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Symmetries, on the other hand, are expected to reduce the randomness of a state. Despite being ubiquitous, the effects of symmetry on quantum state designs remain an outstanding question. The recently introduced projected ensemble framework generates efficient approximate state <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>t<\/mml:mi><\/mml:math>-designs by hinging on projective measurements and many-body quantum chaos. In this work, we examine the emergence of state designs from the random generator states exhibiting symmetries. Leveraging on translation symmetry, we analytically establish a sufficient condition for the measurement basis leading to the state <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>t<\/mml:mi><\/mml:math>-designs. Then, by making use of the trace distance measure, we numerically investigate the convergence to the designs. Subsequently, we inspect the violation of the sufficient condition to identify bases that fail to converge. We further demonstrate the emergence of state designs in a physical system by studying the dynamics of a chaotic tilted field Ising chain with translation symmetry. We find faster convergence of the trace distance during the early time evolution in comparison to the cases when the symmetry is broken. To delineate the general applicability of our results, we extend our analysis to other symmetries. We expect our findings to pave the way for further exploration of deep thermalization and equilibration of closed and open quantum many-body systems.<\/jats:p>","DOI":"10.22331\/q-2024-08-29-1456","type":"journal-article","created":{"date-parts":[[2024,8,29]],"date-time":"2024-08-29T12:39:44Z","timestamp":1724935184000},"page":"1456","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":14,"title":["Unraveling the emergence of quantum state designs in systems with symmetry"],"prefix":"10.22331","volume":"8","author":[{"given":"Naga Dileep","family":"Varikuti","sequence":"first","affiliation":[{"name":"Department of Physics, Indian Institute of Technology Madras, Chennai, India, 600036"},{"name":"Center for Quantum Information, Communication and Computing (CQuICC), Indian Institute of Technology Madras, Chennai, India 600036"}]},{"given":"Soumik","family":"Bandyopadhyay","sequence":"additional","affiliation":[{"name":"Pitaevskii BEC Center, CNR-INO and Dipartimento di Fisica, Universit\u00e0 di Trento, Via Sommarive 14, Trento, I-38123, Italy"},{"name":"INFN-TIFPA, Trento Institute for Fundamental Physics and Applications, Via Sommarive 14, Trento, I-38123, Italy"}]}],"member":"9598","published-online":{"date-parts":[[2024,8,29]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"Joseph Emerson, Robert Alicki, and Karol \u017byczkowski. 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