# Fully non-positive-partial-transpose genuinely entangled subspaces

Owidiusz Makuta  Center for Theoretical Physics, Polish Academy of Sciences,
Aleja Lotników 32/46, 02-668 Warsaw, Poland  Błażej Kuzaka  Center for
Theoretical Physics, Polish Academy of Sciences, Aleja Lotników 32/46, 02-668
Warsaw, Poland  Remigiusz Augusiak  Center for Theoretical Physics, Polish
Academy of Sciences, Aleja Lotników 32/46, 02-668 Warsaw, Poland

###### Abstract

Genuinely entangled subspaces are a class of subspaces in the multipartite
Hilbert spaces that are composed of only genuinely entangled states. They are
thus an interesting object of study in the context of multipartite
entanglement. Here we provide a construction of multipartite subspaces that
are not only genuinely entangled but also fully non-positive-partial-transpose
(NPT) in the sense that any mixed state supported on them has non-positive
partial transpose across any bipartition. Our construction originates from the
stabilizer formalism known for its use in quantum error correction. To this
end, we first introduce a couple of criteria allowing to assess whether any
state from a given non-trivial stabilizer subspace is genuinely multipartite
entangled. We then use these criteria to construct genuinely entangled
stabilizer subspaces for any number of parties and prime local dimension and
conjecture them to be of maximal dimension achievable within the stabilizer
formalism. At the same time, we prove that every genuinely entangled subspace
is fully NPT in the above sense, which implies a quite surprising fact that no
genuinely entangled stabilizer subspace can support PPT entangled states.

##  1 Introduction

Entanglement is one of the most exhilarating features of quantum systems [1].
Its almost magic-like properties defy our real-life intuitions and show that
the laws of the physical at a small scale cannot be reliably modelled with
methods of classical physics. Throughout the years, many have tried, and
succeeded, to use entanglement as a resource for seemingly infinite number of
interesting applications that often have no classical analogue, such as the
well-known quantum teleportation [2] or quantum cryptography [3]. Entanglement
is also vital for quantum steering [4] or Bell nonlocality [5] which represent
other forms of quantum correlations that have been turned into independent
resources themselves (see, e.g., [6, 7]). In particular, Bell nonlocality lies
at the heart of quantum information in the device-independent version [8]. It
is thus not surprising that until now entanglement theory has been a very
lively field of research.

Obviously, in order to study entanglement, one has to identify which states
are entangled in the first place. The separability problem, that is, the
problem of deciding whether a given, in general mixed, quantum state is
entangled, has been thus intensively studied over years [9]. At first, this
problem was considered mostly for bipartite quantum states [10], but it
quickly gained on importance in the many-body regime. This was driven by the
development of experimental techniques that allow one to prepare and control
interesting many-body states (see, e.g., Refs. [11, 12, 13]) but also by the
identification of certain applications of multipartite entanglement such as
quantum metrology [14] or quantum computing [15]. Entanglement turned also
useful in the study of many-body phenomena [16].

Among many forms of entanglement featured by the multipartite scenario, the
strongest and at the same time most valuable from the application point of
view, is arguably the genuine multipartite entanglement [17, 18, 19]. Roughly
speaking, a genuinely entangled multipartite quantum state is one that cannot
be represented as a convex mixture of other states that are separable with
respect to some bipartitions. A significant amount of attention has been
devoted in the literature to characterise the properties of genuine
entanglement and to provide methods of its detection [18, 20, 21, 22, 23].
Yet, the problem is tremendously difficult, certainly more difficult than in
the bipartite case, and our understanding of genuine entanglement still
remains incomplete.

Our aim in this work is to join the effort of characterisation of genuine
entanglement in multipartite systems, taking however, a slightly different
perspective. Instead of focusing on particular multipartite quantum states (in
general mixed) we rather consider entangled subspaces of multipartite Hilbert
spaces. In this way we gain generality: a statement made for a subspace
applies to any mixed state defined on it. While genuinely entangled subspaces
have recently been an object of intensive exploration [24, 25, 26, 27, 28,
29], there are actually not many schemes allowing to judge whether a given
subspace (and thus all states acting on it) is genuinely entangled. Here we
will address this problem, concentrating on a particular class of multipartite
subspaces that originate from the multiqudit stabilizer formalism [30]. The
latter, being known for its use in quantum error correction [31, 32, 33, 34],
provides an easy-to-handle and convenient description of a certain class of
quantum states.

We first provide a simple necessary and sufficient criterion allowing to
decide whether a given stabilizer subspace in an $N$-qudit Hilbert space is
genuinely entangled. Importantly, our criterion reduces to checking certain
commutation relations for operators generating a given stabilizer subspace
which can be done in a finite number of steps, thus allowing us to avoid the
tedious task of assessing whether every state belonging to that subspace is
genuinely entangled. We then generalize the vector formalism introduced for
the multiqubit case in Ref. [35] that provides an efficient description of
entanglement properties of stabilizer subspaces. This formalism is also
particularly useful in constructing stabilizer subspaces that are genuinely
entangled, and thus we employ it to provide a family of such subspaces which
we believe to have the maximal dimension achievable within the stabilizer
formalism. At the same time, we explore the use of partial transposition in
deciding whether stabilizer subspaces are genuinely entangled and show that
any mixed state supported on such a subspace must be NPT with respect to any
bipartition. We thus provide a family of (possibly) maximally-dimensional
genuinely entangled subspaces in $N$-qudit Hilbert spaces that are fully NPT,
thus extending the results of Ref. [36] to the multipartite regime.

##  2 Preliminaries

This section serves as an introduction of commonly used terms within the field
of quantum information. Readers already familiar with concepts of genuine
entanglement, partial transpose and stabilizer formalism are advised to skip
ahead to Section 3.

(1) Genuine entanglement. Let us consider a scenario where $N$ parties share a
pure state $|\psi\rangle\in\mathcal{H}$. We can decompose the Hilbert space
$\mathcal{H}$ in the following manner

$\mathcal{H}=\bigotimes_{i=1}^{N}\mathcal{H}_{i},$ (1)

where $\mathcal{H}_{i}$ is a Hilbert subspace associated with $i$’th party.
Let us then consider a partition of the set of parties
$I_{N}:=\\{1,\ldots,N\\}$ into two disjoint subsets $Q$ and $\overline{Q}$ and
call it a bipartition of $I_{N}$. We say that $|\psi\rangle$ is separable with
respect the bipartition $Q|\overline{Q}$ if

$|\psi\rangle=|\psi\rangle_{Q}\otimes|\psi\rangle_{\overline{Q}},$ (2)

for some $|\psi\rangle_{Q}\in\mathcal{H}^{\prime}=\bigotimes_{i\in
Q}\mathcal{H}_{i}$. Otherwise, we call it entangled with respect to the
bipartition $Q|\overline{Q}$. Then, we call $|\psi\rangle$ genuinely
multipartite entangled iff it is entangled with respect to any nontrivial
bipartition of the set $I_{N}$ [9]. Of course, no state could genuinely
entangled if we were to consider trivial bipartitions, i.e., ones for which
$Q=\emptyset$ or $Q=\\{1,\dots,N\\}$. Thus, whenever we say "all bipartitions"
we only mean the nontrivial ones.

One can also introduce the above notions in the mixed-state case. Precisely,
we say that a density matrix $\rho$ is entangled with respect to a given
bipartition $Q|\overline{Q}$ if it cannot be represented as a convex
combination of pure states that are separable across this bipartition. Then,
$\rho$ is called genuinely multipartite entangled if it cannot be represented
as a convex combination of pure state in which each pure state is separable
across possibly different bipartitions.

In an analogous manner we can also extend these definitions to entire
subspaces of the joint Hilbert space $\mathcal{H}$: a subspace is entangled
with respect to some bipartition or it is genuinely entangled if every state
from that subspace is of the corresponding type [24]. In particular, a
genuinely entangled subspace of $\mathcal{H}$ is one that consists of only
genuinely entangled pure states.

(2) Positive partial transpose states. Consider a density matrix $\rho$ acting
on $\mathcal{H}$. We call it a non-positive-transpose (NPT) state with respect
to a bipartition $Q|\overline{Q}$ if it its partial transposition with respect
to $Q|\overline{Q}$ is not a positive semi-definite matrix,

$\rho^{T_{Q}}\ngeqslant 0,$ (3)

where $T_{Q}$ denotes a transposition over subsystems from $Q$ defined as

$\rho^{T_{Q}}=\left(\bigotimes_{i\in
Q}T^{(i)}\otimes\bigotimes_{j\in\overline{Q}}I^{(j)}\right)[\rho],$ (4)

where $T^{(i)}$ is the transposition in the standard basis performed on the
$i$th subsystem, whereas $I^{(j)}$ is the identity on the subsystem $j$. In
other words, partial transpose is a transpose operation carried out over only
some parties. As an example, let us consider a matrix

$C=\sum_{i,j,k,l}\gamma_{i,j,k,l}|i\rangle_{1}\langle
j|\otimes|k\rangle_{2}\langle l|,$ (5)

where $1,2$ indice indicate the party. Then the partial transpose over the
first party of this matrix is defined as follows

$C^{T_{1}}=\sum_{i,j,k,l}\gamma_{i,j,k,l}|j\rangle_{1}\langle
i|\otimes|k\rangle_{2}\langle l|.$ (6)

Let us finally define fully NPT states to be those for which (3) holds true
for any bipartition $Q|\overline{Q}$ We can also extend the above definitions
of to subspaces of $\mathcal{H}$. Precisely, a subspace $V\subset\mathcal{H}$
is termed NPT with respect to a bipartition $Q|\overline{Q}$ if every mixed
state supported on it is NPT with respect to $Q|\overline{Q}$. Analogously,
$V$ is called fully NPT if any mixed state supported on it is fully NPT.

