# A link between symmetries of critical states and the structure of SLOCC
classes in multipartite systems

Oskar Słowik1 Martin Hebenstreit2 Barbara Kraus2 Adam Sawicki1 1 Center for
Theoretical Physics, Polish Academy of Sciences, 02-668 Warsaw, Poland
2 Institute for Theoretical Physics, University of Innsbruck, A–6020
Innsbruck, Austria

###### Abstract

Central in entanglement theory is the characterization of local
transformations among pure multipartite states. As a first step towards such a
characterization, one needs to identify those states which can be transformed
into each other via local operations with a non-vanishing probability. The
classes obtained in this way are called SLOCC classes. They can be categorized
into three disjoint types: the null-cone, the polystable states and strictly
semistable states. Whereas the former two are well characterized, not much is
known about strictly semistable states. We derive a criterion for the
existence of the latter. In particular, we show that there exists a strictly
semistable state if and only if there exist two polystable states whose orbits
have different dimensions. We illustrate the usefulness of this criterion by
applying it to tripartite states where one of the systems is a qubit.
Moreover, we scrutinize all SLOCC classes of these systems and derive a
complete characterization of the corresponding orbit types. We present
representatives of strictly semistable classes and show to which polystable
state they converge via local regular operators.

##  1 Introduction

Due to the relevance of multipartite entanglement in many areas of physics,
entanglement theory has developed into an important research topic over the
last decades [Nielsen and Chuang(2009), Horodecki  _et al._(2009)Horodecki,
Horodecki, Horodecki, and Horodecki]. Multipartite entanglement does not only
play an essential role in quantum communication [Hillery  _et
al._(1999)Hillery, Bužek, and Berthiaume, Gottesman(2000)], in quantum
computation, e.g., measurement-based quantum computation [Raussendorf and
Briegel(2001)], or quantum metrology (see e.g. [Giovannetti  _et
al._(2011)Giovannetti, Lloyd, and Maccone]), but it does also play a role in
condensed matter physics. There, a connection between entanglement present in
the wave function and quantum phase transitions can be drawn (for a review see
e.g. [Amico  _et al._(2008)Amico, Fazio, Osterloh, and Vedral] and references
therein). Moreover, tensor network methods such as matrix product states
[Fannes  _et al._(1992)Fannes, Nachtergaele, and Werner, Perez-Garcia  _et
al._(2007)Perez-Garcia, Verstraete, Wolf, and Cirac, Vidal(2003)] or projected
entangled pair states [Verstraete and Cirac(2004), Verstraete  _et
al._(2006)Verstraete, Wolf, Perez-Garcia, and Cirac] embody powerful numerical
tools in condensed matter physics (see e.g. [Orús(2014)] and references
therein).

Entanglement theory is a resource theory, which becomes apparent in the
scenario of spatially separated parties sharing a joint quantum state, an
often considered scenario in quantum information theory. There, it is natural
to restrict the allowed operations to Local quantum Operations assisted by
Classical Communication (LOCC). Then, entanglement arises as a resource
allowing to achieve certain tasks that are not possible by LOCC alone.
Moreover, it is apparent that any state $|\psi\rangle$, which can be
deterministically converted to another state $|\phi\rangle$ via LOCC, must be
at least as entangled as $|\phi\rangle$ according to any reasonable measure of
entanglement. The reason for that is that any task starting out with the state
$|\phi\rangle$ could as well be implemented starting out with the state
$|\psi\rangle$ instead (transforming it to $|\phi\rangle$ as a first step).
The partial order that LOCC imposes on the state space must thus be taken into
account by any entanglement measure, which is also known as the LOCC-
monotonicity condition for entanglement measures [Horodecki  _et
al._(2009)Horodecki, Horodecki, Horodecki, and Horodecki].

Studying multipartite entanglement is hence intimately connected to
understanding which state transformations are possible via LOCC. A first step
in studying whether a state transformation is possible via LOCC
deterministically, is studying whether the state transformation is possible
via stochastic LOCC (SLOCC), i.e., with a non-vanishing probability of
success. Clearly, if a state transformation is not possible via SLOCC, it is
not possible via LOCC. It has been shown that if one considers fully entangled
states, i.e., states $|\psi\rangle$ for which all single-particle reduced
density matrices
$\rho_{i}=\text{tr}_{1,\ldots,i-1,i+1,\ldots,N}|\psi\rangle\langle\psi|$ have
full rank (here, $N$ denotes the number of parties), then SLOCC is an
equivalence relation [Dür  _et al._(2000)Dür, Vidal, and Cirac].
Mathematically stated, two $N$-partite states $|\psi\rangle$ and
$|\phi\rangle$ are SLOCC-equivalent if and only if there exist regular
operators $A_{1},\ldots,A_{N}$ such that
$|\psi\rangle=A_{1}\otimes\ldots\otimes A_{N}|\phi\rangle$ [Dür  _et
al._(2000)Dür, Vidal, and Cirac]. The state space thus partitions into so-
called SLOCC classes. Another often considered class of operations are Local
Unitaries (LUs) (for pure states see e.g. [Kraus(2010a), Kraus(2010b)]). Such
operations are reversible and, hence, do not alter the entanglement of a given
state.

For bipartite pure states, necessary and sufficient conditions for when LOCC
transformations are possible have been derived [Nielsen(1999)]. Moreover, for
such states, the concept of SLOCC classes is trivial in the sense that there
exists exactly one SLOCC class containing all fully entangled states, i.e.,
states that have full Schmidt rank. All this is no longer true in the
multipartite scenario. Deciding which state transformations are possible via
LOCC becomes a notoriously difficult problem due to the intricate structure of
multipartite LOCC protocols [Chitambar  _et al._(2014)Chitambar, Leung,
Mančinska, Ozols, and Winter, Chitambar(2011), Chitambar and Hsieh(2017)].
Moreover, considering the simplest multipartite quantum system, three qubits,
there exist two distinct SLOCC classes (considering fully entangled states),
the well-known GHZ- and W-class [Dür  _et al._(2000)Dür, Vidal, and Cirac].
Furthermore, already for systems as simple as four qubits, the Hilbert space
partitions into an infinite number of SLOCC classes [Verstraete  _et
al._(2002a)Verstraete, Dehaene, De Moor, and Verschelde]. Besides small system
sizes, SLOCC classes have been studied for special types of states, such as
permutation-symmetric states [Mathonet  _et al._(2010)Mathonet, Krins,
Godefroid, Lamata, Solano, and Bastin, Bastin  _et al._(2009)Bastin, Krins,
Mathonet, Godefroid, Lamata, and Solano, Migdał  _et al._(2013)Migdał,
Rodriguez-Laguna, and Lewenstein]. The notion of SL-invariant polynomials has
been used to distinguish certain SLOCC classes (see [Luque and Thibon(2003),
Viehmann  _et al._(2011)Viehmann, Eltschka, and Siewert, Eltschka and
Siewert(2014)] and references therein). These are polynomials $f$ in the
coefficients of a state $|\psi\rangle$, which fulfill that
$f(g_{1}\otimes\ldots\otimes g_{N}|\psi\rangle)=f(|\psi\rangle)$ for any
$g_{i}\in SL(d_{i},\mathbb{C})$; a complete set of such polynomials can be
constructed 111For first complete sets of SL-invariant polynomials see [Dür
_et al._(2000)Dür, Vidal, and Cirac] and [Luque and Thibon(2003)].[Gour and
Wallach(2013), Osterloh and Siewert(2005)].

Recently, a notable step in the characterization of state transformations has
been taken for homogeneous systems (systems for which all local dimensions
equal). In particular, it has been shown that in such systems, generically, no
LOCC transformations are possible [Gour  _et al._(2017)Gour, Kraus, and
Wallach, Sauerwein  _et al._(2018a)Sauerwein, Wallach, Gour, and Kraus].
Important in the proof was the fact that within such systems, states for which
all single-particle reduced density matrices $\rho_{i}$ are proportional to
the identity, so-called critical states, exist. A simple criterion that allows
to decide whether a critical state exists for given local dimensions
$d_{1},\ldots,d_{N}$ has been derived in [Bryan  _et al._(2019)Bryan,
Leutheusser, Reichstein, and Raamsdonk].

It has been realized that there is an intrinsic connection between the
existence of a critical state within an SLOCC class and the geometry of that
SLOCC class. Due to the Kempf-Ness theorem, an SLOCC class contains a critical
state if and only if it is closed (wrt. standard complex topology) [Kempf and
Ness(1979)]. We call (states within) SLOCC classes containing a critical state
_polystable_ 222Such states are also known as balanced states [Osterloh and
Siewert(2010)] and critical states are also known as stochastic states
[Verstraete  _et al._(2002a)Verstraete, Dehaene, De Moor, and Verschelde].. It
moreover holds that, within an SLOCC class, critical states are unique (up to
LUs). Moreover, within polystable classes, the norm
$||g_{1}\otimes\ldots\otimes g_{N}|\psi\rangle||$, where $g_{i}\in
SL(d_{i},\mathbb{C})$, does attain its minimum for $g_{1}\otimes\ldots\otimes
g_{N}|\psi\rangle$ being a critical state [Kempf and Ness(1979)]. Polystable
SLOCC classes have been a subject of interest in previous works
[Klyachko(2002), Osterloh and Siewert(2010), Gour  _et al._(2017)Gour, Kraus,
and Wallach, Sauerwein  _et al._(2018a)Sauerwein, Wallach, Gour, and Kraus,
Gour and Wallach(2011), Gour and Wallach(2013), Sauerwein  _et
al._(2018b)Sauerwein, Schwaiger, and Kraus]. It has been shown that such SLOCC
classes can be distinguished by ratios of SL-invariant polynomials [Gour and
Wallach(2013)]. SLOCC classes (states) for which all nonconstant homogeneous
SL-invariant polynomials vanish, form the so-called _null-cone_ , such classes
have been studied e.g. in [Ness and Mumford(1984), Walter  _et
al._(2013)Walter, Doran, Gross, and Christandl, Sawicki  _et
al._(2014)Sawicki, Oszmaniec, and Kuś, Maciążek and Sawicki(2015), Maciążek
and Sawicki(2018), Johansson  _et al._(2014)Johansson, Ericsson, Sjöqvist, and
Osterloh]. These SLOCC classes contain the $0$-vector in their closure and,
obviously, cannot be distinguished with SL-invariant polynomials. SLOCC
classes (states) that are not in the null-cone are called _semistable_. Thus,
the polystable classes are those semistable classes that do contain a critical
state. We call (states within) the remaining classes strictly semistable.
These SLOCC classes do not contain a critical state, however, they do contain
a critical state in their closure [Kempf and Ness(1979)].

Numerical tools which allow distinguishing between certain of the
aforementioned types of SLOCC classes have been developed. In [Verstraete  _et
al._(2003)Verstraete, Dehaene, and De Moor], an algorithm which transforms a
state $|\psi\rangle$ into its so-called normal form has been presented. In
case $|\psi\rangle$ is in the null-cone, this normal form is $0$. Otherwise,
it coincides with the critical state within the SLOCC class of $|\psi\rangle$
(closure of the SLOCC class of $|\psi\rangle$), in case $|\psi\rangle$ is
polystable (strictly semistable), respectively. In [Walter  _et
al._(2013)Walter, Doran, Gross, and Christandl, Sawicki  _et
al._(2014)Sawicki, Oszmaniec, and Kuś], it has been shown that a numerical
algorithm, which follows the gradient of the sum of the linear entropies of
the eigenvalues of $\rho_{i}$, also allows to determine the critical state
within a polystable SLOCC class, or the critical state within the closure of a
strictly semistable SLOCC class. Moreover, the latter method also allows to
distinguish certain SLOCC classes within the null-cone [Ness and
Mumford(1984), Sawicki  _et al._(2014)Sawicki, Oszmaniec, and Kuś, Sawicki
_et al._(2012)Sawicki, Oszmaniec, and Kuś, Maciążek and Sawicki(2015),
Maciążek and Sawicki(2018), Sawicki  _et al._(2018)Sawicki, Maciążek,
Oszmaniec, Karnas, Kowalczyk-Murynka, and Kuś].

Here, we study the existence of semistable states, i.e., states that are
neither in the null-cone, nor SLOCC-equivalent to a critical state using tools
from Geometric Invariant Theory (GIT). One of the main results of this article
is a criterion for the existence of such states in terms of the dimensions of
polystable SLOCC classes. Moreover, we study three-partite systems with local
dimensions $2$, $m$, and $n$ in more details. SLOCC classes in such systems
have been characterized in [Chitambar  _et al._(2010)Chitambar, Miller, and
Shi] (see also [Hebenstreit  _et al._(2018)Hebenstreit, Gachechiladze, Gühne,
and Kraus]) with the help of the theory of matrix pencils [Kronecker(1890)].
We apply the derived criterion for the existence of strictly semistable states
in the context of such systems. Moreover, we not only answer the question of
whether strictly semistable states exist, but also derive a full
characterization of the orbit types of all SLOCC classes within such systems.

The outline of the remainder of the paper is the following. First we introduce
our notation and some preliminary results on SLOCC classes and the normal form
of multipartite states [Verstraete  _et al._(2003)Verstraete, Dehaene, and De
Moor]. We will also recall how SLOCC classes of polystable states can be
distinguished via SL-invariant polynomials [Gour and Wallach(2013)]. In Sec. 3
we will focus on semistable states. Using tools from GIT, we will show that
strictly semistable states exist if and only if there exist two polystable
states whose orbits have different dimensions. In Sec. 4 we illustrate the
usefulness of this criterion by characterizing all SLOCC classes of tripartite
states in $2\times m\times n$. Moreover, we will identify the representatives
of strictly semistable states and will show to which polystable states they
converge via SLOCC.

##  2 Notation and Preliminaries

In this section we first introduce our notation and then review some important
concepts utilized in the characterization of SLOCC classes.

We consider the Hilbert space, ${\cal
H}=\mathbbm{C}^{d_{1}}\otimes\mathbbm{C}^{d_{2}}\otimes\ldots\otimes\mathbbm{C}^{d_{N}}$
with arbitrary local dimensions $d_{i}\geq 2$. We are mainly interested in
pure normalized states in $\mathcal{H}$. Hence, we consider the complex
projective space $\mathbb{P}(\mathcal{H})$, which is obtained from
$\mathcal{H}$ by identifying any two vectors which are proportional to each
other via a non-zero complex number. For any vector
$|\phi\rangle\in\mathcal{H}$ the corresponding quantum state is denoted by
$[\phi]\in\mathbb{P}(\mathcal{H})$. Throughout the paper we will consider
actions of appropriate Lie groups on both $\mathcal{H}$ and
$\mathbb{P}(\mathcal{H}$). The action of a Lie group $H$ on $\cal{H}$ will be
denoted by $H\curvearrowright\cal{H}$ and the action on $\mathbb{P}(\cal{H})$
by $H\curvearrowright\mathbb{P}(\cal{H})$, respectively. Given a Lie group
$H$, the $H$–orbit of a vector $|\Psi\rangle\in{\cal H}$ is defined as

$H|\Psi\rangle=\left\\{h|\Psi\rangle\Big{|}h\in H\right\\}.$ (1)

The stabilizer of a vector $|\Psi\rangle$ with respect to the group $H$ is
defined as

$H_{|\Psi\rangle}=\left\\{h\in H\Big{|}h|\Psi\rangle=|\Psi\rangle\right\\}.$
(2)

Any orbit $H|\Psi\rangle$ is an embedded submanifold of ${\cal H}$ which is
isomorphic to the left coset of $H_{|\Psi\rangle}$ in $H$, namely
$H|\Psi\rangle\simeq H/H_{|\Psi\rangle}$. Hence, the larger the dimension of
the stabilizer of the state, which is equal to the number of linearly
independent generators of the stabilizing Lie subgroup, the smaller the
dimension of the orbit. More precisely, the dimensions satisfy the formula

$\mathrm{dim}\,H|\Psi\rangle+\mathrm{dim}\,H_{|\Psi\rangle}=\mathrm{dim}\,H.$
(3)

The action of a Lie group $H$ on $\mathbb{P}(\mathcal{H})$ is induced by the
action $H\curvearrowright\mathcal{H}$ via

$\displaystyle h[\Psi]\coloneqq[h|\Psi\rangle],\,\,h\in H.$ (4)

Hence, the orbit through any state $[\Psi]$ is given by

$\displaystyle H[\Psi]=[H|\Psi\rangle],$ (5)

i.e. the $H$–orbit of any $[\Psi]$ in $\mathbb{P}(\mathcal{H})$ is the
projection of the orbit $H|\Psi\rangle\subset\mathcal{H}$. Thus, in contrast
to $H[\Psi]$, the $H$–orbit of $|\Psi\rangle$, $H|\Psi\rangle$, might contain
unnormalized states. The stabilizer of a state $[\Psi]$ with respect to the
group $H$ is defined as

$H_{[\Psi]}=\left\\{h\in H\Big{|}h|\Psi\rangle\propto|\Psi\rangle\right\\}.$
(6)

We also have that $H[\Psi]\simeq H/H_{[\Psi]}$ and

$\displaystyle\mathrm{dim}\,H[\Psi]+\mathrm{dim}\,H_{[\Psi]}=\mathrm{dim}\,H.$
(7)

The stabilizer $H_{[\Psi]}$ is the set of symmetries of $[\Psi]$ in $H$. Note
that $H_{\Psi}\subset H_{[\Psi]}$ for any $\Psi$ and any Lie group $H$. The
two groups we will consider are

  1. 1.

