Erratum: “Multipartite entanglement in the diagonal symmetric subspace” [J. Math. Phys 66, 022203 (2025)]

The purpose of this manuscript is to correct the claim made in our previous article1 stating that all three-partite DS states whose M(ρDS) is completely positive are separable. Contrary to our initial claim, there exist states of this type that are nevertheless PPT-entangled.2 This implies that the statements of Theorem III.1, Theorem III.3, and Theorem III.4 provide necessary but not sufficient conditions for separability and should be modified as follows: Theorem III.1. (Formula presented) be a DS with associated matrix M(3)(ρDS). Then, (Formula presented). Theorem III.3. (Formula presented). Theorem III.4. (Formula presented) be a DS state with an associated matrix M(4)(ρDS) which arises from the partial transpose w.r.t. the largest partition. Then, ρDS separable (Formula presented). 2 The non-sufficiency of these theorems can be seen by focusing in the most general form of a three-partite DS separable state, i.e., (Formula presented) and Nijk is given by (Formula presented), Let us recall that any three-partite symmetric separable state can be written as3 (Formula presented) Consider the following parametrization of a single-qudit state, i.e., (Formula presented) Here, k is a vector of size 2d in base 2, w is the 2dth root of the unity, i.e., (Formula presented) and (Formula presented). Inserting Eq. (3) in Eq. (2) we obtain (Formula presented). We proceed by inspecting the different sums in Eq. (4), analyzing case by case. First, it can be easily seen that (Formula presented) since this sum corresponds to a geometrical series. Then, we focus on the following sum, i.e., (Formula presented) It can be noticed that non-zero terms in this sum correspond to those cases where the exponent can be written as a sum of even numbers. For this reason, it is convenient to introduce a partition vector (Formula presented), such that (Formula presented) and each vi counts the number of indices li in the state (Formula presented). In our case we have six indices li that can assume either even or odd values. The only terms that contribute to the sum are those such that the corresponding partition vector v does not contain an odd number in any subset. As such, the only non-zero contributions are given by v = (6, 0, 0), v = (4, 2, 0) and v = (2, 2, 2). For these partitions, the sum leads to a 2d factor. In all other cases, the sum of Eq. (6) evaluates to 0. Intuitively, this can be seen by observing that, in this case, the sum will contain an equal number of +1 and −1 contributions, thus summing up to 0. Combining this with the previous condition from the sum over index s, we realize that the only terms that survive both sums correspond to the matrix elements (Formula presented), partition (Formula presented), where π, π′ correspond to all possible permutations of the parties, and (Formula presented). Finally, we can rewrite Eq. (4) in the Dicke basis as (Formula presented) where the numeric factors derive from the normalization. It can be seen then that the separable symmetric states we have constructed are always DS and therefore depend only on the weights al. Normalizing our state: (Formula presented). (8) This construction allows to generate the extreme points in the set of separable DS states. As a consequence, any point inside this set can be generated by taking a convex combination of such points. This replaces Eq. (A15) as the most generic construction for separable three-partite DS states. Finally, by means of the method based on PPT-symmetric extensions presented in Ref. 4, we provide two examples of PPT-entangled DS states in d = 3 and d = 4, whose corresponding M(ρDS) is CP, along with the entanglement witnesses that detect them. These states lie in the vicinity of the boundary between the set of PPT-entangled and NPT states. Example 1 (N = 3, d = 3). Consider the following three-partite DS state in (Formula presented) where p(3) denotes a probability vector. A PPT-entangled state is obtained by choosing 1 (Formula presented). (10) Its corresponding entanglement witness is given by (Formula presented). Example 2 (N = 3, d = 4). Consider the following three-partite DS state in (Formula presented) where p(4) denotes a probability vector. A PPT-entangled state is obtained by choosing (Formula presented). (12) The corresponding entanglement witness is given by (Formula presented).