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 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 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 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 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.
Entanglement is a key resource in the emerging field of Quantum Information. The strong correlations between systems described by an entangled state allow us to perform certain tasks more efficiently than it would be possible by using only classical resources. This is why the characterization of entanglement is one of the most important problems in Quantum Information. In this thesis, we analyze several aspects of entanglement. First, we introduce a new family of criteria to determine if a bipartite mixed state is entangled or not. This family consists of a sequence of tests that can be implemented efficiently, and has the property that all entangled states can be detected by some test in the sequence. Each test in the family can be stated as a semidefinite program, which is a class of convex optimization problems. The duality structure of these programs allows us to explicitly construct an entanglement witness that proves entanglement of a state, whenever the state fails one of the tests in the sequence. The entanglement witnesses constructed in this manner have well-defined algebraic properties that can be used to give a characterization of the interior of the set of all possible entanglement witnesses, as well as the set of strictly positive bihermitian forms and the set of strictly positive maps. We also study deterministic transformations of three-qubit pure state when only local operations and classical communication (LOCC) are allowed. We derive strong constraints that the operations and states involved must satisfy, and we apply these results to characterize the set of real states that can be obtained from the GHZ state by LOCC.
An additive and semidefinite programming (SDP) computable entanglement measure is introduced to upper bound the amount of distillable entanglement in bipartite quantum states by operations completely preserving the positivity of partial transpose (PPT). This quantity is always smaller than or equal to the logarithmic negativity, the previously best known SDP bound on distillable entanglement, and the inequality is strict in general. Furthermore, a succinct SDP characterization of the one-copy PPT deterministic distillable entanglement for any given state is also obtained, which provides a simple but useful lower bound on the PPT distillable entanglement. Remarkably, there is a genuinely mixed state of which both bounds coincide with the distillable entanglement, while being strictly less than the logarithmic negativity.
Quantum entanglement lies at the heart of quantum information science, yet its reliable detection in high-dimensional or noisy systems remains a fundamental computational challenge. Semidefinite programming (SDP) hierarchies, such as the Doherty-Parrilo-Spedalieri (DPS) and Extension (EXT) hierarchies, offer complete methods for entanglement detection, but it is well known that their practical use is limited by exponential growth in problem size if implemented naively. We make three contributions. First, we introduce a new SDP hierarchy, PST, that is sandwiched between EXT and DP -- offering a tighter approximation to the set of separable states than EXT, while incurring significantly lower computational overhead than DPS. Second, we explicitly construct compact, polynomially-scalable descriptions of EXT and PST using partition mappings and operators. These descriptions in turn yield formulations that satisfy desirable properties such as the Slater condition and are well-suited to both first-order methods (FOMs) and interior-point methods (IPMs). Third, we design a suite of entanglement detection algorithms: three FOMs (Frank-Wolfe, projected gradient, and fast projected gradient) based on a least-squares formulation, and a custom primal-dual IPM based on a conic programming formulation. These methods are numerically stable and capable of producing entanglement witnesses or proximity measures, even in cases where states lie near the boundary of separability. Numerical experiments on benchmark quantum states demonstrate that our algorithms improve the ability to solve deeper levels of the SDP hierarchy.