Special local unitary operators, i.e. elements of $K:=SU(d_{1})\times
SU(d_{2})\times\ldots\times SU(d_{N})$

  2. 2.

SLOCC operators, i.e. elements of $G:=SL(d_{1})\times
SL(d_{2})\times\ldots\times SL(d_{N})$.

Both groups act in a natural way on $\mathcal{H}$ and
$\mathbb{P}(\mathcal{H})$, respectively. The action of local unitary operators
does not alter entanglement and it defines an equivalence relation. Two states
are in the same LU-equivalence class if they belong to the same $K$–orbit in
$\mathbb{P}(\mathcal{H})$. Hence, instead of considering all states in
$\mathbb{P}(\mathcal{H})$, we will consider representatives of LU–equivalence
classes. Similarly the $G$–orbit of a state, $[\Psi]$, is the SLOCC–class
which contains $[\Psi]$ as an element. Two states, $[\Psi],[\Phi]$ are in the
same SLOCC–class, i.e. they are SLOCC–equivalent, if there exist $g_{i}\in
SL(d_{i})$ such that $[|\Psi\rangle]=[\bigotimes_{i=1}^{n}g_{i}|\Phi\rangle]$.

There is a qualitative difference between the actions of $K$ and $G$. The
group $K$ is compact and all $K$–orbits are closed. On the other hand the
group $G$ is not compact and $G$–orbits may not be closed. The closure of an
orbit $\overline{G|\Psi\rangle}\subset\mathcal{H}$ in the standard complex
topology in $\mathcal{H}$ can obviously be obtained by adding to
$G|\Psi\rangle$ the limits of all sequences of vectors in $G|\Psi\rangle$. It
is well known (see e.g. [Wallach(2017)]), that the closure in standard
topology coincides with the closure in Zariski topology. We refer the reader
to [Wallach(2017)] for more details on the Zariski topology and the relations
between those two topologies. Let us just mention here that a set
$X\in\mathbbm{C}^{m}$ is Zariski closed if there exists a set of polynomials,
$S\subset\mathbbm{C}[x_{1},\ldots,x_{m}]$ such that
$X=\mathbbm{C}^{m}(S)=\\{x\in\mathbbm{C}^{m}\mid s(x)=0\forall s\in S\\}$,
where $\mathbbm{C}[x_{1},\ldots,x_{m}]$ denotes the polynomial ring. A Zariski
closed set is called an algebraic variety. As mentioned above, the closures of
$G|\Psi\rangle$ in $\mathcal{H}$ coincide in both topologies. It will become
clear later on, which orbits $G[\Psi]\subset\mathbb{P}(\mathcal{H})$ are
considered to be closed.

States which play a particularly important role in the classification of
multipartite states are so–called critical states. A vector
$|\Psi\rangle\in{\cal H}$ is called critical if all single-particle reduced
states are completely mixed, i.e. $\rho_{i}\propto\mathbbm{1}$ for all $i$. In
the following we will denote the set of critical vectors by
$Crit\subset\mathcal{H}$ and the set of critical states by
$[Crit]\subset\mathbb{P}(\cal{H})$.

###  2.1 Normal form of multipartite states

As mentioned above, $G$-orbits in $\mathcal{H}$ may not be closed. In fact,
the Kempf-Ness theorem [Kempf and Ness(1979)] states that
$G|\Psi\rangle\subset\mathcal{H}$ is closed if and only if it contains a
unique (up to LUs) critical vector $|\Phi\rangle\in Crit\cup\\{0\\}$ 333 Note
that a particular example of a closed $G$-orbit is the $G$-orbit containing
the $0$-vector.. Moreover, $\|g|\Phi\rangle\|\geq\||\Phi\rangle\|$ for any
$g\in G$.

Let us note that $G$-orbit $G|\Psi\rangle$ is open in its closure
$\overline{G|\Psi\rangle}$, therefore the boundary of $G|\Psi\rangle$ can be
defined as $\overline{G|\Psi\rangle}\setminus G|\Psi\rangle$ 444See e.g.
[Brion(2010)] p. 4, Proposition 1.11..

The following lemma describes the structure of the boundary of an orbit and
follows from well-known facts - about the dimensions of orbits in the boundary
of orbit 555See e.g. [Brion(2010)] p. 4, Proposition 1.11. and basic facts in
GIT theory 666See e.g. [Mukai(2003)] p. 161, Theorem 5.3. and p. 162,
Corollary 5.5..

######  Lemma 1.

For each $|\Psi\rangle\in\mathcal{H}$, the boundary of its orbit,
$\overline{G|\Psi\rangle}\setminus G|\Psi\rangle$, is a union of $G$–orbits of
strictly smaller dimension. Each $G$-orbit closure contains a unique closed
$G$–orbit. This closed orbit is the only orbit in the closure which has
minimal dimension, i.e. the closed orbit has the smallest dimension of all
orbits in the closure.

Starting from a vector $|\Psi\rangle\in\mathcal{H}$ one can look for

$\displaystyle\inf_{|\Phi^{\prime}\rangle\in\overline{G|\Psi\rangle}}\||\Phi^{\prime}\rangle\|.$
(8)

According to the above, one of the following must be the case.

  1. (i)

The infimum is $0$.

  2. (ii)

The infimum is not $0$, it is attained at some critical vector
$|\Phi\rangle\in Crit$ within the orbit $G|\Psi\rangle$, and the orbit is
closed.

  3. (iii)

The infimum is not $0$ and it is attained at some critical vector
$|\Phi\rangle\in Crit$, which is not in the orbit $G|\Psi\rangle$, but within
its closure, $\overline{G|\Psi\rangle}$.

In case (i) the vectors are said to be unstable (they form the so-called null-
cone, which will be denoted by ${\cal N}$). For such vectors, there exist
sequences $\\{g_{k}\\}\subset G$ such that
$\lim_{k\rightarrow\infty}g_{k}|\Psi\rangle=0$. Vectors that are not unstable
are called semistable and they form a set
${\mathcal{H}}_{ss}\coloneqq\cal{H}\setminus{\cal N}$. In case (ii) the
vectors are called polystable; they are, hence, elements of an SLOCC class
which contains a critical vector. In case (ii) one can distinguish two
subcases: (ii a) - when the $G$-orbit of a polystable vector is of maximal
dimension; (ii b) - when the $G$-orbit of a polystable state is not of maximal
dimension. Note that maximal dimension refers here and in the following to the
maximal dimension among all polystable orbits. In case (ii a), we call a
vector stable 777Notice that in the literature concerning GIT (see e.g.
[Dolgachev(2010)] Chapter 8), the usual definition of stable vector is
different - a vector is stable if its $G$-orbit is closed (i.e. the state is
polystable) and has a finite stabilizer. Thus, our definition is much weaker
since we only require the dimension of the stabilizer to be minimal (among all
polystable orbits), not necessarily zero. and in case (ii b), we call a vector
strictly polystable. In case (iii), vectors are semistable but not polystable
- we call such vectors strictly semistable. For such states, an infinite
sequence $\\{g_{k}\\}\subset G$ is required in order to reach a critical
state, i.e., $\lim_{k\rightarrow\infty}g_{k}|\Psi\rangle=|\Phi\rangle\in
Crit$. It has been shown that any strictly semistable vector is a
superposition of a polystable vector and a vector from the null-cone (see the
Supplemental Material for [Gour and Wallach(2013)] p. 1, Proposition 2).
Finally, a state in the projective space $[\Psi]\in\mathbb{P}(\mathcal{H})$ is
called unstable/semistable/polystable etc. if it is the projection of some
unstable/semistable/polystable vector $|\Phi\rangle\in\mathcal{H}$,
respectively, i.e. $[\Psi]=[\Phi]$. We denote the set of semistable states by
$\mathbb{P}(\mathcal{H})_{ss}$. In case of a $2\times 2\times 2$ system, the
W–state, $|W\rangle=1/\sqrt{3}(|100\rangle+|010\rangle+|001\rangle)$ is a
prominent example of a state in the null-cone (case (i)), whereas the critical
GHZ–state $|GHZ\rangle=1/\sqrt{2}(|000\rangle+|111\rangle)$ is a stable state
(case (ii a)). In order to give an example of strictly semi- and polystable
states we need to consider systems of larger dimension. In case of a $2\times
4\times 4$ system, a state
$|\Psi_{sps}\rangle=\frac{1}{2}|0\rangle(|22\rangle+|33\rangle)+\frac{1}{2}|1\rangle(|00\rangle+|11\rangle)$
is strictly polystable (case (ii b) and a state
$|\Psi_{sss}\rangle=\frac{1}{\sqrt{5}}|0\rangle(|01\rangle+|22\rangle+|33\rangle)+\frac{1}{\sqrt{5}}|1\rangle(|00\rangle+|11\rangle)$
is strictly semistable (case (iii); cf. Section 4 and Appendix F.7). We
illustrate the relationship between the types of states in Figure 1.

Note that there is a numeric algorithm, which for any state $|\Psi\rangle$,
determines a sequence of SL-operators, which converges, if applied to
$|\Psi\rangle$, to the normal form as discussed above [Verstraete  _et
al._(2003)Verstraete, Dehaene, and De Moor]. If $|\Psi\rangle$ is in the null-
cone, the normal form vanishes. If $|\Psi\rangle$ is polystable, then the
normal form coincides with the critical state within the SLOCC class of
$|\Psi\rangle$. Finally, if $|\Psi\rangle$ is strictly semistable, the
algorithm converges to, but does not reach, a critical state in the closure of
the SLOCC class of $|\Psi\rangle$ 888Let us remark here that for system sizes
we are considering here, numerical errors become relevant. Keeping track of
the state by storing its computational basis coefficients may result in
jumping to another SLOCC class due to precision errors. States that are in the
null-cone may be erroneously identified as semistable, e.g.,
$\left|\psi_{26}\right>$ and $\left|\psi_{27}\right>$ in Appendix F.9. A more
robust implementation can be achieved by keeping track of the accumulated
local operators, instead..

The Kempf-Ness theorem [Kempf and Ness(1979)] is a powerful tool
characterizing polystable states. Any SLOCC–class of a polystable state
contains a unique critical state (up to LUs). It is known that $G[\it{Crit}]$,
is open, dense, and of full measure in $\mathbb{P}({\cal H})$ [Gour and
Wallach(2011), Wallach(2017)], provided that $[Crit]$ is non-empty. We will
use this fact in the proof of our main theorem which gives a criterion for the
existence of strictly semistable states.

Figure 1: Set-theoretic relations between the types of states used in the
paper (notice that e.g. the null-cone is not topologically separated from the
semistable states, as this figure might suggest). The whole space of states
$\mathbb{P}({\cal H})$ is divided into the semistable states $\mathbb{P}({\cal
H})_{ss}$ and the null-cone $\mathcal{N}$. We call those semistable orbits,
that are closed, polystable. We call those polystable orbits, that have
maximal dimension (among all polystable orbits), stable. Moreover, we call
polystable orbits, which are not stable, strictly polystable. Finally, we call
semistable orbits, which are not polystable, strictly semistable. Thus,
$\mathbb{P}({\cal H})$ partitions into four disjoint sets, the null-cone, the
strictly semistable states, the strictly polystable states, and the stable
states.

###  2.2 $G$-invariant polynomials

$G$–invariant polynomials constitute an important tool in the study of
SLOCC–classes. A polynomial, $f:{\cal H}\longrightarrow\mathbbm{C}$ is called
$G$–invariant if $f(g|\Psi\rangle)=f(|\Psi\rangle)$ for any
$|\Psi\rangle\in{\cal H}$ and any $g\in G$. The vector space (over
$\mathbbm{C}$) of all $G$–invariant polynomials is spanned by homogeneous
polynomials of degree $k\in\mathbb{N}$. The set of $G$–invariant polynomials
forms a ring which is finitely generated [Mukai(2003)].

As polynomials are continuous functions, any $G$–invariant polynomial will
have the same value on any two $G$-orbits, $G|\Psi_{1}\rangle$ and
$G|\Psi_{2}\rangle$ in $\mathcal{H}$, whose closures intersect. Hence, as
strictly semistable vectors converge to critical vectors, the SLOCC–class of a
strictly semistable state cannot be differentiated from the one of the
critical state, which is in its closure using $G$-invariant polynomials.
Moreover, for any vector in the null-cone any of these polynomials vanish and
vectors which are in the different SLOCC–classes within the null-cone cannot
be distinguished via $G$–invariant polynomials 999Note however, that some
$G$-orbits in the null-cone can be distinguished by studying the gradient flow
of the function $-\sum_{i=1}^{N}\text{tr}\\{\rho_{i}([\Psi])^{2}\\}$, where
$\rho_{i}([\Psi])$ is the $i$-th reduced state of $[\Psi]$ [Ness and
Mumford(1984), Sawicki  _et al._(2014)Sawicki, Oszmaniec, and Kuś, Sawicki
_et al._(2012)Sawicki, Oszmaniec, and Kuś, Maciążek and Sawicki(2015),
Maciążek and Sawicki(2018)].. However, in [Gour and Wallach(2013)] it has been
shown that SLOCC–classes of polystable states can be distinguished using
ratios of $G$–invariant polynomials, as we recall in the following.

Two polystable states $[{\Psi}],[{\Phi}]$ are SLOCC equivalent if and only if
[Gour and Wallach(2013)]

$\displaystyle\frac{f_{k}(|\Psi\rangle)}{h_{k}(|\Psi\rangle)}=\frac{f_{k}(|\Phi\rangle)}{h_{k}(|\Phi\rangle)},$
(9)

for any $G$–invariant homogeneous polynomials $f_{k},h_{k}$ with
$h_{k}(|\Psi\rangle)\neq 0$ of degree $k$ for any $k$. As mentioned before,
polystable states constitute a full measure set of states in case at least one
critical state exists. Hence, if $[Crit]\neq\emptyset$, the criterion above
solves the SLOCC–equivalence problem for generic states.

The necessary and sufficient condition for the existence of a critical state
has been presented in [Bryan  _et al._(2019)Bryan, Leutheusser, Reichstein,
and Raamsdonk]. For arbitrary dimensional Hilbert spaces, a critical state
exists if and only if

$\displaystyle\prod_{i=1}^{N}d_{i}-\sum_{l=1}^{N}(-1)^{l+1}\sum_{1\leq
i_{1}<\ldots<i_{l}\leq
N}\operatorname{gcd}(d_{i_{1}}^{2},\ldots,d_{i_{l}}^{2})\geq 0.$ (10)

We will elaborate on that for the Hilbert spaces
$\mathbbm{C}^{2}\otimes\mathbbm{C}^{m}\otimes\mathbbm{C}^{n}$, which we denote
by $2\times m\times n$, in Section 4.

In summary, there are many important facts known about polystable states and
also the null-cone has been investigated previously [Ness and Mumford(1984),
Johansson  _et al._(2012)Johansson, Ericsson, Singh, Sjöqvist, and Williamson,
Johansson  _et al._(2014)Johansson, Ericsson, Sjöqvist, and Osterloh, Sawicki
_et al._(2014)Sawicki, Oszmaniec, and Kuś, Maciążek and Sawicki(2018)].
However, little is known about strictly semistable states. In the subsequent
section we will derive a simple necessary and sufficient condition for their
existence. We will use this criterion for the simplest examples of
multipartite systems, namely the Hilbert spaces $2\times m\times n$. There, we
will not only identify the dimensions for which strictly semistable states
exist, but we will scrutinize all SLOCC classes and present a complete
characterization of them.

##  3 The existence of strictly semistable states

In this section we will show that there exists a strictly semistable state if
and only if there exist two critical states whose $G$–orbits do not have the
same dimensions (see Theorem 1). Note that the latter means that there exist a
strictly polystable state (i.e. not all polystable states are stable). To this
end, we will first restrict ourselves to the set of semistable states. We will
then introduce the definition of $c$–equivalence, which leads to a coarser
classification than SLOCC–equivalence. Next, using tools from GIT we will
prove the main theorem of the paper.

###  3.1 The set of semistable states

Our aim is to derive a criterion for the existence of a strictly semistable
state. Hence, we can restrict our attention to the Hilbert space excluding the
null-cone, i.e. ${\cal H}\setminus{\cal N}$, or, more precisely,
$\mathbb{P}({\cal H})_{ss}=\mathbb{P}({\cal H}\setminus{\cal N})$. However, we
will first demonstrate that without excluding the null-cone, taking the
closure in the projective space might become problematic. To give a simple
example, let us consider the 3–qubit GHZ–state,
$|GHZ\rangle=1/\sqrt{2}(|000\rangle+|111\rangle)$ and the regular matrix
$G(\alpha)=\frac{1}{\sqrt{2}}\begin{pmatrix}e^{\alpha}&-e^{\alpha}\\\
e^{-\alpha}&e^{-\alpha}\end{pmatrix}$ with $\det G(\alpha)=1$. Clearly, the
limit $\lim_{\alpha\to\infty}G^{\otimes 3}(\alpha)|GHZ\rangle$ does not exist
in $\mathcal{H}$. However, in the projective space we obtain
$\lim_{\alpha\to\infty}[G^{\otimes 3}(\alpha)|GHZ\rangle]=[|W\rangle]\in[{\cal
N}\setminus\\{0\\}].$ To avoid such situations from now on we will restrict
ourselves to the analysis of semistable states so the full (topological) space
of quantum states is $\mathbb{P}(\mathcal{H})_{ss}$. This restriction is
justified by the following observation which we prove in Appendix A.