(3) Qudit stabilizer formalism. In this work we concentrate on a class of
quantum states (pure or mixed) that originate from the multiqudit stabilizer
formalism. Let us assume that $N$ parties share a pure state
$|\psi\rangle\in\mathcal{H}$. The decomposition of Hilbert space (1) also
holds true in this scenario, however in this case we additionally assume that
for every $i\in\\{1,\dots,N\\}$ we have $\mathcal{H}_{i}=\mathbb{C}^{d}$,
where $d$ is called a dimension of a local Hilbert space, or in other words
each party holds one qudit.

The most important objects in the qudit stabilizer formalism are the
generalised Pauli operators defined as

$X=\sum_{i=0}^{d-1}|i+1\rangle\\!\langle i|,\qquad
Z=\sum_{i=0}^{d-1}\omega^{i}|i\rangle\\!\langle i|,$ (7)

where $\omega=\operatorname{exp}(2\pi\mathbb{i}/d)$ is a primitive $d$th root
of unity, $\mathbb{i}$ denotes the imaginary unit, and the addition is modulo
$d$, meaning that $|d\rangle\equiv|0\rangle$. With a bit of calculation one
can verify that these matrices enjoy the following properties:

$X^{\dagger}X=\mathbb{1},\qquad Z^{\dagger}Z=\mathbb{1},$ (8)
$X^{d}=\mathbb{1},\qquad Z^{d}=\mathbb{1},$ (9)

meaning that they are unitary and their spectra are
$\\{1,\omega,\ldots,\omega^{d-1}\\}$. Moreover, they obey the following
commutation relations

$X^{i}Z^{j}=\omega^{-ij}Z^{j}X^{i}$ (10)

for any $i,j=0,\ldots,d-1$.

Let us now introduce the generalised Pauli group $\mathbbm{P}_{N,d}$ as a set
of all $N$-fold tensor products of matrices $\omega^{r}X^{n}Z^{m}$, where and
$r,n,m\in\\{0,\dots,d-1\\}$, equipped with a matrix multiplication as the
group operation.

Consider then a subgroup $\mathbb{S}\subset\mathbbm{P}_{N,d}$. It is called a
stabilizer if [37]:

  1. 1.

all elements of $\mathbb{S}$ mutually commute,

  2. 2.

given $a\in\mathbb{C}$, we have $a\mathbb{1}\in\mathbb{S}$ if and only if
$a=1$.

We say that a state $|\psi\rangle$ is stabilized by $\mathbb{S}$ if

$G|\psi\rangle=|\psi\rangle$ (11)

holds true for any $G\in\mathbb{S}$. Then, a stabilizer subspace $V$ is a
maximal subspace stabilized by a given stabilizer, i.e., a state
$|\psi\rangle$ is stabilized by $\mathbb{S}$ if and only if $|\psi\rangle\in
V$.

We can also extend the definition of stabilisation to the mixed states: we say
that a state described by a density matrix $\rho$ is stabilized by
$\mathbb{S}$, if for every operator $G\in\mathbb{S}$ we have

$G\rho=\rho G=\rho.$ (12)

For a more in depth discussion of a stabilizer formalism see [38, 39].

Given a stabilizer $\mathbb{S}$, it is often convenient to represent it in
terms of the smallest set of independent elements of $\mathbb{S}$, called
generators, using which one can represent any other element of $\mathbb{S}$,
where ’independent’ means that one generator cannot be represented as a
product of the other generators.

$G_{i}\neq G_{1}^{\alpha_{1}}\dots
G_{i-1}^{\alpha_{i-1}}G_{i+1}^{\alpha_{i+1}}\dots G_{k}^{\alpha_{k}}$ (13)

for $\alpha_{i}\in\\{0,\dots,d-1\\}$. Of course from the definition of a
stabilizer it follows that:

$[G_{i},G_{j}]=0\qquad(i\neq j).$ (14)

For prime $d$, this representation of a stabilizer allows us to easily
determine the dimension of the corresponding stabilizer subspace $V$, namely
$\dim V=d^{N-k}$; for non-prime $d$ the situation is slightly more complicated
(see Ref. [38]).

##  3 Genuinely entangled subspaces in a qudit stabilizer formalism

In this section we aim to characterise genuine entanglement in a qudit
stabilizer formalism and find a genuinely entangled subspace that is as large
as possible. To this end, we generalise the results from [35] that concern
genuine entanglement in a stabilizer formalism for a local Hilbert space of
dimension $d=2$.

Let us consider a nontrivial bipartition $Q|\overline{Q}$ of $\\{1,\dots,N\\}$
and a stabilizer $\mathbb{S}=\langle G_{1},\dots,G_{k}\rangle$ with a
corresponding stabilizer subspace $V\subset\mathcal{H}=\mathbb{C}_{d}^{\otimes
N}$. Each generator $G_{i}$ can be decomposed with respect to the above
bipartition in the following way

$G_{i}=G_{i}^{(Q)}\otimes G_{i}^{(\overline{Q})},$ (15)

where $G_{i}^{(Q)}$, $G_{i}^{(\overline{Q})}$ act on $\bigotimes_{i\in
Q}\mathcal{H}_{i}$, $\bigotimes_{i\in\overline{Q}}\mathcal{H}_{i}$,
respectively. With this we can formulate the following theorem.

######  Theorem 1.

Given a stabilizer $\mathbb{S}=\langle G_{1},\dots,G_{k}\rangle$ of an
arbitrary local dimension, the corresponding stabilizer subspace
$V\neq\\{0\\}$ is entangled with respect to a bipartition $Q|\overline{Q}$ iff
there exists a pair $i,j\in\\{1,\dots,k\\}$ for which the following holds
true:

$\left[G_{i}^{(Q)},G_{j}^{(Q)}\right]\neq 0.$ (16)

###### Proof.

We will begin by proving the "$\Rightarrow$" implication by contradiction. Let
us assume that $V$ is entangled with respect to a bipartition $Q|\overline{Q}$
and that for all $i,j\in\\{1,\dots,N\\}$ we have

$\left[G_{i}^{(Q)},G_{j}^{(Q)}\right]=0.$ (17)

Since all generators commute, this implies that

$\left[G_{i}^{(\overline{Q})},G_{j}^{(\overline{Q})}\right]=0$ (18)

for all $i,j\in\\{1,\dots,N\\}$.

Consequently, there exists a common eigenbasis for all $G_{i}^{(Q)}$ and
another common eigenbasis for all $G_{i}^{(\overline{Q})}$:

$G_{i}^{(Q)}|\phi_{j}\rangle=\lambda_{j}^{(i)}|\phi_{j}\rangle\qquad\textrm{and}\qquad
G_{i}^{(\overline{Q})}|\theta_{j}\rangle=\overline{\lambda}_{j}^{(i)}|\theta_{j}\rangle$
(19)

for $\lambda_{j}^{(i)},\overline{\lambda}_{j}^{(i)}\in\mathbb{C}$. Using this
fact, we can construct a basis of the joint Hilbert space $\mathcal{H}$ by
taking the tensor product of elements of both bases, that is,

$|\varphi_{ij}\rangle=|\phi_{i}\rangle\otimes|\theta_{j}\rangle.$ (20)

Importantly, it follows from Eqs. (15) and (19) that this is a common
eigenbasis of generators $G_{i}$, hence some $|\varphi_{ij}\rangle$ correspond
to the eigenvalue one of all $G_{i}$, or $V$ is empty. The later contradicts
the assumption that $V$ is non-empty, while the former implies that
$|\varphi_{ij}\rangle$ belongs to $V$. Since $|\varphi_{ij}\rangle$ is clearly
separable with respect to the bipartition $Q|\overline{Q}$, this contradicts
the assumption that $V$ is entangled across this bipartition.

From the definition of Pauli group $\mathbb{P}_{N,d}$, $G_{i}$ is a tensor
product of matrices $\omega^{r}X^{n}Z^{m}$ for $r,n,m\in\\{0,\dots,d-1\\}$.

