We extend the algebraic diversity (AD) framework from classical signal processing to quantum measurement theory. The central result -- the Quantum Algebraic Diversity (QAD) Theorem -- establishes that a group-structured positive operator-valued measure (POVM) applied to a single copy of a quantum state produces a group-averaged density matrix estimator that recovers the spectral structure of the true density matrix, analogous to the classical result that a group-averaged outer product recovers covariance eigenstructure from a single observation. We establish a formal Classical-Quantum Duality Map connecting classical covariance estimation to quantum state tomography, and prove an Optimality Inheritance Theorem showing that classical group optimality transfers to quantum settings via the Born map. SIC-POVMs are identified as algebraic diversity with the Heisenberg-Weyl group, and mutually unbiased bases (MUBs) as algebraic diversity with the Clifford group, revealing the hierarchy $\mathrm{HW}(d) \subseteq \mathcal{C}(d) \subseteq S_d$ that mirrors the classical hierarchy $\mathbb{Z}_M \subseteq G_{\min} \subseteq S_M$. The double-commutator eigenvalue theorem provides polynomial-time adaptive POVM selection. A worked qubit example demonstrates that the group-averaged estimator from a single Pauli measurement recovers a full-rank approximation to a mixed qubit state, achieving fidelity 0.91 where standard single-basis tomography produces a rank-1 estimate with fidelity 0.71. Monte Carlo simulations on qudits of dimension $d = 2$ through $d = 13$ (200 random states per dimension) confirm that the Heisenberg-Weyl QAD estimator maintains fidelity above 0.90 across all dimensions from a single measurement outcome, while standard tomography fidelity degrades as $\sim 1/d$, with the improvement ratio scaling linearly with $d$ as predicted by the $O(d)$ copy reduction theorem.

翻译：我们将经典信号处理中的代数多样性(AD)框架扩展至量子测量理论。核心结果——量子代数多样性(QAD)定理——表明：将群结构正算子值测度(POVM)作用于量子态的单副本，可生成一种群平均密度矩阵估计器，其能恢复真实密度矩阵的谱结构；这类似于经典结论中，群平均外积可单次观测恢复协方差特征结构。我们建立了连接经典协方差估计与量子态层析的经典-量子对偶映射，并证明了最优继承定理：经典群最优性可通过玻恩映射迁移至量子场景。我们识别出SIC-POVM为海森堡-魏尔群下的代数多样性，互无偏基(MUB)为克利福德群下的代数多样性，揭示了层级结构$\mathrm{HW}(d) \subseteq \mathcal{C}(d) \subseteq S_d$，其对偶于经典层级$\mathbb{Z}_M \subseteq G_{\min} \subseteq S_M$。双重换位子特征值定理实现了多项式时间的自适应POVM选择。单量子比特实例表明：基于单次泡利测量的群平均估计器可恢复混合量子态的全秩近似，保真度达0.91，而标准单基向量层析仅产生秩1估计，保真度为0.71。维度$d=2$至$d=13$的qudit蒙特卡洛模拟（每维度200个随机态）证实：基于单次测量结果，海森堡-魏尔QAD估计器在所有维度下保真度均高于0.90，而标准层析保真度以$\sim 1/d$衰减，且改进比率随维度线性增长，符合$O(d)$副本缩减定理的预测。