######  Observation 1.

For any vector $|\Psi\rangle\in\mathcal{H}_{ss}$, we have

$\displaystyle\overline{[\;G|\Psi\rangle\;]}\setminus[\;\overline{G|\Psi\rangle}\;]\subset[{\cal
N}\setminus\\{0\\}].$ (11)

Hence, considering $\mathbb{P}(\mathcal{H})_{ss}$ as the full (topological)
space we have that the orbit $G[\Psi]$ is closed (in
$\mathbb{P}(\mathcal{H})_{ss}$) if and only if $G|\Psi\rangle$ is closed in
$\mathcal{H}_{ss}$. Moreover, for any pair of states
$[\Phi],[\Psi]\in\mathbb{P}(\mathcal{H})_{ss}$ it holds that
$[\Phi]\in\overline{G[\Psi]}$ if and only if
$|\Phi\rangle\in\overline{G|\Psi\rangle}$. Thus if we treat
$\mathbb{P}(\mathcal{H})_{ss}$ and $\mathcal{H}_{ss}$ as full topological
spaces, we have a well defined correspondence between closures in
$\mathcal{H}_{ss}$ and $\mathbb{P}(\mathcal{}H)_{ss}$. In the following we
will always consider $\mathbb{P}(\mathcal{H})_{ss}$ and $\mathcal{H}_{ss}$ as
full topological spaces.

Due to the results summarized in Sec. 2 we have that the closure of a
$G$–orbit in $\mathbb{P}(\mathcal{H})_{ss}$ always contains a unique critical
state (up to LUs). It might also contain strictly semistable states of various
SLOCC classes, which converge to this critical state. In order to derive a
criterion for the existence of these strictly semistable states, we adapt now
Lemma 1 for states in $\mathbb{P}(\mathcal{H})_{ss}$. To this end, we use the
following observation, which follows from Corollary 10 of [Gour and
Wallach(2011)].

######  Observation 2.

For any $|\Psi\rangle\in\mathcal{H}_{ss}$ it holds that

$\mathrm{dim}\,G|\Psi\rangle=\mathrm{dim}\,G[\Psi].$

That is, the dimension of the $G$–orbit in the projective space coincides with
the dimension of the $G$–orbit in the Hilbert space for any semistable state.

###### Proof.

As we consider here, in contrast to [Gour and Wallach(2011)], the projective
space, we adapt the proof of [Gour and Wallach(2011)] to our setting. Of
course $G_{|\Psi\rangle}\subset G_{[\Psi]}$. Assume next that there is $g\in
G_{[\Psi]}$ such that $g|\Psi\rangle=\alpha|\Psi\rangle$. When $|\alpha|<1$
this implies that $\lim_{n\rightarrow\infty}\|g^{n}|\Psi\rangle\|=0$ and when
$|\alpha|>1$ it means $\lim_{n\rightarrow\infty}\|g^{-n}|\Psi\rangle\|=0$. In
both cases this is contradiction as $|\Psi\rangle$ is semistable. Thus the
only possibility is $\alpha=e^{i\phi}$. We will show next that $\phi$ belongs
to a finite set. Let $\\{p_{1},p_{2},\ldots,p_{k}\\}$ be a set of homogeneous
generators of $G$-invariant polynomials, $\mathrm{deg}\,p_{j}=d_{j}$. Then

$\displaystyle
p_{j}(|\Psi\rangle)=p_{j}(g|\Psi\rangle)=p_{j}(e^{i\phi}|\Psi\rangle)=e^{id_{j}\phi}p_{j}(|\Psi\rangle).$
(12)

Thus $\forall j\,d_{j}\phi=0\,\mathrm{mod}\,2\pi$. This means that
$G_{[\Psi]}/G_{|\Psi\rangle}$ is a finite set and hence
$\mathrm{dim}\,G|\Psi\rangle=\mathrm{dim}\,G[\Psi]$ 101010Notice that in fact
it suffices to consider any chosen $G$–invariant homogeneous polynomial.. ∎

Due to the relation given in Eqs. (3) and (7) the above statement is
equivalent to the statement that the dimensions of the stabilizers coincide.
That is, the dimension of the set of $SL$ operators, which leave the vector
$|\Psi\rangle$ invariant, $G_{|\Psi\rangle}$, coincides with the dimension of
the set of $SL$ operators, which leave the vector $|\Psi\rangle$ up to a
proportionality factor invariant, $G_{[\Psi]}$.

Combining now Lemma 1 with the Observations 1 and 2 leads to the following
adaptation of Lemma 1 for states in $\mathbb{P}(\mathcal{H})_{ss}$.

######  Lemma 2.

For each $[\Psi]\in\mathbb{P}(\mathcal{H})_{ss}$, the boundary of an orbit
$\overline{G[\Psi]}\setminus G[\Psi]$ in $\mathbb{P}(\mathcal{H})_{ss}$ is a
union of $G$–orbits in $\mathbb{P}(\mathcal{H})_{ss}$ of strictly smaller
dimension. Each $G$-orbit closure contains a unique polystable $G$–orbit. This
polystable orbit is the only orbit in the closure which has minimal dimension.

###  3.2 c–equivalence

The number of SLOCC-classes, i.e. $G$–orbits, is in general infinite [Dür  _et
al._(2000)Dür, Vidal, and Cirac]. Furthermore, as mentioned before, $G$-orbits
may not be closed. This motivates the introduction of closure-equivalence
($c$-equivalence) which is used in GIT. By the Theorem of Nagata and Mumford
it is known that $c$–equivalence can be defined using three equivalent
definitions (see [Mukai(2003)]). As we are concerned with semistable states,
it will suffice to consider the following definition (see also Figure 2).

######  Definition 1.

Two states $[\Psi],[\Phi]\in\mathbb{P}(\mathcal{H})_{ss}$ are called
$c$–equivalent if

$\displaystyle\overline{G[\Psi]}\cap\overline{G[\Phi]}\neq\emptyset.$ (13)

We note that $c$–equivalence is indeed an equivalence relation. As the closure
of a semistable state contains a critical state, which does not belong to its
$G$–orbit, we have that $c$–equivalence induces a coarser classification than
SLOCC-equivalence. As explained in Sec. 2.2, $c$–classes can be distinguished
using the ratios of SL–invariant polynomials. Each $c$–class contains exactly
one closed $G$–orbit. Due to Lemma 2 this $G$–orbit is the only orbit of
minimal dimension (in the closure of the $G$–orbit). This closed orbit lies in
the closure of every $G$–orbit from the $c$–class [Mukai(2003)] and is the one
mentioned in Lemma 2. Moreover, due to the Kempf-Ness theorem [Kempf and
Ness(1979)], each $c$–class is uniquely characterized by the unique (up to LU)
critical state. Hence, $c$–classes are in 1-1 correspondence with critical
states (up to LU) (see Figure 2).

In order to state these facts mathematically rigorously, several definitions
would be in order. As these definitions are cumbersome, we do not state them
here, but refer the reader to Appendix C. We consider the set of classes
$\mathbb{P}(\mathcal{H})_{ss}/\sim_{c}$ given by the $c$–equivalence relation
in $\mathbb{P}(\mathcal{H})_{ss}$. It turns out that the set
$\mathbb{P}(\mathcal{H})_{ss}/\sim_{c}$ can be equipped with the structure of
a projective algebraic variety denoted by $\mathbb{P}(\mathcal{H})_{ss}//G$.
The GIT construction gives a regular (see Definition 6), hence continuous
mapping
$\pi:\mathbb{P}(\mathcal{H})_{ss}\rightarrow\mathbb{P}(\mathcal{H})_{ss}//G$
called the GIT quotient [Brion(2010)]. Thus, $\pi([\Psi])$ is a point in a
quotient variety corresponding to the $c$–class of $[\Psi]$ and
$\pi([\Psi])=\pi([\Phi])$ if and only if $[\Psi]$ and $[\Phi]$ are
$c$–equivalent. The Kempf-Ness theorem gives us the bijection
$\mathbb{P}(\mathcal{H})_{ss}//G\cong[Crit]/K$. Hence, the GIT quotient maps a
semistable state $[\Psi]$ to the unique (up to LUs) critical state in
$\overline{G[\Psi]}$.

In what follows we call a $c$–class trivial if it is a singleton, i.e. if it
consists of exactly one (thus closed/polystable) $G$–orbit. We will say that
the $c$–equivalence in $\mathbb{P}(\mathcal{H})_{ss}$ is trivial if each
$c$–class (if there is any) is trivial, i.e. the GIT quotient does not
identify any two distinct $G$–orbits or there are no critical states (so no
$c$–classes). Otherwise the $c$–equivalence is called non-trivial (see Figure
2). For example, one may check that in case of $2\times 3\times 5$ system, the
left hand side of Formula 10 is negative, what indicates that there are no
critical states in this system (cf. Example F.6 from Appendix F). Hence,
$c$–equivalence in $2\times 3\times 5$ system is trivial. Stated differently,
the $c$–equivalence in $\mathbb{P}(\mathcal{H})_{ss}$ is trivial if each
$c$–class is also a SLOCC–class, i.e. $c$–equivalence is not a coarser
classification than SLOCC. Note that a $c$–class is non-trivial if and only if
there exists a strictly semistable state whose $G$–orbit belongs to this
class. Similarly the $c$–equivalence is non-trivial if and only if there
exists a strictly semistable state (given by some non-trivial $c$–class).

Figure 2: Example of the partition of semistable states into $c$-classes
parametrized by LU–classes of critical states. Two sample $c$–classes 1 and 2
have been drawn. Points of the red line are LU-classes of critical states
($K$–orbits). For each such point we have a unique closed/polystable $G$–orbit
containing it (black horizontal lines) which is also the unique orbit of
smallest dimension in a given class (see Lemma 2). Closed/polystable orbits
correspond one-to-one to $c$-classes of $G$-orbits. Non-closed orbits (which
are of higher dimension) are denoted by dashed gray surfaces. Class 2 contains
only polystable states. Class 1 contains strictly semistable states and are
witnesses of the non-triviality of $c$–equivalence (see Definition 1). States
$[\Psi_{1}]$ and $[\Psi_{2}]$ are strictly semistable and belong to the
$c$–class 1. Hence, they can be transformed in the limit to the critical
states of the LU–class $K[\Phi_{c_{1}}]$ (normal form). The stabilizer of this
critical state has larger dimension, i.e. the $G$–orbit has smaller dimension
than the one of the corresponding strictly semistable state. State
$[\Psi_{3}]$ is polystable and can be transformed by SLOCC–operations to a
critical state in $K[\Phi_{c_{2}}]$ (normal form). As we will show later (see
Observation 3), the dimension of $G[\Phi_{c_{2}}]$ must be maximal and hence
larger than $\mathrm{dim}\,G[\Phi_{c_{1}}]$.

###  3.3 Stability of the action of a group

We briefly discuss here the fact that, in case critical states exist, the
action of $G$ on $\mathbb{P}(\mathcal{H})$ is stable, which is required in the
proof of the main theorem of this paper. Let us first introduce the definition
of stability of an action of a group.

######  Definition 2.

An action $H\curvearrowright\mathbb{P}(\mathcal{H})$ is stable if there exists
an open dense set $U\subset\mathbb{P}(\mathcal{H})$ that is a union of
polystable orbits.

We show in Appendix B that the existence of an open dense set
$U\subset\mathbb{P}(\mathcal{H})$ that is a union of polystable orbits implies
the existence of an open dense set $V\subset U$ that is a union of polystable
orbits of maximal dimension (among all orbits). In particular, $V$ is a union
of stable orbits so in case the action is stable, the maximal dimension among
polystable orbits (i.e. the dimension of stable orbits) coincides with the
maximal dimension among all orbits, which we will denote in the following by
$d_{G}$.

The action $G\curvearrowright\mathbb{P}(\mathcal{H})$ is stable if and only if
critical states exist, as the union of the SLOCC–orbits of critical states
(i.e. polystable orbits) is an open dense subset
$U\subset\mathbb{P}(\mathcal{H})$, whenever $U\neq\emptyset$ [Gour and
Wallach(2011)]. In particular, we have 111111The reason for that is that the
dimension of an open and dense set is the same as the dimension of the whole
space.

$\mathrm{dim}\,G[\it{Crit}]=\mathrm{dim}\,\mathbb{P}(\mathcal{H}).$ (14)

Let us note here that $G[\Psi]$ having maximal dimension does not necessarily
mean that $G_{[\Psi]}$ is finite, i.e. that the dimension of the local
stabilizer is zero. An example would be the $G$–orbit of the 3–qubit GHZ
state, which is of maximal dimension, as it is of full measure. However, its
dimension does not coincide with the dimension of $G$ since it admits
continuous symmetries and its stabilizer has a positive dimension [Bryan  _et
al._(2019)Bryan, Leutheusser, Reichstein, and Raamsdonk] 121212More precisely,
we have that $\operatorname{dim}G[GHZ]=7$ and $\operatorname{dim}G_{[GHZ]}=2$
[Verstraete  _et al._(2002b)Verstraete, Dehaene, and De Moor].. However, we
have that $\mathrm{dim}\,G[GHZ]=\mathrm{dim}\,\mathbb{P}(\mathcal{H})$, as the
$GHZ$–state is the only critical state for three qubits.

###  3.4 Existence of strictly semistable states

In this subsection, we prove the main theorem (Theorem 1), which provides a
necessary and sufficient condition for the existence of strictly semistable
states and discuss its consequences.

In order to do so, we will restate a lemma from [Mumford(1999)], which we will
apply later in the context of the GIT quotient
$\pi:\mathbb{P}(\mathcal{H})_{ss}\rightarrow\mathbb{P}(\mathcal{H})_{ss}//G$
and which is a direct consequence of Corollary 15.5.4 from [Tauvel and
Yu(2015)].

######  Lemma 3 (On the fibres of the mapping [Mumford(1999)]).

Let $X$ and $Y$ be quasi-projective irreducible varieties and
$\pi:X\rightarrow Y$ a regular surjective map. Then, there exists an open and
dense set $V\subset Y$, such that for any points $\alpha\in V$ and $\gamma\in
Y\setminus V$, we have

$\mathrm{dim}\,\pi^{-1}(\alpha)=s\leq\mathrm{dim}\,\pi^{-1}(\gamma)$ (15)

and $s=\mathrm{dim}\,X-\mathrm{dim}\,Y$.

We defer a more detailed discussion of the assumptions of the lemma as well as
proper definitions of quasi-projective varieties, irreducible varieties, and
regular maps to Appendix C. However, for our purposes here, note that Lemma 3
is applicable to the GIT quotient
$\pi:\mathbb{P}(\mathcal{H})_{ss}\rightarrow\mathbb{P}(\mathcal{H})_{ss}//G$.
Let us briefly outline the reason for that. First,
$\mathbb{P}(\mathcal{H})_{ss}$ and $\mathbb{P}(\mathcal{H})_{ss}//G$ are
indeed irreducible quasi-projective varieties (see Appendix C). Moreover,
$\pi$ is a regular map, which follows from the fact that a finite number of
rational polynomials can be used to decide the $c$-class of any given
semistable state $[\psi]$, as explained in Section 3.2.

The sets $\pi^{-1}(\alpha)$ in Lemma 3 are called fibres and the dimension $s$
is called the minimal fibre dimension. Notice that $\pi^{-1}(V)$ is an open
and dense subset of $\mathbb{P}(\mathcal{H})_{ss}$ and generic fibres have
minimal fibre dimension $s$. We are now in the position to prove the main
theorem, which presents a criterion for when a $c$–equivalence is trivial
(i.e. all GIT quotient fibres are single $G$–orbits).

######  Theorem 1 ( Main Theorem ).

There exist no strictly semistable state if and only if the dimensions of all
polystable $G$–orbits are the same (i.e. they are all stable).

###### Proof.

First, note that if there exists no critical state, then there also does not
exist a strictly semistable state and thus the statement holds trivially.

Let us now assume that there does exist a critical state. As mentioned before,
we denote by $d_{G}$ the maximal dimension among all $G$–orbits. As discussed
in Section 3.3, $d_{G}$ coincides with the maximal orbit dimension among all
polystable $G$–orbits, i.e., the dimension of stable orbits. We prove both
directions by contradiction.

If: If there exists a strictly semistable state, $[\Psi]$, we have that
$\overline{G[\Psi]}\setminus G[\Psi]\neq\emptyset$ contains a polystable
$G$–orbit of strictly smaller dimension than $\operatorname{dim}G[\Psi]$
(Lemma 2). Therefore, there has to exist a polystable state whose orbit is of
strictly smaller dimension than $d_{G}$, i.e., there has to exist a strictly
polystable state.

Only if: As explained earlier in this subsection,
$\pi:\mathbb{P}(\mathcal{H})_{ss}\rightarrow\mathbb{P}(\mathcal{H})_{ss}//G$
fulfills the conditions of Lemma 3. Let us first show that the minimal fibre
dimension of $\pi$, $s$, equals the maximal orbit dimension, $d_{G}$. To this
end, let $U$ be the union of all stable orbits. Suppose that $G[\Psi]\subset
U$. Obviously $\mathrm{dim}(G[\Psi])=d_{G}$. Next, by Lemma 2 there exists no
state $[\Phi]\notin G[\Psi]$ such that $G[\Psi]\subset\overline{G[\Phi]}$ as
this would imply that $\mathrm{dim}(G[\Phi])>\mathrm{dim}(G[\Psi])$, which
contradicts the fact that $d_{G}$ is the maximal dimension. This means that
the $c$–class of any $[\Psi]\in U$ is given by $G[\Psi]$, i.e. for each
$[\Psi]\in U$ we have $\pi^{-1}(\pi([\Psi]))=G[\Psi]$. In particular, for each
$\alpha\in\pi(U)$ we have $\mathrm{dim}\,\pi^{-1}(\alpha)=d_{G}$. Recall that
$U$ is open and dense. Let $V$ be the open and dense set from Lemma 3. From
continuity of the GIT quotient $\pi$ we have that $\pi^{-1}(V)$ is an open and
dense set so $O\coloneqq\pi^{-1}(V)\cap U\neq\emptyset$, as two open and dense
sets need to intersect. Thus, we have that $\pi(O)\cap V\neq\emptyset$ and
therefore $s=d_{G}$ [recall that $s$ is the minimal fibre dimension from Eq
(15)]. Hence, the dimension of any $c$–class must be greater than or equal to
$d_{G}$.

