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  • Open Access

Certifying Entanglement Dimensionality by k-Reduction Moments

Changhao Yi1,2,3,4,*,†, Xiaodi Li5,2,3,4,†,‡, and Huangjun Zhu2,3,4

  • *Contact author: cyi@shu.edu.cn
  • †These authors contributed equally to this work.
  • ‡Contact author: lixiaodi@fudan.edu.cn
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PRX Quantum 7, 010356 – Published 20 March, 2026

DOI: https://doi.org/10.1103/cc1n-gmj1

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.

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Popular Summary

Consider two distant particles entangled in a quantum system: a change to one instantaneously influences the other. For a pure quantum state, the Schmidt number quantifies this by counting the nonzero singular values in the entanglement structure. It can be visualized as the number of independent communication channels between two parties—a single channel indicates a basic, separable state, while multiple channels enable richer, high-dimensional entanglement, which is vital for quantum information processing and computing. However, practical techniques for certifying these dimensions in experimental settings remain limited.

This paper introduces a powerful, lab-friendly solution: certifying entanglement dimensionality using k-reduction moments. The core idea is elegant-if a state’s Schmidt number is at most k, applying the k-reduction map keeps it positive; any negativity proves higher entanglement. By pairing this insight with the moment method and randomized measurements, the authors create simple, effective tests. Crucially, the approach requires only 3-designs (like the widely used Clifford group), not 4-designs, and applies to a broad class of states—including unfaithful ones that evade fidelity-based methods.

Simulations show the method strengthens with higher-order moments and can fully certify the Schmidt numbers of many states using just a few moments. The sample complexity of this task scales gently with the system size and requires far fewer measurements than full tomography. In short, it is a robust, assumption free, experimentally practical tool to confirm the high-dimensional entanglement required for next-generation quantum technologies.

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PRX Quantum 7, 010356– Published 20 March, 2026
  • Received 8 February 2025
  • Revised 26 August 2025
  • Accepted 9 February 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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