Physics:Quantum logic - Revision history

WikiHarold: Fix final Quantum red link source

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Fix final Quantum red link source

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	Inspired by [[Biography:Hans Reichenbach\|Hans Reichenbach]]'s recent defence of general relativity, the philosopher [[Biography:Hilary Putnam\|Hilary Putnam]] popularized Mackey's work in two papers in 1968 and 1975,{{sfn\|Maudlin\|2005}} in which he attributed the idea that anomalies associated to quantum measurements originate with a failure of logic itself to his coauthor, physicist [[Biography:David Finkelstein\|David Finkelstein]].{{sfn\|Putnam\|1969}} Putnam hoped to develop a possible alternative to hidden variables or wavefunction collapse in the problem of quantum measurement, but Gleason's theorem presents severe difficulties for this goal.{{sfn\|Maudlin\|2005}}{{sfn\|Wilce}} Later, Putnam retracted his views, albeit with much less fanfare,{{sfn\|Maudlin\|2005}} but the damage had been done. While Birkhoff and von Neumann's original work only attempted to organize the calculations associated with the Copenhagen interpretation of quantum mechanics, a school of researchers had now sprung up, either hoping that quantum logic would provide a viable hidden-variable theory, or obviate the need for one.<ref>T. A. Brody, "On Quantum Logic", ''Foundations of Physics'', vol. 14, no. 5, 1984. pp. 409-430.</ref> Their work proved fruitless, and now lies in poor repute.{{sfn\|Bacciagaluppi\|2009}}		Inspired by [[Biography:Hans Reichenbach\|Hans Reichenbach]]'s recent defence of general relativity, the philosopher [[Biography:Hilary Putnam\|Hilary Putnam]] popularized Mackey's work in two papers in 1968 and 1975,{{sfn\|Maudlin\|2005}} in which he attributed the idea that anomalies associated to quantum measurements originate with a failure of logic itself to his coauthor, physicist [[Biography:David Finkelstein\|David Finkelstein]].{{sfn\|Putnam\|1969}} Putnam hoped to develop a possible alternative to hidden variables or wavefunction collapse in the problem of quantum measurement, but Gleason's theorem presents severe difficulties for this goal.{{sfn\|Maudlin\|2005}}{{sfn\|Wilce}} Later, Putnam retracted his views, albeit with much less fanfare,{{sfn\|Maudlin\|2005}} but the damage had been done. While Birkhoff and von Neumann's original work only attempted to organize the calculations associated with the Copenhagen interpretation of quantum mechanics, a school of researchers had now sprung up, either hoping that quantum logic would provide a viable hidden-variable theory, or obviate the need for one.<ref>T. A. Brody, "On Quantum Logic", ''Foundations of Physics'', vol. 14, no. 5, 1984. pp. 409-430.</ref> Their work proved fruitless, and now lies in poor repute.{{sfn\|Bacciagaluppi\|2009}}

	Most philosophers find quantum logic an unappealing competitor to classical logic. It is far from evident (albeit true<ref>{{harvnb\|Dalla Chiara\|Giuntini\|2002\|p=94}}: "Quantum logics are, without any doubt, logics. As we have seen, they satisfy all the canonical conditions that the present community of logicians require in order to call a given abstract object a logic."</ref>) that quantum logic is a ''logic'', in the sense of describing a process of reasoning, as opposed to a particularly convenient language to summarize the measurements performed by quantum apparatuses.{{sfn\|Maudlin\|2005\|p=159-161}}{{sfn\|Brody\|1984}} In particular, modern philosophers of science argue that quantum logic attempts to substitute metaphysical difficulties for unsolved problems in physics, rather than properly solving the physics problems.{{sfn\|Brody\|1984\|pp=428-429}} [[Biography:Tim Maudlin\|Tim Maudlin]] writes that quantum "logic 'solves' the ~~[[Physics:Measurement problem\|~~[measurement] problem]] by making the problem impossible to state."{{sfn\|Maudlin\|2005\|p=174}}		Most philosophers find quantum logic an unappealing competitor to classical logic. It is far from evident (albeit true<ref>{{harvnb\|Dalla Chiara\|Giuntini\|2002\|p=94}}: "Quantum logics are, without any doubt, logics. As we have seen, they satisfy all the canonical conditions that the present community of logicians require in order to call a given abstract object a logic."</ref>) that quantum logic is a ''logic'', in the sense of describing a process of reasoning, as opposed to a particularly convenient language to summarize the measurements performed by quantum apparatuses.{{sfn\|Maudlin\|2005\|p=159-161}}{{sfn\|Brody\|1984}} In particular, modern philosophers of science argue that quantum logic attempts to substitute metaphysical difficulties for unsolved problems in physics, rather than properly solving the physics problems.{{sfn\|Brody\|1984\|pp=428-429}} [[Biography:Tim Maudlin\|Tim Maudlin]] writes that quantum "logic 'solves' the [measurement] problem by making the problem impossible to state."{{sfn\|Maudlin\|2005\|p=174}}

