A general procedure to construct criteria for identifying genuine multipartite continuous-variable entanglement is presented. It relies on the definition of adequate global operators describing the multipartite system, the positive partial transpose criterion of separability, and quantum-mechanical uncertainty relations. As a consequence, each criterion encountered consists of a single inequality that is nicely computable and experimentally feasible. Violation of the inequality is a sufficient condition for genuine multipartite entanglement. Additionally, we show that the previous work of van Loock and Furusawa [P. van Loock and A. Furusawa, Phys. Rev. A 67, 052315 (2003)] is a special case of our result.
Following previous work, we distinguish between genuine
N-partite entanglement and full
N-partite inseparability. Accordingly, we derive criteria to detect genuine multipartite entanglement using continuous-variable (position and momentum) measurements. Our criteria are similar but different to those based on the van Loock--Furusawa inequalities, which detect full
N-partite inseparability. We explain how the criteria can be used to detect the genuine
N-partite entanglement of continuous variable states generated from squeezed and vacuum state inputs, including the continuous-variable Greenberger-Horne-Zeilinger state, with explicit predictions for up to
N=9. This makes our work accessible to experiment. For
N=3, we also present criteria for tripartite Einstein-Podolsky-Rosen (EPR) steering. These criteria provide a means to demonstrate a genuine three-party EPR paradox, in which any single party is steerable by the remaining two parties.
Genuine entanglement identification of large scale systems is crucial for quantum computation, quantum communication and quantum learning advantage. In contrast to experiments, where noisy intermediate-scale programmable photonic quantum processors have been developed, theoretically very limited results have been achieved for detecting genuine entanglement of continuous variable multipartite systems. We propose a quite general and efficient entanglement detection framework for all kinds of multipartite entanglement of continuous variable systems based on uncertainty relations and the sign matrix technique. Matrix criteria are demonstrated and can be applied to various entanglement depth and k-separability problems of multimode systems. We illustrate the genuine entanglement conditions of continuous variable Greenberger-Horne-Zeilinger states of more than a hundred million modes in a photon loss and noise environment.