# Entangled symmetric states and copositive matrices

Carlo Marconi  Física Teòrica: Informació i Fenòmens Quàntics. Departament de
Física, Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain  Albert
Aloy  ICFO - Institut de Ciències Fotòniques, The Barcelona Institute of
Science and Technology, 08860 Castelldefels (Barcelona), Spain  Jordi Tura
Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Str. 1, 85748 Garching,
Germany  Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA
Leiden, The Netherlands  Anna Sanpera  Física Teòrica: Informació i Fenòmens
Quàntics. Departament de Física, Universitat Autònoma de Barcelona, 08193
Bellaterra, Spain  ICREA, Pg. Lluís Companys 23, 08010 Barcelona, Spain

(December 11, 2020)

###### Abstract

Entanglement in symmetric quantum states and the theory of copositive matrices
are intimately related concepts. For the simplest symmetric states, i.e., the
diagonal symmetric (DS) states, it has been shown that there exists a
correspondence between exceptional (non-exceptional) copositive matrices and
non-decomposable (decomposable) entanglement witnesses (EWs). Here we show
that EWs of symmetric, but not DS, states can also be constructed from
extended copositive matrices, providing new examples of bound entangled
symmetric states, together with their corresponding EWs, in arbitrary odd
dimension.

Entanglement and symmetry lie at the heart of quantum theory. Symmetries
reflect fundamental laws of Nature and are intrinsically present in systems of
physical interest. Moreover, states possessing some symmetry, typically admit
a simplified mathematical description as compared to the one of generic
states, a fact that usually translates into a more feasible way of
characterizing their properties.

Quantum correlations are an intrinsic property of composite systems.
Entanglement, in particular, is regarded as the most significant feature of
quantum physics, not only because it provides unique insights into the
fundamental principles of our physical world, but also because it represents a
resource that allows to perform several tasks that would be, otherwise,
impossible.

Since the birth of quantum information theory, huge efforts have been devoted
to characterize and quantify entanglement (see e.g. [1]). Along the years, it
has become clear that entanglement characterization is a challenging task.
Moreover, it cannot be quantified by a unique measure. The exception lies in
(bipartite) pure entangled states where it is trivial to determine if the
state is entangled and entanglement entropy is the unique measure needed.
Interestingly, in the asymptotic limit, for a sufficient number of copies of
the system, the entanglement entropy measures the resource interconversion
capacity between different pure states, within the paradigm of local
operations and classical communication [2]. However, already in the case of
bipartite mixed states, two of such measures are needed in order to quantify
this interconversion rate: the entanglement of formation and the entanglement
of distillation.

A closely related, although inherently different, approach is the
characterization of entangled states independently of any measure or their
usefulness for a specific task. This problem has been shown to be, in general,
NP-hard [3]. However, partial characterization has been achieved by means of
criteria that provide necessary, but not sufficient, conditions to determine
if a given state is entangled or not. The most powerful of such criteria,
formulated in terms of linear positive maps, is the positivity under partial
transposition (PPT) [4], which is the paradigmatic example of a positive, but
not completely positive, map. States that do not fulfill the PPT criterion are
entangled but the converse is not true, except for few cases. In this regard,
quantum maps and their associated entanglement witnesses (EWs), provide the
strongest criteria for entanglement characterization: a quantum state is
entangled if, and only if, there exists an EW that detects it [5, 6, 7].
Crucially, in order to characterize entanglement in states that do not break
the PPT criterion (PPT entangled states or PPTES), it is necessary to
construct non-decomposable EWs [8]. Interestingly, EWs have been shown to
provide also a measure of entanglement which is upper and lower bounded by
other entanglement measures [9].

Nowadays, it is still unclear whether, in general, the problem of entanglement
characterization remains equally hard for systems displaying some symmetries
[10, 11, 5, 12]. A possible approach in this direction is to investigate if,
and how, symmetries can help to construct EWs for such systems.

In this work, we focus on the entanglement characterization of permutationally
invariant systems and, more specifically, on the class of the so-called
symmetric states. These provide a natural description for sets of
indistinguishable particles, i.e., bosons. To this aim, we derive a method to
construct specific EWs for such states using the theory of copositive
matrices. As shown previously by some of us [13], the characterization of
entanglement for some particularly simple symmetric states that are mixtures
of projectors on symmetric states, also called DS states, can be recast as the
problem of checking membership to the cone of the so-called completely-
positive matrices or to its dual, the cone of copositive matrices. The
equivalence between these two problems allows to establish a correspondence
between entanglement characterization for DS states and non-convex quadratic
optimization [13]. In particular, it provides a very efficient method to
detect states that are PPT entangled. Here, we further extend this mapping by
constructing EWs that detect symmetric PPTES that are not DS. First, we derive
under which conditions a copositive matrix leads to an EW for symmetric
states. Then, we show explicitly how to derive, from such copositive matrices,
both decomposable and non-decomposable EWs. Finally, we use these EWs to
generate several families of symmetric PPTES. It is important to remark that,
to the best of our knowledge, there are not known examples of non-decomposable
EWs for generic symmetric states. The very few examples of symmetric PPTES in
the literature[11], have been found numerically using weaker entanglement
criteria like, e.g., the range criterion applied to edge states[8, 14, 15, 16,
17, 18]. Our work thus, offers a complementary approach to the study of
entanglement characterization in symmetric states. The paper is organized as
follows: in Section 1, we introduce basic concepts concerning the definition
and properties of symmetric states in $\mathbb{C}^{d}\otimes\mathbb{C}^{d}$,
and the mapping between copositive matrices and EWs. In this section, we
review as well some of our previous results for DS states [13] that represent
the starting point of our work. In Section 2, we show how to derive
decomposable and non-decomposable EWs for symmetric states from copositive
matrices and provide some explicit constructive examples. In Section 3, we
focus on symmetric, but not DS, PPTES, showing the existence of PPTES in
arbitrary dimensions and introducing examples of such states in different
dimensions ($d=3,4,5,7$). Finally, in Section 4 we summarize our findings,
present some conjectures derived from our analysis and list some open
questions for further research.

##  1 Basic concepts

We start by introducing the notation used throughout the manuscript along with
some basic concepts and definitions regarding symmetric states, entanglement
witnesses and copositive matrices.

###  1.1 Symmetric states

Henceforth, we focus on bipartite systems. Let
$\mathcal{H}=\mathbb{C}^{d}\otimes\mathbb{C}^{d}$, be the finite dimensional
Hilbert space of two qudits, and $\mathcal{B}(\mathcal{H})$ the set of its
bounded operators. The symmetric subspace $\mathcal{S}\subset\mathcal{H}$, is
the convex set formed by the (normalized) pure states
$\ket{\Psi_{S}}\in\mathcal{H}$ that are invariant under the exchange of
parties. Symmetric states can be mapped to spin systems and, moreover, they
span the subspace of maximum spin in the Schur-Weyl representation. The so-
called Dicke states form a convenient basis of the symmetric subpace, i.e.,
$\mathcal{S}\equiv\text{span}\\{\ket{D_{ij}}\\}$, where
$\ket{D_{ij}}=\ket{D_{ji}}\equiv(\ket{ij}+\ket{ji})/\sqrt{2}$ for $i\neq
j,\;\ket{D_{ii}}\equiv\ket{ii}$ and $\\{\ket{i}\\}_{i=0}^{d-1}$ is an
orthonormal basis of $\mathbb{C}^{d}$. Note the reduced dimensionality of the
symmetric subspace, i.e., $\dim(\mathcal{S})=d(d+1)/2<d^{2}$. In an abuse of
language we denote symmetric quantum states,
$\rho_{S}\in\mathcal{B}(\mathcal{S})$, as the convex hull of projectors onto
pure symmetric normalized states, i.e.,
$\rho_{S}=\sum_{k}p^{(k)}_{S}\ket{\Psi^{(k)}_{S}}\bra{\Psi^{(k)}_{S}}$, with
$p^{(k)}_{S}\geq 0$, $\sum_{k}p^{(k)}_{S}=1$ and
$\ket{\Psi^{(k)}_{S}}=\sum_{ij}c^{(k)}_{ij}\ket{D_{ij}}$,
$c^{(k)}_{ij}\in\mathbb{C}$.
Thus, any $\rho_{S}\in\mathcal{B}(\mathcal{S})$ is a positive semidefinite
operator ($\rho_{S}\succeq 0$) with unit trace ($\mathrm{Tr}(\rho_{S})=1$),
fulfilling $\Pi_{S}\rho_{S}\Pi_{S}=\rho_{S}\Pi_{S}=\Pi_{S}\rho_{S}=\rho_{S}$,
where $\Pi_{S}=\frac{1}{2}(\mathds{1}+F)$ is the projector onto the symmetric
subspace and $F=\sum_{i,j=0}^{d-1}\ket{ij}\\!\bra{ji}$ is the flip operator.
Using the Dicke basis, symmetric quantum states can be compactly expressed as
follows:

