# Symmetries and local transformations of translationally invariant Matrix
Product States

Martin Hebenstreit<EMAIL_ADDRESS>Institute for Theoretical
Physics, University of Innsbruck, 6020 Innsbruck, Austria  David Sauerwein
<EMAIL_ADDRESS>This work was done prior to joining AWS. Amazon Web
Services Europe, Zürich, Switzerland  Andras Molnar
<EMAIL_ADDRESS>University of Vienna, Faculty of Mathematics, 1090
Vienna, Austria  J. Ignacio Cirac<EMAIL_ADDRESS>Max-Planck-
Institut für Quantenoptik, Hans-Kopfermann-Straße 1, 85748 Garching, Germany
Barbara Kraus<EMAIL_ADDRESS>Institute for Theoretical Physics,
University of Innsbruck, 6020 Innsbruck, Austria

###### Abstract

We determine the local symmetries and local transformation properties of
translationally invariant matrix product states (MPS). We focus on physical
dimension $d=2$ and bond dimension $D=3$ and use the procedure introduced in
D. Sauerwein et al., Phys. Rev. Lett. 123, 170504 (2019) to determine all
(including non–global) symmetries of those states. We identify and classify
the stochastic local transformations (SLOCC) that are allowed among MPS. We
scrutinize two very distinct sets of MPS and show the big diversity (also
compared to the case $D=2$) occurring in both, their symmetries and the
possible SLOCC transformations. These results reflect the variety of local
properties of MPS, even if restricted to translationally invariant states with
low bond dimension. Finally, we show that states with non-trivial local
symmetries are of measure zero for d = 2 and D > 3.

##  I Introduction

Entanglement is a unique quantum property that is behind most modern quantum
technologies, such as quantum computers [1, 2], and is key to comprehend
important features of quantum many-body systems [3, 4]. The relevance of
entanglement in many different branches of science has spurred significant
research efforts to understand its properties [2]. While this has led to a
clear understanding of bipartite entanglement, many questions are still open
in the multipartite realm.

At the heart of entanglement theory lies the fact that entanglement is a
resource under local operations assisted by classical communication (LOCC),
which are the most general operations that spatially separated parties can use
to manipulate a shared entangled state. Transformations of entangled states
via LOCC induce a physically meaningful partial order on the set of entangled
states: If $\Psi$ can be deterministically transformed into $\Phi$ via LOCC
this means that $\Psi$ is at least as entangled as $\Phi$ [2]. Furthermore, if
two states cannot even probabilistically be transformed into each other, via
so-called stochastic LOCC (SLOCC), they contain different, incomparable kinds
of entanglement. This entails that they may be useful in different contexts of
quantum information science [5]. Hence, the characterization of LOCC
transformations is central in entanglement theory.

Bipartite LOCC transformations among pure states admit a simple
characterization [6, 7] that led to a clear understanding of bipartite
entanglement and inspired a wide range of applications [2]. A general
characterization of multipartite LOCC is still elusive. This is, among other
reasons, due to the notorious mathematical complexity of multipartite LOCC
[8], the fact that there are multipartite entangled states that cannot even be
transformed into each other via SLOCC [9] and the exponential growth of the
Hilbert space dimension as a function of the number of constituent subsystems.

However, most multipartite (pure) states are not particularily interesting
from the perspective of state transformations. On the one hand, this is
because most of them cannot even be reached in polynomial time, even if
constant-size nonlocal quantum gates are allowed [10]. On the other hand,
generic multiqudit states of $N>4$ $d$-dimensional subsystems cannot be
transformed into nor be obtained from any inequivalent multipartite states of
the same dimensions via LOCC [11, 12] 111Here and in the following we do not
consider local unitary transformations, as they can always be applied locally
and as they do not alter the entanglement contained in the system.. This shows
that investigations can be focused on transformations among states within a
non-generic subset of physically relevant, i.e. naturally occurring in certain
physical contexts, multipartite quantum states.

The starting point of the investigation of entanglement is the
characterization of SLOCC classes: An $N$-partite state $\Psi$ is
SLOCC–equivalent to a state $\Phi$ if there exist local invertible operators
$g_{j}$ ($j=1,\ldots,N$) such that
$|\Psi\rangle=\otimes_{j=1}^{N}g_{j}|\Phi\rangle$. Physically, this means that
one can obtain $\Phi$ with a finite probability (i.e., for certain measurement
outcomes) by applying only local generalized measurements on the state $\Phi$.
As SLOCC–inequivalent states are not related to each other via local
operations, their entanglement, viewed as a resource cannot be compared. This
is why, in the context of entanglement theory, the study of state
transformations and SLOCC classes is central [2]. Whereas the characterization
of SLOCC classes becomes intractable with increasing the number of
constituents and there exist in general infinitely many classes, the problem
has been solved for various physically relevant sets of states. For instance,
SLOCC classes have been characterized for symmetric states [13, 14], i.e.
those that are invariant under permutations, or for certain tripartite ($N=3$)
and four-partite ($N=4$) states [9, 15, 16, 17].

In Ref. [18] we presented a systematic investigation of state transformations
of translationally invariant Matrix Product States (MPS) with periodic
boundary conditions. This family of states is physically relavant, as well as
mathematically tractable. In physics, MPS efficiently describe the ground
state of local, gapped Hamiltonians [19], as well as critical systems. They
also correspond one-to-one to the states that are prepared in the context of
sequential generation [20], where one system sequentially interacts with a set
of subsystems originally in a product state. From the mathematical point of
view, they admit an efficient description in terms of $N$ tripartite
(fiducial) states, or a single one if the state is translationally invariant.
In Ref. [18] we showed how local transformations and SLOCC classes of
translationally invariant MPS can be characterized. We demonstrated that these
properties can be inferred from the corresponding properties of the fiducial
states and certain cyclic structures of operators acting on these fiducial
states. We showed that these properties can be highly size dependent and
revealed many interesting features of prominent many-body states, such as the
cluster [21] and the AKLT state [22]. The methods introduced in [18] can also
be used to identify all local symmetries of MPS (not only corresponding to
global unitary operators [23, 24])222By local (global) symmetries we mean
(in-)homogeneous symmetries, i.e., that a different (the same) action operates
on each physical system.. Such a characterization induces a classification of
zero temperature phases of matter [25, 26, 27].

Whereas we provided in [18] a complete characterization of local symmetries
and SLOCC classes of translationally invaritant MPS with bond dimension $D=2$,
we extend these results here to the case $D=3$. Interestingly, this increment
leads to very different local properties of the corresponding MPS. This is not
only seen in the local symmetries, but also in the possible SLOCC
transformations and the SLOCC classes.

###  I.1 Outline

We summarize our results in Section II, where we also introduce some notation.
In Section III we discuss preliminaries. In the subsequent sections we
characterize the symmetries of normal translationally invariant MPS (TIMPS)
with physical dimension 2 and bond dimension $D=3$. There exist six SLOCC
classes for the fiducial state in this case. We focus on two of them featuring
considerably contrasting properties. In Section IV we discuss a fiducial state
for which only a discrete number of operators $g$ acting on the qubit give
rise to a symmetry of the state. In Section V we discuss a fiducial state for
which any operator $g$ acting on the qubit may give rise to a symmetry of the
state. We characterize the symmetries of the generated (normal) MPS for both
of them. Regarding the SLOCC classification we scrutinize on the latter SLOCC
class of the fiducial states (Section V). All the other classes can be treated
similarly. We show that, in contrast to MPS with bond dimension $2$, a much
larger variety of local and global symmetries occur in this class. Whereas
more local symmetries should enable more SLOCC transformation, we show that
this is not the case. In fact, we show that any SLOCC transformation which is
possible within those states is realizable with a global SLOCC operation.
Finally, in Section VI we study fiducial states represented by diagonal matrix
pencil for a bond dimension $D\geq 3$. In Appendices A to E and G we present
additional details on the concepts used, as well as proofs of claims made in
Sections IV and V. In Appendix F we discuss symmetries of MPS associated to
fiducial states belonging to one of the four remaining SLOCC classes.

##  II Summary of results

Before we summarize our findings, we introduce the following notation and
basic concepts, which are needed in order to formulate our results. MPS are
multipartite states defined in terms of three-partite tensors. We denote the
physical dimension of the MPS by $d$ and the bond dimension by $D$. Given a
rank-three tensor $A$ with respective index ranges $0\ldots d-1$, $0\ldots
D-1$, and $0\ldots D-1$ we often write

$|A\rangle=\sum_{i=0}^{d-1}\sum_{\alpha,\beta=0}^{D-1}A^{i}_{\alpha\beta}|i\rangle\otimes|\alpha\rangle\otimes|\beta\rangle.$
(1)

An MPS on $N$ subsystems is then defined in terms of the (in general site-
dependent) tensors $A_{i}$ ($i\in\\{0,\ldots,N-1\\}$) by

$|\Psi\rangle=\sum_{i_{0},\ldots,i_{N-1}}\text{tr}\left(A_{0}^{i_{0}}\dots
A_{N-1}^{i_{N-1}}\right)|i_{0}\dots i_{N-1}\rangle.$ (2)

If $A=A_{0}=\ldots=A_{N-1}$ the MPS is translationally invariant (TI). In the
TI case we may call $|\Psi\rangle=|\Psi(A)\rangle$ the MPS generated by the
fiducial tripartite state $A$. In fact, $A$ generates a whole family of MPS of
arbitrarily large system size, $N$.

Let us briefly recall some concepts from entanglement theory that are relevant
for this work. Two $N$-partite states $|\psi\rangle$, $|\phi\rangle$ are said
to be local unitary (LU) equivalent ($|\psi\rangle\sim_{LU}|\phi\rangle$) if
there exists a local unitary operator $u=u_{1}\otimes\ldots\otimes u_{N}$ such
that $u|\psi\rangle=|\phi\rangle$. If $|\psi\rangle$ can be transformed into
$|\phi\rangle$ via LOCC with finite probability of success, this
transformation is said to be possible via stochastic LOCC (SLOCC) and we write
$|\psi\rangle\to|\phi\rangle$. Note that $|\psi\rangle\rightarrow|\phi\rangle$
holds if and only if there exists a local operator
$g=g_{0}\otimes\ldots\otimes g_{N-1}$ such that $|\phi\rangle=g|\psi\rangle$
[5]. If $|\psi\rangle\to|\phi\rangle$ and $|\phi\rangle\to|\psi\rangle$ the
two states are said to be SLOCC equivalent. This is the case if and only if
there exists an invertible local operator $g$ such that
$|\phi\rangle=g|\psi\rangle$ [9]. The corresponding equivalence classes are
called SLOCC classes. We denote the group of local symmetries333Note that we
consider here only invertible local symmetries. That is, we do not
characterize those local operators, which annihilate the state. In the cases
investigated here, such symmetries contain necessarily two rank one
projectors. of a state $|\psi\rangle$ by

$\mathcal{S}_{|\psi\rangle}=\\{S:S|\psi\rangle=|\psi\rangle,S=S_{0}\otimes
S_{1}\otimes\ldots\\},$

where $S_{i}\in{GL}({d_{i}},\text{C})$ and $d_{i}$ denotes the local
dimensions of $|\psi\rangle$.

We will focus here on normal (for definition see Section III) Translationally
Invariant MPS (TIMPS) with physical dimension $d=2$ and bond dimension $D=3$
and discuss the higher bond dimensional case in Section VI. We study the local
symmetries of MPS, i.e., for MPS $|\Psi\rangle$ we characterize the set
$\mathcal{S}_{|\Psi\rangle}$. Moreover, we study the SLOCC classes of MPS.
Note that both, the local symmetries as well as the SLOCC classification might
depend on the number of subsystems.

In Ref [18] we showed that the local symmetries, possible SLOCC
transformations, and the SLOCC classes of normal MPS, $|\Psi(A)\rangle$, are
determined by certain cyclic structures of operators that are solely defined
by its fiducial state, $|A\rangle$. More precisely, they are determined by the
properties of the symmetry group of the fiducial state
$\mathcal{S}_{|A\rangle}$, which we also denote by

$G_{A}=\\{h=g\otimes x\otimes y^{T}\ |\ h|A\rangle=|A\rangle\\},$ (3)

where $T$ denotes the transpose in the standard basis. Given $G_{A}$, the only
operators which can occur in a symmetry of the normal MPS are those, which act
on the qubit system, i.e. the operators $g$ in Eq. (3). We call these
symmetries the qubit symmetries. The symmetries of normal MPS are determined
by specific properties of the symmetries of the fiducial state, so-called
_cycles_ (see [18] and Section III). To give an example, a symmetry of the
form $g_{0}\otimes g_{1}\otimes g_{0}\otimes g_{1}\otimes
g_{0}\ldots\in\mathcal{S}_{|\psi\rangle}$ stems from a 2-cycle in the symmetry
group of $|A\rangle$. It should be noted that the concept of cycles allows to
characterize the full symmetry group of normal MPS. For non-normal MPS, the
concept might yield only a subgroup of the symmetry group. Whether a TIMPS is
normal or not depends only on properties of the fiducial state.

Note that if the two fiducial states generating normal MPS are SLOCC
inequivalent, then the same holds for the MPS. The fiducial states
corresponding to the MPS of interest ($D=3$) can be divided into six distinct
SLOCC classes [17]. We focus here on two of them, which show considerably
contrasting properties with respect to their symmetry groups. As the two
classes we focus on can be considered as the two extreme cases, all the other
classes can be treated similarly. The two considered SLOCC classes of the
fiducial states are represented by (the notation used here will become clear
afterwards):

  1. (i)

$|M(\omega)\rangle=|0\rangle(|00\rangle+\omega|11\rangle+\omega^{2}|22\rangle)+|1\rangle(|00\rangle+|11\rangle+|22\rangle)$,
where $\omega=e^{i\frac{2\pi}{3}}$ (see Section IV)

  2. (ii)

$|LLT\rangle=|0\rangle(|01\rangle+|22\rangle)+|1\rangle(|00\rangle+|12\rangle)$,
(see Section V).

Considered SLOCC classes of the fiducial state
---
$M(\omega)$ |  $LLT$
Representatives
$|0\rangle(|00\rangle+\omega|11\rangle+\omega^{2}|22\rangle)+|1\rangle(|00\rangle+|11\rangle+|22\rangle)$ |  $|0\rangle(|01\rangle+|22\rangle)+|1\rangle(|00\rangle+|12\rangle)$
Symmetries of the fiducial state
discrete set of 6 possible operators $g$ appearing on physical site |  3-parametric family, any operator $g$ acting on the physical site gives rise to a symmetry
$g$’s are unitary |  $g$’s are non-compact
Possible (minimal) cycle lengths
1, 2, 3, 4, 6 |  $N$ for any $N\in\mathbb{N}$
Symmetries of normal MPS
unitary (finite) |  both unitary (finite) and non-unitary (non-compact)
diagonalizable (as unitary) |  both diagonalizable and non-diagonalizable
SLOCC among normal MPS
• examples of SLOCC equivalence via global operations • examples where non-global operations required $\rightarrow$ equivalence is $N$-dependent |  global operations suffice $\rightarrow$ equivalence is not $N$-dependent
Results for non-normal MPS
no other than the already identified cycles appear among the non-normal MPS |  The well-known Majumdar–Ghosh states appear as a special case, certain permutation-invariant states appear as another special case. We find SLOCC equivalences through non-global, but not global, operations (in contrast to normal MPS)
Comparison to $D=2$ (GHZ- and W- fiducial states, see [18])
GHZ-states: symmetry group of order $1$, $2$, or $2^{N}$ |  W-states: 2-cycle leading to 1-(complex)-parametric symmetry group
Table 1: Highlights of the obtained results. Out of the six possible SLOCC
classes for fiducial states with $D=3$, we consider two representatives with
considerably contrasting properties. The representative of the class
$M(\omega)$ (see left column) leads to only six different, unitary, operators
appearing as qubit symmetries. The representative of the class $LLT$ has a
symmetry group in which any operator $g$ appears as a qubit symmetry. Note
that the properties of the remaining four representatives lie between those
two extremal cases, as do fiducial states for $D=2$. In this table we
summarize and illustrate how the substantial differences in the symmetry
groups lead to considerably contrasting properties in the generated MPS
regarding the exhibited symmetries and SLOCC equivalence. We also compare to
properties of ($D=2$)-MPS generated by W- and GHZ-states (left and right
bottom).

The SLOCC class represented by some state $|A\rangle$ is given by $a\otimes
b\otimes c|A\rangle$, for any invertible operators $a,b,c$. In order to
determine both, the symmetries as well as the SLOCC classes of all MPS which
correspond to a fiducial state belonging to the SLOCC class represented by
$|A\rangle$, it suffices to consider fiducial states of the form
$\text{1}\otimes b\otimes\text{1}|A\rangle$ only [18].

The main result of the present article is a full characterization of the local
symmetries of normal MPS generated by fiducial states within these two SLOCC
classes, as well as a full characterization of SLOCC equivalence among normal
MPS corresponding to case (ii)—we only outline the procedure for case (i), as
the SLOCC classification is much simpler in that case. In the case (i), the
set of qubit symmetries are finite and unitary (in fact, they form a unitary
representation of the symmetric group $S_{3}$) 444Note that in case there
exist only finitely many symmetries, one can always chose a representative of
the SLOCC class whose symmetry is unitary.. In case (ii), any operator $g$
acting on the qubit system, i.e. on the physical system, leads to a symmetry
of the fiducial state. Stated differently, there is as much freedom in the
qubit symmetries as there could possibly be. A goal of this work is to
illustrate how the contrasting properties of the two considered classes of
fiducial states manifest also in the properties of the associated MPS. We
highlight this in form of a comparison of selected properties of the MPS in
Table 1. The properties of the symmetry group of the remaining four
representatives of the fiducial states lie between the two considered extreme
cases. Note that the symmetry groups of the fiducial states for $D=2$, the
GHZ- and the W-state exhibiting 1- and 2-parametric qubit symmetries (we count
complex parameters here and throughout the remainder of the article),
respectively, do not draw near the here considered extremal cases, either.
This is reflected in the properties of the generated MPS (see Table 1). The
outlined main results are accompanied by several results on non-normal MPS, a
discussion on bond dimension $D>3$, and results on MPS associated to the
remaining SLOCC classes of fiducial states for $D=3$ (see Sections V.5, VI,
and Appendix F).

Let us now discuss the results obtained for the two respective fiducial
states. In the case (i), $|M(\omega)\rangle$, as mentioned above, the set of
qubit symmetries is finite and unitary. This leads to finitely many local
symmetries of the corresponding MPS (for arbitrary system size). We determine
all possible local symmetries (see Table 2). All minimal cycles are of length
$1,2,3,4,6$. Furthermore, we characterize all fiducial states which generate
normal MPS and which lead to the symmetries presented in Table 2. Despite the
fact that deciding whether a family of states is normal or not can be
cumbersome, a complete parametrization of the corresponding fiducial states,
i.e. the operators $b$ can be found in Appendix A.

