We introduce a family of separability criteria that are based on the existence of extensions of a bipartite quantum state \ensuremath{\rho} to a larger number of parties satisfying certain symmetry properties. It can be easily shown that all separable states have the required extensions, so the nonexistence of such an extension for a particular state implies that the state is entangled. One of the main advantages of this approach is that searching for the extension can be cast as a convex optimization problem known as a semidefinite program. Whenever an extension does not exist, the dual optimization constructs an explicit entanglement witness for the particular state. These separability tests can be ordered in a hierarchical structure whose first step corresponds to the well-known positive partial transpose (Peres-Horodecki) criterion, and each test in the hierarchy is at least as powerful as the preceding one. This hierarchy is complete, in the sense that any entangled state is guaranteed to fail a test at some finite point in the hierarchy, thus showing it is entangled. The entanglement witnesses corresponding to each step of the hierarchy have well-defined and very interesting algebraic properties that, in turn, allow for a characterization of the interior of the set of positive maps. Coupled with some recent results on the computational complexity of the separability problem, which has been shown to be NP hard, this hierarchy of tests gives a complete and also computationally and theoretically appealing characterization of mixed bipartite entangled states.
We introduce a family of separability criteria that are based on the existence of extensions of a bipartite quantum state \ensuremath{\rho} to a larger number of parties satisfying certain symmetry properties. It can be easily shown that all separable states have the required extensions, so the nonexistence of such an extension for a particular state implies that the state is entangled. One of the main advantages of this approach is that searching for the extension can be cast as a convex optimization problem known as a semidefinite program. Whenever an extension does not exist, the dual optimization constructs an explicit entanglement witness for the particular state. These separability tests can be ordered in a hierarchical structure whose first step corresponds to the well-known positive partial transpose (Peres-Horodecki) criterion, and each test in the hierarchy is at least as powerful as the preceding one. This hierarchy is complete, in the sense that any entangled state is guaranteed to fail a test at some finite point in the hierarchy, thus showing it is entangled. The entanglement witnesses corresponding to each step of the hierarchy have well-defined and very interesting algebraic properties that, in turn, allow for a characterization of the interior of the set of positive maps. Coupled with some recent results on the computational complexity of the separability problem, which has been shown to be NP hard, this hierarchy of tests gives a complete and also computationally and theoretically appealing characterization of mixed bipartite entangled states.
We show how to design families of operational criteria that distinguish entangled from separable quantum states. The simplest of these tests corresponds to the well-known Peres-Horodecki positive partial transpose (PPT) criterion, and the more complicated tests are strictly stronger. The new criteria are tractable due to powerful computational and theoretical methods for the class of convex optimization problems known as semidefinite programs. We successfully applied the results to many low-dimensional states from the literature where the PPT test fails. As a by-product of the criteria, we provide an explicit construction of the corresponding entanglement witnesses.
We study the application of sums of squares decompositions and semidefinite programming in the formulation of computational criteria for distinguishing entangled from separable quantum states. A hierarchy of tests is obtained, the simplest of which corresponds to the well-known positive partial transpose (PPT) sufficient criterion, with the more complicated tests being strictly stronger. The duality structure of the problem allows us to provide an explicit construction of the entanglement witnesses corresponding to our tests.
We provide a complete mathematical theory for the entanglement of mixtures of Dicke states. These quantum states form an important subclass of bosonic states arising in the study of indistinguishable particles. We introduce a tensor-based parametrization where the diagonal entries of these states are encoded as a symmetric tensor, enabling a direct translation between entanglement properties and well-studied convex cones of tensors. Our results bridge multipartite entanglement theory with semialgebraic geometry and the theory of completely positive and copositive tensors.
This dictionary maps separability to completely positive tensors, the PPT property to moment tensors, entanglement witnesses to copositive tensors, and decomposable witnesses to sum of squares tensors. Using this framework, we construct explicit PPT entangled states in three or more qutrits, disproving a recent conjecture. We establish that PPT entanglement exists for all multipartite systems with local dimension d >= 3 and n >= 3 parties. We also show that, for mixtures of Dicke states, the PPT condition with respect to the most balanced bipartition implies all other PPT conditions.
We further connect bosonic extendibility of mixtures of Dicke states to the duals of known hierarchies for non-negative polynomials, such as the ones by Reznick and Polya. We thus provide semidefinite programming relaxations for separability and entanglement testing in the Dicke subspace.
Ever since entanglement was identified as a computational and cryptographic resource, researchers have sought efficient ways to tell whether a given density matrix represents an unentangled, or \emph{separable}, state. This paper gives the first systematic and comprehensive treatment of this (bipartite) quantum separability problem, focusing on its deterministic (as opposed to randomized) computational complexity. First, I review the one-sided tests for separability, paying particular attention to the semidefinite programming methods. Then, I discuss various ways of formulating the quantum separability problem, from exact to approximate formulations, the latter of which are the paper's main focus. I then give a thorough treatment of the problem's relationship with NP, NP-completeness, and co-NP. I also discuss extensions of Gurvits' NP-hardness result to strong NP-hardness of certain related problems. A major open question is whether the NP-contained formulation (QSEP) of the quantum separability problem is Karp-NP-complete; QSEP may be the first natural example of a problem that is Turing-NP-complete but not Karp-NP-complete. Finally, I survey all the proposed (deterministic) algorithms for the quantum separability problem, including the bounded search for symmetric extensions (via semidefinite programming), based on the recent quantum de Finetti theorem \cite{DPS02,DPS04,qphCKMR06}; and the entanglement-witness search (via interior-point algorithms and global optimization) \cite{ITCE04,IT06}. These two algorithms have the lowest complexity, with the latter being the best under advice of asymptotically optimal point-coverings of the sphere.
Ever since entanglement was identified as a computational and cryptographic resource, researchers have sought efficient ways to tell whether a given density matrix represents an unentangled, or separable, state. This paper gives the first systematic and comprehensive treatment of this (bipartite) quantum separability problem, focusing on its deterministic (as opposed to randomized) computational complexity. First, I review the one-sided tests for separability, paying particular attention to the semidefinite programming methods. Then, I discuss various ways of formulating the quantum separability problem, from exact to approximate formulations, the latter of which are the paper’s main focus. I then give a thorough treatment of the problem’s relationship with NP, NP-completeness, and co-NP. I also discuss extensions of Gurvits ’ NP-hardness result to strong NP-hardness of certain related problems. A major open question is whether the NP-contained formulation (QSEP) of the quantum separability problem is Karp-NP-complete; QSEP may be the first natural example of a problem that is Turing-NP-complete but not Karp-NP-complete. Finally, I survey all the proposed (deterministic) algorithms for the quantum separability problem, including the bounded search for symmetric extensions (via semidefinite programming), based on the recent quantum de Finetti theorem [1, 2, 3]; and the entanglement-witness search (via interior-point algorithms and global optimization) [4, 5]. These two algorithms have the lowest complexity, with the latter being the best under advice of asymptotically optimal point-coverings of the sphere.
We study families of positive and completely positive maps acting on a bipartite system ${\mathbb{C}}^{M}\ensuremath{\bigotimes}{\mathbb{C}}^{N}$ (with $M\ensuremath{\le}N$). The maps have a property that, when applied to any state (of a given entanglement class), result in a separable state or, more generally, a state of another certain entanglement class (e.g., Schmidt number $\ensuremath{\le}\phantom{\rule{0.16em}{0ex}}k$). 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 states, partial transposes, and/or local transformations thereof.