GHZ-W Genuinely Entangled Subspace Verification with Adaptive Local Measurements

Genuinely entangled subspaces (GESs) are valuable resources in quantum information science. Among these, the three-qubit GHZ-W GES, spanned by the three-qubit Greenberger-Horne-Zeilinger (GHZ) and W states, is a universal and crucial entangled subspace resource for three-qubit systems. In this work, we develop two adaptive verification strategies, the XZ strategy and the rotation strategy, for the three-qubit GHZ-W GES using local measurements and one-way classical communication. These strategies are experimentally feasible, efficient and possess a concise analytical expression for the sample complexity of the rotation strategy, which scales approximately as $2.248{\epsilon }^{-1}\ln {\delta }^{-1}$, where $\epsilon$ is the infidelity and $1-\delta$ is the confidence level. Furthermore, we comprehensively analyze the two-dimensional two-qubit subspaces and classify them into three distinct types, which include unverifiable entangled subspaces, revealing intrinsic limitations in local verification of entangled subspaces.