Separability of diagonal symmetric states: a quadratic conic optimization problem

We study the separability problem in mixtures of Dicke states i.e., the separability of the so-called Diagonal Symmetric (DS) states. First, we show that separability in the case of DS in $C^d\otimes C^d$ (symmetric qudits) can be reformulated as a quadratic conic optimization problem. This connection allows us to exchange concepts and ideas between quantum information and this field of mathematics. For instance, copositive matrices can be understood as indecomposable entanglement witnesses for DS states. As a consequence, we show that positivity of the partial transposition (PPT) is sufficient and necessary for separability of DS states for $d \leq 4$. Furthermore, for $d \geq 5$, we provide analytic examples of PPT-entangled states. Second, we develop new sufficient separability conditions beyond the PPT criterion for bipartite DS states. Finally, we focus on $N$-partite DS qubits, where PPT is known to be necessary and sufficient for separability. In this case, we present a family of almost DS states that are PPT with respect to each partition but nevertheless entangled.


Introduction
Entanglement [1] is one of the most striking features of quantum physics, departing entirely from any classical analogy. Furthermore, entanglement is a key resource for quantum information processing tasks, such as quantum cryptography [2] or metrology [3]. Importantly, entanglement is a necessary resource to enable the existence of Bell correlations [4,5], which are the resource device-independent quantum information processing is built upon [6]. Despite its both fundamental and applied interest, the so-called separability problem (i.e., deciding whether a quantum state is entangled or not, given its description) remains open except for very specific cases. Although this problem has been shown to be, in the general case, NP-hard [7], it remains unclear whether this is also the case for physical systems of interest, where symmetries appear in a natural way.
To tackle the separability problem, simple tests have been put forward, which give a partial characterization of entanglement. The most celebrated entanglement detection criterion is the so-called positivity under partial transposition (PPT) criterion [8]. It states that every state that is not entangled must satisfy the PPT criterion. Therefore, states that break the PPT criterion are entangled. Unfortunately, the converse is true only in very low-dimensional systems [9], such as two qubit [10] or qubit-qutrit systems [11]. Examples of entangled states satisfying the PPT criterion have been found for strictly larger-dimensional systems [12].
The paper is organized as follows. In Section 2 we establish the notation and the basic definitions that we are going to use in the next sections. In Section 3 we discuss the separability problem for bipartite DS states of arbitrary dimension, with particular emphasis in their connection to nonconvex quadratic optimization problems. In Section 4 we provide sufficient criteria to certify either separability or entanglement. In Section 5 we present a class of PPT-entangled multipartite qubit almost-diagonal symmetric states. In Section 6 we conclude and discuss further research directions. Finally, in the Appendix we present some proofs, examples and counterexamples that complement the results discussed in the text.

Preliminaries
In this section we set the notation and define the basic concepts that we are going to use throughout the paper.

The separability problem
where p i form a convex combination (p i ≥ 0 and i p i = 1) and ρ A i (ρ B i ) are quantum states acting on Alice's (Bob's) subsystem; i.e., they are positive semidefinite operators of trace one. If a decomposition of ρ of the form of Eq. (1) does not exist, then ρ is entangled.
The separability problem; i.e., deciding whether a quantum state ρ admits a decomposition of the form of Eq. (1) is, in general, an NP-hard problem [7]. However, there exist simple tests that provide sufficient conditions to certify that ρ is entangled [1]. One of the most renowned separability criteria is the positivity under partial transposition (PPT) criterion [8]. It states that, if ρ can be decomposed into the form of Eq. (1), then the state (1 ⊗ T )[ρ] must be positive semi-definite, where T is the transposition with respect to the canonical basis of C d . Such state is denoted ρ T B , the partial transposition of ρ on Bob's side. Because (ρ T B ) T = ρ T A , the PPT criterion does not depend on which side of the bipartite system the transposition operation is applied on. Breaking PPT criterion is a necessary and sufficient condition for entanglement only in the two qubit [10] and qubit-qutrit [11] cases, and there exist counterexamples for states of strictly higher physical dimension [12].
In the multipartite case, the definition of separability given in Eq. (1) naturally generalizes to N subsystems.

Definition 2.2.
A quantum state ρ acting on C d 1 ⊗ · · · ⊗ C d N is fully separable if it can be written as where ρ are quantum states acting on the k-th subsystem and p i form a convex combination.
Therefore, the PPT criterion also generalizes to 2 N/2 criteria, where · is the floor function, depending on which subsystems one chooses to transpose.

