完整乘积态接触集与三量子比特Shifts见证者的最优性
Complete product-state contact set and optimality of the canonical three-qubit Shifts witness
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中文总结 AI 辅助
本研究确定了三量子比特Shifts UPB投影算子的完整乘积态接触集（共八个射线），证明其张成整个希尔伯特空间，从而确立相应见证者的最优性，并给出白噪声下纠缠检测的阈值。
中文摘要 AI 辅助
对于三量子比特Shifts不可扩展乘积基（UPB），已知UPB投影算子的最小乘积态期望值$\lambda_{\mathrm{Shifts}}=1-3\sqrt6/8$，并且此前已展示四个达到该值的乘积态。我们确定了所有复乘积态上的完整等式集，并证明其恰好由八个乘积射线组成：两个在三个量子比特的循环置换下不变的射线和两个大小为三的循环轨道。证明结合了精确相位分析、对所有28个有限实驻点的Gröbner基分类以及对射影边界的完整处理。关键的是，这八个接触射线张成完整的三量子比特希尔伯特空间，而此前已知的四个射线则不能。因此，完整的等式分类确立了关联的投影算子见证者满足张成准则，并且在该意义上是最优的：不存在非零正半定算子可以在保持块正性的前提下被减去。作为推论，对于白噪声族$\rho(p)=(1-p)\rho_{\mathrm{Shifts}}+p\mathbb I/8$，其中$\rho_{\mathrm{Shifts}}$是互补投影Shifts态，同一见证者在$p<2-3\sqrt6/4\approx16.29\\%$时检测到纠缠。
英文摘要
For the three-qubit Shifts unextendible product basis (UPB), the minimum product-state expectation value $λ_{\mathrm{Shifts}}=1-3\sqrt6/8$ of the UPB projector is known, and four attaining product states were exhibited previously. We determine the complete equality set over all complex product states and prove that it consists of exactly eight product rays: two rays invariant under cyclic permutation of the three qubits and two cyclic orbits of size three. The proof combines exact phase analysis, a Gröbner-basis classification of all 28 finite real stationary points, and a complete treatment of the projective boundaries. Crucially, the eight contact rays span the full three-qubit Hilbert space, whereas the four previously known rays do not. The complete equality classification therefore establishes that the associated projector witness satisfies the spanning criterion and is optimal in the sense that no nonzero positive semidefinite operator can be subtracted while preserving block positivity. As a corollary, for the white-noise family $ρ(p)=(1-p)ρ_{\mathrm{Shifts}}+p\mathbb I/8$, where $ρ_{\mathrm{Shifts}}$ is the complementary-projector Shifts state, the same witness detects entanglement for $p<2-3\sqrt6/4\approx16.29\%$.
发表机构
- School of Physics and Electronic Technology, Liaoning Normal University（辽宁师范大学物理与电子科技学院）
- Center for Theoretical and Experimental High Energy Physics, Liaoning Normal University（辽宁师范大学理论与实验高能物理中心）
机构由 AI 辅助整理，请以论文原文为准。
