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Originally put forward as a special case of weak measurement, it has been extended to arbitrary measurement setup. We generalize the idea of \u201cquantum metrology,\u201d where a real-valued phase is estimated, to the estimation of complex-valued phase. We show that it enables to identify the optimal measurements and investigate the fundamental precision limit of the direct tomography. We propose a few experimentally feasible examples of direct tomography schemes and, based on the complex phase estimation formalism, demonstrate that direct tomography can reach the Heisenberg limit.<\/jats:p>","DOI":"10.1007\/s11128-021-03167-0","type":"journal-article","created":{"date-parts":[[2021,7,1]],"date-time":"2021-07-01T09:03:20Z","timestamp":1625130200000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Ultimate precision of direct tomography of wave functions"],"prefix":"10.1007","volume":"20","author":[{"given":"Xuan-Hoai Thi","family":"Nguyen","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6901-5233","authenticated-orcid":false,"given":"Mahn-Soo","family":"Choi","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,7,1]]},"reference":[{"issue":"1","key":"3167_CR1","doi-asserted-by":"publisher","first-page":"299","DOI":"10.1103\/revmodphys.81.299","volume":"81","author":"AI Lvovsky","year":"2009","unstructured":"Lvovsky, A.I., Raymer, M.G.: Continuous-variable optical quantum-state tomography. 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