We solve the open question of the existence of four-qubit entangled symmetric states with positive partial transpositions (PPT states). We reach this goal with two different approaches. First, we propose a half-analytical-half-numerical method that allows to construct multipartite PPT entangled symmetric states (PPTESS) from the qubit-qudit PPT entangled states. Second, we adapt the algorithm allowing to search for extremal elements in the convex set of bipartite PPT states [J. M. Leinaas, J. Myrheim, and E. Ovrum, Phys. Rev. A 76, 034304 (2007)] to the multipartite scenario. With its aid we search for extremal four-qubit PPTESS and show that generically they have ranks (5,7,8). Finally, we provide an exhaustive characterization of these states with respect to their separability properties.Introduction.-Entanglement has become an important notion in modern physics [1]. This striking feature of composite physical systems not only fundamentally distinguishes classical and quantum theories, but it has also developed into a key resource for various applications. For instance, it allows for quantum teleportation [2], quantum cryptography [3], and is a prerequisite for another important resource in quantum information theory (QIT)-nonlocal correlations [4]. Deciding, then, if a given quantum state is entangled (i.e., if it is not a mixture of products of states representing individual subsystems [5]) has become one of the most important problems (the so-called separability problem) in QIT and, even if simple to formulate, it is one of the hardest to solve [6].Due to the recent achievements in experimental implementations of various many-body states such as, for instance, the four-qubit bound entangled Smolin state [7], the six-qubit Dicke state states [8], or the eight-qubit Greenberger-Horne-Zeilinger (GHZ) state [9], the separability problem in quantum systems consisting of more than two constituents has gained importance. Here, the problem becomes even more complicated because one wants to answer not only the simple question of whether a particular state is entangled, but also what type of entanglement it has (see Ref.[10]). Various approaches have been proposed to detect and characterize entanglement in such systems (see, e.g., Refs. [11][12][13][14] and a recent review [15]).With this paper we fit into the above line of research and start a general program of characterization of entanglement properties and correlations of an important class of multipartite states-the so-called symmetric states 1 . These states have already been investigated (see, e.g.,

From both theoretical and experimental points of view symmetric states constitute an important class of multipartite states. Still, entanglement properties of these states, in particular those with positive partial transposition (PPT), lack a systematic study. Aiming at filling in this gap, we have recently affirmatively answered the open question of existence of four-qubit entangled symmetric states with positive partial transposition and thoroughly characterized entanglement properties of such states [J. Tura et al., Phys. Rev. A 85, 060302(R) (2012)] With the present contribution we continue on characterizing PPT entangled symmetric states. On the one hand, we present all the results of our previous work in a detailed way. On the other hand, we generalize them to systems consisting of arbitrary number of qubits. In particular, we provide criteria for separability of such states formulated in terms of their ranks. Interestingly, for most of the cases, the symmetric states are either separable or typically separable. Then, edge states in these systems are studied, showing in particular that to characterize generic PPT entangled states with four and five qubits, it is enough to study only those that assume few (respectively, two and three) specific configurations of ranks. Finally, we numerically search for extremal PPT entangled states in such systems consisting of up to 23 qubits. One can clearly notice regularity behind the ranks of such extremal states, and, in particular, for systems composed of odd number of qubits we find a single configuration of ranks for which there are extremal states.

We reduce the question whether a given quantum mixed state is separable or entangled to the problem of existence of a certain full family of commuting normal matrices whose matrix elements are partially determined by components of the pure states constituting a decomposition of the considered mixture. The method reproduces many known entanglement and/or separability criteria, and provides yet another geometrical characterization of mixed separable states.PACS numbers: 03.67.Mn,03.65.Ud I. INTRODUCTIONEntanglement and separability problem. Entanglement is the most important quantum phenomenon, responsible for genuine, distinct and unique properties of the quantum world, and possibilities this world offers for future technological applications, such as quantum engineering, and quantum information [1]. Despite enormous efforts, many fundamental questions concerning entanglement remain open (for an excellent recent review see [2], and [3] for some general geometric settings of the problem). In the seminal paper in 1989 Werner [4] gave the definition of separable (i.e. non-entangled) states: a state of a bi-partite system is separable iff it is a mixture of pure product states. A simple question: given a state, is it separable or not, is known as the separability problem. Only in very rare instances we know operational sufficient and necessary criteria (SNC) that allow to solve this problem:• for 2×2 (two qubit) and 2×3 (qubit-qutrit) systems the SNC are given by the positive definiteness of the partial transform [5]; this is the famous PPT criterion, introduced by Peres as necessary for separability in Ref.[6].• for 3 qubit symmetric ("bosonic") states PPT criterion is also SNC [7].• for continuous variables 1 × 1 (one mode per party) Gaussian states, PPT criterion (formulated at the level of correlation matrices) is a SNC [8,9].• for continuous variables m × n (all bipartite) Gaussian states there exist an operational SNC based on recursion for correlations matrices [10].• for continuous variables tripartite 1 × 1 × 1 Gaussian states there exist an operational SNC based on "iteration" of PPT condition for correlations matrices [11].In general we have to rely either on only necessary criteria, or only sufficient ones, or on numerical approaches. Although there exist very efficient numerical procedures that employ optimization methods of semi-definite programming [12], the complexity of the problem grows with the dimensionality of the underlying Hilbert spaces: in fact it has been proven that the problem belongs to the complexity NP-class [13].Reformulations of the separability problem. The market for only necessary, or only sufficient criteria is growing constantly, and it is impossible to review it in a non-review style article (for this reason we recommend the readers the review [2]). There are also many attempts to reformulate the problem of separability in different mathematical terms. A paradigm example for such an approach is the formulation of the separability problem in terms of positive maps due to Horodeccy...

Quantum mechanics is already 100 years old, but remains alive and full of challenging open problems. On one hand, the problems encountered at the frontiers of modern theoretical physics like Quantum Gravity, String Theories, etc. concern Quantum Theory, and are at the same time related to open problems of modern mathematics. But even within non-relativistic quantum mechanics itself there are fundamental unresolved problems that can be formulated in elementary terms. These problems are also related to challenging open questions of modern mathematics; linear algebra and functional analysis in particular. Two of these problems will be discussed in this article: a) the separability problem, i.e. the question when the state of a composite quantum system does not contain any quantum correlations or entanglement and b) the distillability problem, i.e. the question when the state of a composite quantum system can be transformed to an entangled pure state using local operations (local refers here to component subsystems of a given system).Although many results concerning the above mentioned problems have been obtained (in particular in the last few years in the framework of Quantum Information Theory), both problems remain until now essentially open. We will present a primer on the current state of knowledge concerning these problems, and discuss the relation of these problems to one of the most challenging questions of linear algebra: the classification and characterization of positive operator maps.

We study families of positive and completely positive maps acting on a bipartite system $\mathbb{C}^M\otimes \mathbb{C}^N$ (with $M\leq N$). The maps have a property that when applied to any state (of a given entanglement class) they result in a separable state, or more generally a state of another certain entanglement class (e.g., Schmidt number $\leq k$). This allows us to derive useful families of sufficient separability criteria. Explicit examples of such criteria have been constructed for arbitrary $M,N$, with a special emphasis on $M=2$. Our results can be viewed as generalizations of the known facts that in the sufficiently close vicinity of the completely depolarized state (the normalized identity matrix), all states are separable (belong to "weakly" entangled classes). Alternatively, some of our results can be viewed as an entanglement classification for a certain family of states, corresponding to mixtures of the completely polarized state with pure state projectors, partially transposed and locally transformed pure state projectors.Comment: 12 pages, 1 figure; V2: some minor typos corrected; comments welcom