Given that the set $G_{A}$ is finite it is straightforward to determine all
possible SLOCC transformations. Stated differently, it is easy to determine
which pairs of fiducial states lead to MPSs $|\Psi\rangle$ and $|\Phi\rangle$
such that $|\Psi\rangle\propto g_{0}\otimes g_{1}\ldots\otimes
g_{N-1}|\Phi\rangle$, where $g_{i}$ denote local regular matrices. We outline
how this question can be answered and how the SLOCC classification can be
derived. In fact, many potential SLOCC transformations can be ruled out due to
the incompatibility of the symmetry groups of the corresponding states. Within
the remaining cases we provide examples of states which are not
SLOCC–equivalent, those which are via global transformations and those, which
require non–global transformations.

The reason for investigating the case (ii), i.e. TIMPS which correspond to
fiducial states within the SLOCC class represented by $|LLT\rangle$, is that
the properties of the fiducial states are in stark contrast to the previously
considered case. Thus, also the properties of the corresponding MPS can be
expected to be. In fact, in this class, the fiducial state has infinitely many
qubit symmetries. As mentioned above, actually, any operator $g$ acting on the
physical system leads to a symmetry of the fiducial state. This is why the
study of this class is particularly interesting. As we show, this leads to a
huge variety of local symmetries of the corresponding MPS. Any possible cycle
length actually appears, moreover, finite as well as infinite symmetry groups
emerge. Among them, we find diagonalizable as well as non-diagonalizable
symmetries. A complete characterization of all possible symmetries for normal
MPS is presented in Figure 4. Although such large symmetry groups could lead
one to believe that there are many possible SLOCC transformation, we show
that, surprisingly, the opposite is true. All possible SLOCC transformations
can be performed via global operations, i.e. for any two states,
$|\Psi\rangle$ and $|\Phi\rangle$, which are SLOCC equivalent, there exits a
global operator $g^{\otimes N}$ such that $|\Psi\rangle\propto g^{\otimes
N}|\Phi\rangle$. Note that, in contrast to case (i) and also the case of TIMPS
with bond dimension $D=2$ MPS generated by GHZ states (see [18]), this implies
that the existence of a SLOCC transformation among two TIMPS does not depend
on the system size. In contrast to case (i), the characterization of the SLOCC
classes is more challenging due to the large symmetry groups. We provide a
parametrization of representatives of each SLOCC class in Figure 5. The fact
that two states are in the same SLOCC class if and only if they are related to
each other via a global operation leads to a huge variety of SLOCC classes.

As mentioned above, only for normal MPS it holds that the whole local symmetry
group of the TIMPS can be determined via the local symmetries of the
corresponding fiducial state. For non-normal MPS, the methodology of cycles is
still useful, but the determined symmetries might form a subgroup of the
symmetry group of the MPS, only. Interestingly, in the case (ii), the whole
symmetry group can also be determined for certain non normal MPS. This is due
to the fact that those states correspond to permutationally invariant states,
for which the local symmetry groups are known [28, 29, 14]. Moreover, in case
(ii) it turns out that certain non-normal MPS are well–known states, the
Majumdar–Ghosh states. We show that non–normal MPS show a distinct behaviour
compared to normal ones (see Section V.5). Not only do we provide examples of
non–normal states which possess a much larger symmetry group than the one
determined by the fiducial state, but also examples of states which are
SLOCC–equivalent, but cannot be transformed into each other via a global
transformation, in contrast to all normal MPS corresponding to case (ii).

###  II.1 Particularly interesting states

In this subsection we present selected fiducial states, which generate MPS
with particularly interesting properties with respect to their symmetry group.
In light of these examples it is evident that bond dimension $D=3$ allows for
much more diverse symmetry groups than is the case for bond dimension $D=2$
[18]. For the examples presented here we use a notation to label the fiducial
states reflecting the properties of the symmetry groups as follows. We use the
notation $|1/2\,G_{(\infty)}\,D^{l_{1},l_{2},\ldots}_{(\infty)}\rangle$ to
label a fiducial state (primarily) according to the properties of the symmetry
group of the generated MPS. The notation should be read in three parts:
"$1/2$", "$G_{(\infty)}$", and "$D^{l_{1},l_{2},\ldots}_{(\infty)}$". It
should be understood as follows. The first part, "$1/2$", should read either
"1", or "2", and indicates whether the fiducial state belongs to the SLOCC
class represented by $|M(\omega)\rangle$ (in case of "1"), or $|LLT\rangle$
(in case of "2"). The second part, "$G_{(\infty)}$", describes the global
symmetries of the generated MPS. It should read "$G$" ("$G_{\infty}$"), in
case finitely many (infinitely many) nontrivial global symmetries are present.
The third part, "$D^{l_{1},l_{2},\ldots}_{(\infty)}$", describes the local
symmetries of the MPS. As before, the presence of the subscript $\infty$
indicates whether there are finitely of infinitely such symmetries. Moreover,
the integers $l_{1},l_{2},\ldots$ are used to indicate that the local part of
the symmetry group changes depending on whether $l_{1},l_{2},\ldots$ divide
the particle number $N$. In case the MPS does not posses any nontrivial local
(any nontrivial global) symmetry, we simply omit the "$D$" ("$G$") part.
Clearly, the naming scheme does not allow to unambiguously identify MPS, but
it suffices to distinguish the examples considered here.

####  II.1.1 Examples of fiducial states within the SLOCC class represented by
$|M(\omega)\rangle$

The first example is given by the fiducial state
$|1\,G\rangle=|012\rangle+|021\rangle+|101\rangle+|102\rangle+\omega(|001\rangle+|010\rangle+|110\rangle+|112\rangle)+\omega^{2}(|002\rangle+|020\rangle+|120\rangle+|121\rangle)$.
The corresponding tensor $A$ reads

$A^{0}=\begin{pmatrix}0&\omega&\omega^{2}\\\ \omega&0&1\\\
\omega^{2}&1&0\end{pmatrix}\quad\text{and}\quad A^{1}=\begin{pmatrix}0&1&1\\\
\omega&0&\omega\\\ \omega^{2}&\omega^{2}&0\end{pmatrix}.$

The generated MPS $|\Psi_{1\,G}\rangle$ has global symmetries only. The
symmetry group is finite and unitary and given by $g^{\otimes N}$, where $g$
is any of the six operators generated by $\sigma_{x}=\begin{pmatrix}0&1\\\
1&0\end{pmatrix}$ and $\operatorname{diag}(\omega,1)$.

The second example is given by the fiducial state
$|1\,G\,D^{3}\rangle=|000\rangle+|002\rangle+|012\rangle+|100\rangle+|101\rangle+|121\rangle+\omega(|001\rangle+|021\rangle+|022\rangle+|102\rangle+|111\rangle+|112\rangle)+\omega^{2}(|010\rangle+|011\rangle+|020\rangle+|110\rangle+|120\rangle+|122\rangle)$.
The corresponding tensor $A$ reads

$A^{0}=\begin{pmatrix}1&\omega&1\\\ \omega^{2}&\omega^{2}&1\\\
\omega^{2}&\omega&\omega\end{pmatrix}\quad\text{and}\quad
A^{1}=\begin{pmatrix}1&1&\omega\\\ \omega^{2}&\omega&\omega\\\
\omega^{2}&1&\omega^{2}\end{pmatrix}.$

The corresponding MPS $|\Psi_{1\,G\,D^{3}}\rangle$ has the same global
symmetries as the first example. If the particle number of the MPS, $N$, is no
multiple of 3, these are the only symmetries and the symmetry group is thus
identical to the first example. If, however, $N$ is a multiple of 3, then
additional local symmetries emerge. Then, the symmetry group is comprised of
18 elements and is generated by repeating sequences (what we will later on
call cycles) of $g_{0}\otimes g_{1}\otimes g_{2}$, as well as by repeating
sequences of $g_{0}\otimes g_{2}\otimes g_{1}$ (and translations thereof),
where $g_{0}=\sigma_{x}$, $g_{1}=\operatorname{diag}(\omega,1)\sigma_{x}$, and
$g_{2}=\operatorname{diag}(\omega,1)^{2}\sigma_{x}$.

The third example is given by the fiducial state
$|1\,G\,D^{2,6}\rangle=|000\rangle+|002\rangle+|022\rangle+|100\rangle+|101\rangle+|111\rangle+\omega(|001\rangle+|011\rangle+|012\rangle+|102\rangle+|121\rangle+|122\rangle)+\omega^{2}(|010\rangle+|020\rangle+|021\rangle+|110\rangle+|112\rangle+|120\rangle)$.
The corresponding tensor $A$ reads

$A^{0}=\begin{pmatrix}1&\omega&1\\\ \omega^{2}&\omega&\omega\\\
\omega^{2}&\omega^{2}&1\end{pmatrix}\quad\text{and}\quad
A^{1}=\begin{pmatrix}1&1&\omega\\\ \omega^{2}&1&\omega^{2}\\\
\omega^{2}&\omega&\omega\end{pmatrix}.$

The generated MPS $|\Psi_{1\,G\,D^{2,6}}\rangle$ possesses the global symmetry
$g^{\otimes N}$ for $g=\sigma_{x}$. In case of an even particle number $N$,
the MPS possesses local symmetries and the symmetry group is generated by
repeating sequences of $g_{0}\otimes g_{1}$, where
$g_{0}=\operatorname{diag}(\omega,1)\sigma_{x}$ and
$g_{1}=\operatorname{diag}(\omega,1)^{2}\sigma_{x}$. Moreover, in case that
$N$ is divisible by 6, additional local symmetries emerge, repeating sequences
of $g_{0}\otimes g_{1}\otimes g_{2}\otimes g_{3}\otimes g_{4}\otimes g_{5}$,
where $g_{0}=g_{5}=\operatorname{diag}(\omega,1)\sigma_{x}$,
$g_{1}=g_{4}=\sigma_{x}$, and
$g_{2}=g_{3}=\operatorname{diag}(\omega^{2},1)\sigma_{x}$ generate the
symmetry group, then. This example is particularly interesting as the MPS
$|\Psi_{1\,G\,D^{2,6}}\rangle$ is SLOCC equivalent to the previous example,
$|\Psi_{1\,G\,D^{3}}\rangle$, for an even particle number $N$. However, this
is not the case if $N$ is odd. This leads to the following possible
situations. If 2 divides $N$, but 3 does not, then
$|\Psi_{1\,G\,D^{3}}\rangle$ and $|\Psi_{1\,G\,D^{2,6}}\rangle$ are SLOCC
equivalent and their symmetry groups are of order 6. If both 2 and 3 divide
$N$, then $|\Psi_{1\,G\,D^{3}}\rangle$ and $|\Psi_{1\,G\,D^{2,6}}\rangle$ are
SLOCC equivalent and their symmetry groups are of order 18. If 3 divides $N$,
but 2 does not, then $|\Psi_{1\,G\,D^{3}}\rangle$ and
$|\Psi_{1\,G\,D^{2,6}}\rangle$ are not SLOCC equivalent. Remarkably, the
orders of the corresponding symmetry groups differ. The symmetry group of
$|\Psi_{1\,G\,D^{3}}\rangle$ is of order 18, while the symmetry group of
$|\Psi_{1\,G\,D^{2,6}}\rangle$ is of order 2. Finally, if neither 2, nor 3
divides $N$, then $|\Psi_{1\,G\,D^{3}}\rangle$ and
$|\Psi_{1\,G\,D^{2,6}}\rangle$ are not SLOCC equivalent, again the orders of
the symmetry groups differ. The symmetry group of $|\Psi_{1\,G\,D^{3}}\rangle$
is of order 6, while the symmetry group of $|\Psi_{1\,G\,D^{2,6}}\rangle$ is
of order 2.

The fourth example is given by the fiducial state
$|1\,D^{3}\rangle=|012\rangle-|021\rangle+|101\rangle+|102\rangle+\omega(|001\rangle-|010\rangle+|022\rangle-|110\rangle-|111\rangle+|112\rangle)+\omega^{2}(|002\rangle-|011\rangle+|020\rangle+|120\rangle-|121\rangle+|122\rangle)$.
The corresponding tensor $A$ reads

$A^{0}=\begin{pmatrix}0&\omega&\omega^{2}\\\ -\omega&-\omega^{2}&1\\\
\omega^{2}&-1&\omega\end{pmatrix}\quad\text{and}\quad
A^{1}=\begin{pmatrix}0&1&1\\\ -\omega&-\omega&\omega\\\
\omega^{2}&-\omega^{2}&\omega^{2}\end{pmatrix}.$

The generated MPS $|\Psi_{1\,D^{3}}\rangle$ does not exhibit any global
symmetry (except the trivial symmetry), however, it does possess local
symmetries if $N$ is a multiple of 3. Its symmetry group then is of order 4
and is generated by repeating sequences of $g_{0}\otimes g_{1}\otimes g_{2}$
(and translations thereof), where $g_{0}=\text{1}$ and
$g_{1}=g_{2}=\sigma_{x}$.

Additional possible symmetry groups (in fact, all possible symmetry groups for
normal MPS generated by a fiducial state that is in the SLOCC class of
$|M(\omega)\rangle$) are displayed in Figure 2. Examples of MPS exhibiting the
corresponding symmetry groups may be easily constructed with the help of Table
2.

####  II.1.2 Examples of fiducial states within the SLOCC class represented by
$|LLT\rangle$

The fifth example is given by the fiducial state
$|2\,G_{\infty}\rangle=|002\rangle-|012\rangle+|021\rangle+|022\rangle-|112\rangle+|120\rangle$.
The corresponding tensor $A$ reads

$A^{0}=\begin{pmatrix}0&0&1\\\ 0&0&-1\\\
0&1&1\end{pmatrix}\quad\text{and}\quad A^{1}=\begin{pmatrix}0&0&0\\\ 0&0&-1\\\
1&0&0\end{pmatrix}.$

The generated MPS $|\Psi_{2\,G_{\infty}}\rangle$ has a one-parametric symmetry
group (counting complex parameters) of global symmetries $g^{\otimes N}$,
where $g=\begin{pmatrix}1&x\\\ 0&1\end{pmatrix}$ for any $x\in\mathbb{C}$.

The sixth example is given by the fiducial state
$|2\,D^{m}_{\infty}\rangle=e^{i\frac{\pi}{m}}|002\rangle+|021\rangle+|022\rangle-e^{-i\frac{\pi}{m}}|112\rangle+|120\rangle$
for some $m\in\mathbb{N}$. The corresponding tensor $A$ reads

$A^{0}=\begin{pmatrix}0&0&e^{i\frac{\pi}{m}}\\\ 0&0&0\\\
0&1&1\end{pmatrix}\quad\text{and}\quad A^{1}=\begin{pmatrix}0&0&0\\\
0&0&-e^{-i\frac{\pi}{m}}\\\ 1&0&0\end{pmatrix}.$

The generated MPS $|\Psi_{2\,D^{m}_{\infty}}\rangle$ has a non-trivial
symmetry group only if the particle number $N$ is a multiple of $m$. Then, the
MPS exhibits a one-parameter symmetry group of local symmetries $g_{0}\otimes
g_{1}\otimes\ldots$, where $g_{k}=\begin{pmatrix}1&xe^{i\frac{2k\pi}{m}}\\\
0&1\end{pmatrix}$, $x\in\mathbb{C}$.

The seventh example is given by the fiducial state
$|2\,G\,D^{2}_{\infty}\rangle=i|002\rangle+|021\rangle+|022\rangle+i|112\rangle+|120\rangle+|122\rangle$.
The corresponding tensor $A$ reads

$A^{0}=\begin{pmatrix}0&0&i\\\ 0&0&0\\\ 0&1&1\end{pmatrix}\quad\text{and}\quad
A^{1}=\begin{pmatrix}0&0&0\\\ 0&0&i\\\ 1&0&1\end{pmatrix}.$

In case of an odd particle number $N$, the generated MPS
$|\Psi_{2\,G\,D^{2}_{\infty}}\rangle$ has a single non-trivial symmetry
$g^{\otimes N}$, where $g=i\sigma_{x}$. In case of an even particle number the
MPS exhibits a continuous symmetry group with symmetries $g_{0}\otimes
g_{1}\otimes g_{0}\otimes g_{1}\otimes\ldots$, where
$g_{0}=\begin{pmatrix}1&x\\\ x&1\end{pmatrix}$ and $g_{1}=g_{0}^{-1}$ for any
$x\in\mathbb{C}\setminus\\{1\\}$.

As an eighth example we give the well-known Majumdar-Ghosh states, which are
non-normal MPS possessing the symmetry group $g^{\otimes N}$ for any $g$. They
appear as a particular case within the SLOCC class of fiducial states
represented by $|LLT\rangle$ (see Section V.5). One such state may be obtained
considering the fiducial state
$|A\rangle=|002\rangle+|021\rangle-|112\rangle+|120\rangle$. The corresponding
tensor $A$ reads

$A^{0}=\begin{pmatrix}0&0&1\\\ 0&0&0\\\ 0&1&0\end{pmatrix}\quad\text{and}\quad
A^{1}=\begin{pmatrix}0&0&0\\\ 0&0&-1\\\ 1&0&0\end{pmatrix}.$

A list of all representatives of normal MPS generated by fiducial states
within the $|LLT\rangle$ class is presented in Figure 5, the emerging symmetry
groups of the generated MPS are displayed in Figure 4.

We briefly discuss the remaining SLOCC classes of the fiducial states in
Appendix F. Moreover, we show that for generic fiducial states of higher bond
dimension, there exists no non–trivial local symmetry. That is the set
$S_{|\Psi\rangle}=\\{\text{1}\\}$, which also implies that the SLOCC
classification is trivial.

##  III Preliminaries

In this section we review relevant concepts from the theory of MPS (Section
III.1) and of Ref. [18] (Section III.2).

###  III.1 Matrix Product States

As injective and normal MPS (not necessarily TI) play a crucial role in the
theory of MPS, we recall here their definitions.

######  Definition 1.

A MPS tensor $A$ is injective if the following map is injective:

$X\mapsto\sum_{i}\text{tr}\left(XA^{i}\right)|i\rangle.$

A MPS is injective if all the defining tensors are injective. An MPS is normal
if there exists an $L$ such that the contraction of any $L$ consecutive
tensors are injective, i.e., the following maps are injective:

$X\mapsto\sum_{j}\text{tr}\left(XA_{i}^{j_{1}}A_{i+1}^{j_{2}}\dots
A_{i+L-1}^{j_{L}}\right)|j_{1}\dots j_{L}\rangle.$

In the following we only consider MPS with $N\geq 2L+1$, if not stated
differently. In this case the following Fundamental Theorem of MPS
characterizes when two normal MPS generate the same state.

