Certifying entanglement dimensionality by $k$-reduction moments

TL;DR

This paper introduces a practical protocol combining k-reduction maps, the moment method, and classical shadows to certify entanglement dimensionality, improving feasibility and applicability over previous methods.

Contribution

It develops a systematic way to construct reduction moment criteria that apply to a broader class of states and requires only a unitary 3-design for implementation.

Findings

Performance improves with higher moment order, validated by numerical simulations.

The method applies to a wider range of states than fidelity-based approaches.

Requires only a unitary 3-design, making it more feasible in practice.

Abstract

In this paper, we combine the k-reduction map, the moment method, and the classical shadow method 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 negative eigenvalues or not. Notably, we propose a systematic method for constructing reduction moment criteria, which apply to a much wider range of states than fidelity-based methods. The performance of our approach gets better and better with the moment order employed, which is corroborated by extensive numerical simulation. 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.