{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,7]],"date-time":"2026-10-07T17:48:40Z","timestamp":1791395320909,"version":"4.3.3"},"reference-count":87,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2015,3,26]],"date-time":"2015-03-26T00:00:00Z","timestamp":1427328000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100002923","name":"CONICET","doi-asserted-by":"publisher","award":["PIP 0512"],"award-info":[{"award-number":["PIP 0512"]}],"id":[{"id":"10.13039\/501100002923","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003947","name":"UNLP","doi-asserted-by":"publisher","award":["Project X725"],"award-info":[{"award-number":["Project X725"]}],"id":[{"id":"10.13039\/501100003947","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We examine the behavior of quantum correlations of spin pairs in a finite anisotropic XY spin chain immersed in a transverse magnetic field, through the analysis of the quantum discord and the conventional and quadratic one-way information deficits. We first provide a brief review of these measures, showing that the last ones can be obtained as particular cases of a generalized information deficit based on general entropic forms. All of these measures coincide with an entanglement entropy in the case of pure states, but can be non-zero in separable mixed states, vanishing just for classically correlated states. It is then shown that their behavior in the exact ground state of the chain exhibits similar features, deviating significantly from that of the pair entanglement below the critical field. In contrast with entanglement, they reach full range in this region, becoming independent of the pair separation and coupling range in the immediate vicinity of the factorizing field. It is also shown, however, that significant differences between the quantum discord and the information deficits arise in the local minimizing measurement that defines them. Both analytical and numerical results are provided.<\/jats:p>","DOI":"10.3390\/e17041634","type":"journal-article","created":{"date-parts":[[2015,3,26]],"date-time":"2015-03-26T12:08:06Z","timestamp":1427371686000},"page":"1634-1659","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["Quantum Discord and Information Deficit in Spin Chains"],"prefix":"10.3390","volume":"17","author":[{"given":"Norma","family":"Canosa","sequence":"first","affiliation":[{"name":"Departamento de F\u00edsica-IFLP, Universidad Nacional de La Plata, C.C.67, La Plata 1900, Argentina"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Leonardo","family":"Ciliberti","sequence":"additional","affiliation":[{"name":"Departamento de F\u00edsica-IFLP, Universidad Nacional de La Plata, C.C.67, La Plata 1900, Argentina"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ra\u00fal","family":"Rossignoli","sequence":"additional","affiliation":[{"name":"Departamento de F\u00edsica-IFLP, Universidad Nacional de La Plata, C.C.67, La Plata 1900, Argentina"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2015,3,26]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"807","DOI":"10.1007\/BF01491891","article-title":"Die gegenw\u00e4rtige situation in der Quantenmechanik","volume":"23","year":"1935","journal-title":"Naturwissenschaften"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"555","DOI":"10.1017\/S0305004100013554","article-title":"Discussion of probability relations between separated systems","volume":"31","year":"1935","journal-title":"Proc. Cambridge Philos. Soc."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"2738","DOI":"10.1103\/PhysRevA.51.2738","article-title":"Quantum coding","volume":"51","author":"Schumacher","year":"1995","journal-title":"Phys. Rev. A"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"2046","DOI":"10.1103\/PhysRevA.53.2046","article-title":"Concentrating partial entanglement by local operations","volume":"53","author":"Bennett","year":"1996","journal-title":"Phys. Rev. A"},{"key":"ref_5","unstructured":"Nielsen, M.A., and Chuang, I. (2000). Quantum Computation and Quantum Information, Cambridge University Press."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Vedral, V. (2006). Introduction to Quantum Information Science, Oxford University Press.","DOI":"10.1093\/acprof:oso\/9780199215706.001.0001"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Haroche, S., and Raimond, J.M. (2007). Exploring the Quantum, Oxford University Press.","DOI":"10.1093\/acprof:oso\/9780198509141.001.0001"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"4277","DOI":"10.1103\/PhysRevA.40.4277","article-title":"Quantum states with Einstein\u2013Podolsky\u2013Rosen correlations admitting a hidden-variable model","volume":"40","author":"Werner","year":"1989","journal-title":"Phys. Rev. A"},{"key":"ref_9","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_10","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"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"050401","DOI":"10.1103\/PhysRevLett.90.050401","article-title":"Classical Correlations and Entanglement in Quantum Measurements","volume":"90","author":"Vedral","year":"2003","journal-title":"Phys. Rev. Lett."