######  Theorem 1 (Fundamental Theorem of MPS [30]).

Two normal MPS given by tensors $A_{0},\ldots,A_{N-1}$ and
$B_{0},\ldots,B_{N-1}$ generate the same state $\Psi$ iff there exist regular
matrices $x_{0},\ldots,x_{N-1}$ such that
$A_{k}^{j}=x_{k}^{-1}B_{k}^{j}x_{k+1}$ for all $k$ and $j$, with $x_{N}\equiv
x_{0}$; that is, iff

$\displaystyle|A_{k}\rangle=\text{1}\otimes x_{k}^{-1}\otimes
x_{k+1}^{T}|B_{k}\rangle\ \forall\ k.$ (4)

The matrices $x_{0},\ldots,x_{N-1}$ are unique up to a multiplicative
constant.

Whenever we refer to MPS in the remainder of this paper we refer to normal TI
MPS, if not stated differently. We call ${\cal N}_{N,D}$ the set of normal,
translationally invariant MPS with bond dimension $D$ and $N\geq 2L+1$ sites.

###  III.2 Review of results on symmetries and local transformations of MPS

As mentioned before, the local symmetries of a normal MPS, $\Psi(A)$, are
determined by certain cyclic structures of operators that are solely defined
by its fiducial state, $A$ [18], $G_{A}$ (see Eq. (3). Two operators
$h_{0},h_{1}\in G_{A}$ with $h_{i}=g_{i}\otimes x_{i}\otimes y_{i}^{T}$ can be
concatenated, denoted as $h_{0}\to h_{1}$, if $y_{0}x_{1}\propto\text{1}$. A
sequence $\\{h_{i}\\}_{i=0}^{k-1}\subseteq G_{A}$ of $k$ elements in $G_{A}$
with

$h_{0}\to h_{1}\to\ldots\to h_{k-1}\to h_{0}$ (5)

is called a $k$-cycle. More explicitly, we have that the sequence
$\\{h_{i}\\}_{i=0}^{N-1}\subseteq G_{A}$ of $N$ elements in $G_{A}$ form a
$N$–cycle if the following conditions hold for any $0\leq k\leq N-2$:

$\displaystyle y_{k}x_{k+1}\propto\text{1}$ $\displaystyle
y_{N-1}x_{0}\propto\text{1}.$ (6)

We showed the following theorem.

######  Theorem 2 ([18]).

The local (global) symmetries of $\Psi(A)\in{\cal N}_{N,D}$ are in one-to-one
correspondence with the $N$-cycles (1-cycles) in $G_{A}$.

The symmetry of the state corresponding to the cycle $h_{0}\to
h_{1}\to\ldots\to h_{N-1}\to h_{0}$ is $g_{0}\otimes\dots\otimes g_{N-1}$.
Hence, one solely has to determine $G_{A}$ and find all $N$-cycles in this set
to characterize the local symmetries of $\Psi(A)$. In fact, this yields the
symmetries of all states in the family of normal MPS generated by $A$. In
practice, it is sufficient to characterize all minimal cycles of $G_{A}$ from
which all others can be obtained by concatenation. For example, a 3-cycle can
always be concatenated with itself to an $N$-cycle if 3 divides $N$. A
symmetry of the form $g^{\otimes N}$ is called global. The global symmetries
are defined in terms of 1-cycles, and thus require that there is a regular $x$
such that $g\otimes x^{-1}\otimes x^{T}|A\rangle=|A\rangle$ [18]. If $g$ is
unitary, this reduces to the well-known characterization of global unitary
symmetries of MPS [23, 24]. However, minimal cycles of length $N>1$ yield
local symmetries of the TI MPS $\Psi(A)$ that are not global and which are
generally not considered.

We often consider fiducial states $\text{1}\otimes b\otimes\text{1}|A\rangle$.
The concatenation conditions [see Eq. (6)] then read

$\displaystyle y_{k}bx_{k+1}b^{-1}\propto\text{1},$ $\displaystyle
y_{N-1}bx_{0}b^{-1}\propto\text{1}.$ (7)

In order to characterize SLOCC transformations among normal MPS, one first
notices that the corresponding fiducial states need to be SLOCC equivalent. We
considered in Ref. [18] the set

$G_{A,B}=\left\\{h=g\otimes x\otimes y^{T}\ |\ h|A\rangle=|B\rangle\right\\},$

As in the case of $G_{A}$ we can define $k$-cycles on $G_{A,B}$. Using the
notation $A\overset{N}{\rightarrow}B$ if an $N$–partite state $A$ can be
transformed via local operations into an $N$–partite state $B$, we proved the
following theorem in Ref. [18].

######  Theorem 3 ([18]).

$A\overset{N}{\rightarrow}B$ with local (global) transformations iff there
exists an $N$-cycle (1-cycle) in $G_{A,B}$.

In this theorem the operators which transform $A$ to $B$ are not necessarily
regular. Here, we focus on SLOCC transformations, i.e. on invertible matrices
on the physical (as well as the virtual) systems. As shown in [18], in order
to solve the problem of SLOCC–equivalence (and also the symmetries), it is
sufficient to consider fiducial state of the form
$|A_{b}\rangle=\text{1}\otimes b\otimes\text{1}|A\rangle$, where $|A\rangle$
denotes a representative of the SLOCC class of the fiducial states. Let us
briefly recall the reason for that. First, it is clear that the two fiducial
states corresponding to SLOCC–equivalent normal TIMPS must be
SLOCC–equivalent. Second, $g^{\otimes N}\Psi(A)$ is obviously SLOCC–equivalent
to $\Psi(A)$ and therefore the operator on the qubit system does not need to
be taken into account. And third, due to the fundamental theorem (see Theorem
1), an operator on the third system can be mapped to an operator on the
second. Clearly, the same argument applies when considering local symmetries.

According to Theorem 3, two normal TIMPS corresponding to the fiducial states
$|A_{b}\rangle$, $|A_{c}\rangle$, respectively, are SLOCC–equivalent iff there
exists an $N$-cycle in $G_{A_{b},A_{c}}$ (or, equivalently, in
$G_{A_{c},A_{b}}$). Using that $G_{A_{b},A_{c}}=(\text{1}\otimes
c\otimes\text{1})G_{A}(\text{1}\otimes b^{-1}\otimes\text{1})$ the existence
of such a cycle can be formulated in terms of the symmetries of the fiducial
state representing the SLOCC class as follows. The operators $h_{0},h_{1}\in
G_{A}$, with $h_{i}=g_{i}\otimes x_{i}\otimes y_{i}^{T}$, are called $(b\to
c)$-concatenated, if $y_{0}bx_{1}\propto c$. In this case we write
$h_{0}\xrightarrow{b\to c}h_{1}$. A sequence $\\{h_{i}\\}_{i=0}^{k-1}\subseteq
G_{A}$ is called a $(b\to c)$-$k$-cycle if

$h_{0}\xrightarrow{b\to c}h_{1}\xrightarrow{b\to c}\dots\xrightarrow{b\to
c}h_{k-1}\xrightarrow{b\to c}h_{0}.$ (8)

Stated explicitly, $\\{h_{i}\\}_{i=0}^{N-1}\subseteq G_{A}$ is a $(b\to
c)$-$N$-cycle if the following concatenation rules are fulfilled for any $k$
such that $0\leq k\leq N-2$

$\displaystyle y_{k}bx_{k+1}\propto c$ (9) $\displaystyle y_{N-1}bx_{0}\propto
c.$ (10)

As in the case of symmetries it might be possible that $G_{A,B}$ only contains
$k$-cycles with $k\geq 2$. Then $\Psi(A)\to\Psi(B)$ only holds if $k$ divides
$N$ and the corresponding SLOCC operator is not global, i.e., not of the form
$g^{\otimes N}$. Note that Theorem 3 was used in [18] to characterize all
SLOCC classes of normal MPS.

###  III.3 The fiducial states of TIMPS with physical dimension $d=2$ and bond
dimension $D=3$

As we reviewed above, the symmetries and the SLOCC classes of TIMPS can be
characterized by considering sets of operators, which are determined by the
three–partite fiducial states.

The fiducial states of TIMPS with physical dimension $d$ and bond dimension
$D$ are $d\times D\times D$ states. We focus on the case $d=2$ and $D=3$ and
discuss extensions of the results presented here in the appendix. For $d=2$
(and arbitrary $D$) the SLOCC classes have been determined using the theory of
Matrix Pencils (MPs) [31, 17, 32]. As this theory is also well suited to
determine the symmetries of the states, we briefly review it here. To the
three-partite state given in Eq. (1) we associate the homogeneous matrix
polynomial (matrix pencil),

$\displaystyle\mathcal{P}_{A}\equiv P_{(A^{0},A^{1})}\equiv\mu A^{0}+\lambda
A^{1}$ (11)

where $\mu,\lambda$ are complex variables and $A^{0},A^{1}\in\text{C}^{D\times
D^{\prime}}$. In Ref. [31] Kronecker showed that every matrix pencil is
strictly equivalent to a matrix pencil in so-called _Kronecker Canonical Form
(KCF)_ , which is characterized by a set of invariants (e.g., the finite and
infinite eigenvalues, eigenvalue size signatures and minimal indices). Stated
differently, for each matrix pencil $\mathcal{P}_{A}$ there exist regular
matrices, $x,y$ such that $x\mathcal{P}_{A}y$ has canonical form. The KCF
together with the results presented in [17, 32] can be used to determine both,
the SLOCC classes of the states as well as the symmetries of the state, as we
will show in the following.

As shown in [17] there is always an operation on the qubit that transforms a
$2\times D\times D$ state $A$ into a state whose matrix pencil has only finite
eigenvalues [17]. Furthermore, the operation on the qubit cannot change the
structure of the matrix pencil, but only its eigenvalues.

More precisely, the action of

$w=\begin{pmatrix}\alpha&\beta\\\ \gamma&\delta\end{pmatrix}\in
GL(2,\text{C})$

on the qubit changes the eigenvalues of the resulting MP from $\\{x_{i}\\}$ to

$x_{i}^{\prime}=\frac{\alpha x_{i}+\beta}{\gamma x_{i}+\delta}.$

Using all that it is then easy to see that the six SLOCC classes of $2\times
3\times 3$ systems are represented by the following states:

  1. (i)

$|M(\omega)\rangle=|0\rangle(|00\rangle+\omega|11\rangle+\omega^{2}|22\rangle)+|1\rangle(|00\rangle+|11\rangle+|22\rangle)$,
with corresponding MP $M^{1}(1)\oplus M^{1}(\omega)\oplus M^{1}(\omega^{2})$,
i.e., a MP with three distinct eigenvalues

  2. (ii)

$|D\rangle=|0\rangle(D\otimes\text{1})(|00\rangle+|11\rangle+|22\rangle)+|1\rangle(|00\rangle+|11\rangle+|22\rangle)$,
with $D$ a diagonal matrix with degenerate eigenvalues, which corresponds to a
MP with degenerate eigenvalues (disregarding biseparable states, there must be
one eigenvalue with degeneracy 1 and one eigenvalue with (algebraic and
geometric) multiplicity 2.)

  3. (iii)

$|J\rangle=|0\rangle(J\otimes\text{1})(|00\rangle+|11\rangle+|22\rangle)+|1\rangle(|00\rangle+|11\rangle+|22\rangle)$,
with $J$ a non-diagonalizable matrix in Jordan normal form, which corresponds
to a MP with degenerate eigenvalues (this case comprises three distinct SLOCC
classes)

  4. (iv)

$|LLT\rangle=|0\rangle(|01\rangle+|22\rangle)+|1\rangle(|00\rangle+|12\rangle)$,
with MP $L_{1}\oplus L_{1}^{T}$. In this case the MP does not have any
eigenvalue.

We will mainly focus here on the cases (i) and (iv) and will discuss the
symmetries of the remaining cases in Appendix F.

Let us already mention here that in order to determine the symmetries in case
(i)-(iii), one first has to ensure that the eigenvalues are at most permuted
by the action on the qubit and then choose the operators $x,y$ such that the
state is again transformed into KCF 555Such a transformation is then always
possible.. This leads to the following Lemma (see [33]).

######  Lemma 1.

Let $A$ be a $2\times D\times D$ state with only finite eigenvalues,
$\\{x_{i}\\}_{i=1}^{l}$. Then $G_{A}$ is characterized as follows. For an
invertible matrix $g\in GL(\text{C},2)$ on the qubit, with

$\displaystyle g=\begin{pmatrix}\alpha&\beta\\\ \gamma&\delta\end{pmatrix},$
(12)

there exist invertible matrices $x,y\in GL(\text{C},D)$ on the qudits such
that $s\otimes x\otimes y^{T}\in G_{A}$ iff there exists a permutation
$\sigma\in S(l)$ of the eigenvalues $\\{x_{i}\\}_{i=1}^{l}$ such that

$\displaystyle\frac{\alpha x_{i}+\beta}{\gamma x_{i}+\delta}=x_{\sigma(i)}\
\forall i$ (13)

and such that $\sigma$ only permutes eigenvalues of matching multiplicities
(i.e., $x_{i}$ and $x_{\sigma(i)}$ have coinciding size signatures for all
$i$).

Let us from now on refer to $g\in GL(\text{C},2)$ in $g\otimes x\otimes
y^{T}\in G_{A}$ as a qubit symmetry and to $x,y$ as qudit symmetries. From
Lemma 1 we have that for a given fiducial state, $A$, the qubit symmetry can
be easily determined via Eq. (13). The corresponding qudit symmetry $x$ and
$y$ can then be computed as explained above (see, e.g., Ref. [34]).

In case the matrix pencil does not possess any eigenvalue (see case (iv)), it
has been shown in [32] that for any operator $g$ there exist operators $x,y$
such that $g\otimes x\otimes y^{T}\in G_{A}$.

Note that the symmetry group of the fiducial state (as of any state) is
generated by symmetries of the form $\text{1}\otimes B\otimes C$ as well as by
symmetries $g\otimes B_{g}\otimes C_{g}$, for predefined operators $B_{g}$ and
$C_{g}$.

In the subsequent sections we will use the results reviewed here to determine
the symmetries of the fiducial states, which we then use to determine the
symmetries and the SLOCC classes of the corresponding TIMPS.

##  IV Symmetries and SLOCC classes of the TIMPS $\Psi(M(\omega))$

We first determine the symmetries of the fiducial state using MP-theory and
then use the results summarized above to determine the symmetries of the
corresponding MPS. As mentioned before, a representative of this SLOCC class
is the state
$|M(\omega)\rangle=|0\rangle(|00\rangle+\omega|11\rangle+\omega^{2}|22\rangle)+|1\rangle(|00\rangle+|11\rangle+|22\rangle)$,
where $\omega=e^{\frac{i2\pi}{3}}$. The corresponding matrix pencil reads

$\displaystyle\mathcal{P}$ $\displaystyle=M^{1}(1)\oplus M^{1}(\omega)\oplus
M^{1}(\omega^{2})$ $\displaystyle=\begin{pmatrix}\mu+\lambda&0&0\\\
0&\omega\mu+\lambda&0\\\ 0&0&\omega^{2}\mu+\lambda\end{pmatrix}.$

Let us remark that an alternative representative is the state
$|0\rangle(|11\rangle+|22\rangle)+|1\rangle(|00\rangle+|11\rangle)$. We
consider the alternative representative when discussing normality in Appendix
B, as it leads to a more sparsely populated tensor $A$. Here, we will stick to
the representative $|M(\omega)\rangle$ though, as the group structure of the
local symmetries of the generated TIMPS will appear more natural for this
representative.

###  IV.1 Symmetries of the fiducial state

In this subsection we discuss the symmetries of the fiducial state
$|M(\omega)\rangle$.

As any symmetry of the form $\text{1}\otimes B\otimes C$ must fulfill that
$B\mathcal{P}C^{T}\propto\mathcal{P}$ (see Sec III), we obtain for any such
symmetry (choosing a convenient normalization) that
$B=\operatorname{diag}(1,B_{11},B_{22})$,
$C=\operatorname{diag}(1,1/B_{11},1/B_{22})$.

The matrix pencil has three distinct eigenvalues,
$(x_{0},x_{1},x_{2})=(1,\omega,\omega^{2})$. It follows that there is a
discrete set of six operators $g$ appearing as the first local operator in the
symmetries of the fiducial state. These operators correspond to all possible
permutations of the eigenvalues of the matrix pencil. We will index these
operators by $\sigma\in S_{3}$, where $\sigma$ describes the permutation of
the eigenvalues. We will use the notation $\sigma=\begin{pmatrix}0&1&2\\\
\sigma(0)&\sigma(1)&\sigma(2)\end{pmatrix}=(\sigma(0),\sigma(1),\sigma(2))$.
Moreover, we use permutation matrices

$\displaystyle P_{\sigma}=\sum_{i}|\sigma_{i}\rangle\langle i|.$

Defining

$\displaystyle{g_{(0,1,2)}}=\text{1},\ {g_{(0,2,1)}}=\begin{pmatrix}0&1\\\
1&0\end{pmatrix},$ $\displaystyle{g_{(2,1,0)}}=\begin{pmatrix}0&\omega^{2}\\\
1&0\end{pmatrix},\ {g_{(1,0,2)}}=\begin{pmatrix}0&\omega\\\ 1&0\end{pmatrix},$
$\displaystyle{g_{(2,0,1)}}=\begin{pmatrix}\omega&0\\\ 0&1\end{pmatrix},\
{g_{(1,2,0)}}=\begin{pmatrix}\omega^{2}&0\\\ 0&1\end{pmatrix}$ (14)

we find symmetries ${g}\otimes B_{g}\otimes C_{g}$ of $|M(\omega)\rangle$ for

$\displaystyle{g}$ $\displaystyle=g_{\sigma}$ $\displaystyle B_{g}$
$\displaystyle=P_{\sigma}^{-1}D_{\sigma}^{-1}$ $\displaystyle C_{g}^{T}$
$\displaystyle=P_{\sigma},$ (15)

where $D_{\sigma}=\begin{cases}\text{1}&\operatorname{sgn}(\sigma)=1\\\
\operatorname{diag}(1,\omega,\omega^{2})&\operatorname{sgn}(\sigma)=-1\end{cases}$,
where $\sigma\in S_{3}$, and where $\operatorname{sgn}$ denotes the signum of
$\sigma$. We will refer to $\sigma\in\\{(0,2,1),(2,1,0),(1,0,2)\\}$ ($\sigma$
with $\operatorname{sgn}(\sigma)=-1$) as transpositions and to
$\sigma\in\\{(1,2,0),(2,0,1)\\}$ ($\sigma$ with
$\operatorname{sgn}(\sigma)=1$, but $\sigma\neq\text{1}$) as cyclic
permutations of length 3. The symmetry group of the state $|M(\omega)\rangle$
is given by (see Section III)

$\displaystyle{g}\otimes B_{g}B\otimes C_{g}C.$ (16)

We will use these symmetries to determine the local symmetries of the
corresponding (normal) TIMPS. As mentioned above, to determine then the TIMPS
which are SLOCC equivalent it is sufficient to consider the fiducial states of
the form $\text{1}\otimes b\otimes\text{1}|M(\omega)\rangle$. The tensor
$A_{b}$ associated to this state reads

$\displaystyle A_{b}^{0}=\begin{pmatrix}b_{00}&b_{01}&b_{02}\\\ \omega
b_{10}&\omega b_{11}&\omega b_{12}\\\
\omega^{2}b_{20}&\omega^{2}b_{21}&\omega^{2}b_{22}\end{pmatrix},\
A_{b}^{1}=b.$ (17)

The corresponding symmetries are obviously of the form

$\displaystyle{g}\otimes bB_{g}Bb^{-1}\otimes C_{g}C.$

###  IV.2 Local symmetries of the TIMPS $\Psi(M(\omega))$

Let us now characterize the symmetries of normal TIMPS generated by
$\text{1}\otimes b\otimes\text{1}|M(\omega)\rangle$. To ease notation, we
denote the local symmetry group of this fiducial state by $G_{b}$ throughout
this whole section. As explained in the preliminaries, to identify the
symmetries of the TIMPS, we first characterize all possible $N$-cycles
$({g}_{\sigma_{k}})_{k=0}^{N-1}$ in $G_{b}$. After identifying all possible
cycles, we characterize all normal TIMPS (i.e. all $b$) admitting each of the
identified cycles.