Entanglement witnesses
Let us denote by D sep the set of separable sates (cf. Eqs. (1), (2)). This set is closed and convex. Therefore it admits a dual description in terms of its dual cone, which we denote where the usual Hilbert-Schmidt scalar product W, ρ = Tr(W † ρ) is taken. The elements of P can be thus viewed as half-spaces containing D sep . Of course, not every operator in P is useful to detect entangled states. In order to be non-trivial, one requires that W has at least one negative eigenvalue. Such operators are called entanglement witnesses (EW) [46] and they form a non-convex set, denoted W = {W ∈ P s. t. W 0}. A state ρ is then separable if, and only if, Tr(W ρ) ≥ 0 for all W ∈ W.
Among EWs, it is worth to make a distinction that relates them to the PPT criterion: decomposable and indecomposable EWs.
with P 0 and Q 0. Indecomposable EWs (IEWs) are those EWs that are not of the form of Eq.
Although DEWs are easier to characterize [47], they do not detect PPT-entangled states, because In Section 4.1 we construct EWs which detect entangled PPTDS states, therefore they correspond to indecomposable witnesses.
3 Separability in diagonal symmetric states acting on C d ⊗ C d .
In this section, we characterize the bipartite diagonal symmetric two-qudit states in terms of the separability and the PPT properties. We establish an equivalence between: (i) separability and the PPT property in DS states and (ii) quadratic conic optimization problems and their relaxations, respectively. We first introduce the Dicke basis in its full generality and then we move to the two particular cases of interest to this paper: the case of N -qubits and the case of 2-qudits. One can think of the space spanned by the Dicke states as the linear subspace of (C d ) ⊗N containing all permutationally invariant states.
where σ runs over all permutations of N elements.
The Dicke state has N k different elements, where the quantity follows from the multinomial combinatorial quantity Finally, recall that there are as many Dicke states as partitions of N into d (possibly empty) subsets; therefore, the dimension of the subspace of (C d ) ⊗N is given by where denotes partition of.
In this paper, we are particularly interested in the case of N -qubits and 2-qudits: For N -qubit states we shall use the notation |D k ≡ |D N k , where k = k 1 denotes the number of qubits in the excited (|1 ) state. Mixtures of Dicke states correspond to • N = 2. For bipartite d-level systems, we are going to denote the Dicke states by |D k ≡ |D ij , where i and j are the indices (possibly repeated) of the non-zero k i and k j . Since the terminology Dicke states is often reserved for the multipartite case, we call |D ij simply symmetric states.
In the computational basis, a DS ρ is a d 2 × d 2 matrix that is highly sparse. Therefore, it will be useful to associate a d × d matrix to ρ that captures all its relevant information. We define the M -matrix of ρ to be which arises from the partially transposed matrix ρ T B .
Notice that, while a DS state ρ is always diagonal in the Dicke basis, its partial transposition (which is defined with respect to the computational basis) scrambles its elements. Then ρ T B is block-diagonal in the Dicke basis and its blocks are 1 × 1 elements corresponding to p ij with i < j, and M (ρ). One can see the effect of the partial transposition operation on a DS state by inspecting the action of T B onto the elements |D ij D ij | that compose Eq. (9): • If i = j, the action of the partial transposition is best seen by expanding |D ij onto the computational basis: (|D ij D ij |) = 1 2 (|ij ij|+|ij ji|+|ji ij|+|ji ji|). Therefore, two of the terms are left invariant and the remaining two are to be mapped as (|ij ji|+|ji ij|) T B = |ii jj|+|jj ii|.
Thus, M (ρ) is the submatrix corresponding to the elements indexed by |ii jj| for 0 ≤ i, j < d of ρ T B . Because there is no mixing between other rows or columns, we have that ρ T B decomposes as the direct sum Since p ij = p ji , we find that the 1 × 1 blocks appear all with multiplicity 2. Therefore, each M (ρ) with non-negative entries summing 1 is in one-to-one correspondence to a DS state ρ. In this section we characterize the separability properties of ρ in terms of equivalent properties of M (ρ), which are naturally related to quadratic conic optimization.
In quadratic conic optimization, one is interested in characterizing the so-called completely positive (CP) matrices, which are defined as follows Matrices which are CP form a proper 3 cone, which is denoted by CP d . Note that the sum of two CP matrices is a CP matrix and the multiplication of a CP matrix by a non-negative scalar is a CP matrix.
Given a non-convex optimization problem over the simplex, which is NP-hard in general, CP matrices translate the complexity of the problem by reformulating it as a linear problem in matrix variables over CP d . Therefore, they allow to shift all the difficulty of the original problem into the cone constraint. Precisely, every non-convex quadratic optimization problem over the simplex (LHS of Eq. (12)) has an equivalent CP formulation (RHS of Eq. (12)): where |u is the unnormalized vector of ones and Q is, without loss of generality 4 , symmetric and positive semi-definite. Therefore, deciding membership in CP d is NP-hard [38].
One can obtain, however, an upper bound on the optimization in Eq. (12) by observing that every CP matrix A is positive semi-definite, because it allows for a factorization A = B · B T . Moreover, it is also entry-wise non-negative: A ij ≥ 0. This motivates Definition 3.5: We are now ready to introduce the equivalences between the separability problem in DS states and quadratic conic optimization. After producing our results, we learned that these equivalences were independently observed by Nengkun Yu in [27]. We nevertheless prove them in a different way.
We prove Theorem 3.1 in Appendix A. By virtue of Theorem 3.1, we recover the result of [27]: Because it is NP-Hard to decide whether a matrix admits a CP decomposition [38], the separability problem in C d ⊗ C d DS states is NP-Hard.
We remark that the NP-hardness result that we obtain holds under polynomial-time Turing reductions 5 , as opposed to poly-time many-one 6 reductions [48]. For instance, this is the case for Gurvits' initial reduction of the weak membership problem 7 in the set of separable states from the NP-complete problem PARTITION 8 [7]. In the case we present here, the reduction holds because the NP-hardness of deciding membership in the CP d set follows via a Turing reduction, which is the result we use as our starting point. The part of the reduction that we provide here, however, is many-one. 3 Closed, convex, pointed and full-dimensional. 4 Q can be assumed to be symmetric because x|Q|x = ( x|Q|x ) T = x|Q T |x . It can be assumed to be positive semi-definite because adding 1 to (Q + Q T )/2 does not change the optimal |x ; it only adds a bias to the maximum. 5 Intuitively speaking, a Turing reduction describes how to solve problem A by running an algorithm for a second problem B, possibly multiple times. 6 A many-one reduction is a special case of a Turing reduction, with the particularity that the algorithm for problem B can be called only one time, and its output is immediately returned as the output of problem A. 7 Weak in the sense that it allows for error in points at a given Euclidean distance from the border of the set. 8 The PARTITION problem is a decision problem corresponding to whether a given set of integer numbers can be partitioned into two sets of equal sum. This problem is efficiently solvable with a dynamic programming procedure [49], but becomes NP-hard when the magnitudes of the input integers become exponentially large with the input size.
We here briefly discuss the steps that would be required to make this result completely rigurous from a computer science point of view. One would need to embed the NP-hardness into the formalism of the weak membership problem [7,50]. This requires, for instance, showing that the convex set of separable DS states has some desirable properties such as being well-bounded or p-centered. We refer the reader to [48] for the technical aspects of these definitions. On the other hand, the completely positive cone is known to be well-bounded and p-centered: in [51] it was proved that the weak membership problem in the completely positive cone is NP-hard. By using the one-to-one correspondence between DS states and CP matrices given by M (ρ) in Def. 3.3, then the result is mapped onto the DS set 9 .
Geometrically, the set of separable DS states is convex. Hence, it is fully characterized by its extremal elements (those that cannot be written as a non-trivial convex combination of other separable DS states). Identifying such elements is of great importance towards the characterization of the separability properties of DS states. For instance, in the set of all separable density matrices, the extremal ones are the rank-1 projectors onto product vectors. However, this property may be lost when restricting our search in a subspace: observe that the set of separable DS states states is obtained as the intersection of the subspace of DS states with the convex set of separable density matrices. Therefore, the set of extremal separable DS states states may contain states that are separable, but not extremal in the set of separable density matrices (see Fig. 1). Theorem 3.1 allows us to fully characterize extremality in the set of separable DS states in terms of extremal CP matrices, thus obtaining the following corollary: Proof. -Since the extremal rays of the CP d cone are the rank- , by normalizing and comparing to Eq. (10) we obtain Eq. (14). Figure 1: Cartoon picture of the set of separable states SEP (cylinder) and its intersection with the subspace of diagonal symmetric states DS (ellipse). The intersection of the subspace of DS states with the set of separable states gives rise to the set of separable DS states, which is represented by the green ellipse, including its interior. Only the states of the form |ii ii| are extremal in both sets (represented by the black dot in the figure). However, states that were in the boundary of SEP, could now be extremal when viewed in DS (represented by the border of the green ellipse in the figure). 9 The technical requirement of full dimensionality [48,51] depends on the set in which one embeds the problem. Recall that we are interested in solving the separability problem within DS states. The set of DS separable states is of course not full-dimensional when embedded in the whole two-qudit Hilbert space. However, it is full-dimensional when viewed in the DS subspace (cf. Figure 1).