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"715","DOI":"10.1103\/RevModPhys.75.715","article-title":"Decoherence, einselection, and the quantum origins of the classical","volume":"75","author":"Zurek","year":"2003","journal-title":"Rev. Mod. Phys."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"012320","DOI":"10.1103\/PhysRevA.67.012320","article-title":"Quantum discord and Maxwells demons","volume":"67","author":"Zurek","year":"2003","journal-title":"Phys. Rev. A"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"080501","DOI":"10.1103\/PhysRevLett.104.080501","article-title":"Unified View of Quantum and Classical Correlations","volume":"104","author":"Modi","year":"2010","journal-title":"Phys. Rev. Lett."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1655","DOI":"10.1103\/RevModPhys.84.1655","article-title":"The classical-quantum boundary for correlations: Discord and related measures","volume":"84","author":"Modi","year":"2012","journal-title":"Rev. Mod. Phys."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"062307","DOI":"10.1103\/PhysRevA.71.062307","article-title":"Local versus nonlocal information in quantum-information theory: Formalism and phenomena","volume":"71","author":"Horodecki","year":"2005","journal-title":"Phys. Rev. A"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"180402","DOI":"10.1103\/PhysRevLett.89.180402","article-title":"Thermodynamical Approach to Quantifying Quantum Correlations","volume":"89","author":"Oppenheim","year":"2002","journal-title":"Phys. Rev. Lett."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"160401","DOI":"10.1103\/PhysRevLett.106.160401","article-title":"Linking Quantum Discord to Entanglement in a Measurement","volume":"106","author":"Streltsov","year":"2011","journal-title":"Phys. Rev. Lett."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"190502","DOI":"10.1103\/PhysRevLett.105.190502","article-title":"Necessary and Sufficient Condition for Nonzero Quantum Discord","volume":"105","author":"Vedral","year":"2010","journal-title":"Phys. Rev. Lett."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"052382","DOI":"10.1103\/PhysRevA.82.052342","article-title":"Generalized entropic measures of quantum correlations","volume":"82","author":"Rossignoli","year":"2010","journal-title":"Phys. Rev. A"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"052329","DOI":"10.1103\/PhysRevA.84.052329","article-title":"Quantum correlations and least disturbing local measurements","volume":"84","author":"Rossignoli","year":"2011","journal-title":"Phys. Rev. A"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"240402","DOI":"10.1103\/PhysRevLett.110.240402","article-title":"Characterizing Nonclassical Correlations via Local Quantum Uncertainty","volume":"110","author":"Girolami","year":"2013","journal-title":"Phys. Rev. Lett."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"032117","DOI":"10.1103\/PhysRevA.85.032117","article-title":"Quantifying correlations via the Wigner-Yanase skew information","volume":"85","author":"Luo","year":"2012","journal-title":"Phys. Rev. A"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"064101","DOI":"10.1103\/PhysRevA.87.064101","article-title":"Geometric quantum discord through the Schatten 1-norm","volume":"87","author":"Paula","year":"2013","journal-title":"Phys. Rev. A"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"032340","DOI":"10.1103\/PhysRevA.87.032340","article-title":"Quantum correlating power of local quantum channels","volume":"87","author":"Hu","year":"2013","journal-title":"Phys. Rev. A"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"012117","DOI":"10.1103\/PhysRevA.88.012117","article-title":"Negativity of quantumness and its interpretations","volume":"88","author":"Nakano","year":"2013","journal-title":"Phys. Rev. A"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"013038","DOI":"10.1088\/1367-2630\/16\/1\/013038","article-title":"Toward computability of trace distance discord","volume":"16","author":"Ciccarello","year":"2014","journal-title":"New J. Phys."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1895","DOI":"10.1103\/PhysRevLett.70.1895","article-title":"Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels","volume":"70","author":"Bennett","year":"1993","journal-title":"Phys. Rev. Lett."},{"key":"ref_29","first-page":"2011","article-title":"On the role of entanglement in quantum computational speed-up","volume":"A459","author":"Josza","year":"2003","journal-title":"Proc. R. Soc."