The fundamental relation between quantum entanglement and convex algebraic geometry has unveiled a set of powerful tools, imported from the former to the study the latter. The space of separable mixed states is convex and so is the space of the corresponding observable values. Therefore, the problem of determining whether a set of observable values come from an entangled state is tantamount to checking for membership in a convex set, of a point with coordinates given by the set of observable values. Here, we use techniques from convex algebraic geometry to develop powerful criteria for entanglement in a many-body system of Bosonic atoms with a non-zero spin. The experimentally accessible observables are the spin expectation values <S_i> and <{S_i, S_j}> which, upon truncating at rank two, are 9 independent quantities. Recently, entanglement criteria in terms of 3 of these 9 quantities have been derived. We develop entanglement criteria using all of these 9 quantities. If we consider these numbers as coordinates of a point in a 9 dimensional space, those with a separable parent state lie within a convex region, also known as the moment cone. Therefore, the problem is to characterize this moment cone. Owing to the Bosonic exchange symmetry, this moment cone is the dual of the cone of non-negative symmetric polynomials. Together with a characterization of this cone, we also develop an entanglement criterion that is asymptotically tight, for large atom numbers.
The fundamental relation between quantum entanglement and convex algebraic geometry has unveiled a set of powerful tools, imported from the former to the study the latter. The space of separable mixed states is convex and so is the space of the corresponding observable values. Therefore, the problem of determining whether a set of observable values come from an entangled state is tantamount to checking for membership in a convex set, of a point with coordinates given by the set of observable values. Here, we use techniques from convex algebraic geometry to develop powerful criteria for entanglement in a many-body system of Bosonic atoms with a non-zero spin. The experimentally accessible observables are the spin expectation values <S_i> and <{S_i, S_j}> which, upon truncating at rank two, are 9 independent quantities. Recently, entanglement criteria in terms of 3 of these 9 quantities have been derived. We develop entanglement criteria using all of these 9 quantities. If we consider these numbers as coordinates of a point in a 9 dimensional space, those with a separable parent state lie within a convex region, also known as the moment cone. Therefore, the problem is to characterize this moment cone. Owing to the Bosonic exchange symmetry, this moment cone is the dual of the cone of non-negative symmetric polynomials. Together with a characterization of this cone, we also develop an entanglement criterion that is asymptotically tight, for large atom numbers.
Although the quantum state of a many-body system (e.g, a system consisting of several atoms) is described in terms of its degrees of freedom, i.e., a large number of variables representing a point on the compact space of quantum states, there is usually a small number of parameters that can be actually measured experimentally. We show that the problem of deciding whether the underlying quantum state is entangled is equivalent to the problem of deciding whether the set of measured parameters are the moments of some measure defined over the compact space of states of the many-body system. By considering quantum entanglement of n identical atoms we arrive at the truncated moment problem defined for symmetric measures over a product of n copies of unit balls in R^d. We consider moments up to degree 2 since these are most readily available experimentally. We derive necessary and sufficient conditions for belonging to the moment cone, which can be expressed via a linear matrix inequality (LMI) of size at most 2d+2, which is independent of n. We can convert the LMIs into a set of explicit semialgebraic inequalities giving necessary and sufficient conditions for membership in the moment cone, and show that the two conditions approach each other in the limit of large n. The inequalities are derived via considering the dual cone of nonnegative polynomials, and its sum-of-squares (SOS) relaxation. We show that the SOS relaxation of the dual cone is asymptotically exact, and using symmetry reduction techniques (J. Pure Appl. Algebra 192, no. 1-3, 95, arXiv:1205.3102), it can be written as a small linear matrix inequality of size at most 2d+2, which is independent of n. For the cone of nonnegative polynomials we also prove an analogue of the half degree principle for globally nonnegative symmetric polynomials (J. Math. Anal. Appl. 284, no. 1, 174, J. Pure Appl. Algebra 216, no. 4, 850).
Randomized measurement protocols give experimental access to low-degree polynomial invariants of a quantum state rather than to the state itself, which raises the question of what the best entanglement criterion is that can be built from finitely many such invariants. In this work we answer this question through convex duality.
For each degree we introduce the cone of local unitary trace inequalities that are valid for separable bipartite states in all local dimensions, together with its one-matrix counterparts for Hermitian and positive semidefinite matrices. Each cone is the polar dual of a moment cone; we also determine the minimal combinatorial parametrization of the separable cone. The framework organizes the known moment criteria: the PPT criterion using the first three moments of the partial transposition is the image under an embedding of a Hankel determinant from the positive semidefinite cone to the bipartite cone, which we show to be an extremal ray, and we determine the first nontrivial separable cone completely, finding exactly four extremal rays.
