Nathan Keenan, John Goold, Alex Nico-Katz (Jun 16 2025).
Abstract: 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.