######  Definition 1.1.

Any bipartite symmetric state, $\rho_{S}\in\mathcal{B}(\mathcal{S})$, can be
written as

$\rho_{S}=\sum_{\begin{subarray}{c}0\leq i\leq j<d\\\ 0\leq k\leq
l<d\end{subarray}}\left(\rho_{ij}^{kl}\ket{D_{ij}}\bra{D_{kl}}+\emph{h.c.}\right)\leavevmode\nobreak\
,$ (1)

with $\rho_{ij}^{kl}\in\mathbb{C}$. Notice that, due to the symmetry of the
Dicke states, it holds that
$\rho_{ij}^{kl}=\rho_{ji}^{kl}=\rho_{ij}^{lk}=\rho_{ji}^{lk}\leavevmode\nobreak\
\forall i,j,k,l$.

Convex mixtures of projectors onto Dicke states are denoted as diagonal
symmetric (DS) states, since they are diagonal in the Dicke basis. They form a
convex subset of $\mathcal{S}$ and are particularly relevant for our analysis.

######  Definition 1.2.

Any DS state, $\rho_{\small{DS}}\in\mathcal{B}(\mathcal{S})$, is of the form

$\rho_{DS}=\sum_{0\leq i\leq j<d}p_{ij}\ket{D_{ij}}\bra{D_{ij}},$ (2)

with $p_{ij}\geq 0,\;\forall\;i,j$ and $\sum_{ij}p_{ij}=1$.

######  Lemma 1.1.

Every symmetric state, $\rho_{S}\in\mathcal{B}(\mathcal{S})$, can be written
as the sum of a DS state, $\rho_{DS}$, and a traceless symmetric contribution,
$\sigma_{CS}$, which contains all coherences between Dicke states, i.e.,

$\rho_{S}=\rho_{DS}+\sigma_{CS}=\sum_{0\leq i\leq
j<d}p_{ij}\ket{D_{ij}}\bra{D_{ij}}+\underset{{(i,j)\neq(k,l)}}{\sum_{ij}\sum_{kl}}\left({\alpha_{ij}^{kl}}\ket{D_{ij}}\bra{D_{kl}}+\emph{h.c.}\right),$
(3)

with ${\alpha_{ij}^{kl}}\in\mathbb{C}$ and
${\alpha_{ij}^{kl}}=({\alpha_{kl}^{ij}})^{*}$.

###  1.2  Separability, EWs and copositivity

######  Definition 1.3.

A bipartite symmetric state $\rho_{S}\in\mathcal{B}(\mathcal{S})$ is separable
(not entangled) if it can be written as a convex combination of projectors
onto pure symmetric product states, i.e.,

$\rho_{S}=\sum_{i}p_{i}\ket{e_{i}e_{i}}\bra{e_{i}e_{i}},$ (4)

with $p_{i}\geq 0$, $\sum_{i}p_{i}=1$ and
$\ket{e_{i}}=\sum_{i}e_{i}^{(k)}\ket{k}$, where $e_{i}^{(k)}\in\mathbb{C}$ and
$\\{\ket{k}\\}_{k=0}^{d-1}$ is an orthonormal basis in $\mathbb{C}^{d}$. If a
decomposition of this form does not exist, then $\rho_{S}$ is entangled.

Figure 1: Pictorial representation of the set of bipartite symmetric separable
states $\mathcal{D}_{S}$ (Sym) embedded into the set of bipartite separable
states. The cylinder represents the separable set $\mathcal{D}$. The
discontinuous (red) line corresponds to the extremal points (of the form
$\ket{e,f}$) generating the set and the continuous (blue and green) lines
corresponds to the respective boundaries (necessarily requiring description as
density matrices with rank $>1$ but not maximal). Both the separable and the
symmetric separable sets share extremal points of the form $\ket{e,e}$, here
represented by the black dots.

We denote by ${\mathcal{D}}$, the compact set of separable quantum states and
by $\mathcal{D}_{S}$, its analogous symmetric counterpart, which is also
compact (see Fig.1). As a consequence of the Hahn-Banach theorem, the set
$\mathcal{D}_{S}$ admits also a dual description in terms of its dual cone,
$\mathcal{P_{S}}$, defined as the set of the operators $W$ fulfilling

$\mathcal{P}_{S}=\\{W=W^{\dagger}\;\;\mbox{s.t}\;\langle W,\rho\rangle\geq
0\leavevmode\nobreak\
,\forall\rho_{S}\in\mathcal{D}_{S}\\}\leavevmode\nobreak\ ,$ (5)

where $\langle W,\rho\rangle\equiv\mbox{Tr}(W^{\dagger}\rho)$ is the Hilbert-
Schmidt scalar product.

######  Definition 1.4.

A Hermitian operator, $W\in\mathcal{P}_{S}$, is an entanglement witness (EW)
of symmetric states if, and only if, it satisfies the following properties:

  * 1.

$\emph{Tr}(W\rho_{S})\geq 0,\leavevmode\nobreak\
\forall\;\rho_{S}\in\mathcal{D}_{S}$ ,

  * 2.

There exists at least one symmetric state $\rho_{S}$ such that
$\emph{Tr}(W\rho_{S})<0$ .

Notice that, by definition, the set of separable symmetric states,
$\mathcal{D}_{S}$, satisfies the inclusion
$\mathcal{D}_{S}\subset\mathcal{D}$, but $\mathcal{P}\subset\mathcal{P}_{S}$,
where $\mathcal{P}$ is the dual cone of the convex set $\mathcal{D}$, i.e.,

$\mathcal{P}=\\{W=W^{\dagger}\;\;\mbox{s.t}.\;\langle W,\rho\rangle\geq
0\leavevmode\nobreak\ ,\forall\rho\in\mathcal{D}\\}\leavevmode\nobreak\ .$ (6)

In other words, any EW acting on $\mathcal{H}$ that detects an entangled state
belongs to $\mathcal{P}_{S}$, but the converse is not necessarily true (see
Fig.2).

Figure 2: Pictorial structure of the quantum states in
$\mathbb{C}^{d}\otimes\mathbb{C}^{d}$ for $d>5$. Each set contains the sets
displayed inside. The colored region (green) represents the set of symmetric
states (SYM). Note that, while for $d>5$ there exist diagonal symmetric (DS)
states that are PPT-entangled, as represented in the figure, for $d<5$ all
PPT-entangled DS state are necessarily separable (SEP) (see the text for
details).

EWs are either decomposable or non-decomposable.

######  Definition 1.5.

An EW, $W$, is said to be decomposable (non-decomposable) if it can (cannot)
be written as

$W=P+Q^{T_{B}},$ (7)

with $P,Q\succeq 0$. Here $T_{B}\equiv\mathds{1}_{A}\otimes T$ denotes the
partial transposition w.r.t. subsystem $B$, where $T$ stands for the usual
matrix tranposition.