Separable
Proof. -Let us assume that ρ is PPT. Note that the partial transposition of ρ can be written as Since ρ is PPT, Eq. (16) implies that M (ρ) 0. Since ρ is a valid quantum state, then p ab ≥ 0 for all is DNN then we have that all its entries are non-negative; i.e., p ab ≥ 0 for 0 ≤ a ≤ b < d. These conditions guarantee that ρ 0. Additionally, as M (ρ) 0, these conditions imply that ρ Γ 0. Hence, ρ is PPT.
Recall (cf. Definitions 3.4 and 3.5, also Fig. 2 [38]. This yields a full characterization of the bipartite separable DS states in terms of the PPT criterion: Proof. -The result follows from the identity CP d = DN N d , which holds for d ≤ 4 [38]. In Example C.1 we provide a constructive proof for d = 3. Finally, we end this section by giving a sufficient separability criteria for any d in terms of the ranks of M (ρ).
Geometrically, every row of V can be seen as a vector in R 2 (or a scalar if the rank of M (ρ) is one). Therefore, M (ρ) can be seen as the Gram matrix of those vectors; each element being their scalar product. Since M (ρ) is doubly non-negative, it implies that all these scalar products must be positive; therefore, the angle between each pair of vectors is smaller or equal than π/2. Thus, the geometrical interpretation is that M (ρ) is CP if, and only if, they can be isometrically embedded into the positive orthant of R k for some k. This is always possible to do for k = 2 (see Fig.   3), which corresponds to M (ρ) having rank at most 2.

