This thesis investigates methods to approximate convex sets which involve minimizing the Hausdorff metric between a set and certain subsets. We begin by giving a lower bound for the Hausdorff metric between a hypersphere and a circumscribed simplex. We show that this bound is achieved by the regular simplex. Next, we form a lower bound on the Hausdorff distance between the convex hull of the joint numerical range of positive operator valued-measures and the probability simplex. An entanglement witness is a linear functional that separates the convex compact set of separable states from certain entangled states in the Hilbert space. We investigate the applications of our methods by exploring the problem of finding a polytope generated by entanglement witnesses that has minimal distance to the set of separable states.
This thesis investigates methods to approximate convex sets which involve minimizing the Hausdorff metric between a set and certain subsets. We begin by giving a lower bound for the Hausdorff metric between a hypersphere and a circumscribed simplex. We show that this bound is achieved by the regular simplex. Next, we form a lower bound on the Hausdorff distance between the convex hull of the joint numerical range of positive operator valued-measures and the probability simplex. An entanglement witness is a linear functional that separates the convex compact set of separable states from certain entangled states in the Hilbert space. We investigate the applications of our methods by exploring the problem of finding a polytope generated by entanglement witnesses that has minimal distance to the set of separable states.
Quantifying entanglement for multipartite quantum state is a crucial task in many aspects of quantum information theory. Among all the entanglement measures, relative entropy of entanglement ER is an important quantity due to its clear geometric meaning, easy compatibility with different system sizes, and various applications in many other related quantity calculations. Lower bounds of ER were previously found based on distance to the set of positive partial transpose states. We propose a method to calculate upper bounds of ER based on the convex hull approximation of the set of separable states. We apply our method to calculate ER for composite systems of various sizes, and compare with the previous known lower bounds, obtaining promising results. Our method adds a new tool for entanglement measure calculation and deepens our understanding for the structure of separable states.
Gilbert proposed an algorithm for bounding the distance between a given point and a convex set. We apply the Gilbert's algorithm to get an upper bound on the Hilbert-Schmidt distance between a given state and the set of separable states. While Hilbert-Schmidt distance does not form a proper entanglement measure, it can nevertheless be useful for witnessing entanglement. We provide a few methods based on the Gilbert's algorithm that can reliably qualify a given state as strongly entangled or practically separable, while being computationally efficient. The method also outputs successively improved approximations to the closest separable state for the given state. We demonstrate the efficacy of the method with examples.
Quantum entanglement is a unique correlation phenomenon in quantum mechanics, and the measurement of quantum entanglement plays an important role in quantum computing and quantum communication. Many mainstream entanglement criteria and measurement methods currently known have shortcomings in certain aspects, such as not being sufficient or necessary conditions for entanglement, or only being effective in simple cases such as 2-qubits or pure states. In this work, we will propose a scheme for measuring quantum entanglement, which starts with treating the set of quantum separable states as a convex hull of quantum separable pure states, and analyzes the properties of the convex hull to obtain a new form of entanglement measurement. Although a large amount of data is required in the measurement process, this method is not only applicable to 2-qubit quantum states, but also a entanglement measurement method that can be applied to any dimension and any fragment. We will provide several examples to compare their results with other entanglement metrics and entanglement determination methods to verify their feasibility.
The problem of determining whether a given quantum state is separable is known to be computationally difficult. We develop an approach to this problem based on approximations of convex polytopes in high dimensions. By showing that a convex polytope constructed from a finite number of hyperplanes approximates the Euclidean ball arbitrarily well in high dimensions, we find evidence that a finite set of approximate entanglement witnesses is potentially sufficient to determine the entanglement of a state with high probability.
Convex optimization problems arise naturally in quantum information theory, often in terms of minimizing a convex function over a convex subset of the space of hermitian matrices. In most cases, finding exact solutions to these problems is usually impossible. As inspired by earlier investigations into the relative entropy of entanglement (REE) (Miranowicz and Ishizaka 2008 Phys. Rev. A 78 032310 ), we introduce a general method to solve the converse problem rather than find explicit solutions. That is, given a matrix in a convex set, we determine a family of convex functions that are minimized at this point. This method allows us find explicit formulae for the REE and the Rains bound, two well-known upper bounds on the distillable entanglement, and yields interesting information about these quantities, such as the fact that they coincide in the case where at least one subsystem of a multipartite state is a qubit.
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.