Using the symmetry of the representative $|M(\omega)\rangle$, $x=B_{g}B$,
$y=\left(C_{g}C\right)^{T}$ [see Eq. (16)] and inserting in the concatenation
conditions for $\text{1}\otimes b\otimes\text{1}|M(\omega)\rangle$ [see Eq.
(III.2)] we obtain

$\displaystyle bP_{\sigma_{k+1}}^{-1}B_{k+1}D_{\sigma_{k+1}}^{-1}b^{-1}\propto
P_{\sigma_{k}}^{-1}B_{k},$ (18)

where $B_{k}$ are arbitrary diagonal matrices stemming from the symmetries of
the form $\text{1}\otimes B\otimes C$, and $P_{\sigma_{k}}$ as well as
$D_{\sigma_{k}}$ stem from the symmetries ${g}\otimes B_{g}\otimes C_{g}$ as
in Eq. (IV.1).

Note that Eq. (18) comprises a similarity transformation among so-called
_monomial matrices_ , which are also called _generalized permutation
matrices_. These are (invertible) matrices that can be written as a product of
a permutation matrix and a diagonal matrix. In the following, we use the
Fourier transform $\mathcal{F}=\begin{pmatrix}1&1&1\\\ 1&\omega&\omega^{2}\\\
1&\omega^{2}&\omega\end{pmatrix}$, as well as Fourier transforms acting on
subspaces $\mathcal{F}_{01}=\begin{pmatrix}1&1&0\\\ 1&-1&0\\\
0&0&1\end{pmatrix}$ ($\mathcal{F}_{02}$ and $\mathcal{F}_{12}$ analogously).

######  Observation 1.

Let $P_{\sigma}$ be a permutation matrix of dimension 3 and
$D=\operatorname{diag}(d_{0},d_{1},d_{2})$,
$d_{i}\in\mathbb{C}\setminus\\{0\\}$. Then the eigenvalues and eigenvectors of
$P_{\sigma}D$ can be determined as follows. In case $\sigma$ is trivial, the
eigenvalues are the entries of $D$ and the eigenvectors are the computational
basis vectors. In case $\sigma$ is a 3-cycle, the eigenvalues read $d$,
$d\omega$, $d\omega^{2}$, where $d=(d_{0}d_{1}d_{2})^{1/3}$. The eigenvectors
are given by $\tilde{D}\mathcal{F}$ with
$\tilde{D}=\operatorname{diag}(\tilde{d}_{0},\tilde{d}_{1},\tilde{d}_{2})$,
where $\tilde{d}$ may be determined via the recurrence relation
$\tilde{d}_{\sigma(i)}=\tilde{d}_{i}d_{i}/d$. Finally, in case $\sigma$ is the
transposition $(0,2,1)$, the eigenvalues read $d_{0}$ and
$\pm\sqrt{d_{1}d_{2}}$, and similarly for the remaining transpositions. The
eigenvectors are given by $\tilde{D}\mathcal{F}_{12}$ with
$\tilde{D}=\operatorname{diag}(1,\sqrt{d_{2}},\sqrt{d_{1}})$, and similarly
for the remaining transpositions.

As an immediate consequence, considering Eq. (18), we observe that if
$\sigma_{k}$ is a transposition for some $k$, then $\sigma_{l}$ cannot be a
three-cyclic permutation for any $l$ (and vice versa), due to the mismatch in
the eigenvalues of the right-hand side and the left-hand side of Eq. (18).

######  Observation 2.

Any cycle in $G_{b}$ involving a transposition, $\sigma_{k}$, cannot involve a
cyclic permutation of length 3, $\sigma_{l}$, and vice versa.

###### Proof.

We make use of Observation 1. Let us assume that $P_{\sigma_{k}}$ is a
transposition. Then, two eigenvalues of $P_{\sigma_{k}}^{-1}B_{k}$ differ by
multiplication with $-1$. However, if $P_{\sigma_{l}}$ is a cyclic permutation
of length 3, then any pair of eigenvalues of $P_{\sigma_{l}}^{-1}B_{l}$
differs by multiplication with $\omega=e^{i\frac{2\pi}{3}}$ (or $\omega^{2}$).
Suppose wlog. that $l>k$ and that $\sigma_{q}$ is identity for
$q\in\\{k+1,\ldots,l-1\\}$ (one may always find such a sequence within any
potential cycle involving a transposition as well as a cyclic permutation of
legnth 3). Recall that $D_{\sigma_{q}}=\text{1}$ unless $\sigma_{q}$ is a
transposition. Then, due to Eq. (18), the eigenvalues of
$P_{\sigma_{k}}^{-1}B_{k}$ and $P_{\sigma_{l}}^{-1}B_{l}$ must coincide (up to
a common scaling factor), leading to a contradiction. ∎

Let us now prove a lemma excluding non-trivial cycles of a certain form. Later
on we will resort to this lemma in order to exclude non-trivial cycles in a
much broader scope.

######  Lemma 2.

Any cycle in $G_{b}$ involving $g_{k}=g_{k+1}=\text{1}$ for some $k$ must be
trivial, i.e., $g_{k}=\text{1}$ for all $k$.

###### Proof.

Let us assume wlog. that $g_{1}=g_{2}=\text{1}$. In order to prove the lemma
it suffices to show that Eq. (18) implies $g_{0}=\text{1}$, as then the
argument may be iterated in order to show $g_{k}=\text{1}$ for all $k$.

Let us now show that $g_{0}=\text{1}$. As $g_{1}=g_{2}=\text{1}$, Eq. (18) for
$k=1$ reads

$\displaystyle bB_{2}b^{-1}\propto B_{1}.$

As Eq. (18) displays a similarity transformation and both $B_{2}$ and $B_{1}$
are diagonal, we have that $B_{2}\propto\tilde{P}B_{1}\tilde{P}^{-1}$ for some
permutation matrix $\tilde{P}$. Let us now distinguish three cases depending
on the degeneracies of the eigenvalues of $B_{1}$. In case
$B_{1}\propto\text{1}$, considering Eq. (18) for $k=0$ immediately yields
$P_{\sigma_{0}}^{-1}B_{0}\propto\text{1}$ and thus $g_{0}=\text{1}$. Let us
now consider the case that all the eigenvalues of $B_{1}$ are non-degenerate.
Then, due to the uniqueness of the spectral decomposition (see Observation 1
for spectral decompositions of the matrices involved in Eq. (18)), $b$ must be
a monomial matrix too. Using this fact in Eq. (18) for $k=0$ shows that
$P^{-1}_{\sigma_{0}}=\text{1}$.

Let us finally consider the case that $B_{1}$ has two distinct eigenvalues
with multiplicities one and two, respectively. Let us assume wlog that the
degenerate subspace is spanned by $|0\rangle,|1\rangle$. Then, due to Eq. (18)
and the uniqueness of the spectral decomposition, $b\tilde{P}$ must be block-
diagonal in the subspace spanned by $\\{|0\rangle,|1\rangle\\}$ and
$|2\rangle$. Let us now consider Eq. (18) for $k=0$. If
$\tilde{P}|2\rangle=|2\rangle$, then $b$ commutes with $B_{1}$, and thus
$P^{-1}_{\sigma_{0}}=\text{1}$ follows immediately. If
$\tilde{P}|2\rangle\neq|2\rangle$, then considering Eq. (18) for $k=0$ shows
that $\sigma_{0}\in\\{\text{1},(1,0,2)\\}$. This can be seen as follows. The
left-hand side of Eq. (18) for $k=0$ reads
$bB_{1}b^{-1}=\left(b\tilde{P}\right)\left(\tilde{P}^{-1}B_{1}\tilde{P}\right)\left(b\tilde{P}\right)^{-1}$.
Combining the fact that $\tilde{P}^{-1}B_{1}\tilde{P}$ is diagonal with the
block-diagonal structure of $\left(b\tilde{P}\right)$ shows that the right-
hand side of Eq. (18) for $k=0$ must have the same block-diagonal structure
and thus $\sigma_{0}\in\\{\text{1},(1,0,2)\\}$. Let us now argue that the case
$\sigma_{0}=(1,0,2)$ cannot occur. To this end, let us assume
$\sigma_{0}=(1,0,2)$ and show that this leads to a contradiction. Let us
consider Eq. (18) for $k=N-1$ and rewrite the left-hand side as
$\left(b\tilde{P}\right)\left(\tilde{P}^{-1}P^{-1}_{(1,0,2)}\tilde{P}\right)D\left(b\tilde{P}\right)^{-1}$
for some diagonal $D$. Due to the right-hand side of Eq. (18),
$\left(b\tilde{P}\right)\left(\tilde{P}^{-1}P^{-1}_{(1,0,2)}\tilde{P}\right)D\left(b\tilde{P}\right)^{-1}$
must be a monomial matrix, which is only possible if $b$ is a monomial matrix
(this can be seen by considering the last row and column of the matrix
expression). However, if $b$ is a monomial matrix, then Eq. (18) for $k=0$
implies that $P_{\sigma_{0}}^{-1}=\text{1}$ (a diagonal matrix conjugated by a
monomial matrix remains diagonal), which is a contradicition. This completes
the proof of the lemma. ∎

In the following, we will make use of the group structure of the symmetries.
Obviously, whenever $S_{1}$ and $S_{2}$ are symmetries of a state, then also
$S_{1}S_{2}$ is. As we consider TIMPS, we additionally have that
$S_{1}\mathcal{T}S_{2}\mathcal{T}^{-1}$ is a symmetry of the MPS, where
$\mathcal{T}$ denotes the translation operator. It can be easily verified that
this structure carries over to cycles. If $G_{b}$ exhibits an $N$-cycle
$g_{0},\ldots,g_{N-1}$, there must also exist an $N$-cycle
$g_{0}g_{1},g_{1}g_{2},\ldots,g_{N-1}g_{0}$, etc. Subsequently, we will also
deal with situations in which we have partial information about a cycle. Given
a string of operators $g_{0},\ldots,g_{N-1}$, we denote by _substring_ any
list of operators which consecutively appear within the string
$g_{0},\ldots,g_{N-1}$. With the considerations above, one may easily convince
oneself of the following. Given a string of operators $g_{0},\ldots,g_{k}$ and
the promise that one may append operators $g_{k+1},\ldots,g_{N-1}$ in order to
obtain an $N$-cycle in $G_{b}$, the promise must also hold for any element-
wise product of substrings of coinciding length such as e.g. the length-3
string $g_{1}g_{3},g_{2}g_{4},g_{3}g_{5}$. Concretely, there must exist $N-3$
additional operators such that an $N$-cycle starting with
$g_{1}g_{3},g_{2}g_{4},g_{3}g_{5}$ is obtained. In the following, it will be
helpful to call the set of all such element-wise products of substrings the
set of _generated substrings_. Using Lemma 2, these considerations allow to
make the following observation.

######  Observation 3.

A given string of operators $g_{0},\ldots,g_{N-1}$, which contains the
substring $A,B,C$ as well as the substring $A,B,D$ for some operators
$A,B,C,D$ such that $C\neq D$, cannot form an $N$-cycle in $G_{b}$.

More generally, the same holds if $A,B,C$ and $A,B,D$ are generated substrings
of $g_{0},\ldots,g_{N-1}$.

###### Proof.

The proof is by contradiction. Suppose that $g_{0},\ldots,g_{N-1}$ forms an
$N$-cycle in $G_{b}$, and $A,B,C,D$ are such as in the statement of the
observation. Due to the group structure, there must also exist an $N$-cycle
described by a string of operators $g^{\prime}_{0},\ldots,g^{\prime}_{N-1}$
containing $A^{-1}A,B^{-1}B,C^{-1}D$ as a substring. Due to Lemma 2,
$C^{-1}D=\text{1}$, which contradicts the assumption $C\neq D$. ∎

We are now in the position to characterize all possible non-trivial cycles
within $G_{b}$. For brevity we use the following shorthand notation for
permutations, $\tau=(0,2,1)$, $\epsilon=(1,0,2)$, $\kappa=(2,1,0)$
(transpositions), and $S=\circlearrowright=(1,2,0)$,
$S^{2}=\circlearrowleft=(2,0,1)$ (cyclic permutations of length 3). We assign
labels
$T_{0}^{\tau/\epsilon/\kappa/\circlearrowleft/\circlearrowright},\ldots,T_{7}^{\tau/\epsilon/\kappa/\circlearrowleft/\circlearrowright}$
and $C_{0},\ldots,C_{4}$ to specific cycles as in Table 2 ("$T$" indicating
that the cycle involves transpositions, and "$C$" indicating that the cycle
involves cyclic permutations of length 3. The superindex differentiates
between subgroups that are of a similar structure, e.g., $T_{1}^{\tau}$ refers
to the 2-cylce $\tau\otimes\text{1}$, while $T_{1}^{\epsilon}$ refers to the
2-cylce $\epsilon\otimes\text{1}$).

######  Theorem 4.

The possible $N$-cycles ${g}_{0},\ldots,{g}_{N-1}$ in $G_{b}$ are given by
$T_{0}^{\tau/\epsilon/\kappa/\circlearrowleft/\circlearrowright},\ldots,T_{7}^{\tau/\epsilon/\kappa/\circlearrowleft/\circlearrowright}$
and $C_{0},\ldots,C_{4}$ as in Table 2 and have lengths $N\in\\{1,2,3,4,6\\}$.

###### Proof.

It turns out that the necessary conditions for a string of operators forming a
cycle as given in Lemma 2 and Observations 2 and 3 are very restrictive. In
fact, all strings of operators satisfying the conditions may be exhaustively
enumerated. We will now argue why this is the case and, in the course of that,
provide such an enumeration. Consider the following tree exploration protocol.
Starting from $N=1$, strings of operators ${g}_{0},\ldots,{g}_{N-1}$ of
increasing length $N$ are generated by appending operators to the previously
considered strings of length $N-1$ (and thus, a tree is formed). A string
(branch) is discarded if it, or any of its generated substrings, violates the
conditions in Lemma 2 or Observations 2 and 3. Moreover, one may stop further
exploring a branch once the substring consisting of the last two operators
${g}_{N-2},{g}_{N-1}$ has appeared previously as a substring within the
considered branch. The reason for this is that the only possibility to
continue from there on (without violating conditions in Lemma 2 or
Observations 2 and 3) is repeating the sequence starting from the first
appearance of the substring ${g}_{N-2},{g}_{N-1}$ over and over again.
Whenever, in some branch, this point is reached, one then either has obtained
a candidate for an $N$-cycle, if the sequence ${g}_{N-2},{g}_{N-1}$ coincides
with ${g}_{0},{g}_{1}$, or one discards the string (and abandons the branch),
otherwise, as in the latter case it is impossible to close the cycle. As the
number of operators to choose from is finite (in fact, 6), this is guaranteed
to happen at a finite $N$ 666A very naive bound would be $N\leq 6^{2}+2=38$
using that a string of length $N$ has $N-1$ substrings of length 2 and there
exist $6^{2}$ distinct strings of length 2. Much better bounds can be
obtained, though.. One may also skip exploring a branch whenever the currently
considered string contains a substring that has been considered already. For
instance, if all branches starting with ${g_{0}},{g_{1}}=\tau,\epsilon$ have
been handled, then one may skip investigating the branch
${g_{0}},{g_{1}},{g_{2}}=\tau,\tau,\epsilon$, as all possibly emerging cycles
will have been identified already.

