Detecting the dimensionality of genuine multi-particle entanglement

Abstract: Complex forms of quantum entanglement can arise in two qualitatively different ways; either between many qubits or between two particles with higher-than-qubit dimension. While the many-qubit frontier and the high-dimension frontier both are well-established, state-of-the-art quantum technology is becoming increasingly able to create and manipulate entangled states that simltaneously feature many particles and high dimension. Here, we investigate generic states that can be considered both genuinely high-dimensional and genuine multi-particle entangled. We consider a natural quantity that characterises this key property. To detect it, we develop three different classes of criteria. These enable us both to probe the ultimate noise tolerance of this form of entanglement and to make detection schemes using sparse or even minimal measurement resources. The approach provides a simple way of benchmarking entanglement dimensionality in the multi-particle regime and general, platform-independent, detection methods that readily apply to experimental use.