Quantum Algebraic Diversity: Single-Copy Density Matrix Estimation via Group-Structured Measurements

Quantum Physics [Submitted on 4 Apr 2026] Quantum Algebraic Diversity: Single-Copy Density Matrix Estimation via Group-Structured Measurements Mitchell A. Thornton 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. Subjects: Quantum Physics (quant-ph); Information Theory (cs.IT); Signal Processing (eess.SP) Cite as: arXiv:2604.03725 [quant-ph]   (or arXiv:2604.03725v1 [quant-ph] for this version)   https://doi.org/10.48550/arXiv.2604.03725 Focus to learn more Submission history From: Mitchell Thornton [view email] [v1] Sat, 4 Apr 2026 13:11:14 UTC (37 KB) Access Paper: HTML (experimental) view license Current browse context: quant-ph < prev   |   next > new | recent | 2026-04 Change to browse by: cs cs.IT eess eess.SP math math.IT References & Citations INSPIRE HEP NASA ADS Google Scholar Semantic Scholar Export BibTeX Citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Demos Related Papers About arXivLabs Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)