Sufficient criteria for entanglement and separability
In this section we further characterize the bipartite DS states by providing sufficient criteria to certify entanglement by means of Entanglement Witnesses for DS states, and by providing sufficient separability conditions in terms of M (ρ).

Entanglement Witnesses for DS states
We begin by introducing the concept of copositive matrix: The set of d × d copositive matrices also forms a proper cone, denoted COP d . The cones CP d and COP d are dual to each other with respect to the trace inner product. It is also easy to see that Therefore, one can view copositive matrices as EWs for DS states. Furthermore, one could think of PSD d + N d as the set of DEWs for DS states, in the sense that they do not detect entangled PPTDS states. In Examples 4.1 and D.1 we provide some M (ρ) ∈ DN N d \ CP d for d ≥ 5, therefore corresponding to entangled PPTDS states. The paradigmatic example of a copositive matrix detecting matrices DNN, but not CP, (i.e., PPT, but entangled DS states) is the Horn matrix [40], which is defined as It is proven that H ∈ COP 5 \ (PSD 5 + N 5 ) in [40]. As Tr(HM (ρ)) = −1 < 0, H corresponds to an (indecomposable) entanglement witness for the state corresponding to M (ρ).
Although the boundary of the set of copositive matrices remains uncharacterized for arbitrary dimensions, COP 5 was fully characterized in [51]: where Furthermore, the extremal rays of COP d have been fully characterized for d ≤ 5, divided into classes, but this also remains an open problem for higher d [38].
In Appendix B we discuss exposedness properties of the sets of completely positive and co-positive matrices and their relation to the separability problem and its geometry.
It can be easily seen using the Range criterion, as in Section C.2, that ρ is entangled. By Theorem Finally, it can be appreciated how the Horn matrix can be used as an EW and certify entanglement Tr(HM (ρ)) = −1 < 0.

Sufficient separability conditions for diagonal symmetric states
In the spirit of best separable approximations (BSA) [24], in this section we provide sufficient separability conditions for bipartite PPTDS states. In the same way that the BSA allows one to express any PPTDS state as a sum of a separable part and an entangled one with maximal weight on the separable one 10 . In this section we introduce Best Diagonal Dominant (BDD) approximations, which give a sufficient criterion to certify that a PPTDS state is separable. The idea is that although checking membership in CP d is NP-hard, it is actually easy to (i) characterize the extremal elements in CP d (cf. Corollary 3.1) and (ii) check for membership in a subset of DD d ⊆ CP d that is formed of those matrices A ∈ N d that are diagonal dominant. In [52] the inclusion DD d ⊆ CP d was proven. Therefore, to show that CP d \ DD d is nonempty we study when the decomposition of a potential element in CP d as a convex combination of an extremal element of CP d and an element of DD d is possible (see Figure  4). Let us start by stating a lemma that gives an explicit separable decomposition of a quantum state.
Proof. -Let |e( ϕ) = |0 + e iϕ 1 |1 + · · · + e iϕ d−1 |d − 1 . A separable decomposition of I is given by Indeed, where δ is the Kronecker delta function. Lemma 4.1 allows us to give a sufficient condition for a state ρ to be separable. The idea is to subtract εI from ρ in such a way that it remains a valid diagonal symmetric state, PPT, and close enough to the interior of the separable set such that it is easy to certify that the state is separable (see Fig. 4).

