{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,31]],"date-time":"2026-03-31T22:15:34Z","timestamp":1774995334192,"version":"3.50.1"},"reference-count":26,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2019,7,3]],"date-time":"2019-07-03T00:00:00Z","timestamp":1562112000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This paper deals with separable and entangled qudits      |   \u03c8 d   \u27e9      (quantum states in dimension d) constructed from Dicke states made of     N = d \u2212 1     qubits. Such qudits present the property to be totally symmetric under the interchange of the N qubits. We discuss the notion of perma-concurrence     P d     for the qudit      |   \u03c8 d   \u27e9     , introduced by the authors (Entropy 2018, 20, 292), as a parameter for characterizing the entanglement degree of      |   \u03c8 d   \u27e9     . For     d = 3    , the perma-concurrence     P 3     constitutes an alternative to the concurrence C for symmetric two-qubit states. We give several expressions of     P d     (in terms of matrix permanent and in terms of unit vectors of     R 3     pointing on the Bloch sphere) and precise the range of variation of     P d     (going from separable to maximally entangled states). Numerous examples are presented for     P d    . Special attention is devoted to states of W type and to maximally entangled states of Bell and Greenberger\u2013Horne\u2013Zeilinger type.<\/jats:p>","DOI":"10.3390\/sym11070875","type":"journal-article","created":{"date-parts":[[2019,7,3]],"date-time":"2019-07-03T11:14:49Z","timestamp":1562152489000},"page":"875","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Generalized Weyl-Heisenberg Algebra, Qudit Systems and Entanglement Measure of Symmetric States via Spin Coherent States. Part II: The Perma-Concurrence Parameter"],"prefix":"10.3390","volume":"11","author":[{"given":"Mohammed","family":"Daoud","sequence":"first","affiliation":[{"name":"Department of Physics, Faculty of Sciences, University Ibn Tofail, BP 133 K\u00e9nitra, Morocco"},{"name":"CNRS\/IN2P3, Institut de Physique Nucl\u00e9aire, 69622 Villeurbanne, France"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8522-4561","authenticated-orcid":false,"given":"Maurice R.","family":"Kibler","sequence":"additional","affiliation":[{"name":"CNRS\/IN2P3, Institut de Physique Nucl\u00e9aire, 69622 Villeurbanne, France"},{"name":"Facult\u00e9 des Sciences et Technologies, Universit\u00e9 Claude Bernard Lyon 1, 69622 Villeurbanne, France"},{"name":"IDEXLYON, Universit\u00e9 de Lyon, 69361 Lyon, France"}]}],"member":"1968","published-online":{"date-parts":[[2019,7,3]]},"reference":[{"key":"ref_1","unstructured":"Nielsen, M.A., and Chuang, I.L. (2000). Quantum Computation and Quantum Information, Cambridge University Press."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"869","DOI":"10.1103\/PhysRevLett.80.869","article-title":"Teleportation of continuous quantum variables","volume":"80","author":"Braunstein","year":"1998","journal-title":"Phys. Rev. Lett."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"575","DOI":"10.1038\/37539","article-title":"Experimental quantum teleportation","volume":"390","author":"Bouwmeester","year":"1997","journal-title":"Nature"},{"key":"ref_4","unstructured":"Bennett, C.H., and Brassard, G. (1984, January 9\u201312). Quantum cryptography: Public key distribution and coin tossing. Proceedings of the IEEE International Conference on Computers, Systems and Signal Processing, Bangalore, India."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"197","DOI":"10.1103\/RevModPhys.74.197","article-title":"The role of relative entropy in quantum information theory","volume":"74","author":"Vedral","year":"2002","journal-title":"Rev. Mod. Phys."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"865","DOI":"10.1103\/RevModPhys.81.865","article-title":"Quantum entanglement","volume":"81","author":"Horodecki","year":"2009","journal-title":"Rev. Mod. Phys."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physrep.2009.02.004","article-title":"Entanglement detection","volume":"474","year":"2009","journal-title":"Phys. Rep."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1655","DOI":"10.1103\/RevModPhys.84.1655","article-title":"Quantum discord and other measures of quantum correlation","volume":"84","author":"Modi","year":"2012","journal-title":"Rev. Mod. Phys."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"2245","DOI":"10.1103\/PhysRevLett.80.2245","article-title":"Entanglement of formation of an arbitrary state of two qubits","volume":"80","author":"Wootters","year":"1998","journal-title":"Phys. Rev. Lett."