We are now in the position to prove that if there exists a polystable
$G$–orbit of smaller than maximal dimension (i.e. a strictly polystable
orbit), then there exists a strictly semistable state. Suppose that there
exists a $[\Phi_{c}]\in[Crit]$ such that $\mathrm{dim}\,G[\Phi_{c}]<d_{G}$.
From the fact that the minimal fibre dimension is $d_{G}$ we know that
$\mathrm{dim}\,\pi^{-1}(\pi([\Phi_{c}]))\geq d_{G}$. Hence, the fibre
$\pi^{-1}(\pi([\Phi_{c}]))$ must contain another $G$–orbit (which cannot be
polystable due to the Kempf-Ness theorem), i.e. there must exist a strictly
semistable state. ∎

As the dimension of the orbits is directly related to the dimension of the
stabilizer (see Eq. (7)) and as the $c$–equivalence is trivial iff there
exists no strictly semistable state, the following statements are equivalent
to Theorem 1.

  1. (i)

The $c$-equivalence is non-trivial if and only if there exists a critical
state whose stabilizer has strictly larger dimension than the stabilizer of a
generic state.

  2. (ii)

There exists a strictly semistable state if and only if there exists a
polystable state whose $G$–orbit has smaller than the maximal dimension
$d_{G}$.

  3. (iii)

The existence of strictly semistable states is equivalent to the existence of
strictly polystable states.

Moreover, note that the proof of Theorem 1 can also be applied to individual
$c$–classes, as stated in the following

######  Observation 3.

The $c$–class of a given polystable orbit $G[\Phi_{c}]$ is non-trivial if and
only if $\mathrm{dim}\,G[\Phi_{c}]$ is smaller than the dimension of a generic
orbit (i.e. $[\Phi_{c}]$ has more continuous symmetries). If this is the case
then the dimension of the $G$–orbit of every strictly semistable state in such
a $c$–class is larger than the dimension of $G[\Phi_{c}]$ (see also Figure 2).

Let us remark here that Theorem 1 can be extended to other groups whose action
is stable and $G$ is the complexification of $K$, i.e. $G=K^{\mathbb{C}}$. In
the subsequent section we demonstrate the usefulness of this criterion for
Hilbert spaces of dimension $2\times m\times n$.

##  4 The categorization of SLOCC classes of states in $2\times m\times n$

In this section, we apply the derived criterion (Theorem 1) for the existence
of strictly semistable states to three partite systems in a Hilbert space
$\mathbb{C}^{2}\times\mathbb{C}^{m}\times\mathbb{C}^{n}$ ($2\leq m\leq n$). We
show that strictly semistable states exist (and thus, c-equivalence is non-
trivial) if and only if $m=n$ and $m\geq 4$. Then, we go beyond that and
provide a full characterization of orbit types of all SLOCC classes, i.e.,
$G$-orbits, for $2\times m\times n$ systems.

The outline of this section is as follows. First, we will briefly recall the
characterization of SLOCC classes in $2\times m\times n$ systems using the
theory of matrix pencils [Chitambar  _et al._(2010)Chitambar, Miller, and Shi]
in Subsection 4.1. Then, in Subsection 4.2, we will recall a characterization
of critical states in $2\times m\times n$ systems obtained in [Bryan  _et
al._(2019)Bryan, Leutheusser, Reichstein, and Raamsdonk]. In Subsection 4.3,
we apply Theorem 1 to $2\times m\times n$ systems. In Subsection 4.4, we
perform a complete characterization of the orbit types of SLOCC classes in
$2\times m\times n$ systems, which we summarize in form of a flowchart in
Subsection 4.5. Finally, we provide tables of SLOCC classes in $2\times
m\times n$ systems equipped with orbit types for small system sizes (up to
$2\times 5\times 5$) in Appendix F.

###  4.1 SLOCC-classes of states in $2\times m\times n$

The SLOCC classification in $2\times m\times n$ systems was obtained via the
Kronecker Canonical Form (KCF) of linear matrix pencils [Chitambar  _et
al._(2010)Chitambar, Miller, and Shi]. As we will use this characterization,
we recall here briefly the idea. The reader is referred to [Gantmacher(1959),
Kronecker(1890), Chitambar  _et al._(2010)Chitambar, Miller, and Shi,
Hebenstreit  _et al._(2018)Hebenstreit, Gachechiladze, Gühne, and Kraus] for
the definitions used in the categorization of matrix pencils. A linear
$m\times n$ matrix pencil $\mathcal{P}$ is a homogeneous matrix polynomial of
degree 1 in variables $\mu$ and $\lambda$,

$\displaystyle\mathcal{P}=\mu R+\lambda S,$ (16)

where $R$ and $S$ are $m\times n$ matrices. The matrix pencil associated to an
arbitrary state $|\psi\rangle$ in
$\mathbb{C}^{2}\times\mathbb{C}^{m}\times\mathbb{C}^{n}$, with

$\displaystyle|\psi\rangle$
$\displaystyle=|0\rangle_{A}|R\rangle_{BC}+|1\rangle_{A}|S\rangle_{BC}$
$\displaystyle=\left[|0\rangle_{A}(R\otimes\mathbbm{1})+|1\rangle_{A}(S\otimes\mathbbm{1})\right]|\phi^{+}_{n}\rangle_{BC},$
(17)

where $|\phi^{+}_{n}\rangle=1/\sqrt{n}\sum_{i=0}^{n-1}|ii\rangle$, is given by
Eq. (16). Let us now recall how the matrix pencil corresponding to a state is
transformed under SLOCC operators $A\otimes B\otimes C$. Let

$\displaystyle A=\begin{pmatrix}\alpha&\beta\\\ \gamma&\delta\end{pmatrix},$
(18)

then the matrix pencil transforms as

$\displaystyle\mathcal{P}$ $\displaystyle=\mu R+\lambda
S\rightarrow\mathcal{P^{\prime}},\text{ where}$ (19)
$\displaystyle\mathcal{P^{\prime}}$
$\displaystyle=B[(\alpha\mu+\gamma\lambda)R+(\beta\mu+\delta\lambda)S]C^{T}.$
(20)

A normal form of matrix pencils, the KCF, under regular operators $B$ and
$C^{T}$ as in Eq. (20) has been derived in [Kronecker(1890)]. It is always
possible to find operators $B$ and $C^{T}$ which transform a matrix pencil
into its KCF. In the following, we only consider fully entangled states, i.e.,
states for which all single-particle reduced density matrices have full rank.
Restricting to matrix pencils that correspond to fully entangled states, the
KCF of a matrix pencil is the (generalized) block-diagonal form

$\displaystyle\mathcal{P}=\operatorname{blockdiag}\left\\{L_{\epsilon_{1}},\dots,L_{\epsilon_{a}},L_{\nu_{1}}^{T},\dots,L_{\nu_{b}}^{T},J\right\\},$
(21)

where

$L_{\epsilon}=\begin{pmatrix}\lambda&\mu&&&\\\ &\lambda&\mu&&\\\
&&\ddots&\ddots&\\\ &&&\lambda&\mu\end{pmatrix}$ (22)

has size $\epsilon\times(\epsilon+1)$ 131313Here and in the following, matrix
elements that are not displayed are $0$. and

$\displaystyle J=\operatorname{blockdiag}\\{$ $\displaystyle
M^{e_{1}^{1}}(x_{1}),\dots,M^{e_{r}^{l}}(x_{l}),$ $\displaystyle\qquad\qquad
N^{e_{1}^{\mu}},N^{e_{2}^{\mu}},\dots,N^{e_{r}^{\mu}}\\},$ (23)

where the pairwise different $x_{i}$ are called the finite eigenvalues of a
matrix pencil, $e_{j}^{i}$ are called size signatures of the eigenvalue
$x_{i}$, and $l$ denotes the number of (distinct) eigenvalues. Moreover, an
eigenvalue may occur several times in Eq. (23) and $m_{i}=\sum_{j}e^{i}_{j}$
is called the multiplicity of the eigenvalue $x_{i}$. $N$-blocks correspond to
an infinite eigenvalue (which we denote by $\infty$). The blocks take the form

$\displaystyle N^{e_{j}^{\mu}}$
$\displaystyle=\begin{pmatrix}\mu&\lambda&&&\\\ &\mu&\lambda&&\\\
&&\ddots&\ddots&\\\ &&&\mu&\lambda\\\ &&&&\mu\end{pmatrix}\text{ and}$
$\displaystyle M^{e_{j}^{i}}(x_{i})$
$\displaystyle=\begin{pmatrix}x_{i}\mu+\lambda&\mu&&&\\\
&x_{i}\mu+\lambda&\mu&&\\\ &&\ddots&\ddots&\\\ &&&x_{i}\mu+\lambda&\mu\\\
&&&&x_{i}\mu+\lambda\end{pmatrix}.$ (24)

The sizes of these matrices are $e^{\mu}_{j}\times e^{\mu}_{j}$ and
$e_{j}^{i}\times e_{j}^{i}$, respectively. The KCF of a matrix pencil is
invariant under the operators $B$, $C$, which makes it a useful tool for
studying SLOCC classes. However, the operator $A$ [given in Eq. (18)] changes
the finite eigenvalues according to

$\displaystyle x_{i}\rightarrow\left\\{\begin{array}[]{lr}\frac{\alpha
x_{i}+\beta}{\gamma x_{i}+\delta},&\text{if }\gamma x_{i}+\delta\neq 0\\\
\infty,&\text{if }\gamma x_{i}+\delta=0\\\ \end{array}\right.,$ (27)

and the infinite eigenvalue according to

$\displaystyle\infty\rightarrow\left\\{\begin{array}[]{lr}\frac{\alpha}{\gamma},&\text{if
}\gamma\neq 0\\\ \infty,&\text{if }\gamma=0\\\ \end{array}\right..$ (30)

Other than that, $A$ does not change the KCF of a matrix pencil 141414In
particular, the occurrence of $L$\- and $L^{T}$-blocks as well as their sizes
are unchanged, as are the size signatures of the eigenvalues.. Note that with
an operator $A$ it is always possible to set three of the distinct eigenvalues
to arbitrary new (distinct) values. Moreover, it is always possible to bring a
matrix pencil into a form without infinite eigenvalues [Chitambar  _et
al._(2010)Chitambar, Miller, and Shi]. Therefore, in the following, we only
consider finite eigenvalues for convenience. Degeneracies of eigenvalues
cannot be changed with any finite $A$. However, we will see later on, it is
indeed possible to create degeneracy when considering the limits of sequences
of $SL$-operators.

All SLOCC classes of states in $2\times m\times n$ can be characterized by the
KCF of the corresponding matrix pencils [Chitambar  _et al._(2010)Chitambar,
Miller, and Shi]. Moreover, the problem of listing all SLOCC classes becomes a
combinatorial problem of listing all possible KCFs. Let us restate here the
according theorem from [Chitambar  _et al._(2010)Chitambar, Miller, and Shi].

######  Theorem 2 ([Chitambar  _et al._(2010)Chitambar, Miller, and Shi]).

Two $2\times m\times n$-dimensional pure states $|\psi\rangle$ and
$|\phi\rangle$ for which their corresponding matrix pencils have only finite
eigenvalues $\\{x_{i}\\}$ and $\\{x_{i}^{\prime}\\}$, respectively, are SLOCC
equivalent if and only if the KCFs of the matrix pencils agree up to a linear
fractional transformation of the eigenvalues,

$\frac{\alpha x_{i}+\gamma}{\beta x_{i}+\delta}=x_{i}^{\prime},$ (31)

for some $\alpha,\beta,\gamma,\delta\in\mathbb{C}$, where
$\alpha\delta-\beta\gamma\neq 0$.

###  4.2 Critical states

The question of the existence of a critical state has been answered for
general multipartite quantum systems in [Bryan  _et al._(2019)Bryan,
Leutheusser, Reichstein, and Raamsdonk]. For $2\times m\times n$ systems, at
least one SLOCC class containing a critical state exists if and only if $m=n$
or $n-m$ divides $m$ [Bryan  _et al._(2019)Bryan, Leutheusser, Reichstein, and
Raamsdonk]. Moreover, in $2\times m\times(m+1)$ systems, there is a unique (up
to LUs) critical state [Bryan  _et al._(2019)Bryan, Leutheusser, Reichstein,
and Raamsdonk], which is given by

$\displaystyle|\psi_{2,m,m+1}\rangle=\frac{1}{\sqrt{m+1}}\sum_{k=0}^{m-1}$
$\displaystyle\left(\sqrt{\frac{m-k}{m}}|0,k,k\rangle\right.$
$\displaystyle\left.\quad+\sqrt{\frac{k+1}{m}}|1,k,(k+1)\rangle\right).$ (32)

More generally, in all $2\times m\times n$ systems for which $(n-m)$ divides
$m$, there is a unique (up to LUs) critical state which is LU-equivalent to
[Bryan  _et al._(2019)Bryan, Leutheusser, Reichstein, and Raamsdonk]

$\displaystyle|\psi_{2,\frac{m}{n-m},\frac{m}{n-m}+1}\rangle_{A\,B_{1}\,C_{1}}\otimes|\phi^{+}_{n-m}\rangle_{B_{2}\,C_{2}},$
(33)

where the system held by the second (third) party is composed of the
subsystems $B_{1}$ and $B_{2}$ ($C_{1}$ and $C_{2}$), respectively. The
corresponding matrix pencil is
$\mathcal{P}=\bigoplus_{i=1}^{(n-m)}L_{\frac{m}{n-m}}$.

Moreover, for $2\times n\times n$ systems, critical states are (up to LUs) of
the form [Bryan  _et al._(2019)Bryan, Leutheusser, Reichstein, and Raamsdonk]

$\displaystyle\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\left(e^{i\phi_{i}}\sin(\theta_{i}/2)|0\rangle+\cos(\theta_{i}/2)|1\rangle\right)|i-1,i-1\rangle,$
$\displaystyle=|0\rangle(D_{0}\otimes\mathbbm{1})|\phi^{+}_{n}\rangle+|1\rangle(D_{1}\otimes\mathbbm{1})|\phi^{+}_{n}\rangle,$
(34)

where
$D_{0}=1/\sqrt{n}\operatorname{diag}(e^{i\phi_{1}}\sin(\theta_{1}/2),\ldots)$
and $D_{1}=1/\sqrt{n}\operatorname{diag}(\cos(\theta_{1}/2),\ldots)$ for
$(\theta_{i},\phi_{i})$ such that

$\displaystyle\sum_{i}\cos(\theta_{i})$ $\displaystyle=0$
$\displaystyle\sum_{i}\sin(\theta_{i})\cos(\phi_{i})$ $\displaystyle=0$
$\displaystyle\sum_{i}\sin(\theta_{i})\sin(\phi_{i})$ $\displaystyle=0.$ (35)

The critical state is unique in cases $n=2$ and $n=3$ [Bryan  _et
al._(2019)Bryan, Leutheusser, Reichstein, and Raamsdonk]. Note that it is
possible to associate a geometrical meaning to the conditions in Eq. (35). The
pairs $(\theta_{i},\phi_{i})$ can be interpreted as spherical coordinates of
unit vectors

$\displaystyle\vec{v}_{i}=\begin{pmatrix}\sin\theta_{i}\cos\phi_{i}\\\
\sin\theta_{i}\sin\phi_{i}\\\ \cos\theta_{i}\end{pmatrix}$ (36)

and the state in Eq. (4.2) is critical if and only if the vectors
$\vec{v}_{i}$ sum to $\vec{0}$ [Bryan  _et al._(2019)Bryan, Leutheusser,
Reichstein, and Raamsdonk].

Note that matrix pencils corresponding to the states given in Eq. (4.2) are
diagonal, i.e. direct sums of $M^{1}(x_{i})$ blocks and the eigenvalues of the
matrix pencil, $x_{i}$, are related to the unit vectors $\vec{v}_{i}$ by
[Bryan  _et al._(2019)Bryan, Leutheusser, Reichstein, and Raamsdonk]

$\displaystyle x_{i}=e^{i\phi_{i}}\tan{\frac{\theta_{i}}{2}}.$ (37)

Recall, however, that the operator on the first system ($A$) may change the
eigenvalues of the matrix pencil 151515Operators $B$ and $C$ do not change the
eigenvalues, though, as the KCF of a matrix pencil and thus the eigenvalues
are invariant under $B$ and $C$.. It will become important later on to
understand how the vectors $\vec{v}_{i}$ are transformed under such an
operator $A$. Let us consider the singular value decomposition,
$A=V\operatorname{diag}\left(\sqrt{\frac{1}{1-\alpha}},\sqrt{1-\alpha}\right)U$,
where $\alpha\in[0,1)$. Writing $U$ in Euler decomposition and absorbing the
third ($\sigma_{z}$) rotation into $V$ (as the rotation commutes with the
diagonal matrix), we have
$U=e^{-i\frac{\Theta}{2}\sigma_{y}}e^{-i\frac{\Phi}{2}\sigma_{z}}$. We are now
interested in how this transformation changes the eigenvalues and thus the
vectors $\vec{v}_{i}$. Note that we will disregard $V$, as it is only a LU
transformation of the corresponding states.