Let us now move on to the "$\Leftarrow$" implication. Let us assume that $V$
is separable with respect to a bipartition $Q|\overline{Q}$ but there exists a
pair of generators $G_{i},G_{j}$ for which (16) holds true. This means that
there exists $|\psi\rangle\in V$ such that for the bipartition
$Q|\overline{Q}$ we have

$|\psi\rangle=|\psi\rangle_{Q}\otimes|\psi\rangle_{\overline{Q}}.$ (21)

This together with (15) implies that the following holds true

$G_{i}^{(Q)}|\psi\rangle_{Q}=e^{\mathbb{i}\phi_{i}}|\psi\rangle_{Q}$ (22)

for all $i$, where $\phi_{i}\in\mathbb{R}$. This formula allows us to write

$\left\\{G_{i}^{(Q)},G_{j}^{(Q)}\right\\}|\psi\rangle_{Q}=2e^{\mathbb{i}(\phi_{i}+\phi_{j})}|\psi\rangle_{Q},$
(23)

where $\\{\cdot,\cdot\\}$ is an anticommutator. On the other hand, from the
definition of Pauli group $\mathbb{P}_{N,d}$, $G_{i}$ is a tensor product of
matrices $\omega^{r}X^{n}Z^{m}$ for $r,n,m\in\\{0,\dots,d-1\\}$, which implies

$\displaystyle\left\\{G_{i}^{(Q)},G_{j}^{(Q)}\right\\}|\psi\rangle_{Q}$
$\displaystyle=$
$\displaystyle\left(1+\omega^{\tau_{i,j}^{(Q)}}\right)G_{j}^{(Q)}G_{i}^{(Q)}|\psi\rangle_{Q}$
$\displaystyle=$
$\displaystyle\left(1+\omega^{\tau_{i,j}^{(Q)}}\right)\mathrm{e}^{\mathbb{i}(\phi_{i}+\phi_{j})}|\psi\rangle_{Q},$

where the value of $\tau_{i,j}^{(Q)}$ depends on the exact form of
$G_{i}^{(Q)}$ and $G_{j}^{(Q)}$. After comparing Eqs. (23) and (3), one
arrives at

$1+\omega^{\tau_{i,j}^{(Q)}}=2,$ (25)

which is fulfilled only if $\tau_{i,j}^{(Q)}=0$, which is to say that for the
bipartition $Q|\overline{Q}$, for every $i,j\in\\{1,\dots,k\\}$ we have

$\left[G_{i}^{(Q)},G_{j}^{(Q)}\right]=0,$ (26)

which contradicts the assumption. ∎

This theorem provides an alternative technique of determining whether a given
subspace originating from the qudit stabilizer formalism is entangled. This
requires far less effort than checking the same property on the explicit form
of the stabilizer subspace, in particular for every pure state belonging to
it. Moreover, by applying Theorem 1 to all nontrivial bipartitions we arrive
at the following corollary.

######  Corollary 1.

Given a stabilizer $\mathbb{S}=\langle G_{1},\dots,G_{k}\rangle$ of an
arbitrary local dimension, the corresponding stabilizer subspace
$V\neq\\{0\\}$ is genuinely entangled iff for each bipartition
$Q|\overline{Q}$ there exists a pair $i,j\in\\{1,\dots,k\\}$ for which the
following holds true:

$\left[G_{i}^{(Q)},G_{j}^{(Q)}\right]\neq 0.$ (27)

Let us notice that the above fact is a direct generalisation of Theorem 1 from
Ref. [35] and it allows us to check whether a stabilizer subspace is genuinely
entangled just by examining generators of the corresponding stabilizer.

Having the above description of genuine entanglement within the qudit
stabilizer formalism at hand, we now aim to find a genuinely entangled
stabilizer subspace of the largest possible dimension. Luckily, we can once
again refer to [35] in which the maximal dimension of a genuinely entangled,
stabilizer subspace for a case of $d=2$ was found. It has been achieved via a
certain novel vector formalism, and so our first step in finding the
aforementioned largest subspace is to generalise this formalism to arbitrary
$d$.

In the remainder of this section we assume that $d$ is a prime number. Let us
consider the set $\mathbb{Z}_{d}=\\{0,\dots,d-1\\}$ equipped with addition and
multiplication modulo $d$. We denote by $F_{N,d}=\mathbb{Z}_{d}^{N}$ a vector
space over $\mathbb{Z}_{d}$, i.e., every $f\in F_{N,d}$ is a $N$-dimensional
vector whose entries are from $\\{0,\dots,d-1\\}$.

We can take advantage of this vector space to find a representation of a
stabilizer $\mathbb{S}=\langle G_{1},\dots,G_{k}\rangle$ that gives us a
clearer view on the entanglement of the subspace $V$. Let us consider a vector

$v_{i,j}=\sum_{n=1}^{N}\tau_{i,j}^{(n)}e_{n},$ (28)

where $e_{n}$ is a unit vector that has an entry $1$ on the $n$’th side and
$0$ elsewhere, and $\tau_{i,j}^{(n)}\in\\{0,\dots,d-1\\}$ is defined as
follows

$G_{i}^{(n)}G_{j}^{(n)}=\omega^{\tau_{i,j}^{(n)}}G_{j}^{(n)}G_{i}^{(n)}.$ (29)

For the definition of a stabilizer, all generators $G_{1},\dots,G_{k}$ have to
mutually commute, which implies that for all $i$, $j$ we have

$\sum_{n=1}^{N}v_{i,j}^{(n)}=0,$ (30)

where $v_{i,j}^{(n)}$ is an $n$’th entry of $v_{i,j}$. Let us denote by
$K(\mathbb{S})$ a subspace of $F_{N,d}$ that is spanned by all the vectors
$v_{i,j}$ (28) corresponding to the stabilizer $\mathbb{S}$. It is not
difficult to see that for any stabilizer $\mathbb{S}$, $K(\mathbb{S})$ is a
proper subspace of $F_{N,d}$: there exist vectors in $F_{N,d}$ that do not
satisfy the condition (30), and, at the same time, any linear combination of
vectors that fulfil (30) fulfils (30) too. In fact, the condition (30) implies
that the maximal dimension of $K(\mathbb{S})$ is $N-1$.

To make full use of the above formalism we need to reformulate Corollary 1 in
terms of vectors $v_{i,j}$. To this end, we need to define a representation of
a bipartition of the set $\\{1,\dots,N\\}$, and so let us consider a vector
$\phi\in F_{N,2}$ (notice that this is $F_{N,2}$ and not $F_{N,d}$). We call
$\phi$ a representation of a bipartition $Q|\overline{Q}$ if the following
holds true

$\phi^{(n)}=1\Longleftrightarrow n\in Q.$ (31)

Notably, this is not an isomorphism, since every $Q|\overline{Q}$ has two
equivalent representation: one where $\phi^{(n)}=1$ for $n\in Q$ and the other
one for $n\in\overline{Q}$.

This representation also includes the trivial bipartitions, and since we
usually want to exclude them, let us denote representations of the trivial
bipartition as

$\phi_{T_{0}}=\sum_{n=1}^{N}0e_{n},\qquad\phi_{T_{1}}=\sum_{n=1}^{N}e_{n},$
(32)

where $T_{0}$ and $T_{1}$ correspond to the cases $Q=\emptyset$ and
$\overline{Q}=\emptyset$, respectively.

The last ingredient needed to rephrase Corollary 1 in this vector formalism is
a function $h(\cdot,\cdot):$ $F_{N,d}\times F_{N,2}\rightarrow\mathbb{Z}_{d}$:

$h(v,\phi)=\sum_{n=1}^{N}v^{(n)}\phi^{(n)}.$ (33)

This function resembles a scalar product in the sense that it is calculated
similarly, however $h(\cdot,\cdot)$ does not meet the necessary conditions
that a scalar product has to obey.

We are finally ready to reformulate Corollary 1 in terms of vectors from
$K(\mathbb{S})$.

######  Lemma 1.

Let the local dimension $d$ be a prime number. Given a stabilizer
$\mathbb{S}=\langle G_{1},\dots,G_{k}\rangle$, the corresponding stabilizer
subspace $V\neq\\{0\\}$ is genuinely entangled iff for every $\phi\in F_{N,2}$
such that $\phi\neq\phi_{T_{0}},\phi_{T_{1}}$ there exists $v_{i,j}\in
K(\mathbb{S})$ for $i\neq j$ such that

$h(v_{i,j},\phi)\neq 0.$ (34)

The proof is very similar to the proof of Theorem 1 and so we do not present
it here. However, for completeness we provide it in Appendix A.

The above lemma gives us a clearer picture of the relations that generators
$G_{i}$ have to fulfil in order for a stabilizer subspace to be genuinely
entangled. But notably in Ref. [35] this lemma only serves as a stepping stone
to a proof of the following theorem.

######  Theorem 2 (for $\boldsymbol{d=2}$).

A subspace $V$ stabilized by $\mathbb{S}$ is genuinely entangled iff $\dim
K(\mathbb{S})=N-1$.

This theorem provides a more convenient criterion than Corollary 1 for
determining if a stabilizer subspace is genuinely entangled. In fact, checking
whether a stabilizer subspace generated by a stabilizer $\mathbb{S}$ is
genuinely entangled boils down to computing the dimension of the corresponding
subspace $K(\mathbb{S})$ which can easily be implemented in standard
mathematical packages. However, the above theorem does not directly generalize
to the case of arbitrary prime $d>2$. More precisely, while the "$\Leftarrow$"
implication remains true, the other one does not hold anymore because there
exist genuinely entangled subspaces for which the dimension of the
corresponding subspaces $K(\mathbb{S})$ is lower than $N-1$. In what follows
we provide an example of such a subspace. We also prove the sufficient
condition for a stabilizer subspace to be genuinely entangled for any $d$,
generalizing the "if part" of Theorem 2 to stabilizer subspace of arbitrary
local dimension $d$.

To demonstrate that Theorem 2 cannot be directly generalized to an arbitrary
$d$ let us consider the following example of stabilizers for $d=3$ and $N=3$:

$G_{1}=X\otimes X\otimes X,\quad G_{2}=Z\otimes Z\otimes Z,$ (35)
$G^{\prime}_{1}=X\otimes Z\otimes\mathbb{1},\quad G^{\prime}_{2}=Z\otimes
X\otimes\mathbb{1}.$ (36)

It is easy to see that the corresponding stabilizers $\mathbb{S}=\langle
G_{1},G_{2}\rangle$ and $\mathbb{S}^{\prime}=\langle
G^{\prime}_{1},G^{\prime}_{2}\rangle$ stabilise three-dimensional subspaces in
$\mathbb{C}_{3}^{\otimes 3}$, which we denote respectively by $V$ and
$V^{\prime}$. Using Corollary 1 we can show that $V$ is genuinely entangled
whereas $V^{\prime}$ is not. At the same time, $\dim K(\mathbb{S})=\dim
K(\mathbb{S}^{\prime})=1$. This is thus certainly a nail in the coffin for the
possibility of a direct generalisation of Theorem 2 to higher dimensional
cases because it is clear that $\dim K(\mathbb{S})$ alone does not determine
in general whether $V$ is genuinely entangled as it does for $d=2$.

