{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:07:49Z","timestamp":1760148469701,"version":"build-2065373602"},"reference-count":34,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2023,5,12]],"date-time":"2023-05-12T00:00:00Z","timestamp":1683849600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"United States Air Force Research Laboratory (AFRL) Summer Faculty Fellowship Program","award":["#FA8750-20-3-1003"],"award-info":[{"award-number":["#FA8750-20-3-1003"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In this paper, we construct the metric tensor and volume for the manifold of purifications associated with an arbitrary reduced density operator \u03c1S. We also define a quantum coarse-graining (CG) to study the volume where macrostates are the manifolds of purifications, which we call surfaces of ignorance (SOI), and microstates are the purifications of \u03c1S. In this context, the volume functions as a multiplicity of the macrostates that quantifies the amount of information missing from \u03c1S. Using examples where the SOI are generated using representations of SU(2), SO(3), and SO(N), we show two features of the CG: (1) A system beginning in an atypical macrostate of smaller volume evolves to macrostates of greater volume until it reaches the equilibrium macrostate in a process in which the system and environment become strictly more entangled, and (2) the equilibrium macrostate takes up the vast majority of the coarse-grained space especially as the dimension of the total system becomes large. Here, the equilibrium macrostate corresponds to a maximum entanglement between the system and the environment. To demonstrate feature (1) for the examples considered, we show that the volume behaves like the von Neumann entropy in that it is zero for pure states, maximal for maximally mixed states, and is a concave function with respect to the purity of \u03c1S. These two features are essential to typicality arguments regarding thermalization and Boltzmann\u2019s original CG.<\/jats:p>","DOI":"10.3390\/e25050788","type":"journal-article","created":{"date-parts":[[2023,5,12]],"date-time":"2023-05-12T09:56:18Z","timestamp":1683885378000},"page":"788","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A Differential-Geometric Approach to Quantum Ignorance Consistent with Entropic Properties of Statistical Mechanics"],"prefix":"10.3390","volume":"25","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7358-4455","authenticated-orcid":false,"given":"Shannon","family":"Ray","sequence":"first","affiliation":[{"name":"Air Force Research Laboratory, Rome, NY 13441, USA"},{"name":"Griffiss Institute, Rome, NY 13441, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Paul M.","family":"Alsing","sequence":"additional","affiliation":[{"name":"Air Force Research Laboratory, Rome, NY 13441, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Carlo","family":"Cafaro","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Physics, SUNY Polytechnic Institute, Albany, NY 12203, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"H S.","family":"Jacinto","sequence":"additional","affiliation":[{"name":"Air Force Research Laboratory, Rome, NY 13441, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,5,12]]},"reference":[{"doi-asserted-by":"crossref","unstructured":"Wilde, M. (2013). Quantum Information Theory, Cambridge University Press.","key":"ref_1","DOI":"10.1017\/CBO9781139525343"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"075021","DOI":"10.1088\/1367-2630\/12\/7\/075021","article-title":"Thermodynamic entropy of a many-body energy eigenstate","volume":"12","author":"Deutsch","year":"2010","journal-title":"New J. Phys."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"010102","DOI":"10.1103\/PhysRevE.86.010102","article-title":"Weak and strong typicality in quantum systems","volume":"86","author":"Santos","year":"2012","journal-title":"Phys. Rev. E"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"042135","DOI":"10.1103\/PhysRevE.87.042135","article-title":"Microscopic origin of thermodynamic entropy in isolated systems","volume":"87","author":"Deutsch","year":"2013","journal-title":"Phys. Rev. E"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"6301","DOI":"10.1126\/science.aaf6725","article-title":"Quantum thermalization through entanglement in an isolated many-body system","volume":"353","author":"Kaufman","year":"2016","journal-title":"Science"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"012103","DOI":"10.1103\/PhysRevA.99.012103","article-title":"Quantum coarse-graining entropy and thermalization in closed systems","volume":"99","author":"Safranek","year":"2019","journal-title":"Phys. Rev. A"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"032113","DOI":"10.1103\/PhysRevA.96.032113","article-title":"Emerging dynamics arising from coarse-grained quantum systems","volume":"96","author":"Duarte","year":"2017","journal-title":"Phys. Rev. A"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"052130","DOI":"10.1103\/PhysRevA.97.052130","article-title":"Quantum coarse graining, symmetries, and reducibility of dynamics","volume":"97","author":"Kabernik","year":"2018","journal-title":"Phys. Rev. A"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"052210","DOI":"10.1103\/PhysRevA.103.052210","article-title":"Macro-to-micro quantum mapping and the emergence of nonlinearity","volume":"103","author":"Correia","year":"2021","journal-title":"Phys. Rev. A"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"022334","DOI":"10.1103\/PhysRevA.100.022334","article-title":"Spin-entanglement wave in a coarse-grained