual examples provided by a human
and organizes related objects from a clutter of objects. This research
demonstrated that my method could be capable of dynamically learning task or
situation specific objects (Ayub and Wagner, 2020e). I also showed that CBCL
is a general approach and can be applied for other tasks, such as RGB-D indoor
scene classification (Ayub and Wagner, 2020a).

In real-world environments, robots must learn from streaming data that lacks
well-defined task boundaries (online learning). The lack of task boundaries
and unknown number of categories makes this problem harder than FSIL. I
developed an updated version of CBCL, termed Centroid-Based Concept Learning
with Pseudorehearsal (CBCL-PR) for online learning. CBCL-PR significantly
outperformed SOTA approaches on a benchmark dataset in terms of detecting
known and unknown scene categories. I then applied CBCL-PR on the Pepper robot
in which the robot wandered in unconstrained real-world environments to learn
new scene categories and detect previously unknown scene categories (Ayub et
al., [n.d.]).

Methods |  iCaRL |  EEIL |  BiC |  CBCL
---|---|---|---|---
Accuracy (%) |  63.75 |  64.02 |  64.84 |  69.85
Table 1. Comparison of CBCL with iCaRL (Rebuffi et al., 2017), EEIL (Castro et
al., 2018) and BiC (Wu et al., 2019) for class-incremental learning with 10
classes per increment on the CIFAR-100 dataset (Krizhevsky, 2009).

In follow-up research, I developed a system that used CBCL-PR for online
learning of scenes/contexts and Dempster Schaeffer theory to represent and
learn appropriate norms related to different scenes in terms of conditional
probabilities. My work was the first of its kind to examine online learning of
norms for social robots. I tested this approach on Pepper in which the robot
wandered around at different scenes and learned norms through simple Q/A
sessions with a human.This research demonstrated that my approach may allow
robots to learn different scene categories and use the recognition of these
scenes to moderate their behavior and decision-making (Ayub and Wagner,
2020f).

I am currently working on the curiosity-driven active online learning (CDAOL)
problem, in which the robot has a large amount of unlabeled objects available
in an environment and it must choose the most informative samples to be
labeled. I am developing a novel approach to assign curiosity scores to new
unlabeled objects in a self-supervised manner using the distance of the new
objects from the previously learned centroids. Preliminary experiments show
that my approach can learn the most informative objects quickly without
forgetting the previously learned objects which results in a dramatic increase
in accuracy over the other approaches, especially in the earlier increments
(Ayub and Wagner, 2020c).

For future work, I plan to apply the above-mentioned approach on a real robot.
However, real-world robots have access to clutters of objects rather than
single object images. Second, capturing multiple views of individual objects
in unconstrained environments through robot’s own cameras without human
assistance is challenging. To deal with this, I plan to develop a complete
system to allow a robot to capture images of cluttered objects, localize all
the objects in the clutter, get labels for the most informative objects, use a
manipulator module to move its hands around the labeled objects to get
different views of the objects and finally train the CNN using the images of
the new objects. I plan to test this system on a real robot in a lab
environment with clutter of objects present at various locations with
different backgrounds. The experiment will be performed over the course of one
month at different times of the day in which the robot will wander around in
the environment and learn about the objects it is curious about by asking a
human teacher. This experiment is the first of its kind, that will demonstrate
a true lifelong learning robot that learns a large number of objects (240
objects) in an unconstrained environment over a long period of time through
limited human supervision.

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