It is easy to show that non-decomposable EWs are the only candidates to detect
PPT entanglement. In fact, given a PPTES $\rho$, for any decomposable EW $W$,
it is

$\mbox{Tr}(W\rho)=\mbox{Tr}(P\rho)+\mbox{Tr}(Q^{T_{B}}\rho)=\mbox{Tr}(P\rho)+\mbox{Tr}(Q\rho^{T_{B}})\geq
0\leavevmode\nobreak\ ,$ (8)

where we have used the properties of the trace and the positive
semidefiniteness of the operators $P$ and $Q$.
In particular from Def.(1.5) it follows that a EW is non-decomposable iff it
detects at least one PPTES.

Remarkably, despite the apparent simplicity of the symmetric subspace due to
its reduced dimensionality ($d(d+1)/2$ instead of $d^{2}$), entanglement
characterization remains, in general, an open problem. For generic symmetric
states, sparsity is preserved when the state is expressed in the computational
basis but it is lost when the partial transposition is performed. However, for
DS states, the corresponding partial transpose remains highly sparse and can
be reduced to an associated matrix, $M_{d}(\rho_{DS})$, of dimensions $d\times
d$, while generically $\rho_{S}^{T_{B}}$ is a matrix of dimension $d^{2}\times
d^{2}$.

######  Definition 1.6.

The partial transpose of every $\rho_{DS}\in\mathcal{B}(\mathcal{S})$ has the
form

$\rho_{DS}^{T_{B}}=M_{d}(\rho_{DS})\bigoplus_{0\leq i\neq
j<d}\left(p_{ij}/2\right)\leavevmode\nobreak\ ,$ (9)

where $M_{d}(\rho_{DS})$, which arises from the partially transposed matrix of
a DS state in the computational basis, is defined as the $d\times d$ matrix
with (non-negative) entries

$M_{d}(\rho_{DS}):=\left(\begin{matrix}p_{00}&p_{01}/2&\cdots&p_{0,d-1}/2\\\
p_{01}/2&p_{11}&\cdots&p_{1,d-1}/2\\\ \vdots&\vdots&\ddots&\vdots\\\
p_{0,d-1}/2&p_{1,d-1}/2&\cdots&p_{d-1,d-1}\end{matrix}\right)\leavevmode\nobreak\
.$ (10)

As shown in previous works [19, 13], deciding if a DS state $\rho_{DS}$ is
separable, is equivalent to check the membership of $M_{d}(\rho_{DS})$ to the
cone of completely positive matrices $\mathcal{CP}_{d}$, i.e., the cone formed
by those $d\times d$ matrices $A_{d}$ that admit a decomposition of the type
$A_{d}=BB^{T}$, where $B$ is a $d\times k$ matrix, for some $k>1$, with
$B_{ij}\geq 0,B_{ij}\in\mathbb{R}$. Thus, if $\rho_{DS}$ is separable, then
its associated matrix of Eq.(10) must satisfy $M_{d}(\rho_{DS})=BB^{T}$ [19].
This correspondence can be recast, equivalently, in the dual cone of
$\mathcal{CP}_{d}$, i.e., in the cone $\mathcal{COP}_{d}$ of copositive
matrices. As a result, copositive matrices act effectively as EWs for DS
states. Below we provide the definition of a copositive matrix together with
some properties that will be useful in the following.

######  Definition 1.7.

A real symmetric matrix, $H$, is copositive if, and only if,
$\vec{x}^{T}H\vec{x}\geq 0,\leavevmode\nobreak\ \forall\vec{x}\geq 0$
component-wise

.

It is easy to see that the diagonal elements of a copositive matrix must be
non negative, i.e., $H_{ii}\geq 0$, while negative elements $H_{ij}$ must
fulfill $\sqrt{H_{ii}H_{jj}}\geq-H_{ij}$. Clearly, every positive semidefinite
matrix is also copositive but the converse is, generically, not true. In fact,
testing membership to the cone of copositive matrices is known to be a co-NP-
hard problem [20]111The co-NP problems are the complementary of the decision
problems in NP., and only for $d\leq 4$, copositivity can be assessed
analytically [21, 22].

Finally, among copositive matrices, we distinguish extreme and exceptional
copositive matrices that stand out for their impossibility to be decomposed.

######  Definition 1.8.

A $d\times d$ copositive matrix $H$ is said to be extreme if $H=H_{1}+H_{2}$
with $H_{1},H_{2}$ copositive, implies $H_{1}=aH,H_{2}=(1-a)H$ for all
$a\in[0,1]$

.

######  Definition 1.9.

A $d\times d$ copositive matrix $H$ is said to be exceptional if, and only if,
$H$ cannot be decomposed as the sum of a positive semidefinite matrix
($\mathcal{PSD}_{d}$), and a symmetric entry-wise non-negative matrix
($\mathcal{N}_{d}$), i.e.,
$H\in\mathcal{COP}_{d}\setminus(\mathcal{PSD}_{d}+\mathcal{N}_{d})$

Remarkably, it has been shown that for $d<5$ there are no exceptional
copositive matrices, meaning that, in this case,
$\mathcal{COP}_{d}=\mathcal{PSD}_{d}+\mathcal{N}_{d}$ [23]. In Fig.3 we
illustrate, schematically, the relation between the aforementioned classes of
copositive matrices.

Figure 3: Pictorial representation of the cone of copositive
$\mathcal{COP}_{d}$ and the cones $\mathcal{PSD}_{d}$ and $\mathcal{N}_{d}$.
The striped region has been overmagnified for clarity and represents the
convex hull of the cones $\mathcal{PSD}_{d}$ and $\mathcal{N}_{d}$, denoted as
$\mathcal{PSD}_{d}+\mathcal{N}_{d}$. Note that exceptional copositive matrices
exist only for $d>5$ (green). Extremal copositive matrices lie at the border
of the cone $\mathcal{COP}_{d}$.

##  2 Copositive matrices as EWs

Using the above definitions we can now show which copositive matrices lead to
EWs. We prove explicitly how to construct a decomposable (non-decomposable) EW
from a non-exceptional (exceptional) copositive matrix. Since decomposable EWs
cannot detect bound entanglement, one is tempted to believe that separability
in the symmetric subspace is equivalent to the analysis of exceptional
copositive matrices. However, as we shall see later, this is not necessarily
the case, and non-exceptional copositive matrices also play a relevant role in
detecting bound entanglement. Our findings are summarized in the following
theorems.

######  Theorem 2.1.

Each copositive matrix $H=\sum_{i,j=0}^{d-1}H_{ij}\ket{i}\\!\bra{j}$, with at
least one negative entry $H_{mn}=H_{nm}<0$ ($m\neq n$), leads to an EW on
$\mathcal{S}$ of the form
$W=(H^{ext})^{T_{B}}=\sum_{{i,j=0}}^{d-1}H_{ij}\ket{ij}\\!\bra{ji}$.

###### Proof.

(i) We extend $H$ to the symmetric subspace as
$H^{ext}=\sum_{i,j=0}^{d-1}H_{ij}\ket{i}\bra{j}\otimes\ket{i}\bra{j}$, and
denote $W=(H^{ext})^{T_{B}}$. It is straightforward to show that
$\mbox{Tr}(W\rho_{S})\geq 0$ for all $\rho_{S}\in\mathcal{D}_{S}$ since
$\bra{ee}W\ket{ee}=\braket{ee^{*}}{H^{ext}}{ee^{*}}=\sum_{ij}|c_{i}|^{2}H_{ij}|c_{j}|^{2}=\vec{x}^{T}H\vec{x}\geq
0$, for every state $\ket{e}=\sum_{i=0}^{d-1}c_{i}\ket{i}$, where
$c_{i}\in\mathbb{C}$, and $\\{\ket{i}\\}_{i=0}^{d-1}$ is orthonormal basis of
$\mathbb{C}^{d}$.

