{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/a-composite-parameterization-of-unitary","title":"A composite parameterization of unitary groups, density matrices and subspaces","arxiv_id":"1004.5252","date":"2010-04-29","proceeding":null,"authors":["Christoph Spengler","Marcus Huber","Beatrix C. Hiesmayr"],"abstract":"Unitary transformations and density matrices are central objects in quantum physics and various tasks require to introduce them in a parameterized form. In the present article we present a parameterization of the unitary group $\\mathcal{U}(d)$ of arbitrary dimension $d$ which is constructed in a composite way. We show explicitly how any element of $\\mathcal{U}(d)$ can be composed of matrix exponential functions of generalized anti-symmetric $\\sigma$-matrices and one-dimensional projectors. The specific form makes it considerably easy to identify and discard redundant parameters in several cases. In this way, redundancy-free density matrices of arbitrary rank $k$ can be formulated. Our construction can also be used to derive an orthonormal basis of any $k$-dimensional subspaces of $\\mathbb{C}^d$ with the minimal number of parameters. As an example it will be shown that this feature leads to a significant reduction of parameters in the case of investigating distillability of quantum states via lower bounds of an entanglement measure (the $m$-concurrence).","url_abs":"https://arxiv.org/abs/1004.5252v2","url_pdf":"https://arxiv.org/pdf/1004.5252v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"a-composite-parameterization-of-unitary","repo_url":"https://gitlab.com/rob106/latxtract-toolbox","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}