Let us first consider the application of $A$ in case $U=V=\mathbbm{1}$, i.e.,
we only have the transformation
$\operatorname{diag}\left(\sqrt{\frac{1}{1-\alpha}},\sqrt{1-\alpha}\right)$,
which acts on all vectors simultaneously as

$\displaystyle f_{\alpha}:[0,\pi]\times[0,2\pi]$
$\displaystyle\rightarrow[0,\pi]\times[0,2\pi]$
$\displaystyle(\theta_{i},\phi_{i})$
$\displaystyle\mapsto(2\arctan\left(\frac{1}{1-\alpha}\tan(\theta_{i}/2)\right),\phi_{i}).$
(38)

Note that—under this non-linear transformation $f_{\alpha}$—$\theta_{i}$
increases strictly with $\alpha$ for all $i$, unless $\theta_{i}=0$ or
$\theta_{i}=\pi$. In the latter cases $\theta_{i}$ is left invariant. The
transformation can be pictorially understood as hinging all $\vec{v}_{i}$
(except those pointing in direction $(0,0,1)^{T}$ or $(0,0,-1)^{T}$) down
towards the direction $(0,0,-1)^{T}$ without changing their azimuthal angle,
$\phi_{i}$.

Let us now consider the full operator $A$ also taking the unitary $U$ into
account. It can be easily seen that unitary transformations by $A$ correspond
to $SO(3)$ rotations of the $\vec{v_{i}}$ [Bryan  _et al._(2019)Bryan,
Leutheusser, Reichstein, and Raamsdonk]. Taking the $SO(3)$ operation stemming
from $U$ into account, the vectors $\vec{v}_{i}$ are simply rotated _before_
hinging them down by $f_{\alpha}$ according to Eq. (4.2) along the $z$-axis.
Equivalently, the $SO(3)$ operation stemming from $U$ can also be understood
as choosing an axis $(\Theta,\Phi)$ along which the “hinging”-transformation
$f_{\alpha}$ is performed (on static vectors).

###  4.3 c-equivalence and existence of strictly semistable states

In this subsection, we will apply Theorem 1 to $2\times m\times n$ systems to
characterize those $m$, $n$ for which strictly semistable states exist.

Recall that in the cases $m\neq n$, or $m=n\leq 3$ [Bryan  _et
al._(2019)Bryan, Leutheusser, Reichstein, and Raamsdonk] there exists either
no critical state, or a unique critical state (up to LUs). Combining this fact
with Theorem 1 we obtain the following corollary.

######  Corollary 2.1.

In $2\times m\times n$ systems, there exists no strictly semistable state if
$m\neq n$, or $m=n\leq 3$.

In the following we will show that in all other cases, i.e. $2\times m\times
n$ for $m=n\geq 4$, strictly semistable states exist and thus c-equivalence is
non-trivial. To this end, it suffices to identify two critical states whose
dimensions of the stabilizer group differ (see Theorem 1).

It can be easily verified that the (complex) dimension of the stabilizer of
states $|\psi\rangle$ of the form given in Eq. (4.2) is given in terms of the
degeneracies of the eigenvalues of the corresponding (diagonal) matrix pencil
and reads

$\displaystyle\dim(G_{|\psi\rangle})=\max\\{3-l,0\\}+\left(-1+\sum_{i=1}^{l}m_{i}^{2}\right),$
(39)

where $l$ denotes the number of distinct eigenvalues and $m_{i}$ their
degeneracies. To this see this, note that symmetries of the form
$\mathbbm{1}\otimes B\otimes C$ for diagonal matrix pencils must satisfy
$C^{T}=B^{-1}$ as it must hold that $B\mathbbm{1}C^{T}=\mathbbm{1}$. Moreover,
$B$ must commute with
$\operatorname{blockdiag}\\{x_{1}\mathbbm{1}_{m_{1}},\ldots,x_{l}\mathbbm{1}_{m_{l}}\\}$.
Hence, the number of free parameters is $-1+\sum_{i=1}^{l}m_{i}^{2}$, which
gives rise to the second summand on the righthand side of Eq. (39). Moreover,
any operator $A$ that gives rise to a symmetry must act in such a way that all
sets of eigenvalues with coinciding multiplicity are mapped into themselfs.
This gives rise to some finite freedom in choosing a permutation of the
eigenvalues in an appropriate way. Then, the image of three distinct complex
numbers uniquely determines an operator $A$, while any number less then three
allows free complex parameters in $A$ as in the first summand on the righthand
side of Eq. (39).

It is clear that $\dim(G_{|\psi\rangle})$ is always minimized by $m_{i}=1\
\forall i$, i.e., by having all distinct eigenvalues, as summarized in the
following observation.

######  Observation 4.

The dimension of the SLOCC orbit of a state corresponding to a diagonal matrix
pencil with all distinct eigenvalues is strictly larger than the orbit
dimension of a state corresponding to a diagonal matrix pencil with any
degeneracy in the eigenvalues.

Using this observation, it is easy to prove the following theorem.

######  Theorem 3.

In $2\times m\times n$ systems, there exist striclty semistable states if and
only if $m=n\geq 4$.

In other words, c-equivalence is non-trivial if and only if $m=n\geq 4$.

###### Proof.

For $m=n\geq 4$, there exist critical states corresponding to diagonal matrix
pencils that do and those which do not have degenerate eigenvalues. This
guarantees the existence of strictly semistable states according to Theorem 1.
This can be easily seen in the geometrical picture reviewed in Section 4.2.
For $n\geq 2$, it is always possible to find $n$ distinct unit vectors in
$\mathbb{R}^{3}$ with multiplicity $m_{i}=1$, which sum to $\vec{0}$.
Moreover, for $m=n\geq 4$, it is always possible to construct one vector
$\vec{v}_{1}$ with $m_{1}=2$ as well as up to $n-2$ additional unit vectors
with multiplicities summing up to $n-2$, such that the total weighted sum of
vectors yields $\vec{0}$. ∎

###  4.4 Characterization of orbit types

In this subsection, we go beyond Theorem 1 and derive a full characterization
of the orbit types of SLOCC classes in $2\times m\times n$ systems with the
help of matrix pencils. More precisely, we provide a systematic way of
checking properties of a matrix pencil (corresponding to a $2\times m\times n$
state) in order to conclude the orbit type of the corresponding SLOCC class.
Let us remark here that for some choices of $m$ and $n$, the considerations
above as well as Theorem 3 (see also [Bryan  _et al._(2019)Bryan, Leutheusser,
Reichstein, and Raamsdonk]) already give the full characterization, e.g.,
cases in which $m\neq n$, or $m=n=3$. However, in case $m=n\geq 4$, the
considerations above guarantee the existence of semistable states, but do not
yet give a full characterization of the orbit type of all SLOCC classes for
the considered system sizes. In the following, we will thus employ two
observations and a theorem which will allow extending Theorem 3 to a full
characterization of the orbit types. Remarkably, we will show that the orbit
type does not depend on the eigenvalues of the matrix pencil, but only on
their multiplicities. If this had not been the case, it would have seemed
unlikely that a self-contained characterization of orbit types is possible. In
the remainder of the section, we will wlog always consider matrix pencils in
KCF and states such that the corresponding matrix pencil is in KCF. As
mentioned before, this is possible as operators $B$ and $C^{T}$ which
transform the matrix pencil to KCF always exist.

Let us start by considering the case $m\neq n$. In this case, Theorem 3 states
that there do not exist strictly semistable states. Moreover, as discussed in
Section 4.2, critical states do only exist if $n-m$ divides $m$ [Bryan  _et
al._(2019)Bryan, Leutheusser, Reichstein, and Raamsdonk]. Furthermore, if
$n-m$ does divide $m$, there exists a unique stable SLOCC class corresponding
to the matrix pencil $\mathcal{P}=\bigoplus_{i=1}^{(n-m)}L_{\frac{m}{n-m}}$.
Thus, for $n\neq m$, there is either exactly one stable SLOCC class or no
stable SLOCC class, which implies that there neither exist any strictly
polystable, nor strictly semistable classes. Thus, all SLOCC classes except
the aforementioned stable class are in the null-cone. Hence, the
characterization of orbit types is already complete for the case $m\neq n$.

In order to tackle the open cases (within $2\times m\times n$ systems such
that $n=m$) let us continue with an observation about matrix pencils
containing an $L_{\epsilon}$\- and an $L^{T}_{\nu}$-block.

######  Observation 5.

An SLOCC class that corresponds to a matrix pencil whose KCF contains an
$L_{\epsilon}$ and an $L^{T}_{\mu}$ block is in the null-cone.

###### Proof.

Let us denote the matrix pencil (in KCF) representing the SLOCC class in
question by $\mathcal{P}$ and the corresponding state by $|\psi\rangle$. We
will prove the observation by explicitly constructing a sequence of SL-
operators of the form $\mathbbm{1}\otimes B_{\alpha}\otimes C_{\alpha}$ such
that $\lim_{\alpha\rightarrow\infty}\mathbbm{1}\otimes B_{\alpha}\otimes
C_{\alpha}|\psi\rangle=0$. Wlog let us assume an ordering of the blocks such
that the matrix pencil has the form $\mathcal{P}=L_{\epsilon}\oplus
L^{T}_{\nu}\oplus\tilde{\mathcal{P}}$, where $\tilde{\mathcal{P}}$ is of size
$[m-(\epsilon+\nu+1)]\times[n-(\epsilon+\nu+1)]$ and contains the remaining
blocks, whose form does not matter.

We define the $(\epsilon+\nu+1)\times(\epsilon+\nu+1)$ matrices

$\displaystyle B_{\alpha}^{\prime}$
$\displaystyle=\sum_{i=0}^{\epsilon+\nu}e^{\left(i-\frac{\epsilon+\nu}{2}\right)\alpha}|i\rangle\langle
i|\text{ and}$ (40) $\displaystyle C_{\alpha}^{\prime}$
$\displaystyle=\sum_{i=0}^{\epsilon+\nu}e^{\left(\frac{\epsilon+\nu}{2}-i\right)\alpha}|i\rangle\langle
i|$ (41)

and the $n\times n$ matrix

$\displaystyle
C_{\alpha}^{\prime\prime}=e^{-\frac{\alpha}{2(n-1)}}\mathbbm{1}_{n}+e^{\frac{n\alpha}{2(n-1)}}|\epsilon\rangle\langle\epsilon|.$
(42)

The matrices of interest are then defined as
$B_{\alpha}=B_{\alpha}^{\prime}\oplus\mathbbm{1}_{m-(\epsilon+\nu+1)}$ and
$C_{\alpha}=C_{\alpha}^{\prime\prime}\left(C_{\alpha}^{\prime}\oplus\mathbbm{1}_{n-(\epsilon+\nu+1)}\right)$.
It can be easily verified that $\det(B_{\alpha})=\det(C_{\alpha})=1$.
Moreover,
$\lim_{\alpha\rightarrow\infty}B_{\alpha}\mathcal{P}C_{\alpha}^{T}=0$. Thus,
also $\lim_{\alpha\rightarrow\infty}\mathbbm{1}\otimes B_{\alpha}\otimes
C_{\alpha}|\psi\rangle=0$. This proves that any such SLOCC class is in the
null-cone. ∎

The intuition behind the construction is the following. The purpose of
$B^{\prime}$ and $C^{\prime}$ is to damp the off-diagonal entries of the
$L_{\epsilon}\oplus L^{T}_{\nu}$ part in $\mathcal{P}$, leaving the diagonal
entries invariant. Thus, the whole $\epsilon$th column of $\mathcal{P}$ is
damped, i.e., multiplied by $e^{-\alpha}$, as in this column, only a non-
diagonal entry is present. The purpose of $C_{\alpha}^{\prime\prime}$ is to
damp all the remaining columns through multiplying them with
$e^{-\frac{\alpha}{2(n-1)}}$ at the cost of boosting the $\epsilon$th column
through multiplying it by $e^{\alpha/2}$. It can be easily seen that
$B_{\alpha}^{\prime}\otimes C_{\alpha}^{\prime}$ damps the entry in the
$\epsilon$th column faster than it is boosted by $C_{\alpha}^{\prime\prime}$.
Hence, overall, the whole matrix pencil $\mathcal{P}$ is damped.

Let us remark here, the also for the previously considered case ($m\neq n$),
it can be analogously seen that matrix pencils which contain an
$L_{\epsilon_{1}}$\- and an $L_{\epsilon_{2}}$-block with
$\epsilon_{1}\neq\epsilon_{2}$ correspond to SLOCC classes in the null-cone
(the same holds for $L^{T}$-blocks of different size). Moreover, the same
holds true for matrix pencils that contain an $M$-block (with arbitrary
eigenvalue and arbitrary size $e$) as well as either an $L_{\epsilon}$\- or an
$L^{T}_{\nu}$-block. Explicit constructions of sequences of SL-operations
that, in the limit, map states corresponding to such matrix pencils to 0 are
provided in Appendix D.

For $m=n$, however, Observation 5 already allows going beyond previous
results. There, whenever an $L$-block is present, it must be accompanied by an
$L^{T}$-block (and vice versa) due to the dimension of the matrix pencil.
Hence, Observation 5 applies and shows that any class corresponding to a
matrix pencil which contains such blocks is in the null-cone. It thus only
remains to classify matrix pencils that are solely composed of $M$-blocks. The
following observation reduces the problem to characterizing diagonal matrix
pencils.

######  Observation 6.

Let $C$ be an SLOCC class that corresponds to a KCF
$\mathcal{P}=\bigoplus_{i,j}M^{e^{i}_{j}}(x_{i})$ with at least one
$e^{i}_{j}>1$. Consider the matrix pencil
$\mathcal{P^{\prime}}=\bigoplus_{i}\bigoplus_{j^{\prime}=1}^{m_{i}}M^{1}(x_{i})$,
which is obtained from $\mathcal{P}$ by deleting all non-diagonal elements of
$\mathcal{P}$ (recall that $m_{i}=\sum_{j}e^{i}_{j}$). Then, the following
holds.

  1. (i)

If the SLOCC class corresponding to $\mathcal{P^{\prime}}$ is semistable, then
$C$ is strictly semistable.

  2. (ii)

If the SLOCC class corresponding to $\mathcal{P^{\prime}}$ is in the null-
cone, then $C$ is in the null-cone.

###### Proof.

Let us denote the state corresponding to $\mathcal{P}$ by $|\psi\rangle$. As
before, we will prove the two statements by explicitly constructing a sequence
of SL-operators of the form $\mathbbm{1}\otimes B_{\alpha}\otimes C_{\alpha}$
such that $\lim_{\alpha\rightarrow\infty}\mathbbm{1}\otimes B_{\alpha}\otimes
C_{\alpha}|\psi\rangle\propto|\psi^{\prime}\rangle$, where by
$|\psi^{\prime}\rangle$ we denote the state corresponding to
$\mathcal{P^{\prime}}$. To this end consider the $n\times n$ matrices

$\displaystyle B_{\alpha}$
$\displaystyle=\sum_{i=0}^{n-1}e^{\left(i-\frac{n-1}{2}\right)\alpha}|i\rangle\langle
i|\text{ and}$ (43) $\displaystyle C_{\alpha}$
$\displaystyle=\sum_{i=0}^{n-1}e^{\left(\frac{n-1}{2}-i\right)\alpha}|i\rangle\langle
i|.$ (44)

It can be easily verified that $\det(B_{\alpha})=\det(C_{\alpha})=1$ and that
$\lim_{\alpha\rightarrow\infty}B_{\alpha}\mathcal{P}C^{T}_{\alpha}=\mathcal{P^{\prime}}$.

Let us first proof statement (i). Due to the considerations above, we have
$G|\psi^{\prime}\rangle\subseteq\overline{G|\psi\rangle}$. Moreover, as the
size signatures of the eigenvalues of $\mathcal{P}$ and of
$\mathcal{P^{\prime}}$ are different, $|\psi\rangle$ and
$|\psi^{\prime}\rangle$ are not in the same SLOCC class. Thus,
$G|\psi^{\prime}\rangle\subseteq\overline{G|\psi\rangle}\setminus
G|\psi\rangle$ and therefore the latter set is not empty and $|\psi\rangle$ is
strictly semistable.

Let us now prove statement (ii). $|\psi^{\prime}\rangle$ being in the null-
cone implies that all SL-invariant polynomials vanish on
$|\psi^{\prime}\rangle$. However, as before we have that
$\det(B_{\alpha})=\det(C_{\alpha})=1$ and
$\lim_{\alpha\rightarrow\infty}B_{\alpha}\mathcal{P}C^{T}_{\alpha}=\mathcal{P^{\prime}}$.
Thus, it also holds that all SL-invariant polynomials vanish on
$|\psi\rangle$, i.e., $|\psi\rangle$ is in the null-cone. ∎

Due to the results above, it remains to characterize the SLOCC classes
corresponding to diagonal matrix pencils. In order to obtain such a
characterization, let us first state a lemma, which we prove in Appendix E.