The framework allows us to analyze the realignment criterion and its centered (or enhanced) variant. We prove that the even singular value moments of the (centered) realignment matrix are trace invariants, associated with explicit fixed-point-free involutions. Extracting the optimal trace-norm bound from m such moments thus becomes a dimension-free truncated moment problem, which we solve in closed form for m=2 and relax to a semidefinite program of size O(m) for arbitrary m. The dual solutions are polynomial minorants of x on [0,1], and two explicit families, binomial and L1-weighted, yield hierarchies of separability inequalities. Finally, we compare the criteria against one another and on PPT 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.
Generalizing the well-known spin-squeezing inequalities, we study the relation between squeezing of collective N -particle s u ( d ) operators and many-body entanglement geometry in multi-particle systems. For that aim, we define the set of pseudo-separable states, which are mixtures of products of single-particle states that lie in the ( d 2 − 1 ) -dimensional Bloch sphere but are not necessarily positive semidefinite. We obtain a set of necessary conditions for states of N qudits to be of the above form. Any state that violates these conditions is entangled. We also define a corresponding s u ( d ) -squeezing parameter that can be used to detect entanglement in large particle ensembles. Geometrically, this set of conditions defines a convex set of points in the space of first and second moments of the collective N -particle s u ( d ) operators. We prove that, in the limit N ≫ 1 , such set is filled by pseudo-separable states, while any state corresponding to a point outside of this set is necessarily entangled. We also study states that are detected by these inequalities: We show that states with a bosonic symmetry are detected if and only if the two-body reduced state violates the positive partial transpose (PPT) criterion. On the other hand, highly mixed states states close to the s u ( d ) singlet are detected which have a separable two-body reduced state and are also PPT with respect to all possible bipartitions. We also provide numerical examples of thermal equilibrium states that are detected by our set of inequalities, comparing the spin-squeezing inequalities with the s u ( 3 ) -squeezing inequalities.
We present solutions to a set of problems that arise in quantum entanglement theory, whose common trait is the use of algebraic methods. The backbone of the thesis consists of two general theorems, pertaining to specific convex sets of quantum maps. They are complemented by solutions of many more specific problems of rather technical character. These problems concerned questions such as the higher rank numerical ranges, the lengths of separable states, the solutions of compression equations, the completely entangled subspaces, and maximally entangled states. The solutions were obtained partly by hand, and partly by using the Groebner basis toolset available in computer algebra systems. The main results, which are those of general nature, consist of a characterization of mapping cones, which is a full analogue of the positive maps criterion for separability, as well as of a full characterization of positive-partial-transpose entangled states of minimum rank. The thesis also contains an introduction to the theory of quantum entanglement and to basic algebraic geometry, whose methods have been used
The main topic of this compilation thesis is the investigation of multipartite entanglement of finite dimensional systems. We developed a numerical algorithm that detects if a multipartite state is entangled or separable in a finite number of steps of a semi-definite optimization task. This method is an extension of previously known semi-definite methods, which are inconclusive when the state is separable. In our case, if the state is separable, an explicit decomposition into a mixture of separable states can be extracted. This was achieved by mapping the entanglement problem onto the mathematically well studied truncated moment problem. Additionally, a new way of writing the partially transposed state for symmetric multi-qubit states was developed which simplifies many results previously known in entanglement theory. This new way of writing the partial transpose criterion unifies different interpretations and alternative formulations of the partial transpose criterion and it is also a part in the aforementioned semi-definite algorithm. The geometric properties of entangled symmetric states of two qubits were studied in detail: We could answer the question of how far a given pure state is from the convex hull of symmetric separable states, as measured by the Hilbert-Schmidt distance, by giving an explicit formula for the closest separable symmetric state. For mixed states we could provide a numerical upper and analytical lower bound for this distance. For a larger number of qubits we investigated the ball of absolutely classical states, i.e. symmetric multi-qubit states that stay separable under any unitary transformation. We found an analytical lower bound for the radius of this ball around the maximally mixed symmetric state and gave a numerical upper bound on this radius, by searching for an entangled state as close as possible to the maximally mixed symmetric state. The tensor representation of a symmetric multi-qubit state, or spin-j state, allowed us to study entanglement properties based on the spectrum of the tensor via tensor eigenvalues. The definiteness of this tensor relates to the entanglement of the state it represents and, hence, the smallest tensor eigenvalue can be used to detect entanglement. However, the tensor eigenvalues are more difficult to determine than the familiar matrix eigenvalues which made the investigation computationally more challenging. The relationship between the value of the smallest tensor eigenvalue and the amount of entanglement in the state was also investigated. It turned out that they are strongly correlated for small system sizes, i.e. for up to six qubits. However, to investigate this correlation we needed an independent way to gauge the amount of entanglement of a state and in order to do so we improved existing numerical methods to determine the distance of a state to the set of separable symmetric states, using a combination of linear and quadratic programming. The tensor representation of symmetric multi-qubit states was also used to formally define a new tensor class of regularly decomposable tensors that corresponds to the set of separable symmetric multi-qubit states.