Certifying Entanglement Dimensionality by k -Reduction Moments

Abstract

In this paper, we combine the k -reduction map, the moment method, and randomized measurements into a practical protocol for certifying the entanglement dimensionality. Our approach is based on the observation that a state with entanglement dimensionality at most k must stay positive under the action of the k -reduction map. The core of our protocol utilizes the moment method to determine whether the k -reduced operator, i.e., the operator obtained after applying the k -reduction map on a quantum state, contains a negative eigenvalue or not. Notably, we propose a systematic method for constructing reduction moment criteria that apply to a broader range of states, including k -unfaithful states, compared with fidelity-based methods. The performance of our approach gets better and better with the moment order employed, which is corroborated by extensive numerical simulations. To apply our approach, it suffices to implement a unitary 3-design instead of a 4-design, which is more feasible in practice than the correlation matrix method. In the course of study, we show that the k -reduction negativity, the absolute sum of the negative eigenvalues of the k -reduced operator, is monotonic under local operations and classical communication for pure states.