	\|Hilary Putnam\|pp. 184-185}}		\|Hilary Putnam\|pp. 184-185}}
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	== Relationship to other logics ==		== 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 ~~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}}		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 ''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>		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>

WikiHarold: Finish Quantum residual red link cleanup

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Finish Quantum residual red link cleanup

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	To illustrate why the distributive law fails, consider a particle moving on a line and (using some system of units where the reduced Planck's constant is 1) let<ref group="Note">Due to technical reasons, it is not possible to represent these propositions as quantum-mechanical operators. They are presented here because they are simple enough to enable intuition, and can be considered as limiting cases of operators that ''are'' feasible. See {{Slink\|\|Quantum logic as the logic of observables}} ''et seq.'' for details.</ref>		To illustrate why the distributive law fails, consider a particle moving on a line and (using some system of units where the reduced Planck's constant is 1) let<ref group="Note">Due to technical reasons, it is not possible to represent these propositions as quantum-mechanical operators. They are presented here because they are simple enough to enable intuition, and can be considered as limiting cases of operators that ''are'' feasible. See {{Slink\|\|Quantum logic as the logic of observables}} ''et seq.'' for details.</ref>
	: ''p'' = "the particle has momentum in the interval [0, +~~{{frac\|~~1\|6}}]"		: ''p'' = "the particle has momentum in the interval [0, +1/6]"
	: ''q'' = "the particle is in the interval "		: ''q'' = "the particle is in the interval "
	: ''r'' = "the particle is in the interval "		: ''r'' = "the particle is in the interval "
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	* D. Cohen, ''An Introduction to Hilbert Space and Quantum Logic'', Springer-Verlag, 1989. Elementary and well-illustrated; suitable for advanced undergraduates.		* D. Cohen, ''An Introduction to Hilbert Space and Quantum Logic'', Springer-Verlag, 1989. Elementary and well-illustrated; suitable for advanced undergraduates.
	* Günther Ludwig, ''Der Grundlagen der Quantenmechanik'' (in German), Springer, 1954. The definitive work. Released in English as: <ul><li>Günther Ludwig, ''[https://archive.org/details/foundations-of-quantum-mechanics-i-g.-ludwig-c.-a.-hein Foundations of Quantum Mechanics]'', vol. 1, trans. Carl A. Hein; Springer-Verlag, 1983.</li><li>Günther Ludwig, ''An Axiomatic Basis for Quantum Mechanics'', vol. 1: "Derivation of Hilbert Space Structure", trans. Leo F. Boron, ed. Karl Just; Springer, 1985. DOI: [http://dx.doi.org/10.1007/978-3-642-70029-3 10.1007/978-3-642-70029-3]. {{ISBN\|978-3-642-70029-3}}.</li></ul>		* Günther Ludwig, ''Der Grundlagen der Quantenmechanik'' (in German), Springer, 1954. The definitive work. Released in English as: <ul><li>Günther Ludwig, ''[https://archive.org/details/foundations-of-quantum-mechanics-i-g.-ludwig-c.-a.-hein Foundations of Quantum Mechanics]'', vol. 1, trans. Carl A. Hein; Springer-Verlag, 1983.</li><li>Günther Ludwig, ''An Axiomatic Basis for Quantum Mechanics'', vol. 1: "Derivation of Hilbert Space Structure", trans. Leo F. Boron, ed. Karl Just; Springer, 1985. DOI: [http://dx.doi.org/10.1007/978-3-642-70029-3 10.1007/978-3-642-70029-3]. {{ISBN\|978-3-642-70029-3}}.</li></ul>
	* ~~{{nlab\|id=quantum+logic\|title=~~Quantum Logic}}		* Quantum Logic
	* C. Piron, ''Foundations of Quantum Physics'', W. A. Benjamin, 1976.		* C. Piron, ''Foundations of Quantum Physics'', W. A. Benjamin, 1976.
	{{DEFAULTSORT:Quantum Logic}}		{{DEFAULTSORT:Quantum Logic}}

