{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:28:36Z","timestamp":1760243316439,"version":"build-2065373602"},"reference-count":32,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2014,9,22]],"date-time":"2014-09-22T00:00:00Z","timestamp":1411344000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Information"],"abstract":"<jats:p>In this essay, I develop order-theoretic notions of determinism and contextuality on domains and topoi. In the process, I develop a method for quantifying contextuality and show that the order-theoretic sense of contextuality is analogous to the sense embodied in the topos-theoretic statement of the Kochen\u2013Specker theorem. Additionally, I argue that this leads to a relation between the entropy associated with measurements on quantum systems and the second law of thermodynamics. The idea that the second law has its origin in the ordering of quantum states and processes dates to at least 1958 and possibly earlier. The suggestion that the mechanism behind this relation is contextuality, is made here for the first time.<\/jats:p>","DOI":"10.3390\/info5030508","type":"journal-article","created":{"date-parts":[[2014,9,22]],"date-time":"2014-09-22T10:39:10Z","timestamp":1411382350000},"page":"508-525","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["An Order-Theoretic Quantification of Contextuality"],"prefix":"10.3390","volume":"5","author":[{"given":"Ian","family":"Durham","sequence":"first","affiliation":[{"name":"Department of Physics, Saint Anselm College, 100 Saint Anselm Drive, Manchester, NH 03102, USA"}]}],"member":"1968","published-online":{"date-parts":[[2014,9,22]]},"reference":[{"key":"ref_1","unstructured":"Martin, K. (2000). A foundation for computation. [Ph.D. Thesis, Tulane University]."},{"key":"ref_2","unstructured":"Zhai, Y., and Erickson, G. (2004). Bayesian Inference and Maximum Entropy Methods in Science and Engineering (AIP Conference Proceedings), American Institute of Physics."},{"key":"ref_3","unstructured":"Sica, G. (2006). What Is Category Theory?, Polimetrica. Volume 30, Advanced Studies in Mathematics and Logic."},{"key":"ref_4","unstructured":"Abramsky, S., and Coecke, B. (2008). Handbook of Quantum Logic and Quantum Structures, Elsevier."},{"key":"ref_5","unstructured":"Halvorsen, H. (2010). Deep Beauty, Cambridge University Press."},{"key":"ref_6","unstructured":"Spivak, D.I. (2013). Category theory for scientists."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Coecke, B., and Martin, K. (2011). New Structures for Physics, Springer. Chapter 10.","DOI":"10.1007\/978-3-642-12821-9"},{"key":"ref_8","unstructured":"Martin, K. (2011). New Structures for Physics, Springer. Chapter 9."},{"key":"ref_9","unstructured":"D\u00f6ring, A., and Isham, C. (2011). New Structures for Physics, Springer. Chapter 13."},{"key":"ref_10","unstructured":"Awodey, S. (2010). Category Theory, Oxford University Press. [2nd ed.]."},{"key":"ref_11","unstructured":"Eddington, A.S. (1939). The Philosophy of Physical Science, Cambridge University Press."},{"key":"ref_12","unstructured":"The notation \u226a is standard, but in order to avoid confusion with other inequalities, we adopt \u2aaf, so as to clearly distinguish it from the usual meaning of \u226a in inequalities."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1090\/S0002-9904-1975-13621-4","article-title":"Some convexity and subadditivity properties of entropy","volume":"81","author":"Lieb","year":"1975","journal-title":"Bull. Am. Math. Soc."},{"key":"ref_14","unstructured":"Nielsen, M.A., and Chuang, I.L. (2000). Quantum Computation and Quantum Information, Cambridge University Press."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"131","DOI":"10.2307\/1426210","article-title":"Why the Shannon and Hartley Entropies are \u2018Natural\u2019","volume":"6","author":"Forte","year":"1974","journal-title":"Adv. Appl. Probab."},{"key":"ref_16","unstructured":"The difference can be better understood by noting that some systems obey the strong subadditivity condition, while others do not [14]."},{"key":"ref_17","unstructured":"It is important to remember that, while the functional form of \u03bc generally depends on an agent, the basic conditions set by (1) and (2) always hold under a neo-realist interpretation."},{"key":"ref_18","first-page":"137","article-title":"Why Shannon\u2019s entropy","volume":"Volume XI","author":"Forte","year":"1975","journal-title":"Symposia Mathematica"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Schumacher, B., and Westmoreland, M. (2010). Quantum Processes, Systems, and Information, Cambridge University Press.","DOI":"10.1017\/CBO9780511814006"},{"key":"ref_20","unstructured":"This example is adapted from one given in [7]."},{"key":"ref_21","unstructured":"There are, of course, other forms of determinism that may allow for invasive measurements. While I do not consider those here, a domain-theoretic definition of such an alternate definition of determinism would an intriguing line of enquiry."},{"key":"ref_22","first-page":"59","article-title":"The problem of hidden variables in quantum mechanics","volume":"17","author":"Kochen","year":"1967","journal-title":"J. Math. Mech."},{"key":"ref_23","unstructured":"Sakurai, J. (1994). Modern Quantum Mechanics, Addison Wesley Longman. [revised ed.]."},{"key":"ref_24","unstructured":"There are many more cases than these three, corresponding to various multiples of \u03c0 and \n        \n          \n            \n              \u03c0\n\t\t\t  2\n\t\t\t\n          \n        \n       . I am simply using these as examples to illustrate how the ordering relations apply in the quantum case."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1078","DOI":"10.1016\/j.aop.2011.10.009","article-title":"Localized qubits in curved spacetimes","volume":"327","author":"Palmer","year":"2012","journal-title":"Ann. Phys."},{"key":"ref_26","unstructured":"Palmer, M.C. (2013). Relativistuc quantum information theory and quantum reference frames. [Ph.D. Thesis, University of Sydney]."},{"key":"ref_27","unstructured":"Landau, L., and Lifshitz, E. (1958). Statistical Physics, Addison Wesley. [2nd ed.]. Volume 5, Course of Theoretical Physics."},{"key":"ref_28","unstructured":"This idea is very similar to one that was relayed to me by Chris Adami, who has suggested that the second law actually refers to relative entropy. He has hinted at the concept in a few of his publications (see, for example, [31,32]), but to my knowledge, has never stated it explicitly in print."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Cerf, N.J., and Adami, C. (1997). Negative Entropy and Information in Quantum Mechanics. Phys. Rev. Lett., 79.","DOI":"10.1103\/PhysRevLett.79.5194"},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Cerf, N.J., and Adami, C. (1999). Quantum extension of conditional probability. Phys. Rev. A, 60.","DOI":"10.1103\/PhysRevA.60.893"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"62","DOI":"10.1016\/S0167-2789(98)00045-1","article-title":"Information theory of quantum entanglement and measurement","volume":"120","author":"Cerf","year":"1998","journal-title":"Physica D"},{"key":"ref_32","unstructured":"Adami, C. (2010). Quantum Mechanics of Consecutive Measurements."}],"container-title":["Information"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2078-2489\/5\/3\/508\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T21:16:10Z","timestamp":1760217370000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2078-2489\/5\/3\/508"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,9,22]]},"references-count":32,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2014,9]]}},"alternative-id":["info5030508"],"URL":"https:\/\/doi.org\/10.3390\/info5030508","relation":{},"ISSN":["2078-2489"],"issn-type":[{"type":"electronic","value":"2078-2489"}],"subject":[],"published":{"date-parts":[[2014,9,22]]}}}