2025/06/13 by Keenan, Nathan, Goold, John, Nico-Katz, Alex · 1 citation
#FOS: Mathematics #FOS: Physical sciences #Quantum Physics (quant-ph) #Statistics Theory (math.ST)
paper · doi:10.48550/arxiv.2506.12010
Quantum state tomography (QST), the process of reconstructing some unknown quantum state ρ from repeated measurements on copies of said state, is a foundationally important task in the context of quantum computation and simulation. For this reason, a detailed characterization of the error Δρ= ρ-ρ^′ in a QST reconstruction ρ^′ is of clear importance to quantum theory and experiment. In this work, we develop a fully random matrix theory (RMT) treatment of state tomography in informationally-complete bases; and in doing so we reveal deep connections between QST errors Δρ and the gaussian unitary ensemble (GUE). By exploiting this connection we prove that wide classes of functions of the spectrum of Δρ can be evaluated by substituting samples of an appropriate GUE for realizations of Δρ. This powerful and flexible result enables simple analytic treatments of the mean value and variance of the error as quantified by the trace distance ‖Δρ‖Tr (which we validate numerically for common tomographic protocols), allows us to derive a bound on the QST sample complexity, and subsequently demonstrate that said bound doesn't change under the most widely-used rephysicalization procedure. These results collectively demonstrate the flexibility, strength, and broad applicability of our approach; and lays the foundation for broader studies of RMT treatments of QST in the future.