	{{Sourceattribution\|Quantum logic}}		{{Sourceattribution\|Quantum logic}}

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WikiHarold: Remove imported red links from Quantum page

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	<div style="flex:1; line-height:1.45; color:#006b45; column-count:2; column-gap:32px; column-rule:1px solid #b8d8c8;">		<div style="flex:1; line-height:1.45; color:#006b45; column-count:2; column-gap:32px; column-rule:1px solid #b8d8c8;">
	~~{{Quantum mechanics\|cTopic=[[Physics:Interpretation of quantum mechanics\|Interpretations]]}}~~
	In the mathematical study of logic and the physical analysis of [[Physics:Quantum foundations\|quantum foundations]], '''quantum logic''' is a set of rules for manipulation of propositions inspired by the structure of [[Physics:Quantum mechanics\|quantum theory]]. The formal system takes as its starting point an observation of [[Biography:Garrett Birkhoff\|Garrett Birkhoff]] and John von Neumann, that 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.		In the mathematical study of logic and the physical analysis of [[Physics:Quantum foundations\|quantum foundations]], '''quantum logic''' is a set of rules for manipulation of propositions inspired by the structure of [[Physics:Quantum mechanics\|quantum theory]]. The formal system takes as its starting point an observation of [[Biography:Garrett Birkhoff\|Garrett Birkhoff]] and John von Neumann, that 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.

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	To illustrate why the distributive law fails, consider a particle moving on a line and (using some system of units where the reduced Planck's constant is 1) let<ref group="Note">Due to technical reasons, it is not possible to represent these propositions as quantum-mechanical operators. They are presented here because they are simple enough to enable intuition, and can be considered as limiting cases of operators that ''are'' feasible. See {{Slink\|\|Quantum logic as the logic of observables}} ''et seq.'' for details.</ref>		To illustrate why the distributive law fails, consider a particle moving on a line and (using some system of units where the reduced Planck's constant is 1) let<ref group="Note">Due to technical reasons, it is not possible to represent these propositions as quantum-mechanical operators. They are presented here because they are simple enough to enable intuition, and can be considered as limiting cases of operators that ''are'' feasible. See {{Slink\|\|Quantum logic as the logic of observables}} ''et seq.'' for details.</ref>
	: ''p'' = "the particle has momentum in the interval {{closed-closed\|0, +{{frac\|1\|6}}}}"		: ''p'' = "the particle has momentum in the interval {{closed-closed\|0, +{{frac\|1\|6}}}}"
	: ''q'' = "the particle is in the interval ~~{{closed-closed\|−1, 1}}~~"		: ''q'' = "the particle is in the interval "
	: ''r'' = "the particle is in the interval ~~{{closed-closed\|1, 3}}~~"		: ''r'' = "the particle is in the interval "
	We might observe that:		We might observe that:
	: ''p'' and (''q'' or ''r'') = ''true''		: ''p'' and (''q'' or ''r'') = ''true''
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	Most philosophers find quantum logic an unappealing competitor to classical logic. It is far from evident (albeit true<ref>{{harvnb\|Dalla Chiara\|Giuntini\|2002\|p=94}}: "Quantum logics are, without any doubt, logics. As we have seen, they satisfy all the canonical conditions that the present community of logicians require in order to call a given abstract object a logic."</ref>) that quantum logic is a ''logic'', in the sense of describing a process of reasoning, as opposed to a particularly convenient language to summarize the measurements performed by quantum apparatuses.{{sfn\|Maudlin\|2005\|p=159-161}}{{sfn\|Brody\|1984}} In particular, modern philosophers of science argue that quantum logic attempts to substitute metaphysical difficulties for unsolved problems in physics, rather than properly solving the physics problems.{{sfn\|Brody\|1984\|pp=428-429}} [[Biography:Tim Maudlin\|Tim Maudlin]] writes that quantum "logic 'solves' the [[Physics:Measurement problem\|[measurement] problem]] by making the problem impossible to state."{{sfn\|Maudlin\|2005\|p=174}}		Most philosophers find quantum logic an unappealing competitor to classical logic. It is far from evident (albeit true<ref>{{harvnb\|Dalla Chiara\|Giuntini\|2002\|p=94}}: "Quantum logics are, without any doubt, logics. As we have seen, they satisfy all the canonical conditions that the present community of logicians require in order to call a given abstract object a logic."</ref>) that quantum logic is a ''logic'', in the sense of describing a process of reasoning, as opposed to a particularly convenient language to summarize the measurements performed by quantum apparatuses.{{sfn\|Maudlin\|2005\|p=159-161}}{{sfn\|Brody\|1984}} In particular, modern philosophers of science argue that quantum logic attempts to substitute metaphysical difficulties for unsolved problems in physics, rather than properly solving the physics problems.{{sfn\|Brody\|1984\|pp=428-429}} [[Biography:Tim Maudlin\|Tim Maudlin]] writes that quantum "logic 'solves' the [[Physics:Measurement problem\|[measurement] problem]] by making the problem impossible to state."{{sfn\|Maudlin\|2005\|p=174}}

