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Our methodology involves analyzing reduced density matrices at different stages of a quantum algorithm&amp;apos;s execution and representing the dominant eigenvalue and von Neumann entropy on a graph, creating an \"entanglement trajectory.\" To establish the trajectory&amp;apos;s boundaries, we employ random matrix theory. Through the examination of examples such as quantum adiabatic computation, the Grover algorithm, and the Shor algorithm, we demonstrate that the entanglement trajectory remains within the established boundaries, exhibiting unique characteristics for each example. Moreover, we show that these boundaries and features can be extended to trajectories defined by alternative entropy measures. The entanglement trajectory serves as an invariant property of a quantum system, maintaining consistency across varying situations and definitions of entanglement. Numerical simulations accompanying this research are available via open access.<\/jats:p>","DOI":"10.22331\/q-2024-03-14-1282","type":"journal-article","created":{"date-parts":[[2024,3,14]],"date-time":"2024-03-14T12:02:36Z","timestamp":1710417756000},"page":"1282","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":0,"title":["Entanglement Trajectory and its Boundary"],"prefix":"10.22331","volume":"8","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9297-7622","authenticated-orcid":false,"given":"Ruge","family":"Lin","sequence":"first","affiliation":[{"name":"Quantum Research Centre, Technology Innovation Institute, United Arab Emirates."},{"name":"Departament de F\u00edsica Qu\u00e0ntica i Astrof\u00edsica and Institut de Ci\u00e8ncies del Cosmos, Universitat de Barcelona, Spain."}]}],"member":"9598","published-online":{"date-parts":[[2024,3,14]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"Richard Jozsa and Noah Linden. 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