Following this procedure, one obtains candidates777Recall that one obtains
strings of operators that satisfy necessary conditions for forming an
$N$-cycle for some MPS. for cycles as in Table 2 as well as the following
additional candidates: $\text{1}\otimes S$ $(\tilde{C}_{0})$, $\text{1}\otimes
S\otimes S$ $(\tilde{C}_{1})$, $\text{1}\otimes S\otimes S\otimes
S^{2}\otimes\text{1}\otimes S^{2}\otimes S^{2}\otimes S$
$(\tilde{C}_{2}^{\circlearrowleft})$, $\text{1}\otimes S\otimes S^{2}\otimes
S^{2}\otimes\text{1}\otimes S^{2}\otimes S\otimes S$
$(\tilde{C}_{2}^{\circlearrowright})$,
$\tau\otimes\tau\otimes\epsilon\otimes\epsilon\otimes\kappa\otimes\kappa({\tilde{T}}_{0}^{\circlearrowleft})$,
${\tilde{T}}_{0}^{\circlearrowright}$ analogously,
$\epsilon\otimes\epsilon\otimes\tau\otimes\epsilon\otimes\kappa\otimes\kappa\otimes\tau\otimes\kappa$
(${\tilde{T}}_{1}^{\tau,\circlearrowleft}$), and
${\tilde{T}}_{1}^{\epsilon,\circlearrowleft}$,
${\tilde{T}}_{1}^{\kappa,\circlearrowleft}$,
${\tilde{T}}_{1}^{\tau,\circlearrowright}$,
${\tilde{T}}_{1}^{\epsilon,\circlearrowright}$,
${\tilde{T}_{1}}^{\kappa,\circlearrowright}$ analogously. Note that any
${\tilde{T}}_{0}$ has $\tilde{C}_{0}$ as a subgroup and moreover, any
${\tilde{T}_{1}}$ has some ${\tilde{C}}_{2}$ as a subgroup. Finally, it is
straightforward to show that for $\tilde{C}_{0}$, $\tilde{C}_{1}$, and
$\tilde{C}_{2}$, there does not exist any $b$ satisfying the concatenation
conditions in Eq. (18). This shows that
$\tilde{C}^{\circlearrowleft/\circlearrowright}_{i}$ and
$\tilde{T}_{i}^{\tau/\epsilon/\kappa/\circlearrowleft/\circlearrowright}$
cannot be cycles, in contrast to the cycles shown in Table 2. ∎

Label | Subgroup(s) | Cycle | Length
---|---|---|---
$C_{0}$ | - | $S$ | 1
$T_{0}^{\tau}$ | - | $\tau$ | 1
$T_{0}^{\epsilon}$ | - | $\epsilon$ | 1
$T_{0}^{\kappa}$ | - | $\kappa$ | 1
$C_{1}$ | - | $S\otimes S^{2}$ | 2
$T_{1}^{\tau}$ | $T_{0}^{\tau}$ | $\text{1}\otimes\tau$ | 2
$T_{1}^{\epsilon}$ | $T_{0}^{\epsilon}$ | $\text{1}\otimes\epsilon$ | 2
$T_{1}^{\kappa}$ | $T_{0}^{\kappa}$ | $\text{1}\otimes\kappa$ | 2
$T_{2}^{\tau}$ |  $T_{0}^{\tau}$, $C_{1}$ | $\epsilon\otimes\kappa$ | 2
$T_{2}^{\epsilon}$ |  $T_{0}^{\epsilon}$, $C_{1}$ | $\tau\otimes\kappa$ | 2
$T_{2}^{\kappa}$ |  $T_{0}^{\kappa}$, $C_{1}$ | $\tau\otimes\epsilon$ | 2
$C_{2}$ | $C_{0}$ | $\text{1}\otimes S\otimes S^{2}$ | 3
$T_{3}^{\tau}$ | - | $\text{1}\otimes\tau\otimes\tau$ | 3
$T_{3}^{\epsilon}$ | - | $\text{1}\otimes\epsilon\otimes\epsilon$ | 3
$T_{3}^{\kappa}$ | - | $\text{1}\otimes\kappa\otimes\kappa$ | 3
$T_{4}^{\circlearrowleft}$ | $C_{0}$ | $\tau\otimes\kappa\otimes\epsilon$ | 3
$T_{4}^{\circlearrowright}$ | $C_{0}$ | $\tau\otimes\epsilon\otimes\kappa$ | 3
$T_{5}^{\circlearrowleft}$ |  $C_{2}$, $C_{0}$ | $\tau\otimes\tau\otimes\kappa$ | 3
$T_{5}^{\circlearrowright}$ |  $C_{2}$, $C_{0}$ | $\tau\otimes\tau\otimes\epsilon$ | 3
$C_{3}$ | - | $\text{1}\otimes S\otimes\text{1}\otimes S^{2}$ | 4
$T_{6}^{\tau}$ |  $C_{3}$, $T_{0}^{\tau}$ | $\kappa\otimes\tau\otimes\epsilon\otimes\tau$ | 4
$T_{6}^{\epsilon}$ |  $C_{3}$, $T_{0}^{\epsilon}$ | $\tau\otimes\epsilon\otimes\kappa\otimes\epsilon$ | 4
$T_{6}^{\kappa}$ |  $C_{3}$, $T_{0}^{\kappa}$ | $\epsilon\otimes\kappa\otimes\tau\otimes\kappa$ | 4
$C_{4}$ | $C_{1}$ | $\text{1}\otimes S\otimes S\otimes\text{1}\otimes S^{2}\otimes S^{2}$ | 6
$T_{7}^{\tau}$ |  $C_{4}$, $C_{1}$, $T_{2}^{\tau}$, $T_{0}^{\tau}$ | $\kappa\otimes\tau\otimes\epsilon\otimes\epsilon\otimes\tau\otimes\kappa$ | 6
$T_{7}^{\epsilon}$ |  $C_{4}$, $C_{1}$, $T_{2}^{\epsilon}$, $T_{0}^{\epsilon}$ | $\tau\otimes\epsilon\otimes\kappa\otimes\kappa\otimes\epsilon\otimes\tau$ | 6
$T_{7}^{\kappa}$ |  $C_{4}$, $C_{1}$, $T_{2}^{\kappa}$, $T_{0}^{\kappa}$ | $\epsilon\otimes\kappa\otimes\tau\otimes\tau\otimes\kappa\otimes\epsilon$ | 6
Table 2: Characterization of all possible cycles in $G_{b}$ (considering
fiducial states $\text{1}\otimes b\otimes\text{1}|M(\omega)\rangle$), and
thus, all possible symmetries of associated normal MPS. Cycles of lengths 1,
2, 3, 4, and 6 are possible. The third column gives the physical operators,
here we use the shorthand notation $\tau$ in place of ${g}_{\tau}$, etc. Note
that the given cycles should be understood as generators, i.e., the cycle
$C_{3}$, $\text{1}\otimes S\otimes\text{1}\otimes S^{2}$, also comprises the
cycle $S\otimes S\otimes S^{2}\otimes S^{2}$, or $T_{6}^{\tau}$ also comprises
$\epsilon\otimes\epsilon\otimes\kappa\otimes\kappa$, etc. In the second
column, we indicate which cycles emerge as a subgroup of the cycle at hand.

Before we characterize the $b$ possessing the cycles mentioned in the theorem,
a few remarks are in order. First, note that some of the identified cycles
lead to symmetry groups that are subgroups of the symmetry group corresponding
to another of the identified cycles, as indicated in the second column of
Table 2. Thus, e.g., it is not possible to find $b$ such that the fiducial
state exhibits the 2-cycle $T_{1}^{\tau}$, but not $T_{0}^{\tau}$. We
illustrate the emerging group structure in Figure 1. However, we will see
later on that not every combination of cycles that is compatible with the
group structure is actually possible. We will see, e.g., that there does not
exist a $b$ exhibiting $C_{3}$, but not any of $T_{6}^{\tau/\epsilon/\kappa}$.

$T_{0}^{\tau}$$T_{0}^{\epsilon}$$T_{0}^{\kappa}$$C_{0}$$C_{1}$$T_{1}^{\tau}$$T_{1}^{\epsilon}$$T_{1}^{\kappa}$$T_{2}^{\tau}$$T_{2}^{\epsilon}$$T_{2}^{\kappa}$$T_{3}^{\tau}$$T_{3}^{\epsilon}$$T_{3}^{\kappa}$$T_{4}^{\circlearrowleft}$$T_{4}^{\circlearrowright}$$C_{2}$$T_{5}^{\circlearrowleft}$$T_{5}^{\circlearrowright}$$C_{3}$$T_{6}^{\tau}$$T_{6}^{\epsilon}$$T_{6}^{\kappa}$$C_{4}$$T_{7}^{\tau}$$T_{7}^{\epsilon}$$T_{7}^{\kappa}$1-cycles2-cycles3-cycles4-cycles6-cycles

Figure 1: Illustration of the group structure of the cycles for normal MPS
generated by the fiducial states $\text{1}\otimes
b\otimes\text{1}|M(\omega)\rangle$. The depicted labels correspond to the
cycles as given in Table 2. A line connecting two labels indicates that the
symmetry group associated to the higher elevated cycle is a subgroup of the
symmetry group associated to the cycle below.

Second, note that we have not restricted our attention to TIMPS that are
normal, so far. In particular, the characterization of allowed cycles in
Theorem 4 holds for normal as well as for non-normal TIMPS. Recall that in
case of normal TIMPS, a characterization of cycles yields a full
characterization of the symmetries of the TIMPS, while non-normal MPS might
possess additional symmetries, which are not captured by the study of cycles.
Thus, Table 2 exhaustively lists all the possible symmetries of normal MPS and
moreover, it exhaustively lists those symmetries of non-normal MPS, that may
be identified via the study of cycles. In the following we will focus on
normal TIMPS, though. We observe that $b$ of a certain form cannot be normal.

######  Observation 4.

If $b$ is such that in any row or column $i$, the entry $b_{ii}$ is the only
non-vanishing entry, or if $b$ is a generalized permutation matrix, then the
TIMPS generated by $\text{1}\otimes b\otimes\text{1}|M(\omega)\rangle$ is non-
normal.

###### Proof.

Let us first consider the case that $b$ is such that in any row or column $i$,
the entry $b_{ii}$ is the only non-vanishing entry. Note that this property is
retained when taking products of matrices of such a form. Moreover, note that
if $b$ is of such a form, then both $A_{b}^{0}$ and $A_{b}^{1}$ as in Eq. (17)
are of this form too. Hence, it is impossible to find more than seven linearly
independent products of $A_{b}^{0}$ and $A_{b}^{1}$ and the tensor $A_{b}$
cannot be normal.

Let us now consider the case that $b$ is a generalized permutation matrix,
$b=P_{\sigma}D$. Then, any product of $A_{b}^{0}$ and $A_{b}^{1}$ comprised of
$L$ factors can be written as $P_{\sigma^{L}}\tilde{D}$ for some diagonal
$\tilde{D}$ (as $A_{b}^{0,1}=bA^{0,1}$ and $A^{0,1}$ is diagonal). Hence, for
any $L$ it is impossible to find more than three linearly independent products
of $A_{b}^{0}$ and $A_{b}^{1}$ with $L$ factors. Hence, the tensor $A_{b}$
cannot be normal. ∎

Thus, we restrict our attention to $b$ which are not of the form given in
observation in the following. In fact, as we will see (see also Observation
5), the remaining $b$’s are either normal or $G_{b}$ does not exhibit any non-
trivial cycles.

With Theorem 4, it is now a straightforward calculation to characterize all
$b$, for which the $G_{b}$ exhibits any specific cycle listed in Table 2. To
this end, one considers Eq. (18) for $\sigma_{k}$ given by the considered
cycle. One may then utilize Observation 1 in order to determine $b$. We
present a full characterization of those $b$ (disregarding $b$ of the form
given in Observation 4) for which $G_{b}$ exhibits non-trivial cycles (see
Table 2) in Table LABEL:tab:M10M11M1infcycleslong in Appendix A. Let us
exemplarily display the calculation for the cycle $C_{0}$. We use Eq. (18)
with $N=1$, $\sigma_{0}=S$ and obtain

$\displaystyle bP_{S}^{-1}B_{0}D_{S}^{-1}b^{-1}\propto P_{S}^{-1}B_{0}.$ (19)

As $D_{S}=\text{1}$, making use of Observation 1 we have
$P_{S}^{-1}B_{0}D_{S}^{-1}=P_{S}^{-1}B_{0}=\tilde{D}\mathcal{F}\operatorname{diag}(1,\omega,\omega^{2})\left(\tilde{D}\mathcal{F}\right)^{-1}$
with
$\tilde{D}=\operatorname{diag}(1,\frac{B^{(0)}_{22}}{\sqrt[3]{B^{(0)}_{11}B^{(0)}_{22}}^{2}},\frac{1}{\sqrt[3]{B^{(0)}_{11}B^{(0)}_{22}}})$.
Due to the uniqueness of the spectral decomposition we then have

$\displaystyle b$
$\displaystyle=\tilde{D}\mathcal{F}P_{\tilde{\sigma}}\operatorname{diag}(x_{0},x_{1},x_{2})\mathcal{F}^{-1}\tilde{D}^{-1}$
$\displaystyle=\tilde{D}\operatorname{diag}(1,\omega,\omega^{2})^{l}\mathcal{F}\operatorname{diag}(x_{0},x_{1},x_{2})\mathcal{F}^{-1}\tilde{D}^{-1}$

for some $x_{i}\in\mathbb{C}$, $l\in\\{0,1,2\\}$, and some permutation
$\tilde{\sigma}$ with $\operatorname{sgn}(\tilde{\sigma})=1$. Here, the
additional permutation matrix $P_{\tilde{\sigma}}$ comes from the fact that
the proportionality factor in Eq. (19) allows to cyclically permute the
eigenvalues. Equivalently, we may write

$\displaystyle
b=D\operatorname{diag}(1,\omega,\omega^{2})^{l}\begin{pmatrix}b_{0}&b_{1}&b_{2}\\\
b_{2}&b_{0}&b_{1}\\\ b_{1}&b_{2}&b_{0}\end{pmatrix}D^{-1},$

where $b_{0},b_{1},b_{2}\in\mathbb{C}$, $l\in\\{0,1,2\\}$, and $D$ is an
arbitrary diagonal matrix. Let us remark that the diagonal matrix $D$ is
actually irrelevant. More precisely, the fiducial states $\text{1}\otimes
b\otimes\text{1}|M(\omega)\rangle$ and $\text{1}\otimes
c\otimes\text{1}|M(\omega)\rangle$ give rise to the same TIMPS, if
$c=DbD^{-1}$ for any diagonal $D$, due to the Fundamental Theorem (Theorem 1)
and the symmetries of the seed state888This may also be easily seen by noting
that such $b$ and $c$ are related by a $(b\rightarrow c)$ 1-cycle as in Eq.
(9) with $g=\text{1}$ for any diagonal matrix $D$..

It is straightforward to perform the calculation for all cycles, leading to
the results in Table LABEL:tab:M10M11M1infcycleslong within Appendix A. We
find that there are continuous families of $b$’s leading to the cycles
$C_{0}$, $T_{0}^{\tau,\epsilon,\kappa}$, $C_{1}$,
$T_{1}^{\tau,\epsilon,\kappa}$, $T_{2}^{\tau,\epsilon,\kappa}$,
$T_{3}^{\tau,\epsilon,\kappa}$, and
$T_{4}^{\circlearrowleft,\circlearrowright}$, while there is a discrete number
of $b$’s leading to the cycles $C_{2}$,
$T_{5}^{\circlearrowleft,\circlearrowright}$, $C_{3}$,
$T_{6}^{\tau,\epsilon,\kappa}$, $C_{4}$, and $T_{7}^{\tau,\epsilon,\kappa}$.

While we defer details on the concrete parametrizations of the sets of
operators $b$ for which $G_{b}$ exhibits the respective cycles to Appendix A,
the relations among these sets are important in order to know which different
symmetry groups can occur simultaneously. Hence, we will discuss these
relations in the following. Let us denote the set of $b$’s leading to a
certain cycle, say $T_{0}^{\tau}$, by $b(T_{0}^{\tau})$, etc. Obviously, the
families of $b$’s satisfy relations imposed by the group structure of the
cycles mentioned above (see Figure 1), e.g., $b(T_{1}^{\tau})$ must be a
subset of $b(T_{0}^{\tau})$. Note, however, that it is not guaranteed that for
every possible combination of cycles, which is compatible with the group
structure, there exists a $b$ such that $G_{b}$ exhibits this combination of
cycles. On the contrary, additional set-theoretic restrictions emerge, which
one may not immediately conclude from the group structure of the cycles. We
display all the relations within an Euler diagram in Figure 2.

(a)

(b)

Figure 2: (a) Set-theoretic sketch (Euler diagram) of $b$’s leading to non-
trivial cycles in $G_{b}$ (considering the fiducial state $\text{1}\otimes
b\otimes\text{1}|M(\omega)\rangle$). The sketch illustrates the inclusion
relations and intersections between the different sets of $b$’s leading to
certain cycles as labelled in Table 2. Let us point out a few of them. For
instance, one sees that the set $b(C_{3})$ is a union of the (pairwise) non-
intersecting sets $b(T_{6}^{\tau})$, $b(T_{6}^{\epsilon})$, and
$b(T_{6}^{\kappa})$, and each of $b(T_{6}^{\sigma})$ is a subset of
$b(T_{0}^{\sigma})$ for $\sigma\in\\{\tau,\epsilon,\kappa\\}$. This
illustrates that, e.g., there does exist a $b$ s.t. $G_{b}$ exhibits the three
cycles $T_{6}^{\tau}$, $T_{0}^{\tau}$, and $C_{3}$, but there does not exist
any $b$ leading to the cycle $C_{3}$ only. We find that $b(T_{0}^{\tau})$,
$b(T_{0}^{\epsilon})$, $b(T_{0}^{\kappa})$, and $b(C_{0})$ intersect. However,
the three sets $b(T_{i}^{\tau})$, $b(T_{j}^{\epsilon})$, and
$b(T_{k}^{\kappa})$ are disjoint for any $i,j,k\in\\{1,2,3,6,7\\}$. In fact,
$b(T_{3}^{\tau,\epsilon,\sigma})$ are completely isolated in the sense that
they do not intersect with any other family. The set $b(T_{1}^{\sigma})$ is a
subset of $b(T_{0}^{\sigma})$ (for $\sigma\in\\{\tau,\epsilon,\kappa\\}$), but
does not intersect with any other family. Note that the sketch does not
correctly represent the geometry or sizes of the sets. As explained in the
main text, the diagram gives as a complete characterization of symmetries of
all normal TIMPS generated by fiducial states $\text{1}\otimes
b\otimes\text{1}|M(\omega)\rangle$. While some of the displayed inclusion
relations follow immediately considering the group structure of the cycles as
in Figure 1, additional relations only reveal themselves considering the full
characterization of $b$ as in Table LABEL:tab:M10M11M1infcycleslong. An
instance of the latter case would be one of the example mentioned above—the
fact that there exists no $b$ s.t. $G_{b}$ possesses the cycle $C_{3}$, but
does not possess any of the cycles $T_{6}^{\tau}$, $T_{6}^{\epsilon}$, and
$T_{6}^{\kappa}$. (b) Detailed view of an aspect of subfigure (a).

In Observation 4, we have given a few conditions on $b$ under which the
generated TIMPS is not normal. In contrast to that, we find that any $b$ s.t.
$G_{b}$ possesses non-trivial cycles and s.t. $b$ does not satisfy one of the
mentioned conditions for non-normality, is actually normal.

######  Observation 5.

All $b$ s.t. there exist non-trivial cycles in $G_{b}$ (see Table
LABEL:tab:M10M11M1infcycleslong) are normal with injectivity length $L=4$,
$5$, or $6$, unless $b$ fulfills the prerequisites of Observation 4.

We present a proof of the observation, as well as additional details on
proving normality in general, in Appendix B.

As a consequence, except for $b$ satisfying the conditions in Observation 4,
the characterization of cycles directly yields the symmetries of the generated
TIMPS. Figure 2 hence shows all possible (non-trivial) symmetries of normal
TIMPS generated by fiducial states of the type $\text{1}\otimes
b\otimes\text{1}|M(\omega)\rangle$ and hence for any normal TIMPS
corresponding to a fiducial state in the SLOCC class of $|M(\omega)\rangle$.
For any given normal $b$, the symmetry group of the generated TIMPS may be
determined by comparing $b$ with the results in Table
LABEL:tab:M10M11M1infcycleslong in Appendix A. Conversely, in order to decide
whether there exists a normal TIMPS (generated by some $\text{1}\otimes
b\otimes\text{1}|M(\omega)\rangle$) possessing a desired symmetry group, one
may simply look up Figure 2 and see whether the corresponding intersection is
non-empty. If it exists, then an appropriate $b$ may be constructed with the
help of Table LABEL:tab:M10M11M1infcycleslong in Appendix A.