for all
Then, ρ is separable.
See the proof in Appendix D.2. A few comments are in order: The first condition on Theorem 4.1 ensures that I can be subtracted from ρ andρ will remain in the DS subspace. The second condition requires that I can be subtracted from ρ while maintaining the PPT property ofρ. If |u / ∈ R(M (ρ)), thenρ would not be PPT for any ε = 0. Therefore, the second condition gives the maximal value of ε that can be subtracted such thatρ remains PPT. Finally, the third condition relies on guaranteeing thatρ is separable, which is ensured by M (ρ) to be diagonal dominant. This means that one might need to subtract a minimal amount of I to accomplish such a property (unless ρ is already diagonal dominant). In Example D.3 we show that CP d \ DD d = ∅ using the approach of Theorem 4.1.
The above result can be now normalized and generalized to any other extremal element I in CP d : Lemma 4.2. Let x ∈ R d with x i ≥ 0 and ||x|| > 0. Let I x be the quantum state defined as Then, the quantum state I x is separable.
See the proof in Appendix D.4. Note that the corresponding M (I x ) is given by |u x u x |, where |u x = x/||x|| 1 . The sum of the elements of M (I x ) is then one; i.e., ||M (I x )|| 1 = 1.
Lemma 4.2 allows us to give a sufficient condition for a state ρ is separable. This time the idea is to decompose ρ as a convex combination between I x , which is a state that is extremal in the set of separable DS states, and a stateρ which is deep enough in the interior of the set of separable states, such that we can certify its separability by other means (by showing that M (ρ) is diagonal dominant and doubly non-negative; therefore completely positive [52]). Example. -In this example we provide a PPTDS state with associated M (ρ) ∈ CP d \ DD d and we show how to apply Theorem 4.2 to guarantee separability. Furthermore, we also apply Theorem 4.1 to illustrate the advantage of Theorem 4.2.

Theorem 4.2. Let ρ be a two-qudit PPTDS state with associated
Take the following DS state ρ ∈ C 3 with associated where it can be checked that the state ρ is normalized. A priori we do not know if this state is separable, the goal is to apply Theorems 4.1 and 4.2 in order to see if separability can be guaranteed. For both Theorems the more restrictive between conditions 1 and 2 provides an upper bound for the corresponding decomposition and condition 3 a lower bound but, as mentioned, Theorem 4.2 offers more flexibility since such bound can be varied by fitting I x . This example illustrates this fact since we will see that Theorem 4. Lets proceed to show that the given M (ρ) and M (I x ) meet the conditions of Theorem 4.2. Condition 1 provides the more restrictive upper bound given by while the lower bound will be given by the following case of Condition 3 Therefore, there exists a range of values λ ∈ [0.7681, 0.8213] that satisfy the conditions of Theorem 4.2 and certifies that the state ρ is separable. Notice that, for illustrative purposes, once we found an M (I x ) fulfilling the conditions we fixed it to find a range of values for λ but we could have allowed for more freedom and find a bigger range of possible decompositions.
Now lets see what happens with Theorem 4.1. In this case the most restrictive upper bound is given by Condition 2 while the most restrictive lower bound will be given by the following case of Condition 3 Thus, for this case there does not exist an ε satisfying the conditions for Theorem 4.1, while there exists a range of λ satisfying conditions for Theorem 4.2 and therefore illustrating its advantage by being able to certify separability.

A class of PPT-entangled quasi-DS states
In this Section, we introduce a uni-parametric class of N -qubit PPTESS, for an odd number of qubits.
As it has been shown in [27,28], N -qubit PPTDS states are fully separable. The class we introduce can be seen as a N -qubit PPTDS state with slight GHZ coherences. Surprisingly, in the family of states we provide, an arbitrarily small weight on the non-diagonal elements (in the Dicke basis) allows the state to be genuinely multipartite entangled while maintaining the PPT property. The procedure we have chosen to derive this class of states is based on the iterative algorithm for finding extremal PPT symmetric states [20,21] (see also [53]), which we briefly recall here in the interest of completeness. One starts with an initial symmetric state ρ 0 that is fully separable; for instance, the symmetric completely mixed state. Then, one picks a random direction σ 0 in the set of quantum states and subtracts it from the initial state while preserving the PPT property, therefore obtaining ρ 0 − x 0 σ 0 , x 0 > 0. One necessarily finds a critical x * 0 such that one arrives at the boundary of the PPT set, where the rank of ρ 0 −x * 0 σ 0 or one of its partial transpositions must have decreased. Hence, at least one new vector appears in the kernel of the state or in the kernel of some of its partial transpositions. This state with lower ranks is set as the initial state for the next iteration ρ 1 = ρ 0 − x * 0 σ 0 . The new direction σ 1 is chosen such that it preserves all the vectors present in the kernels of both the state and its partial transpositions. This process is repeated until no new improving direction can be found, yielding an extremal state ρ k in the PPT set. As the PPT set contains all separable states, we note that if the rank of such extremal PPT state is greater than one, then it cannot be extremal in the set of separable states (because these are pure product vectors, which have rank one), therefore it must be entangled. The study carried out in [20,21] looked for typical extremal PPT states by exploring random directions every time. However, by carefully picking these directions, one can look for classes of states of different forms, such as the ones presented in Theorem 5.1.
In Example E.1 we present a 4-qubit PPT-entangled symmetric state whose density matrix is sparse with real entries when represented in the computational basis and has a closed analytical form.
The class of states we are going to present is furthermore symmetric with respect to the |0 ↔ |1 exchange and, to simplify our proof and take advantage of this symmetry, we shall consider only an odd number of qubits N = 2K + 1, with K > 1.
We split the proof into several lemmas, for better readability and intuition on the above definitions.
Proof. -Let us start with some general considerations on the structure of ρ(Z). In order to efficiently apply the partial transposition operation with respect to m subsystems, we need to express ρ(Z) acting on and GHZ coherences o(σ). Its partial transposition with respect to m subsystems ρ Γm , acting on where A