(ii) The diagonalization of $W$ shows that its eigenvectors are
$\\{\ket{ii},\ket{\psi_{ij}^{\pm}}=(\ket{ij}\pm\ket{ji})/\sqrt{2}\\}$, with
corresponding eigenvalues $\\{H_{ii},\pm H_{ij}\\}$, i.e.,

$\begin{split}W&=(H^{ext})^{T_{B}}=\sum_{i=0}^{d-1}H_{ii}\ket{ii}\bra{ii}+\\\
&+\sum_{i<j}^{d-1}H_{ij}\ket{\psi_{ij}^{+}}\bra{\psi_{ij}^{+}}-\sum_{i<j}^{d-1}H_{ij}\ket{\psi_{ij}^{-}}\bra{\psi_{ij}^{-}},\end{split}$
(11)

where $\ket{\psi_{ij}^{+}}=\ket{D_{ij}}$ and $\ket{ii}=\ket{D_{ii}}$. Notice
that the $d(d-1)/2$ eigenvectors corresponding to the projectors
$\ket{\psi_{ij}^{-}}\bra{\psi_{ij}^{-}}$, are orthogonal to the symmetric
subspace and, therefore, can be discarded by projecting on the symmetric
subspace ($\mathcal{S}$).

$W_{S}=\Pi_{S}W\Pi_{S}=\sum_{i=0}^{d-1}H_{ii}\ket{D_{ii}}\bra{D_{ii}}+\sum_{i<j}^{d-1}H_{ij}\ket{D_{ij}}\bra{D_{ij}}.$
(12)

Finally, since copositivity requires that $H_{ii}\geq 0$ $\forall i$, $W_{S}$
is an EW iff at least one of the remaining eigenvalues is negative, i.e., if
$H$ has at least one negative element $H_{mn}=H_{nm}<0$ for some $m\neq n$. It
is now trivial to see that $W_{S}$ indeed detects, at least, the entangled
state $\ket{\psi_{mn}^{+}}$ since
$\mbox{Tr}(W_{S}\ket{\psi_{mn}^{+}}\bra{\psi_{mn}^{+}})=H_{mn}<0$. To
conclude, if $W_{S}$ is an EW in the symmetric subspace, so it is $W$ given by
Eq. 11. ∎

######  Theorem 2.2.

If $H=H_{\mathcal{N}}+H_{\mathcal{PSD}}$ (i.e., H is non-exceptional) with at
least one negative element, then
$W=\Pi_{S}\left(H^{ext}_{\mathcal{N}}\right)^{T_{B}}\Pi_{S}+\Pi_{S}\left(H^{ext}_{\mathcal{PSD}}\right)^{T_{B}}\Pi_{S}=\Pi_{S}\left(H^{ext}_{\mathcal{N}}\right)^{T_{B}}\Pi_{S}+\left(H^{ext}_{\mathcal{PSD}}\right)^{T_{B}}$
is a decomposable EW. The converse is also true, that is, if
$W_{S}=\Pi_{S}(H^{ext})^{T_{B}}\Pi_{S}=P+Q^{T_{B}}$ with $P,Q\succeq 0$, then
$H=H_{\mathcal{N}}+H_{\mathcal{PSD}}$.

Notice that it is always possible to decompose a copositive matrix in such a
way that the semidefinite positive part $H^{ext}_{\mathcal{PSD}}$ is
symmetric.

###### Proof.

$\Rightarrow$. Assume that $H=H_{\mathcal{N}}+H_{\mathcal{PSD}}$, with
$H_{\mathcal{N}}\in\mathcal{N},\leavevmode\nobreak\
H_{\mathcal{PSD}}\in\mathcal{PSD}$ and $H_{mn}=H_{nm}<0$. Then
$(H_{\mathcal{PSD}})_{mn}=(H_{\mathcal{PSD}})_{nm}<0$ by construction and
$(H^{ext}_{\mathcal{PSD}})^{T_{B}}\nsucceq 0$, but
$\Pi_{S}\left(H^{ext}_{\mathcal{N}}\right)^{T_{B}}\Pi_{S}\succeq 0$. The
operator
$W=\Pi_{S}\left(H^{ext}_{\mathcal{N}}\right)^{T_{B}}\Pi_{S}+\left(H^{ext}_{\mathcal{PSD}}\right)^{T_{B}}$
is a decomposable EW: (i) $\braket{e,e}{W}{e,e}\geq 0$ because $H$ is
copositive, (ii) $W$ has at least one negative eigenvalue associated to the
negative element $(H_{\mathcal{PSD}})_{mn}$, (iii) $W$ is decomposable since
it can be written as $W=P+Q^{T_{B}}$ with
$P=\Pi_{S}\left(H^{ext}_{\mathcal{N}}\right)^{T_{B}}\Pi_{S}\succeq 0$ and
$Q=\Pi_{S}H^{ext}_{\mathcal{PSD}}\Pi_{S}=H^{ext}_{\mathcal{PSD}}\succeq 0$.

$\Leftarrow$ Since $W=\Pi_{S}\left(H^{ext}\right)^{T_{B}}\Pi_{S}$ is an EW,
there exists at least one negative element of $H$, $H_{mn}=H_{nm}<0$. We
construct $Q=H_{mm}|mm\rangle\langle mm|+H_{nn}|nn\rangle\langle
nn|+H_{mn}(|mn\rangle\langle mn|+|nm\rangle\langle nm|)$. By construction $Q$
is symmetric and is $Q\notin\mathcal{N}$, $Q^{T_{B}}\nsucceq 0$ and $Q\succeq
0$, where the last inequality holds because copositivity of $H$ implies
$\sqrt{H_{mm}H_{nn}}\geq-H_{mn}$. Hence, we can identify
$Q=H^{ext}_{\mathcal{PSD}}$. Now, define $P=\Pi_{S}(\sum_{i,j\neq
m,n}H_{ij}|ij\rangle\langle ji|)\Pi_{S}$, which can be expressed as
$P=\sum_{i,j\neq m,n}H_{ii}|ii\rangle\langle ii|+\sum_{i,j\neq
m,n}(H_{ij}/2)(|ij\rangle\langle ji|+|ij\rangle\langle ij|+|ji\rangle\langle
ji|+|ji\rangle\langle ij|)$. Clearly $P\succeq 0,P\in\mathcal{N}$, so that
$P=\Pi_{S}(H^{ext}_{\mathcal{N}})^{T_{B}}\Pi_{S}$ and
$H=H_{\mathcal{N}}+H_{\mathcal{PSD}}$. ∎

Let us illustrate Theorem 2.2 by considering the following copositive matrix
in $d=3$

$H=\begin{pmatrix}[r]1&1&1\\\ 1&1&-1\\\
1&-1&1\end{pmatrix}\leavevmode\nobreak\ .$ (13)

A possible decomposition of $H=H_{\mathcal{PSD}}+H_{\mathcal{N}}$, with
$H_{\mathcal{PSD}}\in\mathcal{PSD}_{3}$ and
$H_{\mathcal{N}}\in\mathcal{N}_{3}$, is given by:

$H_{\mathcal{PSD}}=\begin{pmatrix}[r]0&0&0\\\ 0&1&-1\\\
0&-1&1\end{pmatrix},\quad H_{\mathcal{N}}=\begin{pmatrix}[r]1&1&1\\\ 1&0&0\\\
1&0&0\end{pmatrix}\leavevmode\nobreak\ .$ (14)

The associated EW $W=P+Q^{T_{B}}$, with
$P=\Pi_{S}(H_{\mathcal{N}}^{ext})^{T_{B}}\Pi_{S}$, and
$Q=H_{\mathcal{PSD}}^{ext}$ reads

