
Symmetries are ubiquitous in Nature and they play a fundamental role in finding an efficient de-
scription of physical systems. The so-called symmetric states constitute an important class of quantum
systems to describe systems of indistinguishable particles [13]. Symmetric states can be mapped to
spin systems that are invariant under the exchange of particles and, moreover, they are spanned solely
by the largest-spin subspace in the Schur-Weyl duality representation [14]. The Dicke states [15] pro-
vide a convenient basis to represent symmetric states. Moreover, Dicke states are also experimentally
available [16–18] and they also appear naturally as ground states of physically relevant Hamiltonians,
such as the isotropic Lipkin-Meshkov-Glick model [19]. Much theoretical study has been devoted to
the characterization of entanglement in qubit symmetric states: 3-qubit symmetric states are separable
if, and only if, they satisfy the PPT criterion [13], but this is no longer the case already for N ≥ 4
[20, 21]. Despite diverse separability criteria exist for symmetric states (see e.g. [22]), the separability
problem remains still open.
Mixtures of Dicke states are symmetric states that are diagonal in the Dicke basis. These constitute
an important class of quantum states which naturally arise e.g. in dissipative systems such as photonic
or plasmonic one-dimensional waveguides [23]. Mixtures of Dicke states form a small subclass of
the symmetric states. They are the so-called Diagonal Symmetric (DS) states. In this context, the
separability problem has also gained interest. For instance, the best separable approximation (BSA,
[24]) has been found analytically for DS states for N -qubits [25]. In [26], it was conjectured that
N-qubit DS states are separable if, and only if, they satisfy the PPT criterion with respect to every
bipartition. The conjecture was proven by N. Yu in [27] where, moreover, he observed that PPT is a
sufficient and necessary condition for bipartite DS states of qudits with dimension 3 and 4, but becomes
NP-hard for larger dimensions. Within the N -qubit DS set it has been shown in [28] that there is a
family of states that violate the weak Peres conjecture [29]: those states are PPT-bound entangled
with respect to one partition, but they violate a family of permutationally invariant two-body Bell
inequalities [30–32].
In experiments, PPT-entangled states have also been recently observed. In the multipartite case,
the Smolin state has been prepared with four photons, using the polarization degree of freedom for
the qubit encoding [33, 34]. Very recently, although bound entanglement is the hardest to detect [35],
the Leiden-Vienna collaboration has reported the observation of bound entanglement in the bipartite
case with two twisted photons, combining ideas of complementarity [36] and Mutually Unbiased Bases
(MUBs) [37].
Here, we independently recover the results of N. Yu [27] by reformulating the problem in terms
of optimization in the cone of completely-positive
1
matrices. First, we revisit the problem of deter-
mining separability of two DS qudits in arbitrary dimensions. We show that it can be reformulated
in terms of a quadratic conic optimization problem [38]. In particular, we show that separability in
DS states is equivalent to the membership problem in the set of completely-positive matrices. The
equivalence between these two problems allows us to import/export ideas between entanglement theory
and non-convex quadratic optimization
2
. Second, we provide examples of entangled PPTDS states and
entanglement witnesses detecting them. Third, we give further characterization criteria for separability
in DS states in terms of the best diagonal dominant decomposition. Finally, we present a family of
N-qubit almost-DS states that are PPT with respect to each bipartition, but nevertheless entangled.
The word almost here means that by adding an arbitrarily small off-diagonal term (GHZ coherence)
to a family of separable DS N-qubits, the state becomes PPT-entangled.
1
Throughout this paper, the term completely-positive corresponds to the definition given in Def. 3.4 and it is not to
be confounded with the concept of a completely positive map that arises typically in a quantum information context.
2
Quadratic conic optimization problems appear naturally in many situations (see [38] and references therein). These
include economic modelling [39], block designs [40], maximin efficiency-robust tests [41], even Markovian models of DNA
evolution [42]. Recently, they have found their application in data mining and clustering [43], as well as in dynamical
systems and control [44, 45].
Accepted in Quantum 2017-12-28, click title to verify 2