Difference between revisions of "Quantum Logic"
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==Context== | ==Context== | ||
| + | == Relationship to other logics == | ||
| + | Quantum logic embeds into [[linear logic]]<ref name=linear>Vaughan Pratt, "[http://boole.stanford.edu/pub/ql.pdf Linear logic for generalized quantum mechanics]," in ''Work­shop on Physics and Computation (PhysComp '92)'' proceedings. See also the dis­cuss­ion at [[#{{harvid|nLab}}|''n''Lab]], [http://ncatlab.org/nlab/revision/quantum%20logic/42 Revision 42], which cites G.D. Crown, "On some orthomodular posets of vector bundles," ''Journ. of Natural Sci. and Math.'', vol. 15 issue 1-2: pp. 11–25, 1975.</ref> and the [[modal logic]] ''B''.{{sfn|Dalla Chiara|Giuntini|2002}} Indeed, modern logics for the analysis of quantum computation often begin with quantum logic, and attempt to graft desirable features of an extension of classical logic thereonto; the results then necessarily embed quantum logic.{{sfn|Baltag|Smets|2006}}{{sfn|Baltag|Bergfeld|Kishida|Sack|2014}} | ||
| + | |||
| + | The orthocomplemented lattice of any set of quantum propositions can be embedded into a Boolean algebra, which is then amenable to classical logic.<ref>Jeffery Bub and William Demopoulos, "The Interpretation of Quantum Mechanics," in ''[https://archive.org/details/logicalepistemol0000unse Logical and Epistemological Studies in Contemporary Physics]'', Boston Studies in the Philosophy of Science 13, ed. Robert S. Cohen and Marx W. Wartofsky; D. Riedel, 1974. pp. 92-122. DOI: [http://dx.doi.org/10.1007/978-94-010-2656-7 10.1007/978-94-010-2656-7]. {{ISBN|978-94-010-2656-7}}.</ref> | ||
| + | ==Problems== | ||
| + | Quantum logic admits no reasonable [[material conditional]]; any [[logical connective|connective]] that is [[monotonicity of entailment|monotone]] in a certain technical sense reduces the class of propositions to a [[Boolean algebra (structure)|Boolean algebra]].<ref>{{cite journal | url=https://link.springer.com/content/pdf/10.1007/BF00733278.pdf | doi=10.1007/BF00733278 | title=Quantum logic revisited | year=1991 | last1= Román| first1=L. | last2=Rumbos | first2=B. | journal=Foundations of Physics | volume=21 | issue=6 | pages=727–734 | bibcode=1991FoPh...21..727R | s2cid=123383431 }}</ref> Consequently, quantum logic struggles to represent the passage of time.<ref name=linear /> One possible workaround is the theory of [[Belavkin equation|quantum filtrations]] developed in the late 1970s and 1980s by [[Viacheslav Belavkin|Belavkin]].<ref> | ||
| + | * {{cite journal | ||
| + | | author = V. P. Belavkin | ||
| + | | title = Optimal quantum filtration of Makovian signals | ||
| + | | language=ru | ||
| + | | journal = Problems of Control and Information Theory | ||
| + | | volume = 7 | ||
| + | | number = 5 | ||
| + | | pages = 345–360 | ||
| + | | year = 1978 | ||
| + | | ref = none | ||
| + | }} | ||
| + | * {{cite journal | ||
| + | | author = V. P. Belavkin | ||
| + | | title = Quantum stochastic calculus and quantum nonlinear filtering | ||
| + | | journal = Journal of Multivariate Analysis | ||
| + | | volume = 42 | ||
| + | | number = 2 | ||
| + | | year = 1992 | ||
| + | | pages = 171–201 | ||
| + | | doi = 10.1016/0047-259X(92)90042-E | ||
| + | | arxiv = math/0512362| s2cid = 3909067 | ||
| + | | ref = none | ||
| + | }}</ref><ref name=Bouten2009> | ||
| + | {{cite journal | ||
| + | |author1=Luc Bouten |author2=Ramon van Handel |author3=Matthew R. James | title = A discrete invitation to quantum filtering and feedback control | ||
| + | | journal = SIAM Review | ||
| + | | volume = 51 | ||
| + | |issue=2 | pages = 239–316 | ||
| + | | year = 2009 | ||
| + | | doi = 10.1137/060671504 | ||
| + | | arxiv = math/0606118 | ||
| + | |bibcode = 2009SIAMR..51..239B |s2cid=10435983 | ||
| + | }}</ref> It is known, however, that System [[Noncommutative logic|BV]], a [[deep inference]] fragment of [[linear logic]] that is very close to quantum logic, can handle arbitrary [[causal graph|discrete spacetimes]].<ref>Richard Blute, Alessio Guglielmi, Ivan T. Ivanov, Prakash Panangaden, Lutz Straß­burger, "A Logical Basis for Quantum Evolution and Entanglement" in ''Categories and Types in Logic, Language, and Physics: Essays Dedicated to Jim Lambek on the Occasion of His 90th Birthday''; Springer, 2014. pp. 90-107. DOI: [http://dx.doi.org/10.1007/978-3-642-54789-8_6 10.1007/978-3-642-54789-8_6]. HAL [https://hal.inria.fr/hal-01092279/ 01092279].</ref> | ||
==References== | ==References== | ||
* Also see the wiki page on [[Quantum Mechanics]] | * Also see the wiki page on [[Quantum Mechanics]] | ||
[[Category: Physics]] | [[Category: Physics]] | ||
Latest revision as of 17:54, 17 December 2024
Full Title or Meme
The structure of experimental tests in classical mechanics forms a Boolean algebra, but the structure of experimental tests in quantum mechanics forms a much more complicated structure.
Context
Relationship to other logics
Quantum logic embeds into linear logic[1] and the modal logic B.{{#invoke:Footnotes|sfn}} Indeed, modern logics for the analysis of quantum computation often begin with quantum logic, and attempt to graft desirable features of an extension of classical logic thereonto; the results then necessarily embed quantum logic.{{#invoke:Footnotes|sfn}}{{#invoke:Footnotes|sfn}}
The orthocomplemented lattice of any set of quantum propositions can be embedded into a Boolean algebra, which is then amenable to classical logic.[2]
Problems
Quantum logic admits no reasonable material conditional; any connective that is monotone in a certain technical sense reduces the class of propositions to a Boolean algebra.[3] Consequently, quantum logic struggles to represent the passage of time.[1] One possible workaround is the theory of quantum filtrations developed in the late 1970s and 1980s by Belavkin.[4][5] It is known, however, that System BV, a deep inference fragment of linear logic that is very close to quantum logic, can handle arbitrary discrete spacetimes.[6]
References
- Also see the wiki page on Quantum Mechanics