While it clearly follows from the above considerations that it is in general
not possible to formulate an iff condition for a stabilizer subspace to be
genuinely entangled based solely on the dimension of $K(\mathbb{S})$ (as it is
the case for $d=2$), we believe that the latter can still be somewhat useful
in assessing genuine entanglement of stabilizer subspaces. While the latter
condition implies in particular that $G_{i}^{d}=\mathbbm{1}$ for every
generator, Actually, we conjecture that the following necessary condition
holds true.

######  Conjecture 1.

Given a stabilizer $\mathbb{S}=\langle G_{1},\dots,G_{k}\rangle$, if the local
dimension $d$ is a prime number and the corresponding stabilizer subspace $V$
is genuinely entangled, then

$\dim K(\mathbb{S})\geqslant\left\lceil\frac{N-1}{d-1}\right\rceil.$ (37)

For the particular case of $k=2$ and any $N$ and $d$ the above statement can
actually be rigorously proven. Precisely, the following theorem holds true.

######  Theorem 3.

Consider a stabilizer $\mathbb{S}=\langle G_{1},G_{2}\rangle$. If the local
dimension $d$ is a prime number and the corresponding subspace $V$ generated
by $\mathbb{S}$ is genuinely entangled, then

$\dim K(\mathbb{S})\geqslant\left\lceil\frac{N-1}{d-1}\right\rceil.$ (38)

###### Proof.

The proof is quite technical and therefore it is deferred to Appendix B. ∎

Furthermore, in Appendix C we construct a stabilizer for which $V$ is
genuinely entangled and $\dim K(\mathbb{S})=\lceil(N-1)/(d-1)\rceil$; we
denote this stabilizer $\mathbb{S}_{\max}$, where we added the subscript
’$\max$’ for reasons that will become clear later. This implies that if the
inequality (37) is in general false, the true function $p(N,d)$ that bounds
$\dim K(\mathbb{S})$ from below, that is, one for which $\dim
K(\mathbb{S})\geqslant p(N,d)$ for any genuinely entangled subspace, must
satisfy $p(N,d)\leqslant\lceil(N-1)/(d-1)\rceil$ for any $N$ and $d$.

It is worth pointing out the vector formalism introduced above turned out to
be particularly useful in constructing the stabilizer $\mathbb{S}_{\max}$
presented in Appendix C. Precisely, the generators of this stabilizer are
chosen so that the corresponding vectors $v_{i,j}$ give us the following basis
in $K(\mathbb{S})$:

$u_{i}=\left\\{\begin{array}[]{ll}\displaystyle\sum_{j=(i-1)(d-1)+1}^{i(d-1)+1}e_{j}&\textrm{
for\;}i<\displaystyle\left\lceil\frac{N-1}{d-1}\right\rceil,\\\\[21.52771pt]
\displaystyle\sum_{j=(i-1)(d-1)+1}^{N-1}e_{j}+ae_{N}&\textrm{
for\;}i=\displaystyle\left\lceil\frac{N-1}{d-1}\right\rceil,\end{array}\right.$
(39)

where $a=\lceil(N-1)/(d-1)\rceil(d-1)-(N-1)+1$. For example, for $d=3$ and
$N=5$ our construction gives

$u_{1}=(1,1,1,0,0),\qquad u_{2}=(0,0,1,1,1).$ (40)

To complete our characterization of genuine entanglement within the stabilizer
formalism let us finally demonstrate that the ’$\Leftarrow$’ implication of
Theorem 2 generalizes to any prime $d$. To this end let us first prove the
following lemma.

######  Lemma 2.

Assume $d$ to be prime and consider a stabilizer for which $\dim
K(\mathbb{S})=N-1$. Then, a vector $v\in F_{N,d}$ fulfils

$\sum_{i=1}^{N}v^{(i)}=0,$ (41)

if and only if $v\in K(\mathbb{S})$.

The proof can be found in Appendix A. With this lemma at hand we can prove the
aforementioned implication.

######  Theorem 4.

Given a prime $d$ and a stabilizer $\mathbb{S}=\langle
G_{1},\dots,G_{k}\rangle$, if $\dim K(\mathbb{S})=N-1$ then the corresponding
stabilizer subspace $V$ is genuinely entangled.

###### Proof.

To prove that the stabilizer subspace in genuinely entangled we use the
criterion from Lemma 1. Let us consider a representation $\phi$ of nontrivial
bipartition $Q|\overline{Q}$. Without any loss of generality we can assume
that $n_{0}\in\overline{Q}$ and $n_{1}\in Q$. Consider then a vector
$v=(d-1)e_{n_{0}}+e_{n_{1}}$ for which we have $h(v,\phi)\neq 0$. Moreover,
this vector clearly satisfies the condition (41) which together with the fact
that $\dim K(\mathbb{S})=N-1$ allows one to deduce via Lemma 2 that $v\in
K(\mathbb{S})$. Now, such a vector can be found for every nontrivial $\phi$
and every such vector is in $K(\mathbb{S})$, therefore by the virtue of Lemma
1 the stabilizer subspace $V$ is genuinely entangled. ∎

Let us conclude this section by wrapping up the main statements made in this
section and picturing how the fact whether $V$ is genuinely entangled or not
depends on the dimension of $K(\mathbb{S})$. Assuming that $d$ is prime, we
can divide the possible values of $\dim K(\mathbb{S})$ into three regions:

  * (i)

for $\dim K(\mathbb{S})=N-1$ the corresponding stabilizer subspace is
genuinely entangled,

  * (ii)

for $\dim K(\mathbb{S})<N-1$ but larger or equal to some number $p(N,d)$,
which we suspect to be $\lceil(N-1)/(d-1)\rceil$, we are unable to determine
whether $V$ is genuinely entangled or not just by looking at the dimension of
$K(\mathbb{S})$,

  * (iii)

for $\dim K(\mathbb{S})$ smaller that $p(N,d)$, $V$ cannot be genuinely
entangled. However, since we were not able to prove Conjecture 1, the actual
value of $p(N,d)$ remains unknown.

###  3.1 The maximal dimension of genuinely entangled stabilizer subspaces

Interestingly, the stabilizer $\mathbb{S}_{\max}$ introduced in Appendix C and
discussed above stabilises a genuinely entangled subspace in
$\mathbb{C}_{d}^{\otimes N}$ whose dimension seems to be the maximal
achievable within the stabilizer formalism for prime $d$. While the maximal
dimension of a genuinely entangled stabilizer subspace for any $N$ and $d$ is
unknown, we can speculate on it based on Conjecture 1. Indeed, we can
formulate another conjecture towards this goal.

######  Conjecture 2.

If $d$ is prime then the maximal dimension of genuinely entangled, stabilizer
subspace is given by $d^{N-k_{\min}(N)}$, where

$k_{\min}(N,d)=\left\lceil\frac{1+\sqrt{1+8\lceil(N-1)/(d-1)\rceil}}{2}\right\rceil.$
(42)

Let us now provide a proof that Conjecture 1 implies Conjecture 2.

###### Proof.

Assume that Conjecture 1 is true. We first use the fact, proven in Ref. [38],
that for a given stabilizer $\mathbb{S}=\langle G_{1},\dots,G_{k}\rangle$, the
dimension of $V$ equals $d^{N-k}$ provided that $d$ is prime. Let us then
follow the argumentation of the proof of Theorem 3 in Ref. [35]. Namely,
$k_{\min}(N,d)$ in Eq. (42) is the smallest positive integer that fulfils the
following inequality

$\frac{1}{2}k(k-1)\geqslant\left\lceil\frac{N-1}{d-1}\right\rceil$ (43)

which relates the lower bound on the dimension of $K(\mathbb{S})$ in (37) and
the number of vectors $v_{i,j}$ corresponding to the stabilizer
$\mathbb{S}=\langle G_{1},\dots,G_{k}\rangle$. Indeed, for a stabilizer
subspace $V$ to be genuinely entangled that number of vectors must at least
equal the minimal dimension of $K(\mathbb{S})$ in Conjecture 1. ∎

Since this conjecture is a direct consequence of Conjecture 1 it follows that
the stabilizer constructed in Appendix C gives rise to a genuinely entangled
subspace of the maximal dimension postulated in Conjecture 2. Thus, the
corresponding subspace $V_{\max}$ (see Appendix C) is the one that we were
looking for.

Let us also notice that if the true function $p(N,d)$ that bounds the
dimension of $K(\mathbb{S})$ from below in (37) turns out to be different than
$\lceil(N-1)/(d-1)\rceil$, the reasoning presented in the proof of Conjecture
2 remains valid, one only needs to replace the inequality (43) by
$k(k-1)/2\geqslant p(N,d)$ and modify accordingly $k_{\min}(N,d)$ in Eq. (42).

##  4 Fully NPT subspaces and genuine entanglement

Our aim in the last section of the paper is to explore the relation between
genuine multipartite entanglement in the stabilizer formalism and the
entanglement criterion based on partial transposition. We show that partial
transposition can be used to formulate an iff criterion for stabilizer
subspaces to be genuinely entangled. Precisely, a mixed state supported on a
stabilizer subspace is genuinely entangled if and only if all its partial
transpositions are non-positive. This implies a quite unexpected fact that
genuinely entangled, stabilizer subspaces, independently of how large they
are, do not support mixed states whose partial transpositions are all
positive. In order to achieve this goal we exploit the very simple approach
used in Ref. [40] to prove that no bipartite PPT entangled states violate the
Clauser-Horne-Shimony-Holt Bell inequality.