	{{Quote frame\|The horse of quantum logic has been so thrashed, whipped and pummeled, and is so thoroughly deceased that...the question is not whether the horse will rise again, it is: how in the world did this horse get here in the first place? The tale of quantum logic is not the tale of a promising idea gone bad, it is rather the tale of the unrelenting pursuit of a bad idea. ...Many, many philosophers and physicists have become convinced that a change of logic (and most dramatically, the rejection of classical logic) will somehow help in understanding quantum theory, or is somehow suggested or forced on us by quantum theory. But quantum logic, even through its many incarnations and variations, both in technical form and in interpretation, has never yielded the goods.\|Maudlin\|Maudlin\|2005}}\|Hilary Putnam\|pp. 184-185}}		\|Hilary Putnam\|pp. 184-185}}

	Quantum logic remains in limited use among logicians as an extremely pathological counterexample (Dalla Chiara and Giuntini: "Why quantum logics? Simply because 'quantum logics are there!'").{{sfn\|Dalla Chiara\|Giuntini\|2002}} Although the central insight to quantum logic remains mathematical folklore as an intuition pump for categorification, discussions rarely mention quantum logic.<ref>Terry Tao, "[https://terrytao.wordpress.com/2021/11/07/venn-and-euler-type-diagrams-for-vector-spaces-and-abelian-groups/ Venn and Euler type diagrams for vector spaces and abelian groups]" on ''What's New'' (blog), 2021.</ref>		Quantum logic remains in limited use among logicians as an extremely pathological counterexample (Dalla Chiara and Giuntini: "Why quantum logics? Simply because 'quantum logics are there!'").{{sfn\|Dalla Chiara\|Giuntini\|2002}} Although the central insight to quantum logic remains mathematical folklore as an intuition pump for categorification, discussions rarely mention quantum logic.<ref>Terry Tao, "[https://terrytao.wordpress.com/2021/11/07/venn-and-euler-type-diagrams-for-vector-spaces-and-abelian-groups/ Venn and Euler type diagrams for vector spaces and abelian groups]" on ''What's New'' (blog), 2021.</ref>
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	In the Hilbert space formulation of quantum mechanics as presented by von Neumann, a physical observable is represented by some (possibly unbounded) densely defined self-adjoint operator ''A'' on a Hilbert space ''H''. ''A'' has a spectral decomposition, which is a projection-valued measure E defined on the Borel subsets of '''R'''. In particular, for any bounded Borel function ''f'' on '''R''', the following extension of ''f'' to operators can be made: <math display=block> f(A) = \int_{\mathbb{R}} f(\lambda) \, d \operatorname{E}(\lambda).</math>		In the Hilbert space formulation of quantum mechanics as presented by von Neumann, a physical observable is represented by some (possibly unbounded) densely defined self-adjoint operator ''A'' on a Hilbert space ''H''. ''A'' has a spectral decomposition, which is a projection-valued measure E defined on the Borel subsets of '''R'''. In particular, for any bounded Borel function ''f'' on '''R''', the following extension of ''f'' to operators can be made: <math display=block> f(A) = \int_{\mathbb{R}} f(\lambda) \, d \operatorname{E}(\lambda).</math>

