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

TL;DR

This paper introduces a quantum measurement framework based on algebraic diversity, demonstrating that group-structured measurements can efficiently estimate quantum states with high fidelity from a single copy.

Contribution

It extends classical algebraic diversity concepts to quantum measurement theory, establishing a duality map and demonstrating the effectiveness of group-structured POVMs for quantum state estimation.

Findings

Group-structured POVMs recover spectral structure from a single quantum state copy.

Single-measurement estimators achieve high fidelity (above 0.90) across multiple dimensions.

Classical optimality properties transfer to quantum settings via the Born map.

Abstract

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.