Before stating our results let us recall that we say that a stabilizer
subspace $V$ is entangled with respect to a bipartition $Q|\overline{Q}$ if
every pure state belonging to it (and thus every density matrix supported on
it) is entangled across $Q|\overline{Q}$. We then call $V$ a non-positive
partial transpose (NPT) subspace with respect to $Q|\overline{Q}$ if
$\rho^{T_{Q}}\ngeq 0$ for every mixed state $\rho$ supported on it (i.e., such
that $\mathrm{supp}(\rho)\subseteq V$).

######  Theorem 5.

Consider a stabilizer $\mathbb{S}=\langle G_{1},\dots,G_{k}\rangle$ of an
arbitrary local dimension and a bipartition $Q|\overline{Q}$. The stabilizer
subspace $V$ of $\mathbb{S}$ is entangled with respect to $Q|\overline{Q}$ iff
it is NPT with respect to that bipartition.

###### Proof.

The ’$\Leftarrow$’ implication simply follows from a well known fact that non-
positive partial transpose is a sufficient condition for entanglement [41],
and so we only need to prove the "$\Rightarrow$" implication. Let us assume
that $V$ is entangled with respect to the bipartition $Q|\overline{Q}$, but it
is not NPT. This implies that there exists a state $\rho$, where
$\operatorname{supp}(\rho)\subset V$, such that

$\rho^{T_{Q}}\geqslant 0.$ (44)

The fact that $V$ is entangled across $Q|\overline{Q}$ implies, by virtue of
Theorem 1, that there exists a pair of generators of $\mathbb{S}$, $G_{i}$ and
$G_{j}$, such that:

$G_{i}^{(Q)}G_{j}^{(Q)}=\omega^{\tau_{i,j}^{(Q)}}G_{j}^{(Q)}G_{i}^{(Q)},$ (45)

where $\tau_{i,j}^{(Q)}\neq 0$. Let us introduce an operator $\mathcal{B}$ as
a sum of these two generators $\mathbb{S}$ and their Hermitian conjugates:

$\mathcal{B}=G_{i}+G_{i}^{\dagger}+G_{j}+G_{j}^{\dagger}.$ (46)

Now, following the approach Ref. [40] we derive an upper bound on the
expectation value of the operator $\mathcal{B}$ for any state that obeys (44):

$\displaystyle[\operatorname{tr}(\mathcal{B}\rho)]^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}[\operatorname{tr}(\mathcal{B}\rho)]^{2}+\frac{1}{2}\left[\operatorname{tr}(\mathcal{B}^{T_{Q}}\rho^{T_{Q}})\right]^{2}$
(47) $\displaystyle\leqslant$
$\displaystyle\frac{1}{2}\operatorname{tr}(\mathcal{B}^{2}\rho)+\frac{1}{2}\operatorname{tr}(\mathcal{B}_{Q}^{2}\rho^{T_{Q}})$
$\displaystyle=$
$\displaystyle\frac{1}{2}\operatorname{tr}\left\\{\left[\mathcal{B}^{2}+\left(\mathcal{B}_{Q}^{2}\right)^{T_{Q}}\right]\rho\right\\},$

where $\mathcal{B}_{Q}=\mathcal{B}^{T_{Q}}$ and to obtain the first and the
third equations we have used the fact that under the trace one can add partial
transposition to both operators. Then, to obtain the inequality we exploited
the fact that the variance of a quantum observable is non-negative, together
with the fact that $\rho^{T_{Q}}\geqslant 0$. In order to determine the right-
hand side of (47) we use the explicit forms of the operators $\mathcal{B}^{2}$
and $\left(\mathcal{B}^{2}_{Q}\right)^{T_{Q}}$,

$\displaystyle\mathcal{B}^{2}$ $\displaystyle=G_{i}^{2}+G_{i}^{\dagger
2}+G_{j}^{2}+G_{j}^{\dagger 2}+4\mathbb{1}+2g_{i,j},$ (48)
$\displaystyle\left(\mathcal{B}^{2}_{Q}\right)^{T_{Q}}$
$\displaystyle=G_{i}^{2}+G_{i}^{2\dagger}+G_{j}^{2}+G_{j}^{2\dagger}+4\mathbb{1}+2\cos\left(\phi\right)g_{i,j},$

where $\phi=2\pi\tau_{i,j}^{(Q)}/d$ and

$g_{i,j}=G_{i}G_{j}+G_{i}G_{j}^{\dagger}+G_{i}^{\dagger}G_{j}+G_{i}^{\dagger}G_{j}^{\dagger},$
(50)

and so we have:

$\displaystyle[\operatorname{tr}(\mathcal{B}\rho)]^{2}\leqslant$
$\displaystyle\operatorname{tr}\Big{[}\rho\Big{(}G_{i}^{2}+G_{i}^{2\dagger}+G_{j}^{2}+G_{j}^{2\dagger}$
(51) $\displaystyle\hskip
28.45274pt+4\mathbb{1}+\left(1+\cos(\phi)\right)g_{i,j}\Big{)}\Big{]}.$

Then, taking into account the fact that

$G_{i}\rho=\rho G_{i}$ (52)

holds true for any $i$, the above simplifies to

$\displaystyle\operatorname{tr}(\mathcal{B}\rho)$ $\displaystyle\leqslant$
$\displaystyle 2\sqrt{3+\cos(\phi)},$ (53) $\displaystyle<$ $\displaystyle 4,$

where the last inequality is a consequence of the fact that
$\tau_{i,j}^{(Q)}\neq 0$, which means that $\cos(\phi)<1$. Thus, for any mixed
state stabilized on $V$ obeying (44), the expectation value of $\mathcal{B}$
is strictly smaller than four.

On the other hand, due to Eq. (52) every density matrix $\rho$ for which
$\operatorname{supp}(\rho)\subset V$ satisfies

$\operatorname{tr}(\mathcal{B}\rho)=4$ (54)

which clearly contradicts (53), completing the proof. ∎

This theorem shows that in the qudit stabilizer formalism subspaces in which
all states are entangled with respect to some bipartition cannot support
entangled states that are PPT with respect to that bipartition. Importantly,
based on it one can formulate the following iff criterion for a stabilizer
subspace to be genuinely entangled.

######  Corollary 2.

A stabilizer subspace $V$ is genuinely entangled iff it is fully NPT.

This statement follows from the fact that if $V$ is genuinely entangled then
it is entangled with respect to any bipartition. Then, application of Theorem
5 to every bipartition implies that $V$ is fully NPT. On the other hand, if
every pure state from $V$ is fully NPT, it must also be genuinely entangled
and thus $V$ is genuinely entangled itself.

This result provides us with a convenient way of constructing multipartite
subspaces that are fully NPT: by virtue of Corollary 2 any genuinely entangled
stabilizer subspace does the job. At the same time, it implies a quite
surprising fact that genuinely entangled stabilizer subspaces do not support
genuinely entangled states that are PPT with respect to any bipartition, which
makes us believe that there are actually no PPT genuinely entangled states in
the stabilizer formalism. Let us notice, on the other hand, that while Theorem
5 provides us a method for detecting entanglement in the stabilizer formalism,
the latter role is, however, better served by Theorem 1.

##  5 Conclusions

We have shown that all genuinely entangled stabilizer subspaces are fully NPT,
or, equivalently, that such subspaces do not support PPT entangled states. To
this end, we have introduced a criterion allowing one to judge in a finite
number of steps whether a given stabilizer subspace in an $N$-qudit Hilbert
space is genuinely entangled. We have also generalized the vector formalism
introduced in Ref. [35] in the case of qubit stabilizer formalism for prime
local dimension and used it to construct genuinely entangled stabilizer
subspaces which we conjecture to have the maximal dimension achievable within
the stabilizer formalism. We have thus provided examples of genuinely
entangled fully NPT subspaces, extending in a way the results of Ref. [36] to
the multipartite scenario.

Our work leaves many open questions that might serve as inspiring starting
points for further research. The most obvious one is whether the first
conjecture stated in our paper can be proven rigorously. On the other hand, if
this conjecture fails to be true, an alternative question would be whether one
can find the true function $p(N,d)$ that should appear in Ineq. (38) and that
can later be used to determine the maximal dimension of genuinely entangled
stabilizer subspaces. Another related question is whether one can provide a
general construction of fully NPT genuinely entangled subspaces beyond the
stabilizer formalism that are of maximal dimension; it was show in Ref. [36]
that completely entangled subspaces that are NPT can be as large as any
completely entangled subspaces. While we believe our construction to be of
maximal dimension achievable within the stabilizer formalism, at least for
prime $d$, it is most probably suboptimal when one takes into account
arbitrary genuinely entangled subspaces. On the other hand, it would be highly
interesting to follow the results of Sec. 4 and further explore the relation
between entanglement in the stabilizer formalism and the separability
criterion based on partial transposition. While we prove that genuinely
entangled stabilizer subspaces do not support PPT states, it is at the moment
unclear whether there are no PPT genuinely entangled stabilizer states at all.
In fact, there might exist stabilizer subspaces that are not genuinely
entangled and yet support PPT genuinely entangled states. It would thus be
highly interesting to explore whether there any PPT entangled states in this
formalism, or, alternatively whether partial transposition is a necessary and
sufficient condition for separability for stabilizer states. The last avenue
for further research is to construct Bell inequalities that are maximally
violated by genuinely entangled stabilizer subspaces and see whether they can
be used for self-testing, generalizing the results Refs. [42, 35].

