Optimal qudit overlapping tomography and optimal measurement order

TL;DR

This paper develops optimal measurement schemes and ordering strategies for qudit overlapping tomography, significantly reducing experimental overhead and enabling efficient quantum state characterization beyond qubits.

Contribution

It introduces explicit constructions of optimal measurement schemes for qudits and an algorithm for minimizing measurement switching costs.

Findings

Proves upper bounds on measurement settings for n-qutrit systems

Provides explicit optimal measurement schemes for qudits

Reduces switching costs by approximately 50% with optimized order

Abstract

Quantum state tomography is essential for characterizing quantum systems, but it becomes infeasible for large systems due to exponential resource scaling. Overlapping tomography addresses this challenge by reconstructing all $k$-body marginals using few measurement settings, enabling the efficient extraction of key information for many quantum tasks. While optimal schemes are known for qubits, the extension to higher-dimensional qudit systems remains largely unexplored. Here, we investigate optimal qudit overlapping tomography, constructing local measurement settings from generalized Gell-Mann matrices. By establishing a correspondence with combinatorial covering arrays, we present two explicit constructions of optimal measurement schemes. For $n$-qutrit systems, we prove that pairwise tomography requires at most $8 + 56\left\lceil \log_{8} n \right\rceil$ measurement settings, and provide an explicit scheme achieving this bound. Furthermore, we develop an efficient algorithm to determine the optimal order of these measurement settings, minimizing the experimental overhead associated with switching configurations. Compared to the worst-case ordering, our optimized schedule reduces switching costs by approximately 50\%. These results provide a practical pathway for efficient characterization of qudit systems, facilitating their application in quantum communication and computation.