	In case ''f'' is the indicator function of an interval [''a'', ''b''], the operator ''f''(''A'') is a self-adjoint projection onto the subspace of generalized eigenvectors of ''A'' with eigenvalue in ~~{{closed-closed\|''a'',''b''}}~~. That subspace can be interpreted as the quantum analogue of the classical proposition		In case ''f'' is the indicator function of an interval [''a'', ''b''], the operator ''f''(''A'') is a self-adjoint projection onto the subspace of generalized eigenvectors of ''A'' with eigenvalue in . That subspace can be interpreted as the quantum analogue of the classical proposition
	* Measurement of ''A'' yields a value in the interval [''a'', ''b''].		* Measurement of ''A'' yields a value in the interval [''a'', ''b''].

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	=== Quantum probability measures ===		=== Quantum probability measures ===
	~~{{Main\|Gleason's theorem\|Physics:Quantum statistical mechanics}}~~
	A ''quantum probability measure'' is a function P defined on ''Q'' with values in [0,1] such that P("⊥)=0, P(⊤)=1 and if {''E''<sub>''i''</sub>}<sub>''i''</sub> is a sequence of pairwise-orthogonal elements of ''Q'' then		A ''quantum probability measure'' is a function P defined on ''Q'' with values in [0,1] such that P("⊥)=0, P(⊤)=1 and if {''E''<sub>''i''</sub>}<sub>''i''</sub> is a sequence of pairwise-orthogonal elements of ''Q'' then
	:<math> \operatorname{P}\!\left(\bigvee_{i=1}^\infty E_i\right) = \sum_{i=1}^\infty \operatorname{P}(E_i). </math>		:<math> \operatorname{P}\!\left(\bigvee_{i=1}^\infty E_i\right) = \sum_{i=1}^\infty \operatorname{P}(E_i). </math>
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	* {{wikicite\|Guido Bacciagaluppi, "[http://perso.ens-lyon.fr/jacques.jayez/Cours/Logique_Classique/Handbook_of_Quantum_Logic_2009.pdf Is Logic Empirical?]", in ''Handbook of Quantum Logic and Quantum Structures: Quantum Logic'', ed. K. Engesser, D. M. Gabbay, and D. Lehmann; Elsevier, 2009. pp. 49-78.\|ref={{harvid\|Bacciagaluppi\|2009}}}}		* {{wikicite\|Guido Bacciagaluppi, "[http://perso.ens-lyon.fr/jacques.jayez/Cours/Logique_Classique/Handbook_of_Quantum_Logic_2009.pdf Is Logic Empirical?]", in ''Handbook of Quantum Logic and Quantum Structures: Quantum Logic'', ed. K. Engesser, D. M. Gabbay, and D. Lehmann; Elsevier, 2009. pp. 49-78.\|ref={{harvid\|Bacciagaluppi\|2009}}}}
	* {{wikicite\|[[Biography:Tim Maudlin\|Tim Maudlin]], "The Tale of Quantum Logic" in ''[https://archive.org/details/HilaryPutnam-ContemporaryPhilosophyInFocus Hilary Putnam]''; {{wipe\|Cambridge University Press}} "Contemporary Philosophy in Focus" series, 2005. DOI: [http://dx.doi.org/10.1017/CBO9780511614187.006 10.1017/CBO9780511614187.006] {{ISBN\|9780521012546}}.\|ref={{harvid\|Maudlin\|2005}}}}		* {{wikicite\|[[Biography:Tim Maudlin\|Tim Maudlin]], "The Tale of Quantum Logic" in ''[https://archive.org/details/HilaryPutnam-ContemporaryPhilosophyInFocus Hilary Putnam]''; {{wipe\|Cambridge University Press}} "Contemporary Philosophy in Focus" series, 2005. DOI: [http://dx.doi.org/10.1017/CBO9780511614187.006 10.1017/CBO9780511614187.006] {{ISBN\|9780521012546}}.\|ref={{harvid\|Maudlin\|2005}}}}
	* ~~{{Cite IEP\|qu-logic\|Quantum Logic in Historical and Philosophical Perspective\|author-first1=C.\|author-last1=de Ronde\|author-first2=G.\|author-last2=Domenech\|author-first3=H.\|author-last3=Freytes}}~~		*
	* {{cite SEP \|url-id=qt-quantlog \|title=Quantum Logic and Probability Theory \|last=Wilce \|first=Alexander}}		* {{cite SEP \|url-id=qt-quantlog \|title=Quantum Logic and Probability Theory \|last=Wilce \|first=Alexander}}