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"017901","DOI":"10.1103\/PhysRevLett.88.017901","article-title":"Quantum discord: A measure of the quantumness of correlations","volume":"88","author":"Ollivier","year":"2001","journal-title":"Phys. Rev. Lett."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"6899","DOI":"10.1088\/0305-4470\/34\/35\/315","article-title":"Classical, quantum and total correlations","volume":"34","author":"Henderson","year":"2001","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"052306","DOI":"10.1103\/PhysRevA.61.052306","article-title":"Distributed entanglement","volume":"61","author":"Coffman","year":"2000","journal-title":"Phys. Rev. A"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"044301","DOI":"10.1103\/PhysRevA.63.044301","article-title":"Potential multiparticle entanglement measure","volume":"63","author":"Wong","year":"2001","journal-title":"Phys. Rev. A"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"132","DOI":"10.1007\/s11128-018-1900-3","article-title":"Stochastic local operations and classical communication (SLOCC) and local unitary operations (LU) classifications of n qubits via ranks and singular values of the spin-flipping matrices","volume":"17","author":"Li","year":"2018","journal-title":"Quantum Inf. Process"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"L\u00f3pez-Sald\u00edvar, J.A., Casta\u00f1os, O., Nahmad-Achar, E., L\u00f3pez-Pe\u00f1a, R., Man\u2019ko, M.A., and Man\u2019ko, V.I. (2018). Geometry and entanglement of two-qubit states in the quantum probabilistic representation. Entropy, 20.","DOI":"10.3390\/e20090630"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Daoud, M., and Kibler, M.R. (2018). Generalized Weyl\u2013Heisenberg Algebra, Qudit Systems and Entanglement Measure of Symmetric States via Spin Coherent States. Entropy, 20.","DOI":"10.3390\/e20040292"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"115303","DOI":"10.1088\/1751-8113\/43\/11\/115303","article-title":"Phase operators, temporally stable phase states, mutually unbiased bases and exactly solvable quantum systems","volume":"43","author":"Daoud","year":"2010","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"082101","DOI":"10.1063\/1.3620414","article-title":"Phase operators, phase states and vector phase states for SU3 and SU2,1","volume":"52","author":"Daoud","year":"2011","journal-title":"J. Math. Phys."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1103\/PhysRev.93.99","article-title":"Coherence in spontaneous radiation processes","volume":"93","author":"Dicke","year":"1954","journal-title":"Phys. Rev."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"275","DOI":"10.1364\/JOSAB.24.000275","article-title":"Detection of multipartite entanglement in the vicinity of symmetric Dicke states","volume":"24","year":"2007","journal-title":"J. Opt. Soc. Am. B"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"385304","DOI":"10.1088\/1751-8113\/46\/38\/385304","article-title":"Entanglement criteria for Dicke states","volume":"46","author":"Bergmann","year":"2013","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1007\/BF02960953","article-title":"Atomi orientati in campo magnetico variabile","volume":"9","author":"Majorana","year":"1932","journal-title":"Nuovo Cimento"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"062327","DOI":"10.1103\/PhysRevA.83.062327","article-title":"Casimir invariants for systems undergoing collective motion","volume":"83","author":"Bishop","year":"2011","journal-title":"Phys. Rev. A"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"032330","DOI":"10.1103\/PhysRevA.73.032330","article-title":"Implications of qudit superselection rules for the theory of decoherence-free subsystems","volume":"73","author":"Byrd","year":"2006","journal-title":"Phys. Rev. A"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1142\/S1230161210000084","article-title":"Geometry of pure states of N spin-J system","volume":"17","author":"Kolenderski","year":"2010","journal-title":"Open Syst. Inf. Dyn."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"062314","DOI":"10.1103\/PhysRevA.62.062314","article-title":"Three qubits can be entangled in two inequivalent ways","volume":"62","author":"Vidal","year":"2000","journal-title":"Phys. Rev. A"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/7\/875\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T13:02:18Z","timestamp":1760187738000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/7\/875"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,7,3]]},"references-count":26,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2019,7]]}},"alternative-id":["sym11070875"],"URL":"https:\/\/doi.org\/10.3390\/sym11070875","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,7,3]]}}}