Teo, Yong SiahSánchez Soto, Luis Lorenzo2023-06-162023-06-162021-120219-749910.1142/S0219749921400037https://hdl.handle.net/20.500.14352/4654© 2021 World Scientific Publishing Company. The authors are grateful to previous discussions and collaborations with leading experimental groups from the University of Ottawa, Moscow State University, Pohang University of Science and Technology, Roma Tre University and Paderborn University that successfully brought all theoretical results into experimental fruition. The authors also acknowledge support by the National Research Foundation of Korea (Grant Nos. 2019R1A6A1A10073437, 2019M3E4A1080074, 2020R1A2C1008609, and 2020K2A9A1A06102946), the European Union's Horizon 2020 research and innovation program (ApresSF and STORMYTUNE), and the Ministerio de Ciencia e Innovación (PGC2018-099183-B-I00).This review serves as a concise introductory survey of modern compressive tomography developed since 2019. These are schemes meant for characterizing arbitrary low-rank quantum objects, be it an unknown state, a process or detector, using minimal measuring resources (hence compressive) without any a priori assumptions (rank, sparsity, eigenbasis, etc.) about the quantum object. This paper contains a reasonable amount of technical details for the quantum-information community to start applying the methods discussed here. To facilitate the understanding of formulation logic and physics of compressive tomography, the theoretical concepts and important numerical results (both new and cross-referenced) shall be presented in a pedagogical manner.engjournal articlehttp://dx.doi.org/10.1142/S0219749921400037https://www.worldscientific.comopen access535Signal recoveryDetectorÓptica (Física)2209.19 Óptica Física