$P=\frac{1}{2}\begin{pmatrix}\begin{matrix}2&0&0\\\ 0&1&0\\\
0&0&1\end{matrix}&\vline&\begin{matrix}0&0&0\\\ 1&0&0\\\
0&0&0\end{matrix}&\vline&\begin{matrix}0&0&0\\\ 0&0&0\\\ 1&0&0\end{matrix}\\\
\hline\cr\begin{matrix}0&1&0\\\ 0&0&0\\\
0&0&0\end{matrix}&\vline&\begin{matrix}1&0&0\\\ 0&0&0\\\
0&0&0\end{matrix}&\vline&\begin{matrix}0&0&0\\\ 0&0&0\\\ 0&0&0\end{matrix}\\\
\hline\cr\begin{matrix}0&0&1\\\ 0&0&0\\\
0&0&0\end{matrix}&\vline&\begin{matrix}0&0&0\\\ 0&0&0\\\
0&0&0\end{matrix}&\vline&\begin{matrix}1&0&0\\\ 0&0&0\\\
0&0&0\end{matrix}\end{pmatrix}\leavevmode\nobreak\ ,\quad
Q=\begin{pmatrix}\begin{matrix}[c]0&0&0\\\ 0&0&0\\\
0&0&0\end{matrix}&\vline&\begin{matrix}[c]0&0&0\\\ 0&0&0\\\
0&0&0\end{matrix}&\vline&\begin{matrix}[c]0&0&0\\\ 0&0&0\\\
0&0&0\end{matrix}\\\ \hline\cr\begin{matrix}0&0&0\\\ 0&0&0\\\
0&0&0\end{matrix}&\vline&\begin{matrix}0&0&0\\\ 0&1&0\\\
0&0&0\end{matrix}&\vline&\begin{matrix}\leavevmode\nobreak\ 0&0&0\\\
\leavevmode\nobreak\ 0&0&\leavevmode\hphantom{0}\hbox
to0.0pt{\hss\leavevmode\resizebox{5.00002pt}{0.0pt}{$-$}}1\\\
\leavevmode\nobreak\ 0&0&0\end{matrix}\\\ \hline\cr\begin{matrix}0&0&0\\\
0&0&0\\\ 0&0&0\end{matrix}&\vline&\begin{matrix}0&0&0\\\ 0&0&0\\\
0&\leavevmode\hphantom{0}\hbox
to0.0pt{\hss\leavevmode\resizebox{5.00002pt}{0.0pt}{$-$}}1&0\end{matrix}&\vline&\begin{matrix}0&0&0\\\
0&0&0\\\ 0&0&1\end{matrix}\end{pmatrix}\leavevmode\nobreak\ ,$

but the resulting EW, $W^{\prime}=P^{\prime}+Q^{\prime T_{B}}$, detects
exactly the same states in the symmetric subspace.

The link between non-exceptional copositive matrices and decomposable EWs
extends also to exceptional copositive matrices and non-decomposable EWs in
the symmetric subspace.

######  Theorem 2.3.

Associated to each exceptional copositive matrix $H$ (i.e.,
$H\in\mathcal{COP}\setminus(\mathcal{PSD}+\mathcal{N})$) with at least one
negative entry, there is a non-decomposable EW, $W=(H^{ext})^{T_{B}}$, able to
detect symmetric PPTES.

###### Proof.

For any $H\in\mathcal{COP}\setminus(\mathcal{PSD}+\mathcal{N})$, $H$ always
admits a decomposition of the form $H=H_{\mathcal{N}}+H_{\star}$, where
$H_{\mathcal{N}}$ is a non-negative symmetric matrix and $H_{\star}$ has at
least one negative eigenvalue but is not positive semidefinite. The associated
EW $W=P+Q^{T_{B}}$ with $P=\Pi_{S}(H^{ext}_{\mathcal{N}})^{T_{B}}\Pi_{S}$ and
$Q=H^{ext}_{\star}$, is a non-decomposable EW since $P\succeq 0$ but
$Q\nsucceq 0$. The operator $W=(H^{ext})^{T_{B}}$ is also a non-decomposable
EW. ∎

######  Corollary 2.1 (From [13]).

Since for $d<5$ every copositive matrix is not exceptional (i.e.,
$H=H_{\mathcal{PSD}}+H_{\mathcal{N}}$), all EWs of DS states in $d=3$ and
$d=4$ are decomposable.

The above corollary rephrases the fact that PPT criterion is necessary and
sufficient to assess separability for bipartite DS states
$\rho_{DS}\in\mathcal{B}(\mathbb{C}^{d}\otimes\mathbb{C}^{d})$ for $d<5$.

##  3 Symmetric PPTES

Let us briefly summarize what we have seen so far. The fact that each DS
state, $\rho_{DS}\in\mathcal{B}(\mathbb{C}^{d}\otimes\mathbb{C}^{d})$, is
associated to a matrix $M_{d}(\rho_{DS})$ (see Eq.(10)), allows to reformulate
the problem of entanglement characterization as the equivalent problem of
checking the membership of $M_{d}(\rho_{DS})$ to the cone of completely
positive matrices $\mathcal{CP}_{d}$. Equivalently, according to the dual
formulation, any entangled state $\rho_{DS}$ is detected by an EW $W$ which
can be constructed from a copositive matrix $H$. PPT entangled diagonal
symmetric states (PPTEDS) can only be detected by non-decomposable EWs, which
correspond to exceptional copositive matrices. Since for $d<5$, all copositive
matrices $H$ are of the form $H=H_{\mathcal{PSD}}+H_{\mathcal{N}}$, all EWs
defined as $W=(H^{ext})^{T_{B}}$ are necessarily of the form $W=P+Q^{T_{B}}$,
with $P,Q\succeq 0$, meaning that for $d<5$ there are not PPTEDS.

However, for $d>5$, this is not the case, since there exist exceptional
copositive matrices, i.e., $H\notin\mathcal{PSD}_{d}+\mathcal{N}_{d}$. Thus,
detecting entanglement of $\rho_{DS}$ in $d\geq 5$, is equivalent to checking
membership of the corresponding copositive matrix
$H\in\mathcal{COP}_{d}\setminus(\mathcal{PSD}_{d}+\mathcal{N}_{d})$, which is,
in general, a co-NP-hard problem [20].

What can we say about symmetric PPTES $\rho_{S}$ that are not DS? In this
section, we tackle the problem of entanglement detection for generic states
$\rho_{S}\in\mathcal{B}(\mathbb{C}^{d}\otimes\mathbb{C}^{d})$ in arbitrary
dimension $d$. Following the argument given above, we split our analysis in
two different scenarios, namely when $d\geq 5$ and $d<5$. Remarkably, even
outside of the DS paradigm, we find that copositive matrices lie at the core
of non-decomposable EWs for symmetric PPTES in arbitrary dimensions.

###  3.1 Symmetric PPTES in $d\geq 5$

The fact that for $d\geq 5$ there exist exceptional copositive matrices with
at least one negative entry which lead to non-decomposable EWs in
$\mathcal{S}$, implies that (i) such EW can detect a PPTEDS, and (ii) the same
EW is able to detect other symmetric, but not DS, PPTES "around" it.

######  Theorem 3.1.

Let $\rho_{DS}$ be a PPTEDS. Then any symmetric state
$\rho_{S}=\rho_{DS}+\sigma_{CS}$, such that $\rho_{S}^{T_{B}}\geq 0$, is PPT
entangled.