##  6 acknowledgements

This work was supported by the National Science Center (Poland) through the
SONATA BIS grant no. 2019/34/E/ST2/00369.

##  Appendix A Proofs of Lemmas 1 and 2

For completeness we provide here the proofs of Lemmas 1 and 2.

######  Lemma 1.

Let the local dimension $d$ be prime. Given a stabilizer $\mathbb{S}=\langle
G_{1},\dots,G_{k}\rangle$, the corresponding stabilizer subspace
$V\neq\\{0\\}$ is genuinely entangled iff for every $\phi\in F_{N,2}$ such
that $\phi\neq\phi_{T_{0}},\phi_{T_{1}}$ there exists $v_{i,j}\in
K(\mathbb{S})$ for $i\neq j$ such that

$h(v_{i,j},\phi)\neq 0.$ (55)

###### Proof.

Corollary 1 states that for every nontrivial bipartition $Q|\overline{Q}$ of
the set $\\{1,\dots,N\\}$ there exists $i,j\in\\{1,\dots,k\\}$ such that

$\left[G_{i}^{(Q)},G_{j}^{(Q)}\right]\neq 0.$ (56)

This can be equivalently expressed as

$\tau_{i,j}^{(Q)}\neq 0,$ (57)

where

$\tau_{i,j}^{(Q)}=\sum_{n\in Q}\tau_{i,j}^{(n)}$ (58)

and $\tau_{i,j}^{(n)}$ is defined in Eq. (29). Given the representation $\phi$
of a bipartition $Q|\overline{Q}$, the above equation can be reformulated as

$\tau_{i,j}^{(Q)}=\sum_{n\in
Q}\tau_{i,j}^{(n)}=\sum_{n=1}^{N}\tau_{i,j}^{(n)}\phi^{(n)}.$ (59)

By the definitions of $v_{i,j}$ [cf. Eq. (28)] and of $h(\cdot,\cdot)$ (33),
the above rewrites as

$\tau_{i,j}^{(Q)}=\sum_{n=1}^{N}\tau_{i,j}^{(n)}\phi^{(n)}=h(v_{i,j},\phi),$
(60)

Consequently, from (57) it follows that $h(v_{i,j},\phi)\neq 0$ for all
nontrivial $\phi$ and some $v_{i,j}\in K(\mathbb{S})$ iff $V$ is genuinely
entangled, which ends the proof. ∎

######  Lemma 2.

Assume $d$ to be prime and consider a stabilizer for which $\dim
K(\mathbb{S})=N-1$. Then, a vector $v\in F_{N,d}$ fulfils

$\sum_{i=1}^{N}v^{(i)}=0,$ (61)

if and only if $v\in K(\mathbb{S})$.

###### Proof.

Since the implication "$\Leftarrow$" follows from the definition of
$K(\mathbb{S})$, in order to complete the proof it is sufficient to show that
the number of vectors in $F_{N,d}$ fulfilling (41) equals the number of
vectors in $K(\mathbb{S})$.

Let us start from calculating how many vectors in $F_{N,d}$ satisfy (41).
Clearly, for any such vector we have

$v^{(N)}=-\sum_{n=1}^{N-1}v^{(n)}.$ (62)

Notice that for any choice of $v^{(n)}$ for all $n\in\\{1,\dots,N-1\\}$ we can
find exactly one $v^{(N)}$ that fulfils this equation. This implies that there
are $d^{N-1}$ vectors $v\in F_{N,d}$ fulfilling (41).

Next, let us calculate the number of vectors in $K(\mathbb{S})$. To this end
let us consider the following expression

$\sum_{i=1}^{N-1}a_{i}u_{i}=\sum_{i=1}^{N-1}b_{i}u_{i},$ (63)

where $\\{u_{i}\\}_{i=1}^{N-1}$ is a basis in $K(\mathbb{S})$ and
$a_{i},b_{i}\in\mathbb{Z}_{d}$; notice that by definition each element of this
basis must also satisfy (41). Since $d$ is prime, the above is true only if
$a_{i}=b_{i}$ for all $i\in\\{1,\dots,N-1\\}$. There are $d^{N-1}$ unique
choices of $\\{a_{i}\\}_{i=1}^{N-1}$, hence there are $d^{N-1}$ vectors in
$K(\mathbb{S})$, which ends the proof. ∎

##  Appendix B A proof of Theorem 3

Here we provide a proof of Theorem 3 or, equivalently, of Conjecture 1 for
$k=2$ and arbitrary $d$ and $N$. If someone will attempt to prove Conjecture 1
in its most general form, we hope that the following proof will serve as guide
for how the general proof could be formulated. The main idea of this proof is
to treat bipartition representations $\phi$ as solutions to an equation:

$h(u,\phi)=\beta,$ (64)

where $u\in F_{N,d}$ and $\beta\in\\{0,\dots,d-1\\}$. The first step in our
procedure is to calculate the number of solutions $\phi$ to this equation
depending on $d$ and $N$.

######  Lemma 3.

Let $u\in F_{N,d}$ and let us assume that an equation

$h(u,\phi)=\beta,$ (65)

has at least one solution $\phi\in F_{N,2}$. The number of $\phi$ that fulfil
Eq. (65) is larger or equal than $2^{N-(d-1)}$.

###### Proof.

Let $\sigma(N,\beta)$ denote the number of $\phi\in F_{N,2}$ that fulfil:

$\sum_{i=1}^{N}u^{(i)}\phi^{(i)}\mod d=\beta,$ (66)

where the upper index denotes the $i$’th entry.

Rather than finding a lower bound on $\sigma(N,\beta)$ for each
$\beta\in\\{0,\dots,d-1\\}$, we will find a minimum of $\sigma(N,\beta)$ over
all $\beta\in\\{0,\dots,d-1\\}$. To this end, let us denote by
$\sigma(n,\beta)$ the number of $\phi\in F_{n,2}$ that fulfil:

$\sum_{i=1}^{n}u^{(i)}\phi^{(i)}\mod d=\beta,$ (67)

where $n\in\\{1,\dots,N\\}$. We have introduced a new variable $n$, because in
order to find a minimum of $\sigma(N,\beta)$ over $\beta$ we want to express
$\sigma(n+1,\beta)$ in terms of $\sigma(n,\beta^{\prime})$, namely

$\sigma(n+1,\beta)=\sigma(n,\beta)+\sigma(n,\beta-u^{(n+1)}),$ (68)

however at the end we will set $n=N$. To understand why the above equality is
true let us examine (67), but for $\sigma(n+1,\beta)$:

$\displaystyle\sum_{i=1}^{n+1}u^{(i)}\phi^{(i)}\mod d$ $\displaystyle=$
$\displaystyle\beta,$ $\displaystyle\Downarrow$
$\displaystyle\sum_{i=1}^{n}u^{(i)}\phi^{(i)}\mod d$ $\displaystyle=$
$\displaystyle\beta-u^{(n+1)}\phi^{(n+1)}.$ (69)

As we can see, for $\phi^{(n+1)}=0$ we get an identical equation to (67) and
for $\phi^{(n+1)}=1$ we also have an equation equivalent to (67) but for
$\beta=\beta^{\prime}-u^{(n+1)}$ and so $\sigma(n+1,\beta)$ is a sum of the
number of solution for those two cases, therefore we get (68).

Let us denote by $\sigma_{\min}(n)$ the minimal, nonzero $\sigma(n,\beta)$
over all $\beta\in\\{0,\dots,d-1\\}$. From (68) it follows that either

$\sigma_{\min}(n+1)=\sigma_{\min}(n)$ (70)

or

$\sigma_{\min}(n+1)\geqslant 2\sigma_{\min}(n).$ (71)

If (70) is true, then by the virtue of (68) it implies that for some
$\beta\in\\{0,\dots,d-1\\}$ for which $\sigma(n,\beta)=\sigma_{\min}(n)$ we
have $\sigma(n,\beta-u^{(n+1)})=0$. Let us examine that case further. We can
once again use Eq. (68) to get

$\sigma(n+1,\beta-u^{(n+1)})=\sigma(n,\beta-2u^{(n+1)}).$ (72)

Now, let us assume that $\sigma(n,\beta-ku^{(n+1)})=0$ for all positive
integers $k\leqslant\kappa$, where $\kappa\in\mathbb{Z}_{+}$. Then from Eq.
(68) we have

$\sigma(n+1,\beta-ku^{(n+1)})=\sigma(n,\beta-(k+1)u^{(n+1)}).$ (73)

Notice, that our assumption $\sigma(n,\beta-ku^{(n+1)})=0$ gives us a bound on
the value of $\kappa$, since

$\sigma(n,\beta-du^{(n+1)})=\sigma(n,\beta)=\sigma_{\min}(n)\neq 0,$ (74)

and so $\kappa\in\\{1,\dots d-1\\}$. Importantly, in the case of $k=\kappa$ it
follows from Eq. (68) that

$\sigma(n,\beta-\kappa u^{(n+1)})=0$ (75)

and

$\sigma(n+1,\beta-\kappa u^{(n+1)})\neq 0.$ (76)

In other words, every time (70) is true, it follows that there exists at least
one $\beta^{\prime}\in\\{0,\dots,d-1\\}$ for which equation

$\sum_{i=1}^{n^{\prime}}u^{(i)}\phi^{(i)}\mod d=\beta^{\prime}$ (77)

has no solution when $n^{\prime}=n$ (as in (75)) and has at least one solution
when $n^{\prime}=n+1$ (as in (76)).