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	* {{wikicite\|{{nlab\|id=quantum+logic\|title=Quantum Logic}}\|ref={{harvid\|nLab}}}}		* {{wikicite\|{{nlab\|id=quantum+logic\|title=Quantum Logic}}\|ref={{harvid\|nLab}}}}
	* {{wikicite\|C. Piron, ''Foundations of Quantum Physics'', W. A. Benjamin, 1976.\|ref={{harvid\|Piron\|1976}}}}		* {{wikicite\|C. Piron, ''Foundations of Quantum Physics'', W. A. Benjamin, 1976.\|ref={{harvid\|Piron\|1976}}}}
	~~{{emerging technologies\|quantum=yes\|other=yes}}~~

	{{DEFAULTSORT:Quantum Logic}}		{{DEFAULTSORT:Quantum Logic}}
	~~[[Category:Mathematical logic]]~~
	~~[[Category:Systems of formal logic]]~~
	~~[[Category:Non-classical logic]]~~
	~~[[Category:Quantum mechanics]]~~

	{{Sourceattribution\|Quantum logic}}		{{Sourceattribution\|Quantum logic}}

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 == See also ==
-{{cols|colwidth=26em}}
+{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}
 ==Notes==

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	<div style="width:300px;">		<div style="width:300px;">
	~~<!--~~ No ~~lead~~ image available ~~in existing page~~. ~~-->~~		[[File:File not found.png\|thumb\|280px\|No image available.]]
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 {{Short description|Theory of logic to account for observations from quantum theory}}
 {{Quantum mechanics|cTopic=[[Physics:Interpretation of quantum mechanics|Interpretations]]}}
 In the [[Mathematical logic|mathematical study of logic]] and the [[Physics:Physics|physical]] analysis of [[Physics:Quantum foundations|quantum foundations]], '''quantum logic''' is a set of rules for manip&shy;ulation of [[Proposition|proposition]]s inspired by the structure of [[Physics:Quantum mechanics|quantum theory]].  The formal system takes as its starting point an obs&shy;ervation of [[Biography:Garrett Birkhoff|Garrett Birkhoff]] and John von&nbsp;Neumann, that the structure of experimental tests in classical mechanics forms a [[Boolean algebra (structure)|Boolean algebra]], but the structure of experimental tests in quantum mechanics forms a much more complicated structure.
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 Mathematically, quantum logic is formulated by weakening the distributive law for a Boolean algebra, resulting in an ortho&shy;complemented lattice.   Quantum-mechanical [[Physics:Observable|observable]]s and [[Physics:Quantum state|states]] can be defined in terms of functions on or to the lattice, giving an alternate formalism for quantum computations.
 ==Introduction==