###### Proof.

Since $\rho_{DS}$ is a PPTEDS state there exists an exceptional copositive
matrix $H$ and an associated non decomposable EW $W$ such that
$\mbox{Tr}(W\rho_{DS})<0$. It follows that
$\mbox{Tr}(W\rho_{S})=\mbox{Tr}(W(\rho_{DS}+\sigma_{CS}))=\mbox{Tr}(W\rho_{DS})=\mbox{Tr}(HM_{d}(\rho_{DS}))<0$,
so that $\rho_{S}$ is PPT entangled. ∎

The paradigmatic example of an exceptional copositive matrix in $d=5$, is the
so-called Horn matrix [24] which is the matrix associated to the quadratic
form
$\vec{x}^{T}H\vec{x}=(x_{1}+x_{2}+x_{3}+x_{4}+x_{5})^{2}-4x_{1}x_{2}-4x_{2}x_{3}-4x_{3}x_{4}-4x_{4}x_{5}-4x_{5}x_{1}$
. Exceptional copositive matrices of the Horn type, $H_{\mathcal{H}}$, can be
generated for any odd $d\geq 5$ [25], and are of the form

$H_{\mathcal{H}}=\begin{pmatrix}[r]1&-1&1&1&\cdots&\cdots&1&1&1&-1\\\
-1&1&-1&1&\ddots&&\ddots&1&1&1\\\ 1&-1&1&-1&1&&&\ddots&1&1\\\
1&1&-1&1&-1&\ddots&&&\ddots&1\\\ \vdots&\ddots&1&-1&1&\ddots&&&&\vdots\\\
\vdots&&&\ddots&\ddots&\ddots&\ddots&\ddots&&\vdots\\\
1&\ddots&&&&\ddots&1&-1&1&1\\\ 1&1&\ddots&&&\ddots&-1&1&-1&1\\\
1&1&1&\ddots&&&1&-1&1&-1\\\
-1&1&1&1&\cdots&\cdots&1&1&-1&1\end{pmatrix}\leavevmode\nobreak\ .$ (15)

Since the $H_{\mathcal{H}}$ is exceptional and has negative entries, it leads
to a non-decomposable EW $W=(H^{ext})^{T_{B}}$, that can be used to detect
PPTEDS in any odd dimension $d\geq 5$. Moreover, due to Th.(3.1), by adding
suitable coherences to such states, the same EW can be used to certify PPT-
entanglement also in whole families of symmetric states. Below we provide one
of these families.

######  Corollary 3.1.

Given a PPTEDS state, $\rho_{DS}$, any symmetric state of the form
$\rho_{S}=\rho_{DS}+\sigma_{CS}$, with
$\sigma_{CS}=\sum_{i<j}(\alpha_{ij}\ket{D_{ii}}\bra{D_{jj}}+\emph{h.c.})$ and
$|\alpha_{ij}|\leq\frac{p_{ij}}{2}$ is PPT-entangled.

###### Proof.

The state, $\rho_{S}$, and its partial transpose, $\rho_{S}^{T_{B}}$, can be
cast as

$\displaystyle\rho_{S\;}=\tilde{M}_{d}(\rho_{S})\bigoplus_{i<j}\frac{p_{ij}}{2}\begin{pmatrix}1&1\\\
1&1\end{pmatrix}\leavevmode\nobreak\ ,$ (16)
$\displaystyle\rho_{S}^{T_{B}}=M_{d}(\rho_{DS})\bigoplus_{0\leq
i<j<d}\begin{pmatrix}p_{ij}/2&\alpha_{ij}\\\
\alpha_{ij}^{*}&p_{ij}/2\end{pmatrix}\leavevmode\nobreak\ ,$ (17)

with

$\tilde{M}_{d}(\rho_{S})=\begin{pmatrix}p_{00}&\alpha_{01}&\cdots&\alpha_{0,d-1}\\\
\alpha_{01}^{*}&p_{11}&\cdots&\alpha_{1,d-1}\\\
\vdots&\vdots&\vdots&\ddots&\\\
\alpha_{0,d-1}^{*}&\alpha_{1,d-1}^{*}&\cdots&p_{d-1,d-1}\end{pmatrix}\leavevmode\nobreak\
,$ $M_{d}(\rho_{DS})=\begin{pmatrix}p_{00}&p_{01}/2&\cdots&p_{0,d-1}/2\\\
p_{01}/2&p_{11}&\cdots&p_{1,d-1}/2\\\ \vdots&\vdots&\vdots&\ddots&\\\
p_{0,d-1}/2&p_{1,d-1}/2&\cdots&p_{d-1,d-1}\end{pmatrix}\leavevmode\nobreak\ .$

Positive semidefiniteness of ${\rho_{S}}^{T_{B}}$ implies
$|\alpha_{ij}|\leq\frac{p_{ij}}{2}$, so that the state $\rho_{S}$, generated
from a PPTEDS state, remains PPT-entangled – since it is detected by the same
non-decomposable EW – as long as the coherences respect the condition
$|\alpha_{ij}|\leq\frac{p_{ij}}{2}$. ∎

A further connection between copositive matrices and EWs appears when
considering extreme copositive matrices. For instance, let us consider the
(generalized) Horn matrix $H_{\mathcal{H}}$ of Eq.(15), and the so-called
Hoffmann-Pereira matrix $H_{\mathcal{HP}}$ [25, 26], which, besides of being
exceptional, is also extreme. For $d=7$, such copositive matrices take the
form

$H_{\mathcal{H}}=\begin{pmatrix}[r]1&-1&1&1&1&1&-1\\\ -1&1&-1&1&1&1&1\\\
1&-1&1&-1&1&1&1\\\ 1&1&-1&1&-1&1&1\\\ 1&1&1&-1&1&-1&1\\\ 1&1&1&1&-1&1&-1\\\
-1&1&1&1&1&-1&1\end{pmatrix}\leavevmode\nobreak\ ,$ (18)
$H_{\mathcal{HP}}=\begin{pmatrix}[r]1&-1&1&0&0&1&-1\\\ -1&1&-1&1&0&0&1\\\
1&-1&1&-1&1&0&0\\\ 0&1&-1&1&-1&1&0\\\ 0&0&1&-1&1&-1&1\\\ 1&0&0&1&-1&1&-1\\\
-1&1&0&0&1&-1&1\end{pmatrix}\leavevmode\nobreak\ .$ (19)

Let us inspect the action of both matrices, $H_{\mathcal{H}}$ and
$H_{\mathcal{HP}}$, on a DS state
$\rho_{DS}\in\mathcal{B}(\mathbb{C}^{7}\otimes\mathbb{C}^{7})$, described by
its associated $M_{7}(\rho_{DS})$ (see Eq.(10)):

$M_{d}(\rho_{DS})=\begin{pmatrix}[c]1&1&0&0&0&0&1/8\\\ 1&2&1&0&0&0&0\\\
0&1&2&1&0&0&1/4\\\ 0&0&1&2&1&0&0\\\ 0&0&0&1&2&1&0\\\ 0&0&0&0&1&2&1\\\
1/8&0&1/4&0&0&1&1\end{pmatrix}\leavevmode\nobreak\ .$ (20)

It can be easily checked that
$\mbox{Tr}(H_{\mathcal{HP}}M_{d}(\rho_{DS}))=\mbox{Tr}(\left(H^{ext}_{\mathcal{HP}}\right)^{T_{B}}\rho_{DS})=-\frac{1}{4}$.
Since both $H_{\mathcal{H}}$ and $H_{\mathcal{HP}}$ are exceptional copositive
matrices, $W_{\mathcal{HP}}=\left(H^{ext}_{\mathcal{HP}}\right)^{T_{B}}$ and
$W_{\mathcal{H}}=\left(H^{ext}_{\mathcal{H}}\right)^{T_{B}}$ are non-
decomposable EWs, so that $\rho_{DS}$ is a PPT entangled.

In contrast,
$\mbox{Tr}(H_{\mathcal{H}}M_{d}(\rho_{DS}))=\mbox{Tr}(W_{\mathcal{H}}\rho_{S})=0$,
indicating that $H_{\mathcal{H}}$ fails to detect this state. Moreover, as
stated by Th.(3.1), $W_{\mathcal{HP}}=(H^{ext}_{\mathcal{HP}})^{T_{B}}$,
detects, as well, many other states around the state given by Eq.(20). Given
the relationship between non-decomposable EWs and exceptional matrices, we
conjecture that extremality in the copositive cone correspond to optimality in
the set of EWs. In other words, copositive matrices that are both extreme and
exceptional lead to optimal non-decomposable EWs in the sense of [6]. We
complete this subsection with the following theorem regarding extreme
copositive matrices (see Fig.3)

######  Theorem 3.2.

Let $H$ be an extreme copositive matrix with at least one negative eigenvalue,
and at least one negative element $H_{ij}<0$. Then $H$ must be exceptional.

