We generate and characterise entangled states of a register of 20 individually controlled qubits, where each qubit is encoded into the electronic state of a trapped atomic ion. Entanglement is generated amongst the qubits during the out-of-equilibrium dynamics of an Ising-type Hamiltonian, engineered via laser fields. Since the qubit-qubit interactions decay with distance, entanglement is generated at early times predominantly between neighbouring groups of qubits. We characterise entanglement between these groups by designing and applying witnesses for genuine multipartite entanglement. Our results show that, during the dynamical evolution, all neighbouring qubit pairs, triplets, most quadruplets, and some quintuplets simultaneously develop genuine multipartite entanglement. Witnessing genuine multipartite entanglement in larger groups of qubits in our system remains an open challenge.

Entanglement is an important resource that allows quantum technologies to go beyond the classically possible. There are many ways quantum systems can be entangled, ranging from the archetypal two-qubit case to more exotic scenarios of entanglement in high dimensions or between many parties. Consequently, a plethora of entanglement quantifiers and classifiers exist, corresponding to different operational paradigms and mathematical techniques.However, for most quantum systems, exactly quantifying the amount of entanglement is extremely demanding, if at all possible. This is further exacerbated by the difficulty of experimentally controlling and measuring complex quantum states. Consequently, there are various approaches for experimentally detecting and certifying entanglement when exact quantification is not an option, with a particular focus on practically implementable methods and resource efficiency. The applicability and performance of these methods strongly depends on the assumptions one is willing to make regarding the involved quantum states and measurements, in short, on the available prior information about the quantum system. In this review we discuss the most commonly used paradigmatic quantifiers of entanglement. For these, we survey state-of-the-art detection and certification methods, including their respective underlying assumptions, from both a theoretical and experimental point of view.In the early twentieth century, the phenomenon of quantum entanglement rose to prominence as a central feature of the famous thought experiment by Einstein, Podolsky, and Rosen [1]. Initially disregarded as a mathematical artefact that showcases the incompleteness of quantum theory, the properties of entanglement were largely ignored until 1964, when John Bell famously proposed an experimentally testable inequality able to distinguish between the predictions of quantum mechanics and those of any local-realistic theory [2]. With the advent of the first experimental tests [3], spearheaded by , emerged the realisation that entanglement constitutes a resource for information processing *

High-dimensional encoding of quantum information provides a promising method of transcending current limitations in quantum communication. One of the central challenges in the pursuit of such an approach is the certification of high-dimensional entanglement. In particular, it is desirable to do so without resorting to inefficient full state tomography. Here, we show how carefully constructed measurements in two bases (one of which is not orthonormal) can be used to faithfully and efficiently certify bipartite high-dimensional states and their entanglement for any physical platform. To showcase the practicality of this approach under realistic conditions, we put it to the test for photons entangled in their orbital angular momentum. In our experimental setup, we are able to verify 9dimensional entanglement for a pair of photons on a 11-dimensional subspace each, at present the highest amount certified without any assumptions on the state.Quantum communication offers advantages such as enhanced security in quantum key distribution (QKD) protocols [1] and increased channel capacities [2] with respect to classical means of communication. All of these improvements, ranging from early proposals [3] to recent exciting developments such as fully device-independent QKD [4, 5], rely on one fundamental phenomenon: quantum entanglement. Currently, the workhorse of most implementations is entanglement between qubits, i.e., between two-dimensional quantum systems (e.g., photon polarization). However, it has long been known that higher-dimensional entanglement can be useful in overcoming the limitations of qubit entanglement [6, 7], offering better key rates [8], higher noise resistance [9, 10] and improved security against different attacks [11].Attempting to capitalize on this insight, recent experiments have successfully generated and certified highdimensional entanglement in different degrees of freedom. In particular, the canonical way of generating two-dimensional polarization entanglement in downconversion processes already offers the potential for exploring entanglement in higher dimensions. This can be achieved by exploiting spatial degrees of freedom [12, 13], orbital angular momentum (OAM) [14-16], energy-time based encodings [17][18][19][20], or combinations thereof in hyper-entangled quantum systems [21,22]. High-dimensional quantum systems have recently also been explored in matter-based systems such as Cesium atoms [23] and superconducting circuits [24]. Thus, high- dimensional quantum systems are not only of fundamental interest but are also becoming more readily available.In this context, the certification and quantification of entanglement in many dimensions is a crucial challenge since full state tomography (FST) for bipartite systems of local dimension d requires measurements in (d + 1) 2 global product bases (i.e., tensor product bases for the global state) [25], which quickly becomes impractical in high dimensions. Due to the complexity of realizing measurements in high-dimensional spaces, previous ex...