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"147902","DOI":"10.1103\/PhysRevLett.91.147902","article-title":"Efficient Classical Simulation of Slightly Entangled Quantum Computations","volume":"91","author":"Vidal","year":"2003","journal-title":"Phys. Rev. Lett."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"5672","DOI":"10.1103\/PhysRevLett.81.5672","article-title":"Power of One Bit of Quantum Information","volume":"81","author":"Knill","year":"1998","journal-title":"Phys. Rev. Lett."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"042316","DOI":"10.1103\/PhysRevA.72.042316","article-title":"Entanglement and the power of one qubit","volume":"72","author":"Datta","year":"2005","journal-title":"Phys. Rev. A"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"050502","DOI":"10.1103\/PhysRevLett.100.050502","article-title":"Quantum Discord and the Power of One Qubit","volume":"100","author":"Datta","year":"2008","journal-title":"Phys. Rev. Lett."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"220403","DOI":"10.1103\/PhysRevLett.106.220403","article-title":"All nonclassical correlations can be activated into distillable entanglement","volume":"106","author":"Piani","year":"2011","journal-title":"Phys. Rev. Lett."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"022309","DOI":"10.1103\/PhysRevA.69.022309","article-title":"Monogamy of quantum entanglement and other correlations","volume":"69","author":"Koashi","year":"2004","journal-title":"Phys. Rev. A"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"032323","DOI":"10.1103\/PhysRevA.83.032323","article-title":"Interpreting quantum discord through quantum state merging","volume":"83","author":"Madhok","year":"2011","journal-title":"Phys. Rev. A"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"032324","DOI":"10.1103\/PhysRevA.83.032324","article-title":"Operational interpretations of quantum discord","volume":"83","author":"Cavalcanti","year":"2011","journal-title":"Phys. Rev. A"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"012313","DOI":"10.1103\/PhysRevA.84.012313","article-title":"Conservation law for distributed entanglement of formation and quantum discord","volume":"84","author":"Fanchini","year":"2011","journal-title":"Phys. Rev. A"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"140501","DOI":"10.1103\/PhysRevLett.112.140501","article-title":"Experimental Entanglement Activation from Discord in a Programmable Quantum Measurement","volume":"112","author":"Adesso","year":"2014","journal-title":"Phys. Rev. Lett."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"033027","DOI":"10.1088\/1367-2630\/16\/3\/033027","article-title":"Computing quantum discord is NP complete","volume":"16","author":"Huang","year":"2014","journal-title":"New. J. Phys."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"224413","DOI":"10.1103\/PhysRevB.78.224413","article-title":"Quantum discord and quantum phase transition in spin chains","volume":"78","author":"Dillenschneider","year":"2008","journal-title":"Phys. Rev. B"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"012106","DOI":"10.1103\/PhysRevA.82.012106","article-title":"Quantum and classical thermal correlations in the XY spin-1\/2 chain","volume":"82","author":"Maziero","year":"2010","journal-title":"Phys. Rev. A"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"044101","DOI":"10.1103\/PhysRevA.81.044101","article-title":"Thermal and magnetic quantum discord in Heisenberg models","volume":"81","author":"Werlang","year":"2010","journal-title":"Phys. Rev. A"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"042316","DOI":"10.1103\/PhysRevA.82.042316","article-title":"Quantum discord in finite XY chains","volume":"82","author":"Ciliberti","year":"2010","journal-title":"Phys. Rev. A"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"062334","DOI":"10.1103\/PhysRevA.83.062334","article-title":"Spotlighting quantum critical points via quantum correlations at finite temperatures","volume":"83","author":"Werlang","year":"2011","journal-title":"Phys. Rev. A"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"052112","DOI":"10.1103\/PhysRevA.83.052112","article-title":"Quantum and classical correlations in the one-dimensional XY model with Dzyaloshinskii-Moriya interaction","volume":"83","author":"Liu","year":"2011","journal-title":"Phys. Rev. A"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"052323","DOI":"10.1103\/PhysRevA.83.052323","article-title":"Thermal quantum and classical correlations and entanglement in the XY spin model with three-spin interaction","volume":"83","author":"Li","year":"2011","journal-title":"Phys. Rev. A"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"022104","DOI":"10.1103\/PhysRevA.86.022104","article-title":"Measurements, quantum discord, and parity in spin-1 systems","volume":"86","author":"Rossignoli","year":"2012","journal-title":"Phys. Rev. A"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"043033","DOI":"10.1088\/1367-2630\/15\/4\/043033","article-title":"Global quantum correlations in finite-size spin chains","volume":"15","author":"Campbell","year":"2013","journal-title":"New J. Phys."