###### Proof.

$H$ cannot belong to neither $H_{PSD}$ nor to $H_{N}$ and, while it is
extremal, it cannot be a combination of their elements as well. A more
detailed proof can be found in Appendix B. ∎

###  3.2 Symmetric PPTES in $d<5$

In what remains, we are interested in symmetric PPTES of the form
$\rho_{S}=\rho_{DS}+\sigma_{CS}$ where $\rho_{DS}$ is separable, so that
$\mbox{Tr}(HM_{d}(\rho_{DS}))\geq 0$ for all copositive matrices $H$.
Moreover, since for $d<5$, every copositive matrix is non-exceptional, i.e.,
$H=H_{\mathcal{N}}+H_{\mathcal{PSD}}$, the corresponding witness
$W=(H^{ext})^{T_{B}}$ will always be decomposable. For this reason, coherences
are needed to create PPTES in low dimensional systems. Here we show that such
states symmetric PPTES can nevertheless be detected by EWs which are of the
form $W_{S}=W+W_{CS}$, that is by adding to the decomposable EW, $W$, a
convenient off-diagonal, symmetric contribution $W_{CS}$ which reads the
coherences of $\rho_{S}$.

For the sake of simplicity, we hereby consider symmetric states of the form

$\rho_{S}=\rho_{DS}+\sigma_{CS}=\sum_{ij}p_{ij}\ket{D_{ij}}\bra{D_{ij}}+\sum_{i\neq
j\neq k}(\alpha_{ijk}\ket{D_{ii}}\bra{D_{jk}})+\mbox{h.c.}\leavevmode\nobreak\
,$ (21)

with $p_{ij}\geq 0$ $\forall i,j\;$, $\sum p_{ij}=1$ and
$\alpha_{ijk}\in\mathbb{C}$.

Indeed, in this case, both $\rho_{S}$ and $\rho_{S}^{T_{B}}$ can be cast as a
direct sum of matrices, which highly simplifies our analysis. For instance,
for $d=3$, $\rho_{S}$ and $\rho_{S}^{T_{B}}$ are of the form

$\rho_{S}=\begin{pmatrix}[c]\frac{p_{02}}{2}&\alpha&\frac{p_{02}}{2}\\\
\alpha^{*}&p_{11}&\alpha\\\
\frac{p_{02}}{2}&\alpha^{*}&\frac{p_{02}}{2}\end{pmatrix}\oplus\begin{pmatrix}[c]p_{00}&\beta&\beta\\\
\beta^{*}&\frac{p_{12}}{2}&\frac{p_{12}}{2}\\\
\beta^{*}&\frac{p_{12}}{2}&\frac{p_{12}}{2}\end{pmatrix}\oplus\begin{pmatrix}[c]\frac{p_{01}}{2}&\frac{p_{01}}{2}&\gamma\\\
\frac{p_{01}}{2}&\frac{p_{01}}{2}&\gamma\\\
\gamma^{*}&\gamma^{*}&p_{22}\end{pmatrix}\leavevmode\nobreak\ ,$ (22)

$\rho_{S}^{T_{B}}=M_{d}(\rho_{DS})\oplus\begin{pmatrix}[c]\frac{p_{01}}{2}&\alpha&\beta\\\
\alpha^{*}&\frac{p_{12}}{2}&\gamma\\\
\beta^{*}&\gamma^{*}&\frac{p_{02}}{2}\end{pmatrix}\oplus\begin{pmatrix}[c]\frac{p_{02}}{2}&\beta&\gamma\\\
\beta^{*}&\frac{p_{01}}{2}&\alpha\\\
\gamma^{*}&\alpha^{*}&\frac{p_{12}}{2}\end{pmatrix}\leavevmode\nobreak\ ,$
(23)

where we have defined, for easiness of reading,
$\alpha\equiv\alpha_{120}=\alpha_{102}$,
$\beta\equiv\alpha_{012}=\alpha_{021}$ and
$\gamma\equiv\alpha_{201}=\alpha_{210}$. Such structure, which corresponds to
a particular direct sum decomposition of the total Hilbert space, bears
similitude with the so-called circulant states [27].

In order to investigate the existence of PPTES we focus on edge states, since
their low-dimensional ranks allow for a simpler analysis. By using a notation
common in the literature, we say that an edge state $\rho_{S}$ is of type
$(p,q)$ if $p=r(\rho_{S})$ and $q=r(\rho_{S}^{T_{B}})$ are the ranks of
$\rho_{S}$ and $\rho_{S}^{T_{B}}$, respectively. While symmetric states in
$d=3$ are, generically, of type $(6,9)$, PPT-entangled edge states must have
lower ranks. When dealing with states of the form of Eq.(21), we have found
numerically that at least two coherences must be considered. For instance, we
can set $\gamma=0$ and choose $\alpha$ and $\beta$ in such a way to lower the
value of $(p,q)$. Indeed, a direct inspection of Eqs.(22)-(23), shows that, by
setting $|\alpha|^{2}=p_{11}p_{02}/2$ and
$|\beta|^{2}=p_{02}(p_{01}p_{12}-2p_{02}p_{11})/4p_{12}$, it is possible to
attain a state of type $(5,7)$. Now, starting from a copositve matrix $H$, we
can construct a non-decomposable EW of the form

$W_{S}=\Pi_{S}(H^{ext})^{T_{B}}\Pi_{S}+\sum_{i\neq j\neq
k}W^{i}_{jk}\ket{D_{ii}}\bra{D_{jk}}+\mbox{h.c.}\leavevmode\nobreak\ ,$ (24)

where the coefficients $W^{i}_{jk}$ can be chosen to be real.

Let us illustrate the above results by providing some explicit examples. We
first consider the symmetric edge PPTES provided in [11]. The state is of the
form in Eq.(21) for $d=3$ (i.e., of the form in Eq.(22)), obtained from the DS
state $\rho_{DS}$ with parameters
$p_{00}=0.22,p_{11}=0.234/3,p_{22}=0.183,p_{01}=0.176,p_{02}=0.167/3,p_{12}=0.254$
and coherences $\alpha=0.167/3,\beta=-0.059/\sqrt{2},\gamma=0$. Its
entanglement was previously certified in [11] by means of the PPT symmetric
extension proposed in [28]. With the aid of the same method [29], we can
derive as well the corresponding non-decomposable EW $W$ via semidefinite
programming. The associated copositive matrix, $H$, is easily obtained from
the properly symmetrized EW, i.e., $W_{S}=\Pi_{S}W\Pi_{S}$, and has the form

$H\approx\left(\begin{matrix}0.003&10.39&100.57\\\ 10.39&59.31&-21.02\\\
100.57&-21.02&14.22\end{matrix}\right)\leavevmode\nobreak\ ,$ (25)

while the coefficients $W^{i}_{jk}$ are given by $W^{1}_{02}=23.20$ and
$W^{0}_{12}=-37.40$. If we restrict to the DS part of the state $\rho_{S}$, it
is trivial to check that $\mbox{Tr}(HM_{d}(\rho_{DS}))\geq 0$. This is by no
means a surprise, since for DS states, in $d<5$, the PPT condition implies
separability. For this reason, the coherences provided by the term
$\sigma_{CS}$ are necessary to induce the PPT entanglement. Remarkably, one
can vary the value of the coherences $\alpha$ and $\beta$ to obtain other
symmetric PPTES as certified by the EW, i.e., $\mbox{Tr}(W_{S}\rho_{S})<0$. In
fact, the EW $W_{S}$ can be used to derive families of PPT entangled states
obtained by adding to the state $\rho_{S}$ any coherent contribution
$\sigma_{CS}$ of the form of Eq.(22) that preserves the positivity of both the
state and its partial transpose. Indeed, also in the case $\gamma\neq 0$, as
long as the conditions $\rho_{S}\geq 0$ and $\rho_{S}^{T_{B}}\geq 0$ hold, the
same non-decomposable EW $W_{S}$, is able to detect, for suitable values of
its entries $W^{i}_{jk}$, a whole family of PPTES of the form of Eq.(22), as
depicted in Fig.4.