###  IV.3 SLOCC Classification

As we are dealing here with finitely many symmetries, it is straightforward to
determine the SLOCC classes of the TIMPS. Whereas we will determine all the
classes with infinitely many symmetries in the subsequent section, we will
only outline here the procedure and discuss some examples.

To this end, we consider the concatenation conditions presented in Eq. (9). To
determine for instance all $(b\rightarrow c)$ 1-cycles, we have to only
consider one equation, namely

$\displaystyle bP_{\sigma}^{-1}BD_{\sigma}^{-1}\propto P_{\sigma}^{-1}Bc,$

which immediately lets one construct all $c$ connected to a given $b$ via an
$(b\rightarrow c)$ 1-cycle. Stated differently, normal TIMPSs corresponding to
the fiducial states $\text{1}\otimes b\otimes\text{1}|M(\omega)\rangle$ and
$\text{1}\otimes c\otimes\text{1}|M(\omega)\rangle$ respectively are related
to each other via a global operation iff $b$ and $c$ fulfill the equation
above.

For 2-cycles one would proceed as follows. First, one considers the necessary
condition

$\displaystyle
bP_{\sigma_{1}}^{-1}D_{\sigma_{1}}^{-1}B_{1}^{-1}B_{0}D_{\sigma_{0}}P_{\sigma_{0}}b^{-1}\propto
P_{\sigma_{0}}^{-1}B_{0}^{-1}B_{1}P_{\sigma_{1}}.$ (20)

Obviously, the tools utilized throughout this section so far are applicable
here. Once $g_{0}$ and $g_{1}$ satisfy the necessary conditions for some given
$b$, all $c$ connected to $b$ via the $(b\rightarrow c)$ 2-cycle given by
$g_{0},g_{1}$ may be straightforwardly characterized.

Let us remark that an obvious necessary condition for SLOCC equivalence of two
states $|\psi\rangle$ and $|\phi\rangle$ is that their symmetry group must be
compatible, i.e., $\mathcal{S}_{|\psi\rangle}$ equals
$\mathcal{S}_{|\phi\rangle}$ up to conjugation. For the MPS considered here
this must be a conjugation by some tensor product of $g_{\sigma}$ as in Eq.
(IV.1). Thus, not only the order of the full symmetry group must coincide, in
fact, but also the number of symmetries involving transpositions as well as
the number of symmetries involving cyclic permutations of length 3 must be
retained, each. This immediately rules out SLOCC equivalence among many of the
families of $b$s as in Figure 2.

Let us conclude with some examples. It may be easily verified that the MPS
associated to the family $b(T_{i}^{\tau})$ are SLOCC-equivalent to some MPS
associated to the family $b(T_{i}^{\epsilon})$ and $b(T_{i}^{\kappa})$ (and
vice versa) for any $i\in\\{0,1,2,3,6,7\\}$. This is witnessed by the
$(b\rightarrow c)$ 1-cycles given by $g_{\epsilon}$, or $g_{\kappa}$,
respectively. Moreover, for even $N$, any MPS generated by some fiducial state
belonging to $b(C_{0})\cap b(T_{0}^{\tau})\cap b(T_{0}^{\epsilon})\cap
b(T_{0}^{\kappa})$ is SLOCC equivalent to some MPS associated to $b(C_{1})\cap
b(T_{0}^{\kappa})\cap b(T_{2}^{\kappa})$ and vice versa. This is witnessed by
the $(b\rightarrow c)$ 2-cycle $\text{1}\otimes\kappa$. Conversely, there
exist examples in $b(C_{0})$, which are not related to any $b$ in $b(C_{1})$
despite compatibility of the stabilizer.

##  V Symmetries and SLOCC classes of the TIMPS $\Psi(LLT)$

In this section we discuss MPS generated by fiducial states that are
represented by
$|LLT\rangle=|0\rangle(|01\rangle+|22\rangle)+|1\rangle(|00\rangle+|12\rangle)$.
First, we present the symmetries of the fiducial states. Then, we characterize
the symmetries of normal MPS, which, in contrast to the previous section, are
potentially infinitely many. In the course of that, we give a characterization
of those fiducial states that generate normal MPS. Finally, we characterize
SLOCC equivalence among normal MPS. We conclude with a few remarks on some
non-normal MPS generated by fiducial states represented by $|LLT\rangle$.

The matrix pencil corresponding to $|LLT\rangle$ reads

$\displaystyle\mathcal{P}=L_{1}\oplus
L_{1}^{T}=\begin{pmatrix}\lambda&\mu&0\\\ 0&0&\lambda\\\
0&0&\mu\end{pmatrix}.$

Note that the tensor $A$ corresponding to $|LLT\rangle$ is simply given by
$A^{0}=\mathcal{P}|_{\mu=1,\lambda=0}$ and
$A^{1}=\mathcal{P}|_{\mu=0,\lambda=1}$.

###  V.1 Symmetries of the fiducial state

The symmetries of the state $|LLT\rangle$ are special in the sense that any
invertible operator acting on the first site forms a local symmetry of
$|LLT\rangle$ with appropriate operators acting on the remaining two sites.
That is, for any operator $g$ on the qubit, there exist $3\times 3$ matrices,
which are uniquely determined by $g$, $B_{g}$, and $C_{g}$ such that $g\otimes
B_{g}\otimes C_{g}$ is a symmetry of the state. Moreover, as mentioned above,
any symmetry of ${|LLT\rangle}$ can be written as a product of one symmetry of
the form ${g}\otimes B_{g}\otimes C_{g}$ and symmetries of the form
$\text{1}\otimes B\otimes C$. For

${g}=\begin{pmatrix}\alpha&\beta\\\ \gamma&\delta\end{pmatrix}$

it is easy to show that

$\displaystyle B_{g}$
$\displaystyle=\frac{1}{det(g)}\begin{pmatrix}det(g)&0&0\\\ 0&\alpha&-\beta\\\
0&-\gamma&\delta\end{pmatrix}$ $\displaystyle C_{g}^{T}$
$\displaystyle=\frac{1}{det(g)}\begin{pmatrix}\alpha&-\gamma&0\\\
-\beta&\delta&0\\\ 0&0&det(g)\end{pmatrix}.$

The symmetries of the form $\text{1}\otimes B\otimes C$ are given by

$\displaystyle B$ $\displaystyle=\begin{pmatrix}1&B_{01}&B_{02}\\\
0&B_{11}&0\\\ 0&0&B_{11}\end{pmatrix}$ $\displaystyle C^{T}$
$\displaystyle=\frac{1}{B_{11}}\begin{pmatrix}B_{11}&0&-B_{01}\\\
0&B_{11}&-B_{02}\\\ 0&0&1\end{pmatrix},$

where we use the normalization $B_{00}=1$. The symmetries of $|LLT\rangle$ are
thus given by

$\displaystyle{g}\otimes B_{g}B\otimes C_{g}C$

for any ${g}\in GL(2,\mathbb{C})$, $B_{00},B_{01},B_{02},B_{11}\in\mathbb{C}$.

Clearly, the symmetries of $\text{1}\otimes b\otimes\text{1}|LLT\rangle$ are
given by

$\displaystyle g\otimes
b\underbrace{B_{g}B}_{x}b^{-1}\otimes\underbrace{C_{g}C}_{y^{T}},$ (21)

using the notation $x$ and $y$ for symmetries of the representative as
introduced in Section III. We denote the local symmetry group of this fiducial
state by $G_{b}$ throughout this whole section. The tensor $A_{b}$ associated
to $\text{1}\otimes b\otimes\text{1}|LLT\rangle$ reads

$\displaystyle A_{b}^{0}=\begin{pmatrix}0&b_{00}&b_{02}\\\ 0&b_{10}&b_{12}\\\
0&b_{20}&b_{22}\end{pmatrix}\quad\text{and}\quad
A_{b}^{1}=\begin{pmatrix}b_{00}&0&b_{01}\\\ b_{10}&0&b_{11}\\\
b_{20}&0&b_{21}\end{pmatrix}.$

Let us now introduce the following parametrization for any $3\times 3$
operators $b$ with $b_{20}=1$ (it will become clear later on that $b_{20}\neq
0$ is required in order to obtain normal MPS) in terms of a matrix $T$ and a
vector $\mathbf{v}$, as well as two complex numbers $b_{00},b_{10}$,

$\displaystyle b=\begin{pmatrix}b_{00}\\\ b_{10}\\\
1\end{pmatrix}.\begin{pmatrix}1\\\ v_{1}-b_{00}\\\
v_{0}-b_{10}\end{pmatrix}^{T}-\det
T\begin{pmatrix}&\sigma_{x}T^{-1}\sigma_{z}\\\ 0&\end{pmatrix}.$ (22)

It may be easily veryfied that for a given $b$, $T$ and $\mathbf{v}$ may be
obtained via

$\displaystyle T$
$\displaystyle=\begin{pmatrix}b_{02}-b_{00}b_{22}&b_{10}b_{22}-b_{12}\\\
b_{01}-b_{00}b_{21}&b_{10}b_{21}-b_{11}\end{pmatrix}$ (23)
$\displaystyle\mathbf{v}$ $\displaystyle=\begin{pmatrix}b_{10}+b_{22}\\\
b_{00}+b_{21}\end{pmatrix}.$ (24)

Note that $\det b=\det T$. Despite the fact that this parameterization might
seem a bit arbitrary, we will see that it is particularly useful to
characterize the local symmetries and the SLOCC classes of TIMPS corresponding
to fiducial states of the form $\text{1}\otimes b\otimes\text{1}|L_{1}\otimes
L_{1}^{T}\rangle$

###  V.2 Concatenation conditions

To obtain a physical symmetry, (normal) MPS need to fulfill the conditions
given in Eq. (III.2), which we restate here,

$\displaystyle y_{k}bx_{k+1}b^{-1}$ $\displaystyle\propto\text{1}\quad\forall\
k\in\\{0,\ldots,N-2\\},$ $\displaystyle y_{N-1}bx_{0}b^{-1}$
$\displaystyle\propto\text{1},$

where $x,y$ are such that $g\otimes x\otimes y^{T}$ are a symmetry of the
fiducial state as in Eq. (21). For simplicity, we will in the following only
consider the upper equation for $k\in\\{0,\ldots,N-1\\}$ and consider addition
modulo $N$ in this context, i.e., $N\equiv 0$.

Inserting for $x_{k+1},y_{k}$ the expressions given in Eq. (21) and using the
normalization $B_{00}^{(k)}=1$ as well as $\det{g}_{k}=1$ for all $k$, the
concatenation conditions $bx_{k+1}b^{-1}\propto y_{k}^{-1}$ read

$\displaystyle b\underbrace{\begin{pmatrix}1&B_{01}^{(k+1)}&B_{02}^{(k+1)}\\\
0&\alpha_{k+1}B_{11}^{(k+1)}&-\beta_{k+1}B_{11}^{(k+1)}\\\
0&-\gamma_{k+1}B_{11}^{(k+1)}&\delta_{k+1}B_{11}^{(k+1)}\end{pmatrix}}_{LHS_{k+1}}b^{-1}\propto$
$\displaystyle\underbrace{\begin{pmatrix}\delta_{k}&\gamma_{k}&\gamma_{k}B_{02}^{(k)}+\delta_{k}B_{01}^{(k)}\\\
\beta_{k}&\alpha_{k}&\alpha_{k}B_{02}^{(k)}+\beta_{k}B_{01}^{(k)}\\\
0&0&B_{11}^{(k)}\end{pmatrix}}_{RHS_{k}}.$ (25)

We denote the eigenvalues of $g_{k}$ by $\chi_{k}$ and $1/\chi_{k}$. We use
the convention $|\chi_{k}|\geq 1$ and additionally in case $|\chi_{k}|>1$ (i),
we choose $\operatorname{Im}{\chi_{k}}>0$, or $\operatorname{Im}{\chi_{k}}=0$
and $\operatorname{Re}{\chi_{k}}>0$, while in case $|\chi_{k}|=1$ (ii) we
choose both $\operatorname{Re}{\chi_{k}}\geq 0$ and
$\operatorname{Im}{\chi_{k}}\geq 0$. We denote the domain of $\chi_{k}$ by
$\mathcal{D}$ and show a sketch of $\mathcal{D}$ in Figure 3. This
normalization may be achieved by ordering the eigenvalues appropriately and by
a freedom of multiplying ${g}_{k}$ by $-1$ which still remains after fixing
$\det{g}_{k}=1$. For each $g_{k}$ we then consider the Jordan decomposition

$\displaystyle{g}_{k}=S_{k}J_{k}S_{k}^{-1},$ (26)

where either $J_{k}=\operatorname{diag}(\chi_{k},1/\chi_{k})$ (in case
${g}_{k}$ is diagonalizable), or $J_{k}=\begin{pmatrix}1&1\\\
0&1\end{pmatrix}$ (in case ${g}_{k}$ is not diagonalizable, here
$\chi_{k}=1$).

Figure 3: Sketch of the domain of $chi_{k}$, $\mathcal{D}$. $\mathcal{D}$ is
comprised of complex numbers in the upper half of the complex plane which have
absolute value larger than or equal to one, excluding the negative real axis,
and, moreover excluding numbers with negative real part whose absolute value
equals 1. The imaginary unit $i$ is included in $\mathcal{D}$.

As a simple necessary condition, we see that the set of eigenvalues of
$LHS_{k+1}$ must match the set of eigenvalues of $RHS_{k}$ up to a common
proportionality factor. The eigenvalues read999Note that eigenvalues may
coincide and that $LHS_{k}$, $RHS_{k}$ may be not diagonalizable, even though
$g_{k}$ is diagonalizable.

$\displaystyle\sigma(LHS_{k+1})$
$\displaystyle=\left\\{1,B_{11}^{(k+1)}\chi_{k+1},B_{11}^{(k+1)}/\chi_{k+1}\right\\},$
$\displaystyle\sigma(RHS_{k})$
$\displaystyle=\left\\{B_{11}^{(k)},\chi_{k},1/\chi_{k}\right\\}.$

Let us remark here that considering the concatenation conditions, it is
immediately clear that for any ${g}$, one can find an MPS that has the global
symmetry ${g}^{\otimes N}$. The reason for that is that the concatenation
condition ($bLHS_{0}b^{-1}\propto RHS_{0}$), in case of a global symmetry,
i.e., 1-cycle) reduces to matching the set of eigenvalues (up to a common
proportionality factor) 101010This can be easily seen choosing
$B_{01}^{0}=B_{02}^{0}=0$ and $B_{11}^{0}=0$. For all matrices ${g}$ it is
possible to find a proportionality factor and a choice of $B_{11}$ such that
the eigenvalues match.

###  V.3 Local symmetries of the TIMPS $\Psi(LLT)$

In this subsection, we present a characterization of the symmetries of normal
MPS generated by $\text{1}\otimes b\otimes\text{1}|\Psi_{0}\rangle$ and also
discuss some of the non-normal MPS. Certain details of the derivation will be
deferred to Appendix D.

Let us right away distinguish between the two cases $b_{20}=0$ and $b_{20}\neq
0$. In the former case, $b_{20}=0$, the generated MPS cannot be normal as we
will see in Observation 7. However, despite the fact that the fundamental
theorem does not apply, actually much can be said about the symmetries of the
corresponding MPS as we show in the following observation.

######  Observation 6.

$N$-qubit MPS associated to $\text{1}\otimes b\otimes\text{1}|LLT\rangle$ with
$b_{20}=0$ are either SLOCC-equivalent to $|0\rangle^{\otimes N}$,
$|\mathrm{GHZ}_{N}\rangle$, or they possess only global symmetries.

Let us remark here that using $b=\text{1}$, i.e., using the seed state
$|LLT\rangle$ as fiducial state gives rise to an MPS that is a product state.

###### Proof.

In order to prove the observation, we consider the definition of an MPS as in
Eq. (2) and note that for $b_{20}=0$ it holds that

$\displaystyle\text{tr}\\{A^{j_{1}}A^{j_{2}}\ldots
A^{j_{N}}\\}=b_{22}^{|j|}b_{21}^{N-|j|}+b_{00}^{|j|}b_{10}^{N-|j|},$

where $j=(j_{1},\ldots,j_{N})\in\\{0,1\\}^{N}$ and $|j|$ denotes the Hamming
weight of $j$. In particular, the expression in Eq. (26) does not depend on
the order of the operators $A_{j_{l}}$. Thus, the MPS is, for $b_{20}=0$ not
only translationally invariant, but actually invariant under any particle
permutation. A permutation invariant $N$-qubit state is either SLOCC
equivalent to $|0\rangle^{\otimes N}$, $|\mathrm{GHZ}_{N}\rangle$, or
$|W_{N}\rangle$111111Note that the W-state is not representable by a TIMPS of
bond dimension three., or the state is what was called non–exceptionally
symmetric [35], meaning that all its symmetries are of the form $S^{\otimes
N}$ [28, 29, 14]. In the former cases, the symmetries of the MPS are well-
known and in the latter case (by definition of non–exceptionally symmetric
states) the MPS possesses global symmetries only. ∎

We will focus on the case $b_{20}\neq 0$ for the remainder of this section.
For simplicity, in the following we will assume $b_{20}=1$, as an overall
scaling factor within $b$ is irrelevant. Note, however, that $b_{20}\neq 0$ is
not a sufficient condition to have normal MPS. In fact, normality additionaly
depends on $\mathbf{v}$ as in Eq. (24), as the following observation shows. We
prove the observation in Appendix C (see also Appendix B for a few general
remarks on proving normality).

######  Observation 7.

$N$-qubit MPS associated to $\text{1}\otimes b\otimes\text{1}|LLT\rangle$ are
normal if and only if $b_{20}\neq 0$ and $\mathbf{v}\neq\mathbf{0}$.

We keep this fact in mind, however, in the following we will continue without
narrowing down the considered set of fiducial states any further and postpone
a more detailed discussion on normality till Section V.5.