See the Proof in Appendix E.2.
In what follows we are going to argue the construction of our class of states. Having a state with ranks as low as possible tremendously simplifies the analysis of PPT entanglement [12]. It is the main idea we are going to follow in defining all the elements of our class. Therefore, we first study the condition for which the block of ρ Γm that contains σ has zero determinant. This gives the condition λ m λ N −m = σ 2 = 1. In particular, if we impose this condition for m = K, we obtain λ 2 K = 1, which means (by definition) that f 0 = 1. Now we focus on the block n = −m + 1 with m = K. The determinant of the n-th block is By imposing the determinant of B (K) −K+1 to be Z > 0 we obtain the condition f 1 = 1 + Z with Z > 0. This choice is arbitrary and is the one that characterizes our class.
Finally, we move to the 3 × 3 block determinants, which we shall make 0. These are This condition reads We want to determine a recursive form for the λ m that satisfies Eq. (38). This is an equation in differences which is nonlinear. It is in general extremely hard to solve these kind of equations. Nevertheless, despite the appearance of Eq.
which give c 0 = 1 and c 1 = −2(1 + Z) as the unique solutions. Let us note that one can find the expression for f m in a non-recursive form, with the aid of the Z-transform: By undoing the Z-transform, we obtain the explicit expression for f m : where α := (2 + Z + Z(4 + Z))/2 and β := (2 + Z − Z(4 + Z))/2.
See the proof in Appendix E.3. Proof of Lemma 5.4. -To prove extremality, we use the following theorem from [54]: ρ is extremal in the PPT set if, and only if, every Hermitian matrix H satisfying R(H Γm ) ⊆ R(ρ Γm ) is proportional to ρ. Note that by taking H ∝ ρ we always find a solution satisfying the above conditions, but we have to show that no other exists. Let us consider the subspace E of the (N + 1) × (N + 1) Hermitian matrices spanned by the following matrices: Let us argue that we can assume that H has to live in the same subspace as ρ. Since R(ρ) ⊆ E, H has to be of the form where h k and h are real parameters. The condition R(H Γm ) ⊆ R(ρ Γm ) means that the vectors spanning H Γm must be orthogonal to (at least) the vectors in the kernel of ρ Γm . Fortunately, we have calculated the block-decomposition of ρ Γm : As a side-comment let us observe that (D (m) n . Anyway, being orthogonal to ker B n (m) implies that the coefficients h m must satisfy a recurrence relation of the form which fixes h m ∝ f m . Finally, we fix the value of h by looking at the kernel of the block that goes alone in ρ Γ K , which is spanned by (σ, −1) T . Hence, we have that h K σ − h = 0, which implies that h = σh K . Hence, H = h K ρ ∝ ρ is the only solution to R(H Γm ) ⊆ R(ρ Γm ), proving that ρ is extremal. Since all the states in the PPT set which are separable and extremal have ranks r(ρ Γm ) = 1, an extremal PPT state with a rank r(ρ Γm ) > 1 cannot be separable. Hence, ρ(Z) is a uni-parametric family of PPT-entangled states for all Z ∈ (0, ∞).

