{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,30]],"date-time":"2026-07-30T18:08:02Z","timestamp":1785434882293,"version":"3.56.0"},"reference-count":60,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2026,5,29]],"date-time":"2026-05-29T00:00:00Z","timestamp":1780012800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"crossref","award":["11805010"],"award-info":[{"award-number":["11805010"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]},{"award":["11805010"],"award-info":[{"award-number":["11805010"]}],"id":[{"id":"https:\/\/ror.org\/01h0zpd94","id-type":"ROR","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012236","name":"Beijing Institute of Technology Research Fund Program for Young Scholars","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100012236","id-type":"DOI","asserted-by":"crossref"}]},{"id":[{"id":"https:\/\/ror.org\/01skt4w74","id-type":"ROR","asserted-by":"publisher"}]},{"name":"Ministry of Education and the National Research Foundation of Singapore"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Random samples of quantum states with specific properties are useful for various applications, such as Monte Carlo integration over the state space. In the high-dimensional situations that one already encounters when working with a few qubits, the quantum state space has a very complicated boundary, and it is challenging to incorporate the specific properties into the sampling algorithm. In this paper, we present the Sequentially Constrained Monte Carlo (SCMC) algorithm as a practical and versatile method for sampling quantum states in accordance with properties that can be stated as inequalities. We apply the SCMC algorithm to the generation of samples of bound entangled states; for example, we obtain nearly ten thousand bound, entangled, two-qutrit states in a few minutes, compared with less than ten such states per day from independence sampling in our implementation. In the second application, we draw samples of high-dimensional quantum states from a narrowly peaked target distribution and observe, for the system sizes investigated, that SCMC sampling remains computationally manageable as the dimensions grow. In yet another application, the SCMC algorithm produces uniformly distributed quantum states in regions bounded by values of the problem-specific target distribution; such samples are needed when estimating parameters from the probabilistic data acquired in quantum experiments.<\/jats:p>","DOI":"10.3390\/e28060614","type":"journal-article","created":{"date-parts":[[2026,6,3]],"date-time":"2026-06-03T06:17:31Z","timestamp":1780467451000},"page":"614","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Sampling Quantum States with Inequality Constraints"],"prefix":"10.3390","volume":"28","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8606-5769","authenticated-orcid":false,"given":"Weijun","family":"Li","sequence":"first","affiliation":[{"name":"Department of Physics, University of Oxford, Oxford OX1 3RH, UK"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1282-7075","authenticated-orcid":false,"given":"Rui","family":"Han","sequence":"additional","affiliation":[{"name":"BiQut, Singapore 288564, Singapore"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2332-5882","authenticated-orcid":false,"given":"Jiangwei","family":"Shang","sequence":"additional","affiliation":[{"name":"School of Physics, Beijing Institute of Technology, Beijing 100081, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2397-840X","authenticated-orcid":false,"given":"Hui Khoon","family":"Ng","sequence":"additional","affiliation":[{"name":"Department of Physics, National University of Singapore, Singapore 117542, Singapore"},{"name":"Centre for Quantum Technologies, Singapore 117543, Singapore"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0690-927X","authenticated-orcid":false,"given":"Berthold-Georg","family":"Englert","sequence":"additional","affiliation":[{"name":"School of Physics, Beijing Institute of Technology, Beijing 100081, China"},{"name":"BiQut, Singapore 288564, Singapore"},{"name":"Department of Physics, National University of Singapore, Singapore 117542, Singapore"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2026,5,29]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"R1561","DOI":"10.1103\/PhysRevA.55.R1561","article-title":"Quantum-state estimation","volume":"55","author":"Hradil","year":"1997","journal-title":"Phys. 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