Entanglement Theory

Abstract: Entanglement is an important resource that allows quantum technologies to gobeyond the classically possible. There are many ways quantum systems can beentangled, ranging from the archetypal two-qubit case to more exotic scenariosof entanglement in high dimensions or between many parties. Consequently, aplethora of entanglement quantifiers and classifiers exist, corresponding todifferent operational paradigms and mathematical techniques. However, for mostquantum systems, exactly quantifying the amount of entanglement is extremelydemanding, if at all possible. This is further exacerbated by the difficulty ofexperimentally controlling and measuring complex quantum states. Consequently,there are various approaches for experimentally detecting and certifyingentanglement when exact quantification is not an option, with a particularfocus on practically implementable methods and resource efficiency. Theapplicability and performance of these methods strongly depends on theassumptions one is willing to make regarding the involved quantum states andmeasurements, in short, on the available prior information about the quantumsystem. In this review we discuss the most commonly used paradigmaticquantifiers of entanglement. For these, we survey state-of-the-art detectionand certification methods, including their respective underlying assumptions,from both a theoretical and experimental point of view.