Let us now analyze the concatenation equations in more depth. The (2,0)-matrix
element of the concatenation condition with proportionality factors
$\lambda_{k}$, $bLHS_{k+1}-\lambda_{k}RHS_{k}b=0$, reads
$b_{20}(1-B_{11}^{(k)}\lambda_{k})=0$. As $b_{20}=1$, this matrix element thus
fixes the proportionality factors. We then obtain equality of the following
two sets of eigenvalues as necessary condition,

$\displaystyle\left\\{B_{11}^{(k)},B_{11}^{(k)}B_{11}^{(k+1)}\chi_{k+1},B_{11}^{(k)}B_{11}^{(k+1)}/\chi_{k+1}\right\\}$
$\displaystyle=\left\\{B_{11}^{(k)},\chi_{k},1/\chi_{k}\right\\}.$

Particularly interesting are the trace and the determinant of the matrix
equation in the concatenation condition, i.e., the sum and the product of the
elements in the two sets in Eq. (26). One obtains

$\displaystyle\left(B_{11}^{(k)}\right)^{2}\left(B_{11}^{(k+1)}\right)^{2}$
$\displaystyle=1,$ $\displaystyle
B_{11}^{(k)}B_{11}^{(k+1)}(\chi_{k+1}+1/\chi_{k+1})$
$\displaystyle=\chi_{k}+1/\chi_{k}.$

Considering the chosen normalization, this implies that the sets of
eigenvalues of ${g}_{k}$ must coincide for all $k$, i.e., $\chi_{k}=\chi_{l}$
for all $k,l$. We will thus drop the index $k$ in $\chi_{k}$, in the
following. If $\chi\neq i$, i.e., if $\text{tr}{g}_{k}$ does not vansish, then
$B_{11}^{(k)}B_{11}^{(k+1)}=1$ for all $k$. Considering a cycle of odd length,
one moreover has that either $B_{11}^{(k)}=1$ for all $k$, or
$B_{11}^{(k)}=-1$ for all $k$. To see this, note that using
$B_{11}^{(k)}B_{11}^{(k+1)}=1$ recursively yields
$B_{11}^{(k)}=1/B_{11}^{(k)}$. In the case that $\text{tr}{g}_{k}=0$, we only
have $B_{11}^{(k)}B_{11}^{(k+1)}=\pm 1$, instead. We summarize these findings
in the following observation

######  Observation 8.

Suppose that ${g}_{0},\ldots,{g}_{N-1}$ is an $N$-cycle in $G_{b}$. Then, the
eigenvalues of ${g}_{k}$, $\chi$ and $1/\chi$, coincide for all $k$. If
$\chi\neq i$, we have $B_{11}^{(k)}B_{11}^{(k+1)}=1$. If $\chi=i$, we have
$B_{11}^{(k)}B_{11}^{(k+1)}=\pm 1$ in Eq. (V.2).

Given $b$ with $b_{20}=1$, calculate $T$, $\mathbf{v}$. Write $T=RJR^{-1}$,
where $J$ is JNF of $T$ $T$ diagonalizable? $T^{m}\propto\text{1}$ for some
$m\in\mathbb{N}$? Write $T\propto R\begin{pmatrix}e^{i\frac{r\pi}{m}}&\\\
&e^{-i\frac{r\pi}{m}}\end{pmatrix}R^{-1}$, $r\in\\{0,\ldots,m-1\\}$
$\mathbf{v}=\mathbf{0}$? (I)  1-cycle (1-param.): ${g}=R\begin{pmatrix}x&\\\
&1/x\end{pmatrix}R^{-1}$ for any $x\in\mathbb{C}\setminus\\{0\\}$. $m$-cycle
(3-param.): ${g}_{k}=T^{k}g_{0}T^{-k}$ for any ${g}_{0}$. $m/2$-cycle
(1-param., req. even $n$): ${g}_{k}=R\begin{pmatrix}0&iz\\\
i/z&0\end{pmatrix}R^{-1}$, $z=ye^{i\frac{2k(2r+1)\pi}{m}}$ for any
$y\in\mathbb{C}\setminus\\{0\\}$ (IIa)  1-cycle (1-param.):
${g}=R\begin{pmatrix}1&y\\\ 0&1\end{pmatrix}R^{-1}$ for any $y\in\mathbb{C}$
$\mathbf{v}=\mathbf{0}$? $\mathbf{v}=\mathbf{0}$? (III)  1-cycle (1-param.):
${g}=R\begin{pmatrix}x&\\\ &1/x\end{pmatrix}R^{-1}$ for any
$x\in\mathbb{C}\setminus\\{0\\}$ $T\mathbf{v}\propto\mathbf{v}$? (IV)
$m$-cycle (1-param.): ${g}_{k}=S_{0}\begin{pmatrix}1&e^{i\frac{2kr\pi}{m}}\\\
0&1\end{pmatrix}S_{0}^{-1}$, where $S_{0}=(\mathbf{v},\,\mathbf{w})$ for any
$\mathbf{w}\in\mathbb{C}^{2}$ (IIb)  1-cycle (1-param.):
${g}=R\begin{pmatrix}1&y\\\ 0&1\end{pmatrix}R^{-1}$ for any $y\in\mathbb{C}$
$T\mathbf{v}\propto\mathbf{v}$? (VI) - generic  no sym.
$T^{2}\mathbf{v}\propto\mathbf{v}$, i.e., $T^{2}\propto\text{1}$? (VII)
2-cycle (1-param.): ${g}_{0}$ s.t. ${g}_{0}\mathbf{v}\propto\mathbf{v}$,
${g}_{0}T\mathbf{v}\propto T\mathbf{v}$ with any
$x\in\mathbb{C}\setminus\\{0,i\\}$, ${g}_{1}={g}_{0}^{-1}$. 1-cylce (single):
${g}_{0}$ as above, but with $x=i$. (V)  no sym. (VIII)  no sym.
noyesyesnoyesyesnonoyesnonoyesyesyesnono

Figure 4: Flowchart showing the characterization of the symmetries of all
normal MPS generated by $\text{1}\otimes b\otimes\text{1}|LLT\rangle$, i.e.,
for any $b$ with $b_{20}\neq 0$ (wlog $b_{20}=1$) and
$\mathbf{v}\neq\mathbf{0}$ (shaded rectangles with solid contour), cf.
Observation 7. For any such $b$, the symmetries of the corresponding MPS may
be determined by calculating $T$ and $\mathbf{v}$ as in Eqs. (23) and (24) and
then following the procedure described in the main text, which is shown in the
flowchart. Additionally, the flowchart also shows the cycles in $G_{b}$
obtained for non-normal MPS generated by $b$ such that $b_{20}=1$ and
$\mathbf{v}=\mathbf{0}$ (shaded rectangles with dashed contour). Note that for
non-normal MPS the symmetry group might be larger than displayed, as the
utilized methods may fail to identify the full symmetry group (and yield a
subgroup instead). Here, “no sym.” indicates that the corresponding MPS
possesses only the trivial symmetry. Generic $b$ belong to box (VI), as
indicated in the flowchart.

Building on the observations above and making use of $T$ and $\mathbf{v}$ as
in Eqs. (23) and (24), we derive the following theorem, which gives necessary
and sufficient conditions for ${g}_{0},\ldots,{g}_{N-1}$ forming an $N$-cycle
in $G_{b}$. We prove the theorem in Appendix D. Note that, as we will see
below, this leads to a rich variety of situations involving 1-cycles as well
as $N$-cycles, diagonalizable $g_{k}$ as well as non-diagonalizable $g_{k}$
and single cycles, as well as continuous families of cycles.

######  Theorem 5.

${g}_{0},\ldots,{g}_{N-1}$ is an $N$-cylce in $G_{b}$ 121212Recall that we
consider here $b_{20}=1$. We have dealt with the case $b_{20}=0$ in
Observation 6. if and only if there exist $B_{11}^{(k)}\in\mathbb{C}$ such
that for all $k\in\\{0,\ldots,N-1\\}$,

$\displaystyle{g}_{k+1}$
$\displaystyle=\frac{1}{B_{11}^{(k)}B_{11}^{(k+1)}}T{g}_{k}T^{-1},$ (27)
$\displaystyle\left[{g}_{k}-{B_{11}^{(k)}}\text{1}\right]\mathbf{v}$
$\displaystyle=\mathbf{0},$ (28) $\displaystyle B_{11}^{(k)}B_{11}^{(k+1)}$
$\displaystyle=\begin{cases}\pm 1&\text{if }\chi=i\\\
1&\text{otherwise}\end{cases}$ (29)

With the help of the conditions provided in Theorem 5, the $N$-cycles in
$G_{b}$ may be determined for any given $b$ with $b_{20}=1$. In the following,
we describe the procedure to do so. We defer the details on the derivation of
the procedure to Appendix D. Recall that the considered family of $b$’s also
involves non-normal MPS. For this reason, we have formulated the theorem in
terms of cycles in $G_{b}$, although for normal MPS the theorem directly
characterizes the symmetries of the associated MPS.

First, one calculates the matrix $T$ according to Eq. (23) as well as the
vector $\mathbf{v}$ according to Eq. (24). The symmetries will be completely
determined by $T$ and $\mathbf{v}$, which, as we would like to stress here
again, are merely properties of $b$. Let us denote the similarity
transformation bringing $T$ into its Jordan Normal Form (JNF) $J$ by $R$,
i.e., we have $T=RJR^{-1}$. We now distinguish two cases. We have the case
that $T$ is diagonalizable and the case that $T$ is not diagonalizable. In the
latter case, we obtain only trivial cycles if $\mathbf{v}\neq\mathbf{0}$ and
$T\mathbf{v}\not\propto\mathbf{v}$. In contrast to that, $G_{b}$ exhibits a
one-parametric family of 1-cycles with ${g}=R\begin{pmatrix}1&\eta\\\
0&1\end{pmatrix}R^{-1}$ for any $\eta\in\mathbb{C}$ if
$T\mathbf{v}\propto\mathbf{v}$ (or $\mathbf{v}=\mathbf{0}$).

Let us now discuss the case that $T$ is diagonalizable. We now distinguish two
further cases depending on whether there exists an $m\in\mathbb{N}$ such that
$T^{m}\propto\text{1}$, or not. In case such an $m$ does not exist, we
distinguish several subcases depending on $\mathbf{v}$. If
$\mathbf{v}=\mathbf{0}$ we obtain a one-parametric family of 1-cycles with
${g}=R\begin{pmatrix}\chi&\\\ &1/\chi\end{pmatrix}R^{-1}$ for any
$\chi\in\mathbb{C}\setminus\\{0\\}$. In contrast, if
$T\mathbf{v}\not\propto\mathbf{v}$, but $T^{2}\mathbf{v}\propto\mathbf{v}$ we
obtain a single 1-cycle with ${g}=R\begin{pmatrix}i&\\\
&-i\end{pmatrix}R^{-1}$. We obtain only trivial cycles for all other
$\mathbf{v}$. Note that a generic $b$ falls into this category.

Let us now discuss the case that there exists an $m\in\mathbb{N}$ such that
$T^{m}\propto\text{1}$. In this case, we may write $T\propto
R\begin{pmatrix}e^{i\frac{r\pi}{m}}&\\\
&e^{i\frac{r\pi}{m}}\end{pmatrix}R^{-1}$ for some $r\in\\{0,\ldots,m-1\\}$.
Again, we distinguish several subcases depending on the vector $\mathbf{v}$.
First, let us consider the case that $\mathbf{v}=\mathbf{0}$. In this case we
obtain a rich set of cycles in $G_{b}$. Actually, we obtain $m$-cycles with
${g}_{k}=T^{k}{g}_{0}T^{-k}$ for any ${g}_{0}$. This is effectively a three-
parametric family of cycles including both, instances in which the ${g}_{k}$
are diagonalizable, as well as instances in which ${g}_{k}$ are not
diagonalizable. In case $m$ is even, in addition to that a one-parametric
family of $m/2$-cycle emerges. There, we have ${g}_{k}=R\begin{pmatrix}0&i\eta
e^{i\frac{2k(2r+1)\pi}{m}}\\\ i/\eta
e^{-i\frac{2k(2r+1)\pi}{m}}&0\end{pmatrix}R^{-1}$, for any
$y\in\mathbb{C}\setminus\\{0\\}$. Note that $\text{tr}{g}_{k}=0$. Let us now
discuss the case that we have a non-vanishing $\mathbf{v}$ with
$T\mathbf{v}\propto\mathbf{v}$. In this case we obtain an effectively one-
parametric $m$-cycle of non-diagonalizable cycles with
${g}_{k}=S_{0}\begin{pmatrix}1&e^{i\frac{2kr\pi}{m}}\\\
0&1\end{pmatrix}S_{0}^{-1}$, where $S_{0}=\left(\mathbf{v},\mathbf{w}\right)$
for any $\mathbf{w}\in\mathbb{C}^{2}$. In other words, ${g}_{0}$ may be chosen
as any non-diagonalizable matrix whose eigenvector is given by $\mathbf{v}$,
the remaining matrices are then determined. Let us now discuss the case that
we have a non-vanishing $\mathbf{v}$ with $T\mathbf{v}\not\propto\mathbf{v}$,
but $T^{2}\mathbf{v}\propto\mathbf{v}$. Note that this implies
$T^{2}\propto\text{1}$. In this case, we obtain cycles with diagonalizable
${g}_{k}$. The eigenvectors of each ${g}_{k}$ are given by $\mathbf{v}$ and
$T\mathbf{v}$. We obtain global cycles, in which the eigenvalues of ${g}$ are
$\pm i$. Moreover, in case of an even particle number, we obtain two-cycles
with ${g}_{1}={g}_{0}^{-1}$ and a freely choosable eigenvalue $\chi\neq 0,\pm
i$. Finally, in case of a non-vanishing $\mathbf{v}$ with
$T^{2}\mathbf{v}\not\propto\mathbf{v}$ we obtain no non-trivial cycles. This
completes the characterization of cycles within $G_{b}$ considering the
fiducial states $\text{1}\otimes b\otimes\text{1}|LLT\rangle$. We present a
summary of the findings in terms of a flowchart in Figure 4.

Let us conclude with remarking that for any specified $T$ and $\mathbf{v}$,
there is a two-parametric family of $b$’s (with $b_{20}=1$) leading to the
specified $T$, $\mathbf{v}$, as in Eq. (22). Thus, it is possible to construct
a $b$ possessing any desired symmetry presented in Figure 4 using the
appropriate $T$ and $\mathbf{v}$. Moreover, for normal MPS, Theorem 5
characterizes all possible symmetries.

###  V.4 SLOCC Classification

In order to identify the different SLOCC classes emerging within the normal
MPS associated to $|LLT\rangle$, we consider $(b\rightarrow c)$ cycles within
the symmetry group of the fiducial state. More precisely, we study the
relation

$\displaystyle y_{k}bx_{k+1}\propto c\text{ for all }k\in\\{0,\ldots,N-1\\}$
(30)

in order to decide whether the MPS generated by $\text{1}\otimes
b\otimes\text{1}|LLT\rangle$ and $\text{1}\otimes c\otimes\text{1}|LLT\rangle$
are SLOCC equivalent to each other. As shown in [18] (see also Sec. III and
Sec. IV), they are SLOCC equivalent to each other iff it is possible to
identify an $N$-cycle (or an $M$-cycle, where $M$ divides the total particle
number $N$). We will first characterize 1-cycles. Then, we will introduce a
(non-unique) standard form for $b$ and $c$ up to global SLOCC operations.
Finally, we complete the classification by considering non-global operations.
Note that we charcterize the $(b\rightarrow c)$ cycles for all $b$, $c$ with
$b_{20},c_{20}\neq 0$, however, we keep in mind that certain such $b,c$ lead
to non-normal MPS. We present the SLOCC classification in the flowchart shown
in Figure 5, following the same structure as in Figure 4.

Recall that two states $|\psi\rangle$ and $|\phi\rangle$ can only be SLOCC
equivalent if their symmetry group is compatible, i.e.,
$\mathcal{S}_{|\psi\rangle}$ equals $\mathcal{S}_{|\phi\rangle}$ up to
conjugation. This immediately shows that, e.g., states belonging to the box
(IV) cannot be SLOCC equivalent to states belonging to box (V) in Figure 4.
However, this necessary condition is not strong enough to reveal anything
about SLOCC equivalence between, e.g., states belonging to boxes (V) and (VI)
within the figure, yet.

####  V.4.1 Global SLOCC operations and standard form

As a first step, we investigate $(b\rightarrow c)$ 1-cycles, which allow to
characterize equivalence of normal MPS under global operations. For normal
MPS, stated differently, we characterize here all $b$ for which there exists
an operator $g$ such that $\Psi_{b}(LLT)=g^{\otimes n}\Psi_{c}(LLT)$ for a
given $c$. Using the symmetry of the fiducial state [see Eq. (21)] this leads
to the following. A $(b\rightarrow c)$ 1-cycle exists if

$\displaystyle b\propto x^{-1}cy^{-1},$

where

$\displaystyle x^{-1}$
$\displaystyle=\begin{pmatrix}\delta&\gamma&B_{02}\gamma+B_{01}\delta\\\
\beta&\alpha&B_{02}\alpha+B_{01}\beta\\\ 0&0&B_{11}\end{pmatrix}$
$\displaystyle y^{-1}$
$\displaystyle=\frac{1}{B_{11}}\begin{pmatrix}B_{11}&-(B_{02}\gamma+B_{01}\delta)&-(B_{02}\alpha+B_{01}\beta)\\\
0&\delta&\beta\\\ 0&\gamma&\alpha\end{pmatrix},$

where $\alpha,\beta,\gamma,\delta,B_{11},B_{01},B_{02}\in\mathbb{C}$ such that
$\det g=1$, where $g=\begin{pmatrix}\alpha&\beta\\\
\gamma&\delta\end{pmatrix}$. We use the parametrization and normalization of
$b$ and $c$ as in Eq. (22) and write

$\displaystyle b=b[T_{b},\mathbf{v}_{b},\begin{pmatrix}b_{10}\\\
b_{00}\end{pmatrix}]$

and similarly for $c$. We obtain all $b$ that are connected to $c$ via a
$(b\rightarrow c)$ 1-cycle through

$\displaystyle
b=b[\frac{1}{B_{11}^{2}}gT_{c}g^{-1},\frac{1}{B_{11}}g\mathbf{v}_{c},\frac{1}{B_{11}}g\begin{pmatrix}c_{10}+B_{02}\\\
c_{00}+B_{01}\end{pmatrix}]$ (31)

for $g$ with $\det g=1$ and $B_{11},B_{01},B_{02}\in\mathbb{C}$. Here, $g$ is
the global physical operation relating the two MPS. Note that even
$g=\text{1}$ leads to a freedom in $b$, which is due to symmetries of the
fiducial state that have the form $\text{1}\otimes B\otimes C$. Note also that
in Eq. (31) we have equality and not proportionality as $b_{20}=c_{20}=1$
fixes the proportionality factor to 1. Note further that Eq. (31) allows to
easily identify global LU-invariant quantities.

We introduce a standard form for $b$, $c$ up to global operations (1-cycles).
It then suffices to study SLOCC equivalence for MPS associated to $b$, $c$
which are in standard form in order to provide a full characterization of
SLOCC equivalence. We choose the following standard form:

$\displaystyle b=b[T_{b},\mathbf{v}_{b},0],$

where $T_{b}$ is in JNF, $\det T_{b}=1$. Moreover, we use the same convention
for the ordering and possible sign flip of the eigenvalues as earlier in this
section. More precisely, for diagonalizable $T_{b}$ we write
$T_{b}=\operatorname{diag}(\sigma_{b},\sigma_{b}^{-1})$, where
$\sigma_{b}\in\mathcal{D}$ (see Figure 3). Note that the standard form is not
unique as we may flip the direction of $\mathbf{v}$ via a sign change in
$B_{11}$ and moreover, special forms of $T_{b}$ such as e.g. $T_{b}=\text{1}$
leave even more freedom to choose the direction of $\mathbf{v}$. Let us stress
here that $b$ with coinciding $T$ and non-vanishing $\mathbf{v}$ whose
directions coincide, but whose norms differ, lead to MPS that share the same
symmetry group, but are not necessarily related by a global SLOCC operation.
Clearly, if $b$ and $c$ which are connected by a $(b\rightarrow c)$ 1-cycle
are in standard form, we necessarily have that $T_{c}=T_{b}$.