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"062305","DOI":"10.1103\/PhysRevA.88.062305","article-title":"Criticality, factorization, and long-range correlations in the anisotropic XY model","volume":"88","author":"Campbell","year":"2013","journal-title":"Phys. Rev. A"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"012119","DOI":"10.1103\/PhysRevA.88.012119","article-title":"Discord and Information Deficit in XX chains","volume":"88","author":"Ciliberti","year":"2013","journal-title":"Phys. Rev. A"},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"054410","DOI":"10.1103\/PhysRevB.89.054410","article-title":"Scaling of quantum discord in spin models","volume":"89","author":"Huang","year":"2014","journal-title":"Phys. Rev. B"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"235","DOI":"10.1016\/0378-4371(82)90217-5","article-title":"Antiferromagnetic long-range order in the anisotropic quantum spin chain","volume":"112","author":"Kurmann","year":"1982","journal-title":"Physica A"},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"022322","DOI":"10.1103\/PhysRevA.74.022322","article-title":"Divergence of the entanglement range in low-dimensional quantum systems","volume":"74","author":"Amico","year":"2006","journal-title":"Phys. Rev. A"},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"052322","DOI":"10.1103\/PhysRevA.77.052322","article-title":"Entanglement of finite cyclic chains at factorizing fields","volume":"77","author":"Rossignoli","year":"2008","journal-title":"Phys. Rev. A"},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"062325","DOI":"10.1103\/PhysRevA.80.062325","article-title":"Factorization and entanglement in general xyz spin arrays in nonuniform transverse fields","volume":"80","author":"Rossignoli","year":"2009","journal-title":"Phys. Rev. A"},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"054415","DOI":"10.1103\/PhysRevB.81.054415","article-title":"Separability and entanglement in finite dimer-type chains in general transverse fields","volume":"81","author":"Canosa","year":"2010","journal-title":"Phys. Rev. B"},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"197201","DOI":"10.1103\/PhysRevLett.100.197201","article-title":"Theory of ground state factorization in quantum cooperative systems","volume":"100","author":"Giampaolo","year":"2008","journal-title":"Phys. Rev. Lett."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"224434","DOI":"10.1103\/PhysRevB.79.224434","article-title":"Separability and ground-state factorization in quantum spin systems","volume":"79","author":"Giampaolo","year":"2009","journal-title":"Phys. Rev. B"},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"207202","DOI":"10.1103\/PhysRevLett.104.207202","article-title":"Probing quantum frustrated systems via factorization of the ground state","volume":"104","author":"Giampaolo","year":"2010","journal-title":"Phys. Rev. Lett."},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"1413","DOI":"10.1103\/PhysRevLett.77.1413","article-title":"Separability criterion for density matrices","volume":"77","author":"Peres","year":"1996","journal-title":"Phys. Rev. Lett."},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/S0375-9601(96)00706-2","article-title":"Separability of mixed states: Necessary and sufficient conditions","volume":"223","author":"Horodecki","year":"1996","journal-title":"Phys. Lett. A"},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"3824","DOI":"10.1103\/PhysRevA.54.3824","article-title":"Mixed-state entanglement and quantum error correction","volume":"54","author":"Bennett","year":"1996","journal-title":"Phys. Rev. A"},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"355","DOI":"10.1080\/09500340008244048","article-title":"Entanglement monotones","volume":"47","author":"Vidal","year":"2000","journal-title":"J. Mod. Opt."},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"5022","DOI":"10.1103\/PhysRevLett.78.5022","article-title":"Entanglement of a pair of quantum bits","volume":"78","author":"Hill","year":"1997","journal-title":"Phys. Rev. Lett."},{"key":"ref_66","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_67","doi-asserted-by":"crossref","first-page":"1838","DOI":"10.1103\/PhysRevA.54.1838","article-title":"Information-theoretic aspects of inseparability of mixed states","volume":"54","author":"Horodecki","year":"1996","journal-title":"Phys. Rev. A"},{"key":"ref_68","doi-asserted-by":"crossref","first-page":"5184","DOI":"10.1103\/PhysRevLett.86.5184","article-title":"Separable states are more disordered hlobally than locally","volume":"86","author":"Nielsen","year":"2001","journal-title":"Phys. Rev. Lett."