optical lattice","volume":"100","author":"Correia","year":"2019","journal-title":"Phys. Rev. A"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"032217","DOI":"10.1103\/PhysRevA.102.032217","article-title":"Decay of quantumness in a measurement process: Action of a coarse-graining channel","volume":"102","author":"Carvalho","year":"2020","journal-title":"Phys. Rev. A"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"042218","DOI":"10.1103\/PhysRevA.104.042218","article-title":"Fuzzy measurement and coarse graining in quantum many-body systems","volume":"104","author":"Pineda","year":"2021","journal-title":"Phys. Rev. A"},{"unstructured":"Zalta, E.N. (2017). The Stanford Encyclopedia of Philosophy, Metaphysics Research Lab, Stanford University. Spring 2017 ed.","key":"ref_13"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1600301","DOI":"10.1002\/andp.201600301","article-title":"Macroscopic and microscopic thermal equilibrium","volume":"529","author":"Goldstein","year":"2017","journal-title":"Ann. Phys."},{"doi-asserted-by":"crossref","unstructured":"Goldstein, S. (2001). Boltzmann\u2019s Approach to Statistical Mechanics, Springer.","key":"ref_15","DOI":"10.1007\/3-540-44966-3_3"},{"doi-asserted-by":"crossref","unstructured":"Goldstein, S., Lebowitz, J.L., Tumulka, R., and Zanghi, N. (2020). Gibbs and Boltzmann Entropy in Classical and Quantum Mechanics, World Scientific Publishing Co.","key":"ref_16","DOI":"10.1142\/9789811211720_0014"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"050403","DOI":"10.1103\/PhysRevLett.96.050403","article-title":"Canonical Typicality","volume":"96","author":"Goldstein","year":"2006","journal-title":"Phys. Rev. Lett."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1063\/1.3569494","article-title":"On the approach to thermal equilibrium of macroscopic quantum systems","volume":"1332","author":"Goldstein","year":"2011","journal-title":"AIP Conf. Proc."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"937","DOI":"10.1007\/s10955-016-1511-2","article-title":"Typicality of thermal equilibrium and thermalization in isolated macroscopic quantum systems","volume":"163","author":"Tasaki","year":"2016","journal-title":"J. Stat. Phys."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"754","DOI":"10.1038\/nphys444","article-title":"Entanglement and the foundations of statistical mechanics","volume":"2","author":"Popescu","year":"2006","journal-title":"Nat. Phys."},{"unstructured":"Boltzmann, L. (1964). Vorlesungen \u00fcber Gastheorie. Leipzi: Barth (Part I 1896, Part II 1898), University of California Press.","key":"ref_21"},{"unstructured":"Landford, O.E. (1973). Entropy and Equilibrium States in Classical Statistical Mechanics, Springer-Verlag.","key":"ref_22"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1103\/RevModPhys.81.1","article-title":"Colloquium: The physics of Maxwell\u2019s demon and information","volume":"81","author":"Maruyama","year":"2009","journal-title":"Rev. Mod. Phys."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"334","DOI":"10.1063\/1.1699951","article-title":"Maxwell\u2019s demon cannot operate: Information and entropy: I","volume":"22","author":"Brillouin","year":"1951","journal-title":"J. App. Phys."},{"unstructured":"Schr\u00f6dinger, E. (1944). What Is Life? The Physical Aspect of the Living Cell, Cambridge University Press.","key":"ref_25"},{"unstructured":"Brillouin, L. (2013). Science and Information Theory, Dover Publications. [2nd ed.].","key":"ref_26"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"620","DOI":"10.1103\/PhysRev.106.620","article-title":"Information theory and statistical mechanics","volume":"106","author":"Jaynes","year":"1957","journal-title":"Phys. Rev."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"4235","DOI":"10.1088\/0305-4470\/27\/12\/028","article-title":"Random unitary matrices","volume":"27","author":"Zyczkowski","year":"1994","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"2","DOI":"10.1103\/PhysRevA.58.883","article-title":"Volume of the set of separable states","volume":"58","author":"Zyczkowski","year":"1998","journal-title":"Phys. Rev. A"},{"doi-asserted-by":"crossref","unstructured":"Bengtsson, I., and Zyczkowski, K. (2017). Geometry of Quantum States: An Introduction to Quantum Entanglement, Cambridge University Press. [2nd ed.].","key":"ref_30","DOI":"10.1017\/9781139207010"},{"key":"ref_31","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_32","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1007\/s00220-006-1535-6","article-title":"Aspects of generic entanglement","volume":"265","author":"Hayden","year":"2006","journal-title":"Commun. Math. Phys."},{"unstructured":"Lloyd, S. (1988). Black Holes, Demons and the Loss of Coherence: How Complex System Get Information, and What They Do with It, Rockefeller University. Chapter 3.","key":"ref_33"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"063013","DOI":"10.1088\/1367-2630\/18\/6\/063013","article-title":"Foundations of statistical mechanics from symmetries of entanglement","volume":"18","author":"Deffner","year":"2016","journal-title":"New J. Phys."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/5\/788\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T19:33:34Z","timestamp":1760124814000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/5\/788"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,5,12]]},"references-count":34,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2023,5]]}},"alternative-id":["e25050788"],"URL":"https:\/\/doi.org\/10.3390\/e25050788","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2023,5,12]]}}}