Finally, we are ready to set a bound on $\sigma_{\min}(n)$. In the case of
$n=1$ we have exactly two values of $\beta$ for which we have a solution to
Eq. (66), so there are $d-2$ values of $\beta$ with no solution. If we assume
that $\sigma_{\min}(2)=\sigma_{\min}(1)=1$, then we know that there are $d-3$
values of $\beta^{\prime}$ for which (66) has no solutions. Following this
pattern we conclude that if for all $n\in\\{1,\dots,d-2\\}$ we have
$\sigma_{\min}(n+1)=\sigma_{\min}(n)=1$, then for $n=d-1$ and for all
$\beta\in\\{0,\dots,d-1\\}$ the following holds true

$\sigma(d-1,\beta)\neq 0.$ (78)

From this we can infer that for $n>d-1$, $\sigma_{\min}(n)$ will be bounded
according to (71). Moreover, since in general the bound from (71) for the case
$n^{\prime}=n+1$ is a linear function of the bound for the case
$n^{\prime}=n$, the exact order of applying bounds (70) and (71) does not
matter - only the total number of times that each bound is applied matters.
Therefore, the assumption $\sigma_{\min}(n+1)=\sigma_{\min}(n)=1$ for
$n\in\\{1,\dots,d-2\\}$ yields a lower bound on on $\sigma_{\min}(n)$:

$\sigma_{\min}(n)\geqslant\begin{cases}1\quad\textrm{for}\quad n\leqslant
d-1,\\\ 2^{n-(d-1)}\quad\textrm{for}\quad n>d-1.\end{cases}$ (79)

so we can simply write

$\sigma_{\min}(N)\geqslant 2^{N-(d-1)}$ (80)

which ends the proof. ∎

This leads us strait to Theorem 3 which for completeness we restate here.

######  Theorem 3.

Consider a stabilizer $\mathbb{S}=\langle G_{1},G_{2}\rangle$. If the local
dimension $d$ is prime and the corresponding stabilizer subspace $V$ is
genuinely entangled then $\dim
K(\mathbb{S})\geqslant\lceil\frac{N-1}{d-1}\rceil$.

###### Proof.

From the number of generators it directly follows that $\dim
K(\mathbb{S})\leqslant 1$, but since we assume that $V$ is genuinely entangled
we know that $v_{1,2}\neq 0$, and so we can just write $\dim K(\mathbb{S})=1$.
We need to show that

$1\geqslant\left\lceil\frac{N-1}{d-1}\right\rceil.$ (81)

By the virtue of Lemma 1, if $V$ is genuinely entangled, then

$h(v_{1,2},\phi)=0$ (82)

is only true for the representations of trivial bipartitions
$\phi\in\\{\phi_{T_{0}},\phi_{T_{1}}\\}$, i.e., the number of $\phi\in
F_{N,2}$ that fulfil (82) equals $2$. Then, from Lemma 3 we have

$2\geqslant 2^{N-(d-1)},$ (83)

from which it easily follows that

$1\geqslant\frac{N-1}{d-1}.$ (84)

If this inequality is true, then (81) is as well, which ends the proof. ∎

Now that this theorem has been proven, we can discus what are the difficulties
of proving Conjecture 1 in its most general form. The issue lays in the proof
of a version of Lemma 3 for an arbitrary $k$. The number of solutions in the
most general case would be $2^{N-(d-1)\cdot\dim K(\mathbb{S})}$, but trying to
prove it by using the same arguments runs into problem when discussing the
number of $\sigma(n,\beta)$ that are zero for $n=n^{\prime}$ and are nonzero
for $n=n^{\prime}+1$. For arbitrary $k$, the procedure from Lemma 3 would give
us the number of $\phi$ to be larger or equal to $2^{N-(d^{\dim
K(\mathbb{S})}-1)}$. This is a result of the fact that for higher $\dim
K(\mathbb{S})$ a situation where increase in $n$ makes only one
$\sigma(n,\beta)$ nonzero is really rare. For example, if increasing $n$ from
0 to $d-1$ resulted in the change from zero to nonzero for $d-1$ sigmas, then
each increase after that would make at least $d$ sigmas nonzero (this
situation occurs when on the first $d-1$ sites, one vector $v_{i,j}$ has ones
and the other vectors have zeros). To conclude, in order to prove Conjecture 1
using the same methods as we did, one would need to find a better way of
estimating how many $\sigma(n,\beta)$ become nonzero with each increase in
$n$.

##  Appendix C Genuinely entangled, stabilizer subspace of maximal dimension

In this section we show an example of a genuinely entangled, stabilizer
subspace of dimension that we claim is maximal for such subspaces (see
Conjecture 2). More specifically we construct the generators of stabilizer of
the aforementioned subspace. Since the expressions on the generators are
recursive equations dependent on the number of qubits $N$, we will use a
notation $G_{i}(N)$ for an $i$’th generator for $N$ qubits. Similarly, we
denote by $v_{i,j}(N)$ the vector defined as in (28) for generators $G_{i}(N)$
and $G_{j}(N)$. Moreover, the number of generators $k$ will also depend on
$N$, however in this case we will use $k$ and $k(N)$ interchangeably depending
on whether the dependence on $N$ is important to a specific case.

Let $k=k_{min}(N)$ be defined by Eq. (42) and let us consider a set of
generators $\\{G_{i}(N)\\}_{i=1}^{k}$. For $N=2$ we construct the generators
as

$\displaystyle G_{1}(2)$ $\displaystyle=X\otimes X^{d-1},$ $\displaystyle
G_{2}(2)$ $\displaystyle=Z\otimes Z.$ (85)

For arbitrary $N$ we give a recurrence relation for the generators and we
distinguish two cases: if $N$ fulfils $k(N-1)=k(N)-1$, then

$G_{i}(N)=\begin{cases}G_{i}(N-1)\otimes\mathbb{1}&i\notin\\{k-1,k\\},\\\
G_{i}(N-1)\otimes P_{k-1}^{d-1}&i=k-1,\\\ \mathbb{1}^{\otimes(N-2)}\otimes
P_{k}\otimes P_{k}&i=k,\end{cases}$ (86)

and otherwise

$G_{i}(N)=\begin{cases}G_{i}(N-1)\otimes\mathbb{1}&i\notin\\{l,k\\},\\\
\tilde{G}_{i}(N-1)\otimes P_{k-1}^{d-1-m}&i=l,\\\ G_{i}(N-1)\otimes
P_{k}&i=k,\end{cases}$ (87)

where we define $\tilde{G}_{i}(N-1)$ as

$\tilde{G}_{i}(N-1)=G_{i}(N-1)\left(\mathbb{1}^{\otimes(N-2)}\otimes
P_{k-1}^{m+1}\right),$ (88)

parameter $m$ is equal to

$\displaystyle m$ $\displaystyle=$ $\displaystyle\sup(\mathcal{M}_{N,d}),$
$\displaystyle\mathcal{M}_{N,d}$ $\displaystyle=$
$\displaystyle\Bigg{\\{}n\in\mathbb{N}:\left\lceil\frac{N-1-n}{d-1}\right\rceil=\left\lceil\frac{N-1}{d-1}\right\rceil\Bigg{\\}},$
(89)

matrix $P_{i}$ equals

$P_{i}=\left\\{\begin{array}[]{ll}X&\textrm{ for odd\;}i,\\\\[4.30554pt]
Z&\textrm{ for even\;}i,\end{array}\right.$ (90)

and

$l(N)=\frac{k(k-1)}{2}+1-\left\lceil\frac{N-1}{d-1}\right\rceil.$ (91)

The above function $l(N)$ is defined on the set of $N$ for which
$k(N)=\operatorname{const}$. Let $\\{N_{\min},\dots,N_{\max}\\}$ be a set of
$N$ for which $k(N)=\operatorname{const}$, i.e.,

$\displaystyle k(N_{\min}-1)$ $\displaystyle=k(N_{\min})-1,$ $\displaystyle
k(N_{\max}+1)$ $\displaystyle=k(N_{\max})+1.$ (92)

From (43) we have

$\left\lceil\frac{N_{\min}-2}{d-1}\right\rceil=\frac{1}{2}[k(N_{\min})-1][k(N_{\min})-2],$
(93)
$\left\lceil\frac{N_{\max}-1}{d-1}\right\rceil=\frac{1}{2}k(N_{\min})\;[k(N_{\min})-1].$
(94)

Substituting them into (91) we get a set $\\{1,\dots,k-1\\}$, which is the set
of all $l(N)$ for $k=\operatorname{const}.$. Let us note here that for $d=2$
this set is effectively $\\{1,\dots,k-2\\}$, because in that case equation
$l=k-1$ is equivalent to $k(N-1)=k(N)-1$ which is the condition for (86).

######  Theorem 4.

The stabilizer $\mathbb{S}_{\max}$ generated by the generators defined in (C),
(86) and (87) stabilize a genuinely entangled, stabilizer subspace, where
$k_{\min}(N)$ is defined in (42). Moreover, if $d$ is prime then the dimension
of the stabilizer subspace equals $d^{N-k_{\min}(N)}$.

Before we start the proof of the above theorem, let us introduce a convenient
convention, namely for even $k$ instead of calculating vectors $v_{i,k}$ we
will calculate

$v^{\prime}_{i,k}=(d-1)v_{i,k}.$ (95)

This makes it so that the vast majority of vector elements equals $0,1$, which
simplifies our calculations. Moreover, this does not pose any problems, since
in this proof we only care about the general properties of
$K(\mathbb{S}_{\max})$, which both $v_{i,k}$ and $v^{\prime}_{i,k}$ are a part
of, and not the vectors in particular. In practise, it is as if we assumed
that $k$ is always odd while calculating the elements of $v_{i,k}$. With that,
we can move to the main part of the proof.

