{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,11]],"date-time":"2025-11-11T13:54:26Z","timestamp":1762869266824,"version":"build-2065373602"},"reference-count":20,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2024,1,12]],"date-time":"2024-01-12T00:00:00Z","timestamp":1705017600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100000015","name":"U.S. Department of Energy","doi-asserted-by":"publisher","award":["DE-SC0024631","DE-AC02-06CH11357"],"award-info":[{"award-number":["DE-SC0024631","DE-AC02-06CH11357"]}],"id":[{"id":"10.13039\/100000015","id-type":"DOI","asserted-by":"publisher"}]},{"name":"U.S. Office of Science\u2019s Advanced Scientific Computing Research FAIR program","award":["DE-SC0024631","DE-AC02-06CH11357"],"award-info":[{"award-number":["DE-SC0024631","DE-AC02-06CH11357"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>To estimate the degree of quantum entanglement of random pure states, it is crucial to understand the statistical behavior of entanglement indicators such as the von Neumann entropy, quantum purity, and entanglement capacity. These entanglement metrics are functions of the spectrum of density matrices, and their statistical behavior over different generic state ensembles have been intensively studied in the literature. As an alternative metric, in this work, we study the sum of the square root spectrum of density matrices, which is relevant to negativity and fidelity in quantum information processing. In particular, we derive the finite-size mean and variance formulas of the sum of the square root spectrum over the Bures\u2013Hall ensemble, extending known results obtained recently over the Hilbert\u2013Schmidt ensemble.<\/jats:p>","DOI":"10.3390\/e26010068","type":"journal-article","created":{"date-parts":[[2024,1,12]],"date-time":"2024-01-12T07:47:16Z","timestamp":1705045636000},"page":"68","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Square Root Statistics of Density Matrices and Their Applications"],"prefix":"10.3390","volume":"26","author":[{"given":"Lyuzhou","family":"Ye","sequence":"first","affiliation":[{"name":"Department of Computer Science, Texas Tech University, Lubbock, TX 79409, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6568-2706","authenticated-orcid":false,"given":"Youyi","family":"Huang","sequence":"additional","affiliation":[{"name":"Department of Computer Science, Texas Tech University, Lubbock, TX 79409, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7843-7622","authenticated-orcid":false,"given":"James C.","family":"Osborn","sequence":"additional","affiliation":[{"name":"Computational Science Division, Argonne National Laboratory, Argonne, IL 60439, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8065-0956","authenticated-orcid":false,"given":"Lu","family":"Wei","sequence":"additional","affiliation":[{"name":"Department of Computer Science, Texas Tech University, Lubbock, TX 79409, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,1,12]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1291","DOI":"10.1103\/PhysRevLett.71.1291","article-title":"Average entropy of a subsystem","volume":"71","author":"Page","year":"1993","journal-title":"Phys. Rev. Lett."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1028","DOI":"10.1063\/1.523763","article-title":"Entropy of an n-system from its correlation with ak-reservoir","volume":"19","author":"Lubkin","year":"1978","journal-title":"J. Math. Phys."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"080501","DOI":"10.1103\/PhysRevLett.105.080501","article-title":"Entanglement entropy and entanglement spectrum of the Kitaev model","volume":"105","author":"Yao","year":"2010","journal-title":"Phys. Rev. Lett."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"032314","DOI":"10.1103\/PhysRevA.65.032314","article-title":"Computable measure of entanglement","volume":"65","author":"Vidal","year":"2002","journal-title":"Phys. Rev. A"},{"key":"ref_5","first-page":"363001","article-title":"Entanglement typicality","volume":"47","author":"Dahlsten","year":"2014","journal-title":"Phys. Rev. A"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"034206","DOI":"10.1103\/PhysRevE.107.034206","article-title":"Random density matrices: Analytical results for mean fidelity and variance of squared Bures distance","volume":"107","author":"Laha","year":"2023","journal-title":"Phys. Rev. E"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"022342","DOI":"10.1103\/PhysRevA.97.022342","article-title":"Axiomatic and operational connections between the l1-norm of coherence and negativity","volume":"97","author":"Zhu","year":"2018","journal-title":"Phys. Rev. A"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"2315","DOI":"10.1080\/09500349414552171","article-title":"Fidelity for Mixed Quantum States","volume":"41","author":"Jozsa","year":"1994","journal-title":"J. Mod. Opt."},{"key":"ref_9","unstructured":"Akemann, G., Baik, J., and Di Francesco, P. (2011). The Oxford Handbook of Random Matrix Theory, Oxford University Press."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Bengtsson, I., and Zyczkowski, K. (2006). Geometry of Quantum States: An Introduction to Quantum Entanglement, Cambridge University Press.","DOI":"10.1017\/CBO9780511535048"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"295203","DOI":"10.1088\/1751-8121\/ab2675","article-title":"Bures\u2013Hall ensemble: Spectral densities and average entropies","volume":"52","author":"Sarkar","year":"2019","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1016\/S0375-9601(98)00190-X","article-title":"Random quantum correlations and density operator distributions","volume":"242","author":"Hall","year":"1998","journal-title":"Phys. Lett. A"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"7111","DOI":"10.1088\/0305-4470\/34\/35\/335","article-title":"Induced measures in the space of mixed quantum states","volume":"34","author":"Zyczkowski","year":"2001","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"10083","DOI":"10.1088\/0305-4470\/36\/39\/308","article-title":"Bures volume of the set of mixed quantum states","volume":"36","author":"Sommers","year":"2003","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"062128","DOI":"10.1103\/PhysRevE.102.062128","article-title":"Exact variance of von Neumann entanglement entropy over the Bures-Hall measure","volume":"102","author":"Wei","year":"2020","journal-title":"Phys. Rev. E"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"105201","DOI":"10.1088\/1751-8121\/ac4e20","article-title":"Second-order statistics of fermionic Gaussian states","volume":"55","author":"Huang","year":"2022","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"151","DOI":"10.1007\/s00220-015-2435-4","article-title":"Relating the Bures Measure to the Cauchy Two-Matrix Model","volume":"342","author":"Forrester","year":"2016","journal-title":"Comm. Math. Phys."},{"key":"ref_18","unstructured":"Prudnikov, A., Brychkov, Y., and Marichev, O. (1989). Integrals and Series. Volume 3: More Special Functions, Gordon and Breach."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"235203","DOI":"10.1088\/1751-8121\/ab8d07","article-title":"Proof of Sarkar\u2013Kumar\u2019s conjectures on average entanglement entropies over the Bures\u2013Hall ensemble","volume":"53","author":"Wei","year":"2020","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1007\/s00220-013-1833-8","article-title":"Cauchy\u2013Laguerre Two-Matrix Model and the Meijer-G Random Point Field","volume":"326","author":"Bertola","year":"2014","journal-title":"Commun. Math. Phys."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/26\/1\/68\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T13:45:31Z","timestamp":1760103931000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/26\/1\/68"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,1,12]]},"references-count":20,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2024,1]]}},"alternative-id":["e26010068"],"URL":"https:\/\/doi.org\/10.3390\/e26010068","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2024,1,12]]}}}