We address the question of quantifying entanglement in pure graph states. We demonstrate how solving one problem in graph theory, namely the identification of maximum independent set, allows for analytic evaluation of three multipartite entanglement measures for pure graph states. We achieve this by explicit construction of relevant closest separable states and minimal linear decompositions into product states. We show how these separable states can be described using stabiliser formalism as well as PEPs-like construction opening the possibility to applying same techniques to more general states, such as weighted graph states.
We address the question of quantifying entanglement in pure graph states. We demonstrate how solving one problem in graph theory, namely the identification of maximum independent set, allows for analytic evaluation of three multipartite entanglement measures for pure graph states. We achieve this by explicit construction of relevant closest separable states and minimal linear decompositions into product states. We show how these separable states can be described using stabiliser formalism as well as PEPs-like construction opening the possibility to applying same techniques to more general states, such as weighted graph states.
We address the question of quantifying entanglement in pure graph states. The evaluation of multipartite entanglement measures is extremely difficult for most pure quantum states. In this paper, we demonstrate how solving one problem in graph theory, namely the identification of a maximum independent set, allows us to evaluate three multipartite entanglement measures for pure graph states. We construct the minimal linear decomposition into product states for a large group of pure graph states, allowing us to evaluate the Schmidt measure. Furthermore, we show that the computation of distance-like measures such as relative entropy of entanglement and geometric measure becomes tractable for these states by the explicit construction of the closest separable and closest product states, respectively. We show how these separable states can be described using stabilizer formalism as well as projected entangled pairs-like construction. Finally, we discuss how introducing noise into the system can optimally destroy entanglement.
We address the question of quantifying entanglement in pure graph states. The evaluation of multipartite entanglement measures is extremely difficult for most pure quantum states. In this paper, we demonstrate how solving one problem in graph theory, namely the identification of a maximum independent set, allows us to evaluate three multipartite entanglement measures for pure graph states. We construct the minimal linear decomposition into product states for a large group of pure graph states, allowing us to evaluate the Schmidt measure. Furthermore, we show that the computation of distance-like measures such as relative entropy of entanglement and geometric measure becomes tractable for these states by the explicit construction of the closest separable and closest product states, respectively. We show how these separable states can be described using stabilizer formalism as well as projected entangled pairs-like construction. Finally, we discuss how introducing noise into the system can optimally destroy entanglement.
This thesis is an attempt to enhance understanding of the following questions A- Given a multipartite quantum state (possibly mixed), how to find out whether it is entangled or separable? (Detection of entanglement.) B- Given an entangled state, how to decide how much entangled it is? (Measure of entanglement.), in the context of multipartite quantum states. We have explored two approaches. In the first approach, we assign a weighted graph with multipartite quantum state and address the question of separability in terms of these graphs and various operations involving them. In the second approach we use the so called Bloch representation of multipartite quantum states to establish new criteria for detection of multipartite entangled states. We further give a new measure for entanglement in N-qubit entangled pure state and formally extend it to cover N-qubit mixed states.
We present a general theory for the construction of witnesses that detect genuine multipartite entanglement in graph states. First, we present explicit witnesses for all graph states of up to six qubits which are better than all criteria so far. Therefore, lower fidelities are required in experiments that aim at the preparation of graph states. Building on these results, we develop analytical methods to construct two different types of entanglement witnesses for general graph states. For many classes of states, these operators exhibit white noise tolerances that converge to 1 when increasing the number of particles. We illustrate our approach for states such as the linear and the 2D cluster state. Finally, we study an entanglement monotone motivated by our approach for graph states.
Abstract Computationally feasible multipartite entanglement measures are essential for advancing our understanding of complex quantum systems. Entanglement distance (ED), introduced by Cocchiarella et al (2020 Phys. Rev. A 101 042129), based on the Fubini–Study metric, offers several advantages over existing methods, including ease of computation, a profound geometrical interpretation, and applicability to multipartite entanglement. Although ED has been successfully applied to systems of qudits, an explicit formulation for continuous quantum variables, particularly for pure Gaussian states, remains unexplored. In this work, we address this limitation by deriving the analytical expression for the Gaussian entanglement measure (GEM), a multipartite entanglement monotone for multimode pure Gaussian states based on the purity of fragments of the whole system, through a generalization of ED to group-theoretic coherent states. We show the efficacy of GEM across various scenarios, including the analysis of a two-mode Gaussian state under beam-splitter and squeezing transformations, and exploring graph states involving three and four modes. Notably, comparing GEM values for states with different graph topologies reveals insights into the connectivity of the underlying graphs. Additionally, we illustrate how GEM provides insights into free bosonic field theory on R t × S 1 beyond standard bipartite entanglement entropy, paving the way for quantum information-theoretic tools to probe the topological properties of quantum field theories.
Several inequivalent definitions of the geometric measure of entanglement (GM) have been introduced and studied in the past. Here we review several known and new definitions, with the qualifying criterion being that for pure states the measure is a linear or logarithmic function of the maximal fidelity with product states. The entanglement axioms and properties of the measures are studied, and qualitative and quantitative comparisons are made between all definitions. Streltsov et al. [New J. Phys. 12, 123004 (2010)] proved the equivalence of two linear definitions of GM, whereas we show that the corresponding logarithmic definitions are distinct. Certain classes of states such as ``maximally correlated states'' and isotropic states are particularly valuable for this analysis. A little-known GM definition is found to be the first one to be both normalized and weakly monotonous, thus being a prime candidate for future studies of multipartite entanglement. We also find that a large class of graph states, which includes all cluster states, have a ``universal'' closest separable state that minimizes the quantum relative entropy, the Bures distance, and the trace distance.