Conclusions and Outlook
In this work we have studied the separability problem for diagonal symmetric states that are positive under partial transpositions. In the bipartite case, we have explored its connection to quadratic conic optimization problems, which naturally appear in a plethora of situations. Via this equivalence, we have been able to translate results from quantum information to optimization and vice-versa. For instance, we have characterized the extremal states of the set of separable DS states, defined entanglement witnesses for PPTDS states in terms of copositive matrices and we have rediscovered that the separability problem is NP-hard even in this highly symmetric and simplified case. We have shown that PPT is equivalent to separability in this context only for states of physical dimension not greater than 4. We have complemented our findings with a series of analytical examples and counterexamples. Furthermore, the state of the art in quadratic conic optimization allows us to see which are going to be the forthcoming challenges, in which insights developed within the quantum information community might contribute in advancing the field. Second, we have provided a set of tools to certify separability of a bipartite PPTDS state in arbitrary dimensions, by decomposing it as a combination of an extremal DS state and a diagonal dominant DS state.. A natural further research direction is to study whether more sophisticated decompositions are possible, in terms of various extremal elements in CP d and by understanding how the facial structure of CP d plays a role in this problem.
Third, we have shown that, although N -qubit DS states are separable if, and only if, they are PPT with respect to every bipartition, just adding a new GHZ-like coherence can entangle the state while mantaining the PPT property for every bipartition. We have characterized analytically this class and we have shown that its ranks are much lower than those typically found in previous numerical studies [21]. In this search, we have also found an analytical example of a 4-qubit PPT-entangled symmetric state, whose density matrix is sparse with real entries, when expressed in the computational basis, contrary to previous numerical examples [20]. A natural following research direction is to connect the recently developed concept of coherence [55] to the Dicke basis, and to explore further its connection to PPT-entangled symmetric states. Furthremore, it seems plausible that one can find further connections between the properties of M (ρ) and those of MUBs since it has been shown that one can construct EWs based on MUBs that are capable of detecting bound entangled states (See [56] for a recent construction or [57] for an application to magic simplex states in an experimentally feasible way). Whether there is a clear connection between EWs from MUBs and EWs for bound entangled DS states in terms of the properties of M (ρ) remains an open research direction.
A Proof of Theorem 3.1 Proof. Let us assume that ρ is separable. Since it is symmetric, it admits a convex decomposition into product vectors of the following form: where λ i form a convex combination and |e i := d−1 j=0 e ij |j , e ij ∈ C. It follows that we have the By projecting Eq. (47) onto the Dicke basis we obtain the following conditions 11 : We can now construct M (ρ), which has the form It is clear from Eq. Conversely, let us assume that M (ρ) is CP. Note that, as ρ is DS, M (ρ) is in one-to-one correspondence with ρ. Since M (ρ) is CP, we can write M (ρ) = B · B T , with B being a d × k matrix fulfilling B ij ≥ 0. We have to give a separable convex combination of the form of Eq. (46) that produces the DS ρ matching the given M (ρ). As we shall see, this separable decomposition is by no means unique. We begin by writing where b i are the columns of B, so all the coordinates of b i are non-negative. Let {z ij } ij be a set of complex numbers satisfying |z ij | 2 = ( b i ) j ≥ 0 and let us define Note that if we naively make the convex combination Eq. (46) with the vectors introduced in Eq.
(52), we shall produce a state with the corresponding M (ρ), but it will not be DS in general. In order 11 There are, of course, more conditions that follow from Eq. (47), such as i λi(eir) 2 (e * is ) 2 = 0, but we do not need them for this implication.
to ensure that the ρ we are going to construct is DS we have to build it in a way that we eliminate all unwanted coherences. To this end, let us consider the more general family of vectors where k l is the l-th digit of k in base 2 and ω is a primitive 2d−th root of the unity (for instance, ω = exp(2πi/2d)). Let us now construct the (unnormalized) quantum state By expanding Eq. (54) we obtain which we can rewrite as (56) Let us inspect the possible values for the sums in parenthesis in Eq. (56). The expression involving k will be zero whenever l 1 , l 2 , l 3 , l 4 are all different (half of the sum will come with a plus sign and the other half with a minus sign). If only two of them are equal, the expression is still zero by the same argument. If two of the indices are equal to the other two, then the value of the expression is 2 d . Hence, we have that (57) where δ x is the Kronecker Delta function (note that the last term prevents that the case l 1 = l 2 = l 3 = l 4 is counted more than once). The second parenthesis in Eq. (56) is a geometrical series, so we have that it is d if, and only if, l 1 + l 2 ≡ l 3 + l 4 mod 2d (because ω is taken to be primitive); otherwise it is 0. As 0 ≤ l 1 + l 2 , l 3 + l 4 < 2d, this can only happen if l 1 + l 2 = l 3 + l 4 . Thus, 0≤j<d ω (l 1 +l 2 −l 3 −l 4 )j = dδ l 1 +l 2 −(l 3 +l 4 ) .