For normal tensors, the characterization of $(b\rightarrow c)$ 1-cycles allows
to characterize equivalence of the associated MPS under global SLOCC
operations (for non-normal tensors, additional MPS might turn out to be
equivalent, which are not identified as equivalent by considering
$(b\rightarrow c)$ cycles) [18]. Due to the considerations of $(b\rightarrow
c)$ 1-cycles above, we obtain such a characterization as stated in the
following lemma.

######  Lemma 3.

Consider fiducial states $\text{1}\otimes b\otimes\text{1}|LLT\rangle$ and
$\text{1}\otimes c\otimes\text{1}|LLT\rangle$ which correspond to normal MPS
(i.e., $b_{20}=c_{20}=1$ and additionally
$\mathbf{v}_{b},\mathbf{v}_{c}\neq\mathbf{0}$). Then, the MPS are related via
a global SLOCC operation if and only if there exists a $g\in SL(2,\mathbb{C})$
such that $\mathbf{v}_{b}\propto g\mathbf{v}_{c}$ and
$T_{b}=\frac{\mathbf{v}_{b}^{T}\mathbf{v}_{b}}{(g\mathbf{v}_{c})^{T}g\mathbf{v}_{c}}gT_{c}g^{-1}$.

###### Proof.

The statement follows from the considerations of $(b\rightarrow c)$ 1-cycles
above. ∎

####  V.4.2 Non-global SLOCC operations

Let us now also take non-global SLOCC operations into account. Considering

$\displaystyle b\propto x_{k}^{-1}cy_{k+1}^{-1},$

and imposing that both $b$ and $c$ are in standard form we obtain
$B_{01}^{(k)}=B_{02}^{(k)}=0$ for all $k$ and

$\displaystyle
b=b[\frac{1}{B_{11}^{(k)}B_{11}^{(k+1)}}g_{k+1}T_{c}g_{k}^{-1},\frac{1}{B_{11}^{(k+1)}}g_{k+1}\mathbf{v}_{c},0],$
(32)

where we use the normalization $\det g_{k}=1$. We obtain as a simple necessary
condition for having an $N$-cycle

$\displaystyle T_{b}^{N}=\pm g_{k}T_{c}^{N}g_{k}^{-1}$ (33)

for all $k$ (with a positive sign in case of even $N$, $\pm$ in case of odd
$N$).

Using Eqs. (32) and (33) it is straightforward to establish that within Figure
5, fiducial states that belong to different boxes do allow for a
$(b\rightarrow c)$ cycle (see Observation 10 in Appendix E).

Let us now complete the characterization of $(b\rightarrow c)$ cycles. In the
case that $T_{c}^{N}\not\propto\text{1}$, considering Eq. (33), the standard
form for $T_{b},T_{c}$, and the uniqueness of the Jordan decomposition
straightforwardly leads to the fact that all $g_{k}$ must be in Jordan normal
form (special care needs to be taken in case $\operatorname{tr}T_{c}=0$).
Then, using Eq. (32) in addition, a tedious calculation shows that $g_{k}=g$
for all $k$. However, the case $T_{c}^{N}\propto\text{1}$ is more involved as
in this case, the condition in Eq. (33) is not helpful. Let us thus take
intermediate steps in completing the characterization of $(b\rightarrow c)$
cycles. To this end, we will introduce two lemmata, which we prove in Appendix
E. It is obvious that whenever we have $b$ and $c$ in standard form allowing
for a $(b\rightarrow c)$ 1-cycle, it holds that $T_{b}=T_{c}$. The first lemma
shows that the same is true for $(b\rightarrow c)$ $N$-cycles if
$\mathbf{v}_{b}\neq\mathbf{0}$, $\mathbf{v}_{c}\neq\mathbf{0}$.

######  Lemma 4.

Consider $b$ and $c$ in standard form which correspond to normal MPS (i.e.,
$b_{20}=c_{20}=1$ and additionally
$\mathbf{v}_{b},\mathbf{v}_{c}\neq\mathbf{0}$). If there exists a
$(b\rightarrow c)$ $N$-cycle, then, $T_{b}=T_{c}$.

Building on Lemma 4, the next lemma shows that whenever such $b$ and $c$ are
connected by a $(b\rightarrow c)$ $N$-cycle, there also exists a
$(b\rightarrow c)$ $1$-cycle.

######  Lemma 5.

Consider $b$ and $c$ which correspond to normal MPS (i.e., $b_{20}=c_{20}=1$
and additionally $\mathbf{v}_{b},\mathbf{v}_{c}\neq\mathbf{0}$). If there
exists a $(b\rightarrow c)$ $N$-cycle, then, there also exists a
$(b\rightarrow c)$ $1$-cycle.

We are now in the position to state simple necessary and sufficient conditions
for SLOCC equivalence of normal MPS generated by fiducial states within the
$LLT$ class.

######  Theorem 6.

Consider fiducial states $\text{1}\otimes b\otimes\text{1}|LLT\rangle$ and
$\text{1}\otimes c\otimes\text{1}|LLT\rangle$ which correspond to normal MPS
(i.e., $b_{20}=c_{20}=1$ and additionally
$\mathbf{v}_{b},\mathbf{v}_{c}\neq\mathbf{0}$). Then, the MPS are SLOCC
equivalent if and only if they are related via a global operation, i.e., there
exists a $g\in SL(2,\mathbb{C})$ such that $\mathbf{v}_{b}\propto
g\mathbf{v}_{c}$ and
$T_{b}=\frac{\mathbf{v}_{b}^{T}\mathbf{v}_{b}}{(g\mathbf{v}_{c})^{T}g\mathbf{v}_{c}}gT_{c}g^{-1}$.

Let us remark here that the operator $g$ in the Theorem is such that
$g^{\otimes N}$ transforms one state into the other.

###### Proof.

The statement of the theorem follows directly from Lemma 5 together with the
considerations on global SLOCC operations [$(b\rightarrow c)$ 1-cycles] in
Lemma 3. ∎

A straightforward consequence of the theorem is that SLOCC equivalence of the
considered MPS is not particle-number dependent, in spite of all the variety
within their ($N$-dependent) symmetry group. Note that this is not true in
general, see e.g. the SLOCC classes for MPS with bond dimension $D=2$ that are
generated by fiducial states within the GHZ class [18].

####  V.4.3 Representatives and parametrization of SLOCC classes

Due to Theorem 6 we have that two MPS are SLOCC equivalent iff they are
related by a global transformation. Here, we parameterize all SLOCC classes by
introducing a more precise standard form for the various $b$’s, i.e. the
fiducial states. This standard form is then also useful to obtain
representative MPS for the different SLOCC classes. The resulting
representatives of the SLOCC classes are presented in the flowchart in Fig. 5.
This completes the characterization of all SLOCC classes of TIMPS
corresponding to the fiducial states which are SLOCC equivalent to the
$|LLT\rangle$ state.

As mentioned before (see Observation 10 in Appendix E) SLOCC equivalent normal
TIMPS must belong to the same box in Fig. 4. We obtain the parameterization of
the SLOCC classes by introducing a precise standard form of all operators $b$
corresponding to the individual boxes. To this end we consider the operators
$b$ and $c$ which have the same standard form as given above. We have seen
that $T$ may be brought into Jordan normal form, and normalized to determinant
1. Let us now further specify $T$ and the vector $\mathbf{v}$ for the various
cases (boxes).

Let us first consider the scenario that $T_{b}$ and $T_{c}$ are not
diagonalizable in more details. Due to the chosen standard form, we then have
$T_{b}=T_{c}=\begin{pmatrix}1&1\\\ 0&1\end{pmatrix}$. Moreover, due to
$T_{b}\propto gT_{c}g^{-1}$ we have that

$\displaystyle g=\pm\begin{pmatrix}1&\beta\\\ 0&1\end{pmatrix}.$

Note that these are the only global transformations, which map normal MPSs
with fiducial states $\text{1}\otimes b\otimes\text{1}|LLT\rangle$, with $b$
such that $T_{b}$ is non–diagonalizable and in standard from into each other.
We hence have that $b$ and $c$ (in standard form) such that $T_{b}$ and
$T_{c}$ are not diagonalizable lead to MPS that are SLOCC related if and only
if

$\displaystyle\mathbf{v}_{b}=\begin{pmatrix}v_{b,0}\\\
v_{b,1}\end{pmatrix}=\pm\begin{pmatrix}v_{c,0}+\beta v_{c,1}\\\
v_{c,1}\end{pmatrix}$

for some $\beta\in\mathbb{C}$. In order to take into account this freedom, we
amend the definition of the standard form of $b$ by additionally requiring
that either $\mathbf{v}\propto(1,0)^{T}$, or $\mathbf{v}\propto(0,1)^{T}$ (or
$\mathbf{v}=\mathbf{0}$). Then, we have that $b$ and $c$ in standard form with
non-diagonlizable $T_{b}$, $T_{c}$ correspond to MPS that are related by a
global SLOCC operation if and only if $\mathbf{v}_{b}=\pm\mathbf{v}_{c}$. The
standard form for $b$ may now be used to obtain MPS that are representatives
for the present SLOCC classes. Contemplating the characterization of
symmetries, we have that there is a one-parametric131313Recall that we count
complex parameters. family of SLOCC classes exhibiting a non-trivial global
symmetry with $\mathbf{v}_{b}\propto(1,0)^{T}$ (the proportionality factor is
the free complex parameter). More precisely, all states which belong to the
SLOCC class can be transformed into the standard form with $\mathbf{v}_{b}=\pm
x(1,0)^{T}$, and they belong to different SLOCC classes for different values
of $x\in\text{C}$.

Supposing $\mathbf{v}\neq\mathbf{0}$, these classes correspond to the box
(IIb) in Figure 4. Moreover, we find a one-parametric family with trivial
symmetry group for $\mathbf{v}\propto(0,1)^{T}$, $\mathbf{v}\neq\mathbf{0}$,
which corresponds to box (V) within Figure 4. More precisely, all states with
$\mathbf{v}_{c}=\pm x(\beta,1)^{T}$ belong to the SLOCC class with
$\mathbf{v}_{b}=x(0,1)^{T}$ for arbitrary $\beta\in\text{C}$ and fixed
$x\in\text{C}$.

Let us now discuss the case that $T_{c}$ and $T_{b}$ are diagonalizable, i.e.,
$T_{c}=T_{b}=\operatorname{diag}(\sigma,1/\sigma)$ for some
$\sigma\in\mathbb{C}$. In the case that $\sigma=1$, the MPS are SLOCC-
equivalent if and only if there exists a $g$ with $\det g=1$ such that
$\mathbf{v}_{b}=g\mathbf{v}_{c}$. We thus choose the standard form
$\mathbf{v}=(1,0)^{T}$. For $\sigma=i$, we obtain a $(b\rightarrow c)$ 1-cycle
if and only if there exists an $g=\operatorname{diag}(\alpha,1/\alpha)$ such
that $\mathbf{v}_{b}=\pm g\mathbf{v}_{c}$, or $\mathbf{v}_{b}=\pm
ig\sigma_{x}\mathbf{v}_{c}$. For $\sigma\neq 1,i$, we obtain a $(b\rightarrow
c)$ 1-cycle if and only if there exists an
$g=\operatorname{diag}(\alpha,1/\alpha)$ such that $\mathbf{v}_{b}=\pm
g\mathbf{v}_{c}$. In case we have $T\mathbf{v}\propto\mathbf{v}$, we thus
choose the standard form such that either $\mathbf{v}=(1,0)^{T}$, or
$\mathbf{v}=(0,1)^{T}$. Otherwise, we choose the standard form
$\mathbf{v}=(v_{1},1/v_{1})^{T}$ for $v_{1}\in\mathbb{C}\setminus\\{0\\}$. Due
to these considereations, there is a two-parametric family of SLOCC classes
corresponding to box (VI) within Figure 4 (one free parameter within $T$ plus
one free parameter within $\mathbf{v}$). For the remaining boxes containing
normal MPS, i.e., boxes (IV), (VII), and (VIII), let us suppose that
$T^{m}\propto\text{1}$ for some fixed $m$. Then, there is a discrete number of
possible $T$s, namely $\lfloor m/2\rfloor+1$ (see Figure 3). For a fixed $m$,
there are exactly $m$ different SLOCC classes corresponding to box (IV), as
for each of the $\lfloor m/2\rfloor+1$ possible $T$’s we either have
$\mathbf{v}=(1,0)^{T}$, or $\mathbf{v}=(0,1)^{T}$, except for $T=\text{1}$ and
$T=\operatorname{diag}(-i,i)$, where these possibilities are equivalent and we
hence choose $\mathbf{v}=(1,0)^{T}$. Particularly interesting will be the
subfamily corresponding box (IV) for which $m=N$, as this subfamily
encompasses all type-(IV) MPS with non-trivial symmetry group. Due to the
reasonings above, there are exactly $N$ of them. Finally, there are one-
parametric families of SLOCC classes corresponding to boxes (VII) and (VIII),
respectively, with the free parameter stemming from $\mathbf{v}$. We summarize
these findings in Figure 5 following the same structure as in Figure 4, which
displays the corresponding symmetries.

Given $b$ with $b_{20}=1$, calculate $T$, $\mathbf{v}$. Write $T=RJR^{-1}$,
where $J$ is JNF of $T$ $T$ diagonalizable? $T^{m}\propto\text{1}$ for some
$m\in\mathbb{N}$? Write $T\propto R\begin{pmatrix}e^{i\frac{r\pi}{m}}&\\\
&e^{-i\frac{r\pi}{m}}\end{pmatrix}R^{-1}$, $r\in\\{0,\ldots,m-1\\}$
$\mathbf{v}=\mathbf{0}$? (I)  2 SLOCC classes (even $N$):
$T\in\\{\text{1},i\sigma_{z}\\}$, $\mathbf{v}=\mathbf{0}$ (IIa)  single SLOCC
class (even $N$): $T=\begin{pmatrix}1&1\\\ 0&1\end{pmatrix}$,
$\mathbf{v}=\mathbf{0}$ $\mathbf{v}=\mathbf{0}$? $\mathbf{v}=\mathbf{0}$?
(III)  1-param. SLOCC family (even $N$): $T=\begin{pmatrix}\sigma&0\\\
0&1/\sigma\end{pmatrix}$, $\mathbf{v}=\mathbf{0}$,
$\sigma\in\mathbb{C}\setminus\\{0\\}$ s.t. there exists no $m\in\mathbb{N}$
s.t. $\sigma^{m}=\pm 1$ $T\mathbf{v}\propto\mathbf{v}$? (IV)  $m$ SLOCC
classes for a fixed $m$: $T=\begin{pmatrix}e^{i\frac{r\pi}{m}}&0\\\
0&e^{-i\frac{r\pi}{m}}\end{pmatrix}$ for
$r\in\\{1,\ldots,\lfloor\frac{m-1}{2}\rfloor\\}$,
$\mathbf{v}\in\\{(1,0)^{T},(0,1)^{T}\\}$. Moreover,
$T\in\\{\text{1},i\sigma_{z}\\}$ ($T=\text{1}$) for even $m$ (odd $m$),
respectively, and $\mathbf{v}=(1,0)^{T}$. (IIb)  1-param. SLOCC family:
$T=\begin{pmatrix}1&1\\\ 0&1\end{pmatrix}$, $\mathbf{v}=(x,0)^{T}$,
$x\in\mathbb{C}\setminus\\{0\\}$. $T\mathbf{v}\propto\mathbf{v}$? (VI) -
generic  2-param. SLOCC family: $T=\begin{pmatrix}\sigma&0\\\
0&1/\sigma\end{pmatrix}$, $\sigma\in\mathbb{D}$ s.t. $\sigma^{m}\neq 1$ for
any $m\in\mathbb{N}$, $\mathbf{v}=(1,0)^{T}$, $\mathbf{v}=(0,1)^{T}$, or
$\mathbf{v}=(v_{1},1/v_{1})^{T}$, $v_{1}\in\mathbb{C}\setminus\\{0\\}$.
$T^{2}\mathbf{v}\propto\mathbf{v}$, i.e., $T^{2}\propto\text{1}$? (VII)
1-param. SLOCC family: $T=\begin{pmatrix}i&0\\\ 0&-i\end{pmatrix}$,
$\mathbf{v}=(v_{1},1/v_{1})^{T}$, $v_{1}\in\mathbb{C}\setminus\\{0\\}$. (V)
1-param. SLOCC family: $T=\begin{pmatrix}1&1\\\ 0&1\end{pmatrix}$,
$\mathbf{v}=(0,x)^{T}$, $x\in\mathbb{C}\setminus\\{0\\}$. (VIII)  1-param.
SLOCC family: $T=\begin{pmatrix}e^{i\frac{s\pi}{m}}&0\\\
0&e^{-i\frac{s\pi}{m}}\end{pmatrix}$ for $s\in\\{0,\ldots,\lfloor
m/2\rfloor\\}$, $\mathbf{v}=(v_{1},1/v_{1})^{T}$,
$v_{1}\in\mathbb{C}\setminus\\{0\\}$. noyesyesnoyesyesnonoyesnonoyesyesyesnono

Figure 5: Summary of the SLOCC classification of all normal MPS generated by
$\text{1}\otimes b\otimes\text{1}|LLT\rangle$ (shaded rectangles with solid
contour) plus partial results on SLOCC classes of some non-normal MPS (shaded
rectangles with dashed contour). We display the number of SLOCC classes
corresponding to each type and give representatives for every SLOCC class (we
count complex parameters). To simplify the presentation not all redundancies
in the representatives of the continuous SLOCC families are avoided.
Redundancy may be removed straightforwardly, though. The flowchart is
following the same structure as the one in Figure 4, in particular, the
displayed type labels agree. Note that additional non-normal MPS that have not
been identified as SLOCC equivalent might in fact be equivalent. The displayed
non-normal MPS vanish in case of odd $N$.

###  V.5 Non-normal MPS

We have seen in Observation 7 that an MPS associated to $b$ is normal if and
only if $b_{20}=1$ and $\mathbf{v}_{b}\neq\mathbf{0}$. In Observation 6 we
have analyzed (non-normal) MPS associated to $b$ with $b_{20}=0$. In this
section, we discuss the remaining non-normal MPS, i.e., MPS generated by $b$
such that $b_{20}=1$ and $\mathbf{v}=\mathbf{0}$, i.e. the three boxes (I),
(IIa), and (III) in Fig. 4. In particular, we present non-normal MPS belonging
to box (I) and show that the symmetries determined in this section might