######  Lemma 4.

Let $\\{\vec{v}_{i}\\}_{i=1}^{l}$ be a set of $l$ distinct unit vectors in
$\mathbb{R}^{3}$ with associated multiplicities $m_{i}\in\mathbb{N}$ sorted in
descending order such that $\sum_{i=1}^{l}m_{i}=n$ for some $n\geq 3$ and
$m_{1}<\sum_{i=2}^{l}m_{i}$. Let $(\theta_{i},\phi_{i})$ denote the spherical
coordinates of $\vec{v}_{i}$, i.e.,

$\displaystyle\vec{v_{i}}=\begin{pmatrix}\sin\theta_{i}\cos\phi_{i}\\\
\sin\theta_{i}\sin\phi_{i}\\\ \cos\theta_{i}\end{pmatrix}.$ (45)

Moreover, let $f_{\alpha}$ be a (non-linear) transformation acting on the
spherical coordinates of the vectors as

$\displaystyle f_{\alpha}:[0,\pi]\times[0,2\pi]$
$\displaystyle\rightarrow[0,\pi]\times[0,2\pi]$
$\displaystyle(\theta_{i},\phi_{i})$
$\displaystyle\mapsto(2\arctan\left(\frac{1}{1-\alpha}\tan(\theta_{i}/2)\right),\phi_{i}),$
(46)

where $\alpha\in[0,1)$. Then, it is always possible to transform the vectors
$\\{\vec{v}_{i}\\}_{i=1}^{l}$ to new set of vectors
$\\{\vec{v^{\prime}}_{i}\\}_{i=1}^{l}$ such that
$\sum_{i}m_{i}\vec{v^{\prime}}_{i}=0$ through some simultaneous $SO(3)$
rotation of the vectors $\vec{v}_{i}$ followed by $f_{\alpha}$.

Note that the operation considered in the lemma is exactly how an operator $A$
(up to a final local unitary) transforms a state whose corresponding matrix
pencil in KCF has eigenvalues that correspond to the vectors
$\\{\vec{v}_{i}\\}_{i=1}^{l}$ (as in the geometrical picture discussed in
Section 4.2) into a new state whose corresponding matrix pencil in KCF has
eigenvalues that correspond to the vectors
$\\{\vec{v^{\prime}}_{i}\\}_{i=1}^{l}$. Let us remark here, that this lemma
applied in our context shows that diagonal matrix pencils whose eigenenvalues
with degeneracies $m_{i}$ are such that every $m_{i}$ is strictly smaller than
the sum of the remaining $m_{j}$, correspond to SLOCC classes that contain a
critical state. With the help of this lemma, we can now perform the
characterization of SLOCC classes corresponding to diagonal matrix pencils as
shown in the following theorem. Note, that as we only consider fully entangled
states, matrix pencils that we consider have at least two distinct eigenvalues
(otherwise, the state would be bi-separable).

######  Theorem 4.

Let $C$ be an SLOCC class that corresponds to a diagonal matrix pencil of size
$n\times n$ with $l\geq 2$ distinct eigenvalues $x_{i}$. Let $m_{i}$ denote
the degeneracies of the eigenvalues $x_{i}$ in descending order. Then, the
following holds.

  1. (i)

If $m_{1}>\sum_{i=2}^{l}m_{i}$, then $C$ is in the null-cone.

  2. (ii)

If $l=2$ and $m_{1}=m_{2}$, then $C$ is strictly polystable (except when
$n=2$, which corresponds to the 3-qubit GHZ state, which is stable).

  3. (iii)

If $l>2$ and $m_{1}=\sum_{i=2}^{l}m_{i}$, then $C$ is strictly semistable.

  4. (iv)

Otherwise, i.e., if $l>2$ and $m_{1}<\sum_{i=2}^{l}m_{i}$, $C$ is polystable.
Moreover, $C$ is stable if and only if all eigenvalues are distinct ($l=n$)
and $C$ is strictly polystable if and only if there is any degeneracy ($l<n$).

###### Proof.

We will first prove statements (i), (ii), and (iii) by explicit construction
of sequences of $SL$-operators transforming the state of interest into $0$, a
critical state, or asymptotically into a critical state, respectively.
Utilizing Lemma 4, we will then show statement (iv).

In order to prove statement (i), let us first apply an operator $A$, which
brings the state to a form such that the eigenvalue with highest degeneracy is
$x_{1}=0$. Then the matrix pencil is of the form

$\displaystyle\mathcal{P}=\begin{pmatrix}\lambda&&&&\\\ &\ddots&&&\\\
&&\lambda&&\\\ &&&\lambda+x_{2}\mu&\\\ &&&&\ddots\end{pmatrix}.$ (47)

Let us denote the corresponding state by $|\psi\rangle$, which is given by
$|\psi\rangle=|0\rangle(\operatorname{diag}(0,\ldots,0,x_{2},\ldots)\otimes\mathbbm{1})|\phi^{+}_{n}\rangle+|1\rangle|\phi^{+}_{n}\rangle$.
Note that $n-m_{1}<m_{1}$ due to the assumption. Consider the matrices

$\displaystyle A_{\alpha}$
$\displaystyle=\operatorname{diag}(e^{\alpha},e^{-\alpha})\text{ and}$ (48)
$\displaystyle B_{\alpha}$
$\displaystyle=e^{\frac{2(n-m_{1})}{n}\alpha}\mathbbm{1}_{m_{1}}\oplus
e^{-\frac{2m_{1}}{n}\alpha}\mathbbm{1}_{n-m_{1}}.$ (49)

Then it holds that $\det(A_{\alpha})=\det(B_{\alpha})=1$ and moreover
$\lim_{\alpha\rightarrow\infty}A_{\alpha}\otimes
B_{\alpha}\otimes\mathbbm{1}|\psi\rangle=0$, which proves that the
corresponding SLOCC class is in the null-cone.

Let us now prove statement (ii). In this case there are exactly two
eigenvalues and these have the same multiplicity. It is possible to apply an
operator $A$ such that the matrix pencil takes the form
$\mathcal{P}=\lambda\mathbbm{1}_{n/2}\oplus\mu\mathbbm{1}_{n/2}$, which is
easy to see as up to three eigenvalues of a matrix pencil can be chosen freely
by an operator $A$ [Chitambar  _et al._(2010)Chitambar, Miller, and Shi]. It
can be easily verified that the corresponding state is critical, i.e., all
single-particle reduced density matrices are proportional to $\mathbbm{1}$.
Thus, the SLOCC class in question is polystable. Moreover, unless $n=2$, the
multiplicity of the eigenvalues is larger then 1. In Observation 4 we have
seen that this implies that the dimension of the orbit is strictly smaller
than the maximal dimension. This implies that, unless $n=2$, the considered
SLOCC class is strictly polystable. In case $n=2$, the class in question is
the 3-qubit GHZ class, which is stable.

Let us now prove statement (iii). As in the proof of statement (i) above, let
us first bring the matrix pencil to a form such that $x_{1}=0$. The matrix
pencil is then of the form as in Eq. (47). Let us denote the corresponding
state (after the operation) by $|\psi\rangle$. Considering now the matrices

$\displaystyle A_{\alpha}$
$\displaystyle=\operatorname{diag}(e^{\alpha},e^{-\alpha})\text{ and}$ (50)
$\displaystyle B_{\alpha}$
$\displaystyle=\left(\prod_{i=2}^{l}x_{i}^{m_{i}}\right)^{\frac{1}{n}}\left(e^{\alpha}\mathbbm{1}_{m_{1}}\oplus
e^{-\alpha}\bigoplus_{i=2}^{l}\frac{1}{x_{i}}\mathbbm{1}_{m_{i}}\right),$ (51)

we have $\det(A_{\alpha})=\det(B_{\alpha})=1$ and moreover
$\lim_{\alpha\rightarrow\infty}A_{\alpha}\otimes
B_{\alpha}\otimes\mathbbm{1}|\psi\rangle=\left(\prod_{i=2}^{l}x_{i}^{m_{i}}\right)^{\frac{1}{n}}|\psi^{\prime}\rangle$,
where $|\psi^{\prime}\rangle$ is a critical state corresponding to an SLOCC
class belonging to a matrix pencil of the form given in (ii). This proves
statement (iii).

For proving statement (iv), let us now consider the mapping of the problem to
a geometrical problem in $\mathbb{R}^{3}$ as in Section 4.2. Recall that
rewriting the eigenvalues of the matrix pencil as
$x_{i}=e^{i\phi_{i}}\tan{\frac{\theta_{i}}{2}}$ and constructing unit vectors
with $(\theta_{i},\phi_{i})$ as spherical coordinates [see Eq. (36)], then the
state $|\psi\rangle$ corresponding to the matrix pencil is related to a
critical state via operators $B$ and $C$ iff
$\sum_{i}m_{i}\vec{v}_{i}=\vec{0}$ [Bryan  _et al._(2019)Bryan, Leutheusser,
Reichstein, and Raamsdonk]. At this point we can apply Lemma 4, which
guarantees the existence of an operator $A$ applied on the first system, which
transforms the eigenvalues such that the condition is met. Let us remark here,
that the assumptions in the lemma are indeed such that the lemma only applies
for $m_{i}$ as in case (iv). As the existence of a critical state is
guaranteed, the considered SLOCC class is polystable. Moreover, strictly
polystable classes can be distinguished from stable classes by utilizing
Observation 4 in Section 4.3, stating that the dimension of the orbit is
maximal if and only if there is no degeneracy in the eigenvalues. Hence,
statement (iv) follows. ∎

This completes the classification of orbit types in $2\times m\times n$
systems. Let us remark that cases (ii) and (iii) can only occur in case $n$ is
even.

Let us also remark here that in the geometrical picture utilized to prove
statement (iv) of Theorem 4, statements (i)-(iii), which we have proven
differently, also become very transparent. In case (i) we are dealing with a
vector $\vec{v}_{1}$, which is pointing with a majority of the weight,
$m_{1}$, into a single direction. Even if all the remaining vectors could be
freely rotated independent of each other, their cumulative weight is at most
$m_{1}-1$. Hence the weighted sum of the vectors must always maintain a
distance from the origin of at least one. Hence, in case (i), $C$ must be in
the null-cone. In case (ii), we are dealing with exactly two vectors
$\vec{v_{1}}$ and $\vec{v_{2}}$ with equal weight $m_{1}=m_{2}$. These vectors
can be transformed in such a way that they point in opposite directions and
hence can clearly be transformed such that they sum up to $\vec{0}$. Hence, a
critical state exists. In case (iii) we are dealing with a vector
$\vec{v}_{1}$ pointing in some direction with weight $m_{1}$ and at least two
more vectors $\vec{v}_{i}$ with cumulative weight $m_{1}$ pointing in
different directions. In order to make the weighted sum of the vectors equal
$\vec{0}$, all the remaining vectors would have to point in opposite direction
of $\vec{v}_{1}$. As $A$ cannot change the multiplicities of the eigenvalues,
this is not possible by some finite $A$. However, applying the non-linear
transformation from Lemma 4, by first rotating all vectors such that
$\vec{v}_{1}$ points in z-direction and then performing the non-linear hinging
operation (see also Figure 4), it can be seen that in the limit
$\alpha\rightarrow 1$ all vectors $\vec{v}_{i}$ for $i\in\\{2,\ldots,l\\}$
point into $(-z)$-direction. Thus the weighted vector sum equals $\vec{0}$
(only) in the limit and thus, case (iii) corresponds to a strictly semistable
SLOCC class.

###  4.5 Algorithm to determine the orbit type

In this subsection, we present a flowchart (Figure 3) which can be followed in
order to decide the orbit type of an SLOCC class in
$\mathbb{C}^{2}\otimes\mathbb{C}^{m}\otimes\mathbb{C}^{n}$ given in terms of
the representing matrix pencil $\mathcal{P}$. The flowchart summarizes the
results obtained in this section. We present tables of the orbit types of
SLOCC classes for system sizes up to $2\times 5\times 5$ in Appendix F.

Figure 3: Characterization of orbit type of any SLOCC class in
$\mathbb{C}^{2}\otimes\mathbb{C}^{m}\otimes\mathbb{C}^{n}$ (wlog $m\leq n$).
$\mathcal{P}$ denotes the $m\times n$ matrix pencil whose corresponding SLOCC
class we are interested in. Wlog $\mathcal{P}$ is in KCF with $0\leq l\leq n$
distinct eigenvalues. Their multiplicities are denoted by $m_{i}$, and are
sorted in descending order. Then the orbit type can be decided by following
the flowchart. Remarkably, the concrete values which the eigenvalues of the
matrix pencil take do not play a role when deciding the orbit type, only their
multiplicities matter. A non-rectangular matrix pencil (i.e., $m\neq n$)
either equals $\mathcal{P}=\bigoplus_{i=1}^{m}L_{\frac{m}{n-m}}$ (which is
only possible if $n-m$ divides $m$), in which case the class is stable, as
shown in [Bryan  _et al._(2019)Bryan, Leutheusser, Reichstein, and Raamsdonk],
or is in the null-cone otherwise, as either [Bryan  _et al._(2019)Bryan,
Leutheusser, Reichstein, and Raamsdonk] together with Theorem 3, or,
alternatively, Observation 5 and the considerations in Appendix D show. In
case of square matrix pencils (i.e. $m=n$), the class is in the null-cone if
any $L$-block is present. The reason for that is that such a block must always
be accompanied by an $L^{T}$-block and thus Observation 5 applies. Theorem 4
and Observation 6 then allow classification of square matrix pencils that do
not contain any $L$\- or $L^{T}$-block depending on the multiplicities of the
eigenvalues.

##  5 Conclusion

In this work we have derived a necessary and sufficient criterion for the
existence of strictly semistable states in terms of the orbit dimensions of
polystable states using GIT methods. In particular, we have shown that
strictly semistable states exist if and only if there exist polystable states
whose orbit dimensions differ. We have applied the criterion to three-partite
states of local dimensions $2$, $m$, and $n$ and characterized those $m$ and
$n$, for which strictly semistable states do exist. Going beyond that, we have
scrutinized all SLOCC classes in such systems and characterized their orbit
types. It turns out that (for different values of $m$ and $n$), a rich variety
of behaviors manifests. More precisely, there are examples where only one
SLOCC class containing fully entangled states exists (which is stable); there
are examples in which all SLOCC classes are in the null-cone (as characterized
in [Bryan  _et al._(2019)Bryan, Leutheusser, Reichstein, and Raamsdonk]); in
some systems, stable states and states in the null-cone exist, but no strictly
semistable states; and finally, there are systems in which all of the
discussed orbit types are present. Moreover, all of these situations become
already apparent within small system sizes, that is $m\leq n\leq 5$ as shown
in Appendix F.

It would be interesting to see whether one can identify a difference between
stable states, strictly polystable states, strictly semistable states, and
states in the null-cone for practical applications. A well-known result in
this vein is that the entanglement in the W-state, which is in the null-cone,
is more robust than the entanglement in the three-qubit GHZ-state, which is
stable [Dür  _et al._(2000)Dür, Vidal, and Cirac].

Finally, let us remark that $2\times m\times n$-states find application as
fiducial states in the context of matrix product states [Sauerwein  _et
al._(2019)Sauerwein, Molnar, Cirac, and Kraus]. In particular, properties of
$2\times m\times n$-states can be used to characterize properties of the
matrix product state they give rise to, such as local symmetries, or SLOCC
classes.

O.S. and A.S. acknowledge financial support from National Science Centre,
Poland under the grant SONATA BIS: 2015/18/E/ST1/00200. M.H. and B.K.
acknowledge financial support from the Austrian Science Fund (FWF) grant DK-
ALM: W1259-N27 and the SFB BeyondC (Grant No. F7107-N38). Furthermore, B.K.
acknowledges support of the Austrian Academy of Sciences via the Innovation
Fund “Research, Science and Society”.

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##  Appendix A Proof of Observation 1

In this appendix we prove Observation 1. In order to do so, we will first
state and prove a lemma and another observation.

######  Lemma 5.

Suppose $|\Psi\rangle\in\mathcal{H}_{ss}$ and we have a sequence
$(g_{n})_{n},\,g_{n}\in G$ such that $\lVert
g_{n}|\Psi\rangle\rVert\xrightarrow{n}\infty$. Then for any sequence
$(\phi_{n})_{n}$,

$\lim_{n\to\infty}\frac{g_{n}|\Psi\rangle}{\lVert
g_{n}|\Psi\rangle\rVert}e^{i\phi_{n}}\in\mathcal{N}$

whenever this limit exists.

