Analytically continuing the Randomized measurement Toolbox

Recently, a new protocol for measuring integer moments of a density matrix was developed by using randomized measurements [1]. We prescribe an extension of this protocol to obtain fractional moments of the system’s density matrix. This extension is an important step in obtaining information-theoretic quantities such as Petz and sandwiched Rényi mutual informations. A crucial step in our procedure is the analytic continuation from integer to fractional moments. We explore various analytic continuation techniques, including conventional fitting-based methods, and optimization-based methods. We then adapt these techniques to the study of primarily the Von Neumann entanglement entropy, and then the Petz Rényi mutual information and others, with the primary objective of identifying the most efficient protocol that simultaneously minimizes errors in analytic continuation and data requirements. We also discuss the adaptability of our protocol in realistic experiments.

[1] Elben et al, Nature Review Physics 5, 9 (2023)