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arXiv:2507.11204

$d+1$ Measurement Bases are Sufficient for Determining $d$-Dimensional Quantum States: Theory and Experiment

Xiao, Tianqi, Wang, Yaxin, Xia, Ying, Li, Zhihao, Xiaoqi Zhou

Published Jul 15, 2025• 0 citations
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Abstract

A long-standing problem in quantum physics is to determine the minimal number of measurement bases required for the complete characterization of unknown quantum states, a question of particular relevance to high-dimensional quantum information processing. Here, we propose a quantum state tomography scheme that requires only $d+1$ projective measurement bases to fully reconstruct an arbitrary $d$-dimensional quantum state. As a proof-of-principle, we experimentally verified this scheme on a silicon photonic chip by reconstructing quantum states for $d=6$, in which a complete set of mutually unbiased bases does not exist. This approach offers new perspectives for quantum state characterization and measurement design, and holds promise for future applications in quantum information processing.

DOI
10.48550/arxiv.2507.11204

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We present a quantum state tomography method that enables the reconstruction of \emph{arbitrary} d−d-d−dimensional quantum states encoded in the discretized transverse momentum of photons, by using \emph{only} d+1d+1d+1 experimental settings. To this end, we identify a family of bases with the property that the outcomes of a projective measurement are \emph{spatially multiplexed} on the interference pattern of the projected state. Using the proposed scheme we performed, as a proof-of-principle, an experimental reconstruction of d=6−d=6-d=6−dimensional states, for which a complete set of mutually unbiased bases does not exist. We obtained fidelity values above 0.97 for both pure and mixed states, reducing the number of experimental settings from 424242 to only 777.
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