Figure 4:  The PPT entangled states detected by $W_{S}$ of Eq.(24) with
coefficients $|W^{1}_{02}|=23.20,|W^{0}_{12}|=-37.40$ (dark orange) as
compared to the whole family of PPT states $\rho_{S}$ of Eq.(22) with
$p_{00}=0.22,p_{11}=0.234/3,p_{22}=0.183,p_{01}=0.176,p_{02}=0.167/3,p_{12}=0.254$
(light orange).  Figure 5: The PPT entangled states detected by $W_{S}$ of
Eq.(24) with coefficients
$|W^{1}_{02}|=\frac{4595}{191},|W^{0}_{12}|=\frac{6114}{113}$ (dark orange) as
compared to the whole family of PPT states $\rho_{S}$ of Eq.(22) with
$p_{00}=p_{11}=p_{12}=\frac{1848}{7625}$, $p_{22}=\frac{464}{7625}$,
$p_{01}=\frac{231}{1525}$, $p_{02}=\frac{462}{7625}$ (light orange).

In Fig.5, we display a new example of a symmetric PPTES $\rho_{S}$ of the form
of Eq.(22), found by semidefinite programming. Also in this case, we have
found a non-decomposable EW $W_{S}$ of the form of Eq.(24), with coefficients
$W^{1}_{02}=\frac{4595}{191}$ and $W^{0}_{12}=-\frac{6114}{113}$ and whose
associated copositive matrix is given by

$H=\left(\begin{matrix}1/172&1009/151&11025/68\\\ 1009/151&1803/22&-5829/65\\\
11025/68&-5829/65&1224/7\end{matrix}\right)\leavevmode\nobreak\ .$ (26)

Similarly, we can use the same procedure to derive families of PPT entangled
symmetric states for $d=4$. In this case, we have found, numerically, that at
least three different coherences of the form of Eq.(21) are needed in order to
get a low-dimensional PPT entangled edge state. To the best of our knowledge,
there are no explicit examples of symmetric PPT entangled states in $d=4$. We
provide an example in Appendix A together with the non-decomposable EW that
detects it and its associated copositive matrix.

##  4 Conclusions

In this work we have studied the connection between EWs and copositive
matrices, showing how this class of matrices can be effectively used to detect
entanglement in bipartite symmetric states. First, we have proved that non-
decomposable (decomposable) EWs can be derived from exceptional (non-
exceptional) copositive matrices. Second, we have tackled the problem of the
entanglement detection for two symmetric qudits. In this context, our analysis
has shown that, on the one hand, for dimension $d\geq 5$, it is possible to
certify bound entanglement in any family of symmetric states constructed by
adding coherences to a PPTEDS state. Indeed, we have provided the explicit
expression of a symmetric PPTES in $d=7$, along with the exceptional matrix
that detects it. On the other hand, for dimension $d<5$, every copositive
matrix is not exceptional, so that a different approach is needed in order to
construct a non-decomposable EW. Nevertheless, we have shown that, also in
this case, an EW of this type can be derived from a not exceptional copositive
matrix by adding suitable coherent-like contributions. Indeed, we conjecture
that any symmetric PPTES of two qudits can be detected by a non-decomposable
EW of the form of Eq.(24). Our conjecture seems to be well grounded as pointed
out by the several examples we have found of symmetric PPTES detected with
this method, both for $d=3$ and $d=4$ (see Appendix A for further details).

##  Appendix A A symmetric PPTES in $d=4$

Here we present an explicit example of a PPTES
$\rho_{S}=\rho_{DS}+\sigma_{CS}$ of the form of Eq.(21) for $d=4$. The state
is given by: i) a DS state $\rho_{DS}$ with
$p_{00}=p_{02}=p_{03}=p_{11}=p_{22}=\frac{172+16\sqrt{2}}{1817}$,
$p_{01}=p_{13}=\frac{32+172\sqrt{2}}{1817}$,
$p_{11}=p_{12}=p_{23}=\frac{86+8\sqrt{2}}{1817}$,
$p_{33}=\frac{721-440\sqrt{2}}{1817}$; and ii) a coherence term $\sigma_{CS}$
with $\alpha=p_{00},\beta=-p_{01}/2$ and $\gamma=p_{01}/4$.

Again, to certify its entanglement we have used the symmetric extension [29],
which provides a non-decomposable EW, $W_{S}$, via semidefinite programming.
Such EW is of the form of Eq.(24), with coefficients
$W^{0}_{23}=\frac{6526}{321},W^{1}_{03}=-\frac{1896}{107},W^{2}_{13}=-\frac{549}{1238}$
and has an associated copositive matrix

$H=\left(\begin{matrix}21/3590&9425/1571&4853/464&1111/28\\\
9425/1571&1293/88&2122/145&220/323\\\ 4853/464&2122/145&6/5951&1355/3014\\\
1111/28&220/323&1355/3014&862/7403\end{matrix}\right)\leavevmode\nobreak\ .$
(27)

Let us observe that, despite the fact that $H$ of Eq.(27) does not have any
negative matrix element, the corresponding $EW$ has nevertheless a negative
eigenvalue. This observation makes clearer, once more, the fact that, in
$d<5$, differently from the case $d\geq 5$, the possibility to detect a PPTES
relies exclusively on a convenient choice of the coherences $W^{i}_{jk}$.

##  Appendix B An alternative proof of Theorem 3.2

###### Proof.

Since $H$ is an extreme copositive matrix, it only admits the trivial
decomposition $H=H_{1}+H_{2}$ with $H_{1}=aH$ and $H_{2}=(1-a)H$ copositive,
for every $a\in[0,1]$. Let $H=H_{\mathcal{PSD}}+H_{\mathcal{N}}$ where
$H_{\mathcal{PSD}}$ is a positive semidefinite matrix and $H_{\mathcal{N}}$ is
a non-negative matrix. We are left with three possibilities: i)
$H_{1}=H_{\mathcal{PSD}}$ and $H_{2}=H_{\mathcal{N}}$, ii)
$H_{1}=H_{\mathcal{N}}$ and $H_{2}=H_{\mathcal{PSD}}$, iii)
$H=H_{1}+H_{2}=H_{\mathcal{N}}$. i) Since $H$ has at least one negative
eigenvalue, the same holds true also for $H_{1}=aH$, so that $H_{1}$ is not
positive semidefinite. ii) An analogous consideration on $H_{2}$ leads to the
same conclusion also in this case, so that $H$ is exceptional. iii)
$H=H_{1}+H_{2}=H_{\mathcal{N}}$. Since $H$ has at least one negative entry
$H_{ij}<0$ it cannot be a non-negative matrix. ∎

We acknowledge financial support from ERC-AdG NOQIA; Spanish Agencia Estatal
de Investigación (AEI): Severo Ochoa Grant No. CEX2019-000910-S,
PID2019-107609GB-100, PID2019-106901GB-I00/10.13039 / 501100011033, MINCIN-EU
QuantERA MAQS PCI2019-111828-2, 10.13039/501100011033; Generalitat de
Catalunya: AGAUR 2017-SGR-1341,2017-SGR-1127, CERCA Program and QuantumCAT
001-P-001644, QuantumCAT U16-011424 co-funded by ERDF Operational Program of
Catalonia 2014-2020; Fundació Privada Cellex; Fundació Mir-Puig; EU Horizon
2020 FET-OPEN OPTOLogic: Grant No. 899794; and the National Science Centre
Poland-Symfonia Grant No. 2016-20-W-ST4-00314. J.T. thanks the Alexander von
Humboldt foundation for support.

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