B Exposedness
Convex sets are completely determined by their extremal elements (those that cannot be written as a proper convex combination of other elements in the set). An important step in characterizing the extremal elements of convex sets is understanding their facial structure.
Definition B.1. Given a convex cone K, a face of K is a subset F ⊆ K such that every line segment in the cone with an interior point in F must have both endpoints in F.
Note that every extreme ray of K is a one-dimensional face. To understand the facial structure of cones, one is interested in learning whether K is facially exposed. Facial exposedness is an important property that is exploited in optimization, allowing to design facial reduction algorithms [58].
Definition B.2. Let K be a cone in the space of real, symmetric matrices and let F ⊆ K be a nonempty face. F is defined as an exposed face of K if, and only if, there exists a non-zero real symmetric matrix A such that and Hence, a face is exposed if it is the intersection of the cone with a non-trivial supporting hyperplane.
A cone is facially exposed if all of its faces are exposed. Although every extreme ray of CP d is exposed [59], it remains unknown whether CP d is facially exposed. In the case of COP d , the extreme rays corresponding to |ii ii| are not exposed [59], implying that PSD d + N d (the set of DEWs for PPTDS states) is not facially exposed. However, the set DN N d of PPTDS states is facially exposed, because both PSD d and N d are facially exposed [60] and the intersection of facially exposed cones is facially exposed.

C Examples and counterexamples
C.1 Every PPTDS state acting on C 3 ⊗ C 3 is separable In this example, we prove that every PPTDS state ρ acting on C 3 ⊗ C 3 is separable. This follows from Theorem 3.3, which is usually proven [27] invoking results from quadratic non-convex optimization [38]. We prove it here using quantum information tools solely: by building a convex separable decomposition of ρ of the form of Eq. (1). We do this in two steps. First, we provide a three-parameter class of PPTDS states that are separable. Then, by performing a Cholesky decomposition of ρ Γ we see that ρ can be expressed as a convex combination of the family we introduced, for some parameters. The PPT conditions directly relate to the existence of such a Cholesky decomposition.
Recall that ρ is written as where |D ii = |ii and |D ij = (|ij + |ji )/ √ 2 if i < j. Short algebra shows that ρ and its partial transpose ρ Γ have the form and where | · | denotes the complex modulus.
Proof. The proof follows from expressing σ x,y,z in the computational basis. After some elementary algebra, one arrives at the form of Eq. (65).
The following Lemma allows us to find a decomposition of a positive semi-definite matrix A of the form A = B · B T . To this end, we apply Cholesky's decomposition.
Proof. The idea of the proof is to use the rank-1 matrix A 1 := (a, b, c) T (a, b, c)/a to fix the elements of A that lie on the first column and first row. Then, the second summand will adjust the elements of the second row, second column of A and the last summand will fix the bottom-right element of A. Therefore, we have where the · are terms that are not yet fixed. When we add the second term to A 1 we have and adding the last term to A 2 we recover A.
Now we have the tools to prove that every DNN 3 × 3 matrix is CP: Lemma C.3. If A is a 3 × 3 positive-semidefinite matrix, and it is entry-wise non-negative, then there exists a Cholesky decomposition of A with non-negative vectors (i.e., the vectors' coordinates are non-negative).
Proof. The only problematic term in Eq (73) If any of the numbers in (77) is non-negative, then we can pick the Cholesky decomposition for that particular order and we obtain the result. The alternative is that all of them are strictly negative. We are going to see now that this would contradict the fact that A 0.
Note that all the numbers in (77) being strictly negative imply that d > 0. Otherwise, if d = 0, since A 0, this would imply that b = e = 0. Then, all the numbers in (77) would be zero. Therefore, d must be strictly positive. Similarly, b > 0. Otherwise, if b = 0, then we would have cd < 0 and ae < 0, contradicting the fact that A is entry-wise non-negative. Therefore, b > 0.
It is sufficient to find a contradiction just with a subset of the conditions given by (77): Let us assume that ae < bc and cd < be. Then, we have that where we used ae < bc and d > 0 in the first inequality and cd < be and b > 0 in the second. Therefore, aed < b 2 e. Hence, it must be that e > 0 (otherwise we would have 0 < 0). Then, we deduce that ad < b 2 , but this directly contradicts A 0, as the latter implies ad ≥ b 2 .
Now we have the necessary tools to prove the result claimed in the example. Consider ρ to be a PPTDS state. Then, we have that p ij ≥ 0 and M 0. We want to write ρ as a convex combination of some elements σ x,y,z introduced in Lemma C.1 by appropriately picking x, y, z as functions of p ij . Observe that for all x, y, z ∈ C, the entries of σ x,y,z will be non-negative.