###### Proof.

To complete the proof we first need to show that $\mathbb{S}_{\max}=\langle
G_{1}(N),\dots,G_{k}(N)\rangle$ is the stabilizer, after which we need to
prove that the stabilizer subspace is genuinely entangled. The subgroup
$\mathbb{S}$ is a stabilizer if $a\mathbb{1}\notin\mathbb{S}$ for $a\neq 1$
and if all the elements of $\mathbb{S}$ commute. This can be shown by proving
the following conditions

$\forall_{i\in\\{1,\dots,k\\}}\quad G_{i}(N)^{d}=\mathbb{1},$ (96)
$\forall_{i,j\in\\{1,\dots,k\\}}\quad\left[G_{i}(N),G_{j}(N)\right]=0,$ (97)
$\forall_{a\neq
1}\;\forall{\alpha_{i}\in\\{0,\dots,d-1\\}}\quad\prod_{i=1}^{k}G_{i}^{\alpha_{i}}(N)\neq
a\mathbb{1}.$ (98)

A proof of (96) is trivial, since by definition the generators $G_{i}(N)$ are
a tensor product of matrices (7).

Next, proving (98) is also fairly simple: from Eq. (C) and Eq. (86) it follows
that $G^{(1)}_{i}(N)=X$ implies $i=1$ and similarly $G^{(2)}_{i}(N)=Z$ implies
$i=2$. Moreover, from Eq. (86) and Eq. (87) we can conclude that for $n$ for
which $k(n)=k(n+1)-1$, $G_{i}^{(n)}(N)=P_{k(n)}$ implies $i=k(n)$.
Consequently, for every generator $G_{i}(N)$ we can find $n$ such that
$G_{i}^{(n)}(N)\neq G_{j}^{(n)}(N)$ for $i\neq j$, which in turn implies that

$\prod_{i=1}^{k}G_{i}^{\alpha_{i}}(N)\sim\mathbb{1},$ (99)

iff for all $i\in\\{1,\dots,k\\}$, $\alpha_{i}=0$, but for that case
$\prod_{i=1}^{k}G_{i}^{0}(N)=\mathbb{1}$, which proves (98).

Proving Eq. (97) is more complex, but thankfully we can make use of the
vectors $v_{i,j}(N)$ defined in Eq. (28). In this formalism, the condition
(97) translates to

$\sum_{n=1}^{N}v_{i,j}^{(n)}=0\mod d.$ (100)

Let us begin with $N=2$. From Eq. (C) it is clear that $v_{1,2}(2)=(1,d-1)$
and so (100) is fulfilled. The proof for an arbitrary $N$ will be done by
induction, i.e., we assume that (100) is fulfilled for $N^{\prime}=N-1$ and we
show that it is fulfilled for $N^{\prime}=N$. For clarity, we will consider
two separate cases: first (86) and then (87).

From (86) we can easily conclude that for $i,j\neq k$ we have

$v_{i,j}(N)=v_{i,j}(N-1)\oplus(0).$ (101)

As for the vectors $v_{i,k}(N)$, from (86) it is clear that

$\displaystyle v_{i,k}^{(N)}(N)=0$ $\displaystyle i\neq k-1,$ $\displaystyle
v_{i,k}^{(N)}(N)=d$ $\displaystyle-1\qquad$ $\displaystyle i=k-1,$
$\displaystyle v_{i,k}^{(n)}(N)=0$ $\displaystyle n\in\\{1,\dots,N-2\\}.$
(102)

The only unknown elements are $v_{i,k}^{(N-1)}$, however they also can be
easily determined, by analysing generators for $N^{\prime}=N-1$. There are two
subcases: for $N=3$ we have to refer to (C) and for $N>3$ to (87). However,
according to both (C) and (87) we have

$\displaystyle G_{i}^{(N-1)}(N-1)$ $\displaystyle\in$
$\displaystyle\\{\mathbb{1},P_{k^{\prime}-1}^{q}\\}\quad
i\in\\{1,\dots,k^{\prime}-1\\},$ $\displaystyle G_{k^{\prime}}^{(N-1)}(N-1)$
$\displaystyle=$ $\displaystyle P_{k^{\prime}},$ (103)

where $k^{\prime}=k(N-1)=k-1$ and $q\in\mathbb{Z}_{+}$. This implies that

$\displaystyle v_{i,k}(N)$ $\displaystyle=$ $\displaystyle 0\qquad
i\in\\{1,\dots,k-2\\},$ $\displaystyle v_{k-1,k}(N)$ $\displaystyle=$
$\displaystyle e_{N-1}+(d-1)e_{N}.$ (104)

Clearly from induction, all vectors $v_{i,j}(N)$ fulfil (100).

Next, let us consider the case (87). It is easy to see that

$v_{i,j}(N)=v_{i,j}(N-1)\oplus(0)$ (105)

is true for $i,j\neq l$. Moreover, the above is also true for $i\neq l,k$ and
$j=l$, since from (86) and (87) it follows that for such $i$ we have

$G_{i}^{(N-1)}(N-1)\in\left\\{\mathbb{1},P_{k-1}^{q}\right\\}$ (106)

for some $q\in\mathbb{Z}_{+}$. This means that we have one vector left to
consider, namely for $i=l$ and $j=k$ we have

$v_{l,k}(N)=v_{l,k}(N-1)\oplus(0)+(m+1)e_{N-1}+(d-1-m)e_{N}.$ (107)

Clearly, if for $N^{\prime}=N-1$ (100) is fulfilled then all $v_{i,j}(N)$ also
fulfil (100). This proves (97) for all $N$.

To reiterate, we proved that (96), (97) and (98) are fulfilled by
$G_{1},\dots,G_{k}$, hence $\mathbb{S}_{\max}$ is a stabilizer. Moreover,
since we have shown that $\prod_{i=1}^{k}G_{i}^{\alpha}(N)=\mathbb{1}$ is only
true if all $\alpha_{i}=0$, the generators are independent and so if $d$ is
prime then the dimension of the stabilizer subspace equals $d^{N-k_{\min}(N)}$
[38]. This leaves only the question of genuine entanglement of the stablizer
subspace $V$.

For $N=2$ (C) it is easy to see from Corollary 1 that $V$ is genuinely
entangled, since the only nontrivial bipartition is $\\{1\\}|\\{2\\}$. For
arbitrary $N$ we again use a prove by induction, i.e., we assume that for
$N^{\prime}=N-1$, the subspace $V(N-1)$ is genuinely entangled and we show
that this implies that $V(N)$ is also genuinely entangled.

First, consider a case $k(N-1)=k(N)-1$, which corresponds to generators (86).
From our assumption and from (105) it follows that if we only consider
nontrivial bipartitions of $\\{1,\dots,N-1\\}$, then for all such bipartitions
there exist $i,j\in\\{1,\dots,k-1\\}$ such that

$h(v_{i,j}(N),\phi)\neq 0,$ (108)

where $\phi$ is a representation of a bipartition $Q|\overline{Q}$ and
$h(\cdot,\cdot)$ is defined in Eq. (33). Moreover, from (101) and (105) it
follows that $v_{i,j}^{(N)}(N)=0$ for $i,j\neq k$, and so by the above
argument, for all $Q\subset\\{1,\dots,N\\}$ such that $Q$ is nontrivial and
$Q\neq\\{N\\},\\{1,\dots,N-1\\}$ there exist $i,j\in\\{1,\dots,k-1\\}$ for
which (108) is fulfilled. As for $Q=\\{N\\},\\{1,\dots,N-1\\}$, from (C) we
have that

$h(v_{k-1,k}(N),\phi)\neq 0,$ (109)

hence for all nontrivial bipartitions of the set $\\{1,\dots,N\\}$ there exist
a pair $i,j\in\\{1,\dots,k\\}$ such that

$h(v_{i,j}(N),\phi)\neq 0,$ (110)

which by the virtue of Lemma 1 shows that $V(N)$ is genuinely entangled.

Next, let us consider a case (87) for $l(N)=l(N-1)+1$. Since $l(N)$ is a
nonincreasing function, we know that $v_{l,k}(N-1)=0$. Then, we can use the
same argumentation as for the previous case but with a vector $v_{l,k}(N)$
instead of $v_{k-1,k}(N)$.

Lastly, let us consider a case (87) for $l(N)=l(N-1)$. By iterating (107) from
$m^{\prime}=m$ to $m^{\prime}=0$ we can derive an explicit formula for
$v_{l,k}(N)$:

$v_{l,k}(N)=\sum_{m^{\prime}=0}^{m}e_{N-1-m^{\prime}}+(d-1-m)e_{N}.$ (111)

Moreover, the same iteration of (105) implies that for all pairs
$(i,j)\neq(l,k)$ and for all $n\in\\{N-m,\dots,N\\}$

$v_{i,j}^{(n)}=0.$ (112)

Therefore, our induction assumption implies that for every nontrivial subset
$Q\subset\\{1,\dots N-1-m\\}$ there exist a pair $(i,j)\neq(l,k)$ for which
(108) is fulfilled, which by the virtue of Lemma 1 implies, that $V(N)$ is
genuinely entangled. This leaves us with bipartitions $Q|\overline{Q}$ for
which $Q\subset\\{N-m,\dots,N\\}$ or $\overline{Q}\subset\\{N-m,\dots,N\\}$.
However, from (111) it follows that for every such $Q$ (or $\overline{Q}$),
(108) is fulfilled by $v_{l,k}(N)$, which ends the proof. ∎

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