###### Proof.

Let $\\{p_{1},\ldots,p_{k}\\}$ be the set of homogeneous polynomials
$p_{i}:\mathcal{H}\rightarrow\mathbb{C}$ of degree $\mathrm{\deg}\,p_{i}$ that
generate the ring of $G$–invariant polynomials. Then for
$|\Phi\rangle=\displaystyle\lim_{n\to\infty}\frac{g_{n}|\Psi\rangle}{\lVert
g_{n}|\Psi\rangle\rVert}e^{i\phi_{n}}$ we have

$\displaystyle\forall_{i}\;p_{i}(|\Phi\rangle)$
$\displaystyle=\lim_{n\to\infty}p_{i}\left(\frac{g_{n}|\Psi\rangle}{\lVert
g_{n}|\Psi\rangle\rVert}e^{i\phi_{n}}\right)$
$\displaystyle=p_{i}(|\Psi\rangle)\lim_{n\to\infty}\left(\frac{e^{i\phi_{n}}}{\lVert
g_{n}|\Psi\rangle\rVert}\right)^{\mathrm{deg}\,p_{i}}=0,$

since the limit exists and all $p_{i}$ are continuous, homogeneous and
$G$–invariant. Thus $\forall_{i}\;p_{i}(|\Phi\rangle)=0$ so
$|\Phi\rangle\in\mathcal{N}$. ∎

######  Observation 7.

If $[\Psi_{n}]\xrightarrow{n\to\infty}[\Phi]$ in $\mathbb{P}(\mathcal{H})$,
then the sequence $(|\tilde{\Psi}_{n}\rangle)_{n}$ in $\mathcal{H}$ with
$|\tilde{\Psi}_{n}\rangle=\frac{|\Psi_{n}\rangle}{\lVert|\Psi_{n}\rangle\rVert}e^{i\phi_{n}}$
and $\phi_{n}=\mathrm{arg}(\langle\Psi_{n}|\Phi\rangle)$ converges to
$|\tilde{\Phi}\rangle=\frac{|\Phi\rangle}{\lVert|\Phi\rangle\rVert}$, i.e.,

$\displaystyle|\tilde{\Psi}_{n}\rangle\xrightarrow{n\to\infty}|\tilde{\Phi}\rangle.$

###### Proof.

Let us introduce the Fubini-Study distance in $\mathbb{P}(\mathcal{H})$,

$d([\Psi],[\Phi])=\mathrm{arccos}\frac{\lvert\langle\Psi|\Phi\rangle\rvert}{\lVert|\Psi\rangle\rVert\lVert|\Phi\rangle\rVert}\mathrm{,}$

which defines a metric compatible with the standard topology on
$\mathbb{P}(\mathcal{H})$. The convergence
$[\Psi_{n}]\xrightarrow{n\to\infty}[\Phi]$ implies that
$d([\Psi_{n}],[\Phi])\xrightarrow{n\to\infty}0$ which means that

$\frac{\lvert\langle\Psi_{n}|\Phi\rangle\rvert}{\lVert|\Psi_{n}\rangle\rVert\lVert|\Phi\rangle\rVert}\xrightarrow{n\to\infty}1\text{.}$

On the other hand

$\displaystyle\lVert|\tilde{\Psi}_{n}\rangle-|\tilde{\Phi}\rangle\rVert$
$\displaystyle=2\left(1-\frac{\Re\left(e^{-i\phi_{n}}\langle\Psi_{n}|\Phi\rangle\right)}{\lVert|\Psi_{n}\rangle\rVert\lVert|\Phi\rangle\rVert}\right)=$
$\displaystyle=2\left(1-\frac{\lvert\langle\Psi_{n}|\Phi\rangle\rvert}{\lVert|\Psi_{n}\rangle\rVert\lVert|\Phi\rangle\rVert}\mathrm{cos}(\psi_{n}-\phi_{n})\right),$

where $\psi_{n}=\mathrm{arg}(\langle\Psi_{n}|\Phi\rangle)$. Hence, with the
choice $\phi_{n}=\psi_{n}$ one obtains

$\lVert|\tilde{\Psi}_{n}\rangle-|\tilde{\Phi}\rangle\rVert=2\left(1-\frac{\lvert\langle\Psi_{n}|\Phi\rangle\rvert}{\lVert|\Psi_{n}\rangle\rVert\lVert|\Phi\rangle\rVert}\right)\xrightarrow{n\to\infty}0\text{.}$

∎

Now we are ready to prove Observation 1, which we restate here in order to
increase readability.

Observation 1 . For any vector $|\Psi\rangle\in\mathcal{H}_{ss}$, we have

$\displaystyle\overline{[\;G|\Psi\rangle\;]}\setminus[\;\overline{G|\Psi\rangle}\;]\subset[{\cal
N}\setminus\\{0\\}].$ (52)

###### Proof.

Notice that $\overline{[\;G|\Psi\rangle\;]}$ is $G$–invariant and the
projection $[\cdot]$ is continuous in the standard topologies on $\mathcal{H}$
and $\mathbb{P}(\mathcal{H})$, thus

$[\;\overline{G|\Psi\rangle}\;]\subset\overline{[\;G|\Psi\rangle\;]}\text{.}$

Let $|\Psi\rangle\in\mathcal{H}_{ss}$. Choose any state
$[\Phi]\in\overline{[\;G|\Psi\rangle\;]}$. We will show that either
$[\Phi]\in[\mathcal{N}\setminus\\{0\\}]$ or
$[\Phi]\in[\;\overline{G|\Psi\rangle}\;]$. Let $(g_{n})_{n},g_{n}\in G$, be a
sequence of elements in $G$ such that
$[g_{n}|\Psi\rangle]\xrightarrow{n\to\infty}[\Phi]$. Due to Observation 7
there exists a sequence $(|\Psi_{n}\rangle)_{n}$ such that
$|\Psi_{n}\rangle\xrightarrow{n\to\infty}|\tilde{\Phi}\rangle$ where

$|\Psi_{n}\rangle=\frac{g_{n}|\Psi\rangle}{\lVert
g_{n}|\Psi\rangle\rVert}e^{i\phi_{n}},\quad|\tilde{\Phi}\rangle=\frac{|\Phi\rangle}{\lVert|\Phi\rangle\rVert}\text{.}$

Let us denote

$r_{n}\coloneqq\lVert g_{n}|\Psi\rangle\rVert\in\mathbb{R},\quad
c_{n}\coloneqq r_{n}e^{-i\phi_{n}}\in\mathbb{C}\text{,}$

so $c_{n}|\Psi_{n}\rangle=g_{n}|\Psi\rangle$ and $\lvert c_{n}\rvert=r_{n}$.
Suppose that the sequence $(r_{n})_{n}$ is not bounded from above. Then we can
pick a subsequence $(r_{n_{k}})_{k}$ such that
$r_{n_{k}}\xrightarrow{k\to\infty}\infty$. It holds that
$|\Psi_{n_{k}}\rangle\xrightarrow{k\to\infty}|\tilde{\Phi}\rangle$ and from
Lemma 5 we get that $[\Phi]=[\tilde{\Phi}]\in[\mathcal{N}\setminus\\{0\\}]$
161616Note that the projection of this diverging sequence of vectors (i.e. the
corresponding sequence of states) may actually converge in
$\mathbb{P}(\mathcal{H})$ so that this sequence of states corresponds to a
state from the closure of $[G|\Psi\rangle]$ in $\mathbb{P}(\mathcal{H})$.
Suppose that $(r_{n})_{n}$ is bounded from above by some $b\in\mathbb{R}_{+}$.
Because $|\Psi\rangle\in\mathcal{H}_{ss}$, the sequence $(r_{n})_{n}$ has to
be bounded from below by a non-zero norm infimum $a\in\mathbb{R}_{+}$. Thus,
for every $n$ we have $r_{n}\in[a,b]$, i.e., $(c_{n})_{n}$ is bounded with
$\lvert c_{n}\rvert=r_{n}\in[a,b]\subset\mathbb{R}_{+}$. From the Bolzano-
Weierstrass theorem for $\mathbb{C}$, we can pick a convergent subsequence
$(c_{n_{k}})_{k}$ such that $c_{n_{k}}\xrightarrow{k\to\infty}c$ and the
corresponding subsequence $(|\Psi_{n_{k}}\rangle)_{k}$. Moreover
$c\in\mathbb{C}\setminus\\{0\\}$, as $\lvert
c\rvert\in[a,b]\subset\mathbb{R}_{+}$. Since
$|\Psi_{n}\rangle\xrightarrow{n\to\infty}|\tilde{\Phi}\rangle$ then also
$|\Psi_{n_{k}}\rangle\xrightarrow{k\to\infty}|\tilde{\Phi}\rangle$. The
product $c_{n_{k}}|\Psi_{n_{k}}\rangle$ of convergent sequences $c_{n_{k}}$
and $|\Psi_{n_{k}}\rangle$ converges to $c|\tilde{\Phi}\rangle$. Thus
$g_{n_{k}}|\Psi\rangle\xrightarrow{k\to\infty}c|\tilde{\Phi}\rangle\in\overline{G|\Psi\rangle}$.
But $[\Phi]=[c|\tilde{\Phi}\rangle]$ since $c\neq 0$, so
$[\Phi]\in[\;\overline{G|\Psi\rangle}\;]$. This completes the proof. ∎

##  Appendix B An equivalent definition of stability

In this appendix, we show that the stability of the action can be defined in
an alternative way, which is equivalent to the definition introduced in the
main text, Definition 2. In the course of that, it will become clear that the
maximal orbit dimension among all polystable orbits equals the maximal orbit
dimension among all orbits, $d_{G}$, in case the action is stable.

Let us now consider the following alternative definition of the stability of
the action.

######  Definition 3.

An action $H\curvearrowright\mathbb{P}(\mathcal{H})$ is stable if there exists
an open dense set $U\subset\mathbb{P}(\mathcal{H})$ that is a union of stable
orbits.

We introduce the definition of a function $\mathrm{dim}\,G\textunderscore$ as
follows: the value of $\mathrm{dim}\,G\textunderscore$ at $x$ is equal to
$\mathrm{dim}\,G_{x}$. In order to prove the equivalence of Definition 3 to
Definition 2, we use the following lemma, which is a particular version of
Proposition 2.27. from [Hoskins(2012)].

######  Lemma 6 ([Hoskins(2012)]).

In our setting, the stabilizer dimension function
$\mathrm{dim}\,G\textunderscore:\mathbb{P}(\mathcal{H})\rightarrow\mathbb{N}_{0}$
is upper semi-continuous, i.e. for each $n\in\mathbb{N}_{0}$, the set

$X_{n}:=\\{x\in\mathbb{P}(\mathcal{H}):\mathrm{dim}\,G_{x}\geq n\\}\text{,}$
(53)

is closed in $\mathbb{P}(\mathcal{H})$.

Below we prove the equivalence of Definitions 2 and 3.

###### Proof of the equivalence of Defs. 2 and 3..

Let $n_{G}$ denote the minimal stabilizer dimension for
$G\curvearrowright\mathbb{P}(\mathcal{H})$ and suppose that the action is
stable. Consider a set $X:=X_{n_{G}}\setminus X_{n_{G}+1}$. The set $X$
contains exactly those orbits whose stabilizers have dimension $n_{G}$, i.e.,
orbits with maximal dimension. As $X_{n_{G}}=\mathbb{P}(\mathcal{H})$ and
$X_{n_{G}+1}$ is closed (Lemma 6), the set $X$ is open (and dense). From the
stability of action we know that there is a set $U$ which is the open (and
dense) set of closed orbits mentioned in Definition 2. Consider the set
$Y:=X\cap U$. Then, $Y$ has all of the defining properties of the set $U$ and,
in addition, it contains only orbits of the maximal dimension. Thus, we can
always choose the set $U$ in Definition 2 in such a way that it contains
orbits of the maximal dimension. ∎

Moreover, from the proof it follows that in case of stable action, the
dimension of stable orbits is maximal among all orbits, not only among
polystable ones.

##  Appendix C Regularity of the GIT-quotient $\pi$

In this appendix, we elaborate in more details on the assumptions of Lemma 3
from Section 3.4. In particular, we show that the lemma applies to the GIT
quotient
$\pi:\mathbb{P}(\mathcal{H})_{ss}\rightarrow\mathbb{P}(\mathcal{H})_{ss}//G$.

Let us first introduce the following definitions. Here and in the following,
$\mathbb{P}^{q}$ denotes a $q$-dimensional projective space, e.g.,
$\mathbb{P}(\mathcal{H})$ for $q=\mathrm{dim}(\mathcal{H})-1$.

######  Definition 4 ([Smith  _et al._(2000)Smith, Kahanpää, Kekäläinen, and
Traves] Ch. 4, p. 49).

An algebraic variety is quasi-projective if it is an intersection of open
and/or closed subsets of $\mathbb{P}^{q}$ considered with the Zariski topology
induced from $\mathbb{P}^{q}$.

######  Definition 5.

  1. (i)

A topological space $X$ is irreducible if $X$ cannot be written as the union
of two closed and proper subsets $F_{1}\subsetneq X$, $F_{2}\subsetneq X$,
i.e., $X\neq F_{1}\cup F_{2}$.

  2. (ii)

A subset $A\subset X$ of a topological space $X$ is irreducible if it is
irreducible as a topological space with topology induced from $X$.

With these definitions, the following observation can be obtained.

######  Observation 8.

  1. (i)

A topological space $X$ is irreducible if and only if any non-empty open
subset $U\subset X$ is dense in $X$.

  2. (ii)

Any open subset $U\subset X$ of an irreducible topological space $X$ is
irreducible.

  3. (iii)

If $f:X\rightarrow Y$ is a continuous map between topological spaces then if
$X$ is irreducible so is $f(X)\subset Y$.

Central in Lemma 3 is also the definition of a regular map, which is given in
the following.

######  Definition 6 ([Smith  _et al._(2000)Smith, Kahanpää, Kekäläinen, and
Traves]).

We say that a map $f:X\rightarrow Y$ between a quasi-projective variety
$X\subset\mathbb{P}^{q}$ and a projective variety $Y\subset\mathbb{P}^{r}$ is
regular (or a morphism) if for every $x\in X$ there exist homogeneous
polynomials of the same degree $F_{0},\ldots,F_{r}$ in $q+1$ variables
(homogeneous coordinates of $x$) such that the map
$X\rightarrow\mathbb{P}^{r}$ given by $x\mapsto[F_{0}(x):\ldots:F_{r}(x)]$ is
well-defined and agrees with $f$ on some open neighborhood of $x$. In
particular for every $x\in X$ there has to be the local expression for $f$ of
the above form such that ${\ F_{j}(x)\neq 0}$ for some $j$ .

It can be seen that one can also use $r$ rational functions
$f_{i}=F_{i}/F_{j}$, where $i\neq j$ to define regular maps.

The basic fact states that regular maps between quasi-projective varieties are
continuous (see [Shafarevich(2013)]). Using the discussion above in the
context of our GIT quotient
$\mathbb{P}(\mathcal{H})_{ss}\rightarrow\mathbb{P}(\mathcal{H})_{ss}//G$, we
have

  1. (i)

Projective spaces (e.g. $\mathbb{P}(\mathcal{H})$) are irreducible.

  2. (ii)

The set of semistable states $\mathbb{P}(\mathcal{H})_{ss}$ is an irreducible
quasi-projective variety. Indeed, since $[\mathcal{N}\setminus\\{0\\}]$ is a
closed set, $\mathbb{P}(\mathcal{H})_{ss}$ is an open subset of irreducible
projective variety $\mathbb{P}(\mathcal{H})$. Obviously
$\mathbb{P}(\mathcal{H})_{ss}=\mathbb{P}(\mathcal{H})_{ss}\cap\mathbb{P}(\mathcal{H})$
and $\mathbb{P}(\mathcal{H})$ is closed.

  3. (iii)

The GIT quotient
$\pi:\mathbb{P}(\mathcal{H})_{ss}\rightarrow\mathbb{P}(\mathcal{H})_{ss}//G$
is a regular map between quasi-projective algebraic varieties, i.e. a morphism
(see [Brion(2010)] p. 10). In particular $\pi$ is continuous.

  4. (iv)

The set $\mathbb{P}(\mathcal{H})_{ss}//G$ is a projective variety (see
[Brion(2010)] p. 11, Proposition 1.29.). Moreover
$\mathbb{P}(\mathcal{H})_{ss}//G$ is irreducible. Thus,
$\mathbb{P}(\mathcal{H})_{ss}//G$ is the irreducible quasi-projective variety.

##  Appendix D Additional details on $2\times m\times n$ classes that are in
the null-cone.

In this appendix, we explicitly construct sequences of SL-operators
$A_{\alpha}$, $B_{\alpha}$, and $C_{\alpha}$, such that
$\lim_{\alpha\rightarrow\infty}A_{\alpha}\otimes B_{\alpha}\otimes
C_{\alpha}|\psi\rangle=0$ for
$|\psi\rangle\in\mathbb{C}^{2}\otimes\mathbb{C}^{m}\otimes\mathbb{C}^{n}$
corresponding to matrix pencils in KCF, in which certain combinations of
blocks are present. Observation 5 in the main text does exactly that for
matrix pencils (in KCF) that contain an $L_{\epsilon}$\- as well as an
$L^{T}_{\nu}$-block and thus shows that SLOCC classes corresponding to such
matrix pencils are in the null-cone. Here, we show that a similar reasoning
applies to matrix pencils whose KCF contains an $L_{\epsilon_{1}}$\- and an
$L_{\epsilon_{2}}$-block of different size, i.e.,
$\epsilon_{1}\neq\epsilon_{2}$. Moreover, the reasoning also applies to matrix
pencils that contain an $L_{\epsilon}$\- and an $M$-block. Note however, that
those SLOCC classes correspond to the nullcone follows already from [Bryan
_et al._(2019)Bryan, Leutheusser, Reichstein, and Raamsdonk] together with
Theorem 3, as discussed in the main text.