},{"key":"ref_69","doi-asserted-by":"crossref","first-page":"042306","DOI":"10.1103\/PhysRevA.66.042306","article-title":"Generalized entropic criterion for separability","volume":"66","author":"Rossignoli","year":"2002","journal-title":"Phys. Rev. A"},{"key":"ref_70","doi-asserted-by":"crossref","first-page":"042302","DOI":"10.1103\/PhysRevA.67.042302","article-title":"Violation of majorization relations in entangled states and its detection by means of generalized entropic forms","volume":"67","author":"Rossignoli","year":"2003","journal-title":"Phys. Rev. A"},{"key":"ref_71","doi-asserted-by":"crossref","first-page":"221","DOI":"10.1103\/RevModPhys.50.221","article-title":"General properties of entropy","volume":"50","author":"Wehrl","year":"1978","journal-title":"Rev. Mod. Phys."},{"key":"ref_72","doi-asserted-by":"crossref","first-page":"042318","DOI":"10.1103\/PhysRevA.90.042318","article-title":"Generalized conditional entropy optimization for qudit-qubit states","volume":"90","author":"Gigena","year":"2014","journal-title":"Phys. Rev. A"},{"key":"ref_73","doi-asserted-by":"crossref","first-page":"015302","DOI":"10.1088\/1751-8113\/47\/1\/015302","article-title":"Generalized conditional entropy in bipartite systems","volume":"47","author":"Gigena","year":"2014","journal-title":"J. Phys. A"},{"key":"ref_74","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_75","doi-asserted-by":"crossref","first-page":"170401","DOI":"10.1103\/PhysRevLett.88.170401","article-title":"Generalized Nonadditive Entropies and Quantum Entanglement","volume":"88","author":"Canosa","year":"2002","journal-title":"Phys. Rev. Lett."},{"key":"ref_76","doi-asserted-by":"crossref","unstructured":"Bhatia, R. (1997). Matrix Analysis, Springer.","DOI":"10.1007\/978-1-4612-0653-8"},{"key":"ref_77","unstructured":"Marshall, A., and Olkin, I. (1979). Inequalities: Theory of Majorization and its Applications, Academic Press."},{"key":"ref_78","doi-asserted-by":"crossref","first-page":"062320","DOI":"10.1103\/PhysRevA.65.062320","article-title":"Overlap and entanglement-witness measurements","volume":"65","author":"Filip","year":"2002","journal-title":"Phys. Rev. A"},{"key":"ref_79","doi-asserted-by":"crossref","first-page":"042316","DOI":"10.1103\/PhysRevA.85.042316","article-title":"Measurement scheme for purity based on two two-body gates","volume":"85","author":"Nakazato","year":"2012","journal-title":"Phys. Rev. A"},{"key":"ref_80","doi-asserted-by":"crossref","first-page":"012303","DOI":"10.1103\/PhysRevA.87.012303","article-title":"Possibility of a Minimal Purity-Measurement Scheme Critically depends on the parity of dimension of the quantum system","volume":"87","author":"Tanaka","year":"2013","journal-title":"Phys. Rev. A"},{"key":"ref_81","doi-asserted-by":"crossref","first-page":"479","DOI":"10.1007\/BF01016429","article-title":"Possible generalization of Boltzmann-Gibbs statistics","volume":"52","author":"Tsallis","year":"1988","journal-title":"J. Stat. Phys."},{"key":"ref_82","unstructured":"Tsallis, C. (2009). Introduction to non-Extensive Statistical Mechanics, Springer."},{"key":"ref_83","doi-asserted-by":"crossref","first-page":"034101","DOI":"10.1103\/PhysRevA.86.034101","article-title":"Problem with the geometric discord","volume":"86","author":"Piani","year":"2012","journal-title":"Phys. Rev. A"},{"key":"ref_84","doi-asserted-by":"crossref","first-page":"095303","DOI":"10.1088\/1751-8113\/45\/9\/095303","article-title":"Quantum Discord for qubit-qudit systems","volume":"45","author":"Vinjanampathy","year":"2012","journal-title":"J. Phys. A"},{"key":"ref_85","doi-asserted-by":"crossref","first-page":"407","DOI":"10.1016\/0003-4916(61)90115-4","article-title":"Two soluble models of an antiferromagnetic chain","volume":"16","author":"Lieb","year":"1961","journal-title":"Ann. Phys."},{"key":"ref_86","doi-asserted-by":"crossref","first-page":"032102","DOI":"10.1103\/PhysRevA.80.032102","article-title":"XY model on the circle: Diagonalization, spectrum, and forerunners of the quantum phase transition","volume":"80","author":"Facchi","year":"2009","journal-title":"Phys. Rev. A"},{"key":"ref_87","doi-asserted-by":"crossref","first-page":"032350","DOI":"10.1103\/PhysRevA.75.032350","article-title":"Entanglement between distant qubits in cyclic XX chains","volume":"75","author":"Canosa","year":"2007","journal-title":"Phys. Rev. A"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/17\/4\/1634\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T20:43:58Z","timestamp":1760215438000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/17\/4\/1634"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,3,26]]},"references-count":87,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2015,4]]}},"alternative-id":["e17041634"],"URL":"https:\/\/doi.org\/10.3390\/e17041634","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2015,3,26]]}}}