Let us first consider an SLOCC class that corresponds to a matrix pencil whose
KCF contains an $L_{\epsilon_{1}}$\- and an $L_{\epsilon_{2}}$-block of
different size, i.e., $\epsilon_{1}\neq\epsilon_{2}$. Wlog we assume an
ordering of the blocks such that the matrix pencil has the form
$\mathcal{P}=L_{\epsilon_{1}}\oplus
L_{\epsilon_{2}}\oplus\tilde{\mathcal{P}}$, where $\tilde{\mathcal{P}}$ is of
size $[m-(\epsilon_{1}+\epsilon_{2})]\times[n-(\epsilon_{1}+\epsilon_{2}+2)]$
and contains the remaining blocks. As discussed above, such an SLOCC classes
is in the null-cone (the same holds for two $L^{T}$ blocks of different size).
Let us now explicitly construct a sequence of SL-operations which, in the
limit, transforms states corresponding to such $\mathcal{P}$ to $0$. Let us
define

$\displaystyle B_{\alpha}^{\prime}$
$\displaystyle=e^{-\frac{\epsilon_{2}}{\epsilon_{2}-\epsilon_{1}}\alpha}\mathbbm{1}_{\epsilon_{1}}\oplus
e^{\frac{\epsilon_{1}}{\epsilon_{2}-\epsilon_{1}}\alpha}\mathbbm{1}_{\epsilon_{2}}\text{
and}$ (54) $\displaystyle C_{\alpha}^{\prime}$
$\displaystyle=e^{\frac{\epsilon_{2}}{\epsilon_{2}-\epsilon_{1}}\alpha}\mathbbm{1}_{\epsilon_{1}+1}\oplus
e^{-\frac{\epsilon_{2}}{\epsilon_{2}-\epsilon_{1}}\alpha}\mathbbm{1}_{1}\oplus
e^{-\frac{\epsilon_{1}}{\epsilon_{2}-\epsilon_{1}}\alpha}\mathbbm{1}_{\epsilon_{2}}$
(55)

of sizes $(\epsilon_{1}+\epsilon_{2})\times(\epsilon_{1}+\epsilon_{2})$ and
$(\epsilon_{1}+\epsilon_{2}+2)\times(\epsilon_{1}+\epsilon_{2}+2)$,
respectively. We take $C_{\alpha}^{\prime\prime}$ as in the proof of
Observation 5, replacing $\epsilon$ with $\epsilon_{1}+1$, in order to
construct
$B_{\alpha}=B_{\alpha}^{\prime}\oplus\mathbbm{1}_{m-(\epsilon_{1}+\epsilon_{2})}$
and
$C_{\alpha}=C_{\alpha}^{\prime\prime}\left(C_{\alpha}^{\prime}\oplus\mathbbm{1}_{n-(\epsilon_{1}+\epsilon_{2}+2)}\right)$.
Then, $\lim_{\alpha\rightarrow\infty}B_{\alpha}\mathcal{P}C_{\alpha}^{T}=0$.
For two $L^{T}$ blocks instead of two $L$ blocks, the matrices can be
constructed analogously.

The intuition behind this construction is similar as in Observation 5. As in
Observation 5, $B^{\prime}$ and $C^{\prime}$ are constructed in order to damp
one of the columns of the matrix pencil. Here, it is the $(\epsilon_{1}+1)$st
column of the $L_{\epsilon_{1}}\oplus L_{\epsilon_{2}}$ part in $\mathcal{P}$,
leaving the remaining columns invariant. Moreover, $C_{\alpha}^{\prime\prime}$
is used to damp all the remaining columns at the cost of boosting the
$(\epsilon_{1}+1)$st column in such a way that the damping overcompensates the
boosting as in Observation 5.

Let us finally consider an SLOCC class that corresponds to a matrix pencil
whose KCF contains an $L_{\epsilon}$\- and an $M$-block (with arbitrary
eigenvalue and arbitrary size $e$). Again, such an SLOCC classes is in the
null-cone, as discussed above (the same holds for an $L^{T}_{\nu}$\- together
with an $M$-block). A sequence of SL-operations which, in the limit,
transforms states corresponding to such matrix pencils to $0$ can be
constructed as follows. Consider an ordering of the blocks such that the
matrix pencil has the form $\mathcal{P}=L_{\epsilon}\oplus
M\oplus\tilde{\mathcal{P}}$, where $\tilde{\mathcal{P}}$ contains the
remaining blocks. Let us define

$\displaystyle B_{\alpha}^{\prime}$
$\displaystyle=e^{-\alpha}\mathbbm{1}_{\epsilon}\oplus
e^{\frac{\epsilon}{e}\alpha}\mathbbm{1}_{e}\text{ and}$ (56) $\displaystyle
C_{\alpha}^{\prime}$
$\displaystyle=e^{\alpha}\mathbbm{1}_{\epsilon}\oplus\mathbbm{1}_{1}\oplus
e^{-\frac{\epsilon}{e}\alpha}\mathbbm{1}_{e}$ (57)

of sizes $(\epsilon+e)\times(\epsilon+e)$ and
$(\epsilon+1+e)\times(\epsilon+1+e)$, respectively. We take
$C_{\alpha}^{\prime\prime}$ as in the proof of Observation 5 in order to
construct $B_{\alpha}=B_{\alpha}^{\prime}\oplus\mathbbm{1}_{m-(\epsilon+e)}$
and
$C_{\alpha}=C_{\alpha}^{\prime\prime}\left(C_{\alpha}^{\prime}\oplus\mathbbm{1}_{n-(\epsilon+1+e)}\right)$.
Then, $\lim_{\alpha\rightarrow\infty}B_{\alpha}\mathcal{P}C_{\alpha}^{T}=0$,
which proves the statement. In case of an $L^{T}$\- together with an
$M$-block, the matrices can be constructed analogously.

##  Appendix E Proof of Lemma 4

In this appendix, we restate and then prove Lemma 4.

Lemma 4. Let $\\{\vec{v}_{i}\\}_{i=1}^{l}$ be a set of $l$ distinct unit
vectors in $\mathbb{R}^{3}$ with associated multiplicities
$m_{i}\in\mathbb{N}$ sorted in descending order such that
$\sum_{i=1}^{l}m_{i}=n$ for some $n\geq 3$ and $m_{1}<\sum_{i=2}^{l}m_{i}$.
Let $(\theta_{i},\phi_{i})$ denote the spherical coordinates of $\vec{v}_{i}$,
i.e.,

$\displaystyle\vec{v_{i}}=\begin{pmatrix}\sin\theta_{i}\cos\phi_{i}\\\
\sin\theta_{i}\sin\phi_{i}\\\ \cos\theta_{i}\end{pmatrix}.$ (58)

Moreover, let $f_{\alpha}$ be a (non-linear) transformation acting on the
spherical coordinates of the vectors as

$\displaystyle f_{\alpha}:[0,\pi]\times[0,2\pi]$
$\displaystyle\rightarrow[0,\pi]\times[0,2\pi]$
$\displaystyle(\theta_{i},\phi_{i})$
$\displaystyle\mapsto(2\arctan\left(\frac{1}{1-\alpha}\tan(\theta_{i}/2)\right),\phi_{i}),$
(59)

where $\alpha\in[0,1)$. Then, it is always possible to transform the vectors
$\\{\vec{v}_{i}\\}_{i=1}^{l}$ to new set of vectors
$\\{\vec{v^{\prime}}_{i}\\}_{i=1}^{l}$ such that
$\sum_{i}m_{i}\vec{v^{\prime}}_{i}=0$ through some simultaneous $SO(3)$
rotation of the vectors $\vec{v}_{i}$ followed by $f_{\alpha}$.

Let us first discuss some properties of the considered operation before we
prove the lemma. First, note that the described operation is exactly the
operation described in Section 4.2. Recall that the non-linear part can be
geometrically understood as hinging all $\vec{v}_{i}$ (except those pointing
in direction $(0,0,1)^{T}$ or $(0,0,-1)^{T}$) down towards direction
$(0,0,-1)^{T}$ and the $SO(3)$ operation can be understood as choosing some
axis $(\Theta,\Phi)$ with respect to which the non-linear transformation is
performed.

Let us now define a function $f$ for any set of distinct vectors
$\\{\vec{v}_{i}\\}_{i=1}^{l}$ (with mutliplicities $m_{i}$) fulfilling the
premises of Lemma 4 as follows.

$\displaystyle f:D$ $\displaystyle\rightarrow[0,n^{2}]$ (60)
$\displaystyle(\alpha,\Theta,\Phi)$ $\displaystyle\mapsto
f(\alpha,\Theta,\Phi)=\left|\sum_{i}m_{i}\vec{v}^{\prime}_{i}\right|^{2},$
(61)

where $D=[0,1)\times[0,\pi]\times[0,2\pi]$ and $\vec{v}^{\prime}_{i}$ are
obtained by transforming the vectors $\vec{v}_{i}$ as follows. First, the
vectors $\vec{v}_{i}$ are rotated around the z-axis by the angle $\Phi$. Then
they are rotated around the y-axis by $\Theta$. Finally, $f_{\alpha}$ is
performed on them. See Figure 4 for a sketch of the operation.

Figure 4: Sketch of the operation considered in Lemma 4. We depict a case in
which $l=3$, $m_{i}=1$, and for simplicity all vectors (arrows) as well as the
$z$-axis (dashed line) lie in the same plane. Initial vectors
$\\{\vec{v}_{i}\\}$ of unit length (dotted arrows) are first jointly rotated
(yielding the dashed arrows) followed by performing the non-linear operation
$f_{\alpha}$ yielding the vectors $\\{\vec{v^{\prime}}_{i}\\}$ (solid arrows),
an operation that can be pictorially understood as hinging the vectors down
towards negative $z$-direction. In the limit $\alpha\rightarrow 1$,
$\\{\vec{v^{\prime}}_{i}\\}$ become perfectly antiparallelly aligned with the
$z$-axis. Vectors which would already be perfectly aligned with the $z$-axis
after the initial rotation are exceptions to this, they would remain unchanged
under $f_{\alpha}$ (not shown in the graphic).

Obviously, $f$ is bounded from below and above respectively by 0 (when the
vectors $\vec{v}^{\prime}_{i}$ sum to 0) and $n^{2}$ (when all vectors
$\vec{v}^{\prime}_{i}$ are aligned, which is not possible via an invertible
operation, but nevertheless upper bounds $f$). In order to prove the lemma, we
will show that $f$ actually attains 0, regardless of the initial orientations
of the vectors, as long as the premises of Lemma 4 are fulfilled. To this end,
let us define $\tilde{f}$ extending the function $f$ to the domain
$\tilde{D}=[0,1]\times[0,\pi]\times[0,2\pi]$ through

$\displaystyle\tilde{f}(\alpha,\Theta,\Phi)=\begin{cases}f(\alpha,\Theta,\Phi),&\text{if
}\alpha<1\\\ (n-2m_{i})^{2}&\text{if }\alpha=1\text{ and }(\Theta,\Phi)=\\\
&\qquad(\theta_{i},\phi_{i})\text{ for some }i\\\
n^{2}&\text{otherwise}.\end{cases}$ (62)

Here, note that the condition $(\Theta,\Phi)=(\theta_{i},\phi_{i})$ means that
the axis with respect to which the hinging operation is performed is exactly
aligned with the vector $\vec{v}_{i}$. Moreover, note that by construction
$\tilde{f}(1,\Theta,\Phi)=\lim_{\alpha\rightarrow 1}f(\alpha,\Theta,\Phi)$ for
all $\Theta$, $\Phi$.

Let us now state some properties of the functions $f$ and $\tilde{f}$ in the
following observation, which will allow us to prove Lemma 4.

######  Observation 9.

The following statements hold.

  1. (i)

$f$ is continuous on $D$.

  2. (ii)

For all $x\in D\setminus L_{0}(f)$ there exists some $x^{\prime}$ in $D$ such
that $f(x^{\prime})<f(x)$, where $L_{0}(f)=\\{x\in D:f(x)=0\\}$ is the zero-
level set of the function.

  3. (iii)

For a fixed pair $\Theta$, $\Phi$ there exists an $\alpha_{*}$ such that
$f(\alpha,\Theta,\Phi)$ is strictly monotonically increasing in $\alpha$ for
all $\alpha>\alpha_{*}$. Note however, that there does not exist an
$\alpha_{*}$ such that the statement holds for all $\Theta$, $\Phi$.

  4. (iv)

$\tilde{f}$ is lower semi-continuous on $\tilde{D}$. That is, for all
$x\in\tilde{D}$ it holds that for all $\delta>0$ there exists some $R>0$ such
that for all $x^{\prime}\in\tilde{D}$ with $d(x,x^{\prime})\leq R$,
$\tilde{f}(x^{\prime})\geq\tilde{f}(x)-\delta$. Here, $d$ is a distance on
$\tilde{D}$.

  5. (v)

For any fixed $\Theta$ and $\Phi$, there exists some $\alpha\in[0,1)$ such
that $\tilde{f}(\alpha,\Theta,\Phi)<\tilde{f}(1,\Theta,\Phi)$.

With this observation, we are now in the position to prove Lemma 4.
Afterwards, we will prove the observation.

###### Proof of Lemma 4.

As discussed above, in order to prove the lemma it suffices to show that $f$
as defined in Eq. (60) attains zero. To this end, we also make use of
$\tilde{f}$ as defined in Eq. (62) and Observation 9. As $\tilde{f}$ is a
lower semi-continuous function on a compact domain $\tilde{D}$ according to
Observation 9 (iv), Weierstrass’ extreme value theorem guarantees that
$\tilde{f}$ attains its infimum on $\tilde{D}$. Let us denote the point at
which the infimum is attained by
$x_{0}=(\alpha_{0},\Theta_{0},\Phi_{0})\in\tilde{D}$. Observation 9 (v)
guarantees that $\alpha_{0}<1$ and therefore $x_{0}\in D$. Hence, also $f$
attains its infimum at $x_{0}$. Lastly, Observation 9 (ii) guarantees that
this infimum is $0$. ∎

###### Proof of Observation 9.

Proof of statement (i). To see that $f$ is continuous on $D$, it can first be
verified that $f$ can be written as

$\displaystyle f(\alpha,\Theta,\Phi)$
$\displaystyle=\left|\sum_{i}m_{i}\frac{2\Im\\{(ax_{i}+b)(cx_{i}+d)^{*}\\}}{|cx_{i}+d|^{2}+|ax_{i}+b|^{2}}\right|^{2}$
$\displaystyle\qquad+\left|\sum_{i}m_{i}\frac{2\Re\\{(ax_{i}+b)(cx_{i}+d)^{*}\\}}{|cx_{i}+d|^{2}+|ax_{i}+b|^{2}}\right|^{2}$
$\displaystyle\qquad+\left|\sum_{i}m_{i}\frac{|cx_{i}+d|^{2}-|ax_{i}+b|^{2}}{|cx_{i}+d|^{2}+|ax_{i}+b|^{2}}\right|^{2},$
(63)

where $a=\sqrt{\frac{1}{1-\alpha}}e^{-I\frac{\Phi}{2}}\cos\frac{\Theta}{2}$,
$b=-\sqrt{\frac{1}{1-\alpha}}e^{I\frac{\Phi}{2}}\sin\frac{\Theta}{2}$,
$c=\sqrt{1-\alpha}e^{-I\frac{\Phi}{2}}\sin\frac{\Theta}{2}$, and
$d=\sqrt{1-\alpha}e^{I\frac{\Phi}{2}}\cos\frac{\Theta}{2}$. Let us remark
here, that $a,b,c,d$ are actually the matrix elements of the corresponding
operator $A$ as in Section 4.2 (which were denoted by $\alpha$, $\beta$,
$\gamma$, and $\delta$, there). Note that $f$ is built by concatenation,
addition, and multiplication of elementary functions that are all continuous,
as well as rational functions, which are continuous except at points where the
denominator is 0. However, it can be easily verified that none of the
denominators appearing in Eq. (63) can attain 0. Thus, $f$ is continuous on
$D$. Let us remark here, however, that $f$ is not uniformly continuous.

Proof of statement (ii). Let us first show the statement for $x=(0,0,0)$,
which corresponds to $A=\mathbbm{1}$. Consider
$x^{\prime}=(\frac{\epsilon}{1+\epsilon},\Theta,\Phi)$, where $\epsilon>0$,
$\Theta=\arccos\sum_{i}m_{i}\cos\theta_{i}$,
$\Phi=\arcsin\frac{\sum_{i}m_{i}\sin\phi_{i}\sin\theta_{i}}{\sin\Theta}$.
$x^{\prime}$ corresponds to first rotating the vectors $\vec{v_{i}}$ such that
the with multiplicities weighted vector sum of $\vec{v}_{i}$ points into
direction $(0,0,1)^{T}$ followed by an $\epsilon$-hinging transformation.
Taylor expansion of $f$ around $\epsilon=0$ shows

$\displaystyle f(\frac{\epsilon}{1+\epsilon},\Theta,\Phi)$
$\displaystyle=f(0,\Theta,\Phi)$
$\displaystyle\qquad-2\epsilon\left(\sum_{i}m_{i}\cos\tilde{\theta}_{i}\right)\left(\sum_{i}m_{i}\sin^{2}\tilde{\theta}_{i}\right)$
$\displaystyle\qquad+\mathcal{O}(\epsilon^{2}),$ (64)

where by $\tilde{\theta}_{i}$ we denote the spherical coordinates of the
vectors after the rotation. It is clear that there exists an $\epsilon>0$ such
that $f(x^{\prime})<f(x)$, unless either
$\sum_{i}m_{i}\cos\tilde{\theta}_{i}=0$, or
$\sum_{i}m_{i}\sin^{2}\tilde{\theta}_{i}=0$. In case
$\sum_{i}m_{i}\cos\tilde{\theta}_{i}=0$ the vectors already sum to 0, i.e.,
$f(x)=0$. In the latter case, all vectors are pointing either in direction
$(0,0,1)^{T}$ or $(0,0,-1)^{T}$. This cannot be the case as by assumption the