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	Inspired by [[Biography:Hans Reichenbach\|Hans Reichenbach]]'s recent defence of general relativity, the philosopher [[Biography:Hilary Putnam\|Hilary Putnam]] popularized Mackey's work in two papers in 1968 and 1975,{{sfn\|Maudlin\|2005}} in which he attributed the idea that anomalies associated to quantum measurements originate with a failure of logic itself to his coauthor, physicist [[Biography:David Finkelstein\|David Finkelstein]].{{sfn\|Putnam\|1969}} Putnam hoped to develop a possible alternative to hidden variables or wavefunction collapse in the problem of quantum measurement, but Gleason's theorem presents severe difficulties for this goal.{{sfn\|Maudlin\|2005}}{{sfn\|Wilce}} Later, Putnam retracted his views, albeit with much less fanfare,{{sfn\|Maudlin\|2005}} but the damage had been done. While Birkhoff and von Neumann's original work only attempted to organize the calculations associated with the Copenhagen interpretation of quantum mechanics, a school of researchers had now sprung up, either hoping that quantum logic would provide a viable hidden-variable theory, or obviate the need for one.<ref>T. A. Brody, "On Quantum Logic", ''Foundations of Physics'', vol. 14, no. 5, 1984. pp. 409-430.</ref> Their work proved fruitless, and now lies in poor repute.{{sfn\|Bacciagaluppi\|2009}}		Inspired by [[Biography:Hans Reichenbach\|Hans Reichenbach]]'s recent defence of general relativity, the philosopher [[Biography:Hilary Putnam\|Hilary Putnam]] popularized Mackey's work in two papers in 1968 and 1975,{{sfn\|Maudlin\|2005}} in which he attributed the idea that anomalies associated to quantum measurements originate with a failure of logic itself to his coauthor, physicist [[Biography:David Finkelstein\|David Finkelstein]].{{sfn\|Putnam\|1969}} Putnam hoped to develop a possible alternative to hidden variables or wavefunction collapse in the problem of quantum measurement, but Gleason's theorem presents severe difficulties for this goal.{{sfn\|Maudlin\|2005}}{{sfn\|Wilce}} Later, Putnam retracted his views, albeit with much less fanfare,{{sfn\|Maudlin\|2005}} but the damage had been done. While Birkhoff and von Neumann's original work only attempted to organize the calculations associated with the Copenhagen interpretation of quantum mechanics, a school of researchers had now sprung up, either hoping that quantum logic would provide a viable hidden-variable theory, or obviate the need for one.<ref>T. A. Brody, "On Quantum Logic", ''Foundations of Physics'', vol. 14, no. 5, 1984. pp. 409-430.</ref> Their work proved fruitless, and now lies in poor repute.{{sfn\|Bacciagaluppi\|2009}}

	Most philosophers find quantum logic an unappealing competitor to classical logic. It is far from evident (albeit true<ref>{{harvnb\|Dalla Chiara\|Giuntini\|2002\|p=94}}: "Quantum logics are, without any doubt, logics. As we have seen, they satisfy all the canonical conditions that the present community of logicians require in order to call a given abstract object a logic."</ref>) that quantum logic is a ''logic'', in the sense of describing a process of reasoning, as opposed to a particularly convenient language to summarize the measurements performed by quantum apparatuses.{{sfn\|Maudlin\|2005\|p=159-161}}{{sfn\|Brody\|1984}} In particular, modern philosophers of science argue that quantum logic attempts to substitute metaphysical difficulties for unsolved problems in physics, rather than properly solving the physics problems.{{sfn\|Brody\|1984\|pp=428-429}} [[Biography:Tim Maudlin\|Tim Maudlin]] writes that quantum "logic 'solves' the ~~[[Physics:Measurement problem\|~~[measurement] problem]] by making the problem impossible to state."{{sfn\|Maudlin\|2005\|p=174}}		Most philosophers find quantum logic an unappealing competitor to classical logic. It is far from evident (albeit true<ref>{{harvnb\|Dalla Chiara\|Giuntini\|2002\|p=94}}: "Quantum logics are, without any doubt, logics. As we have seen, they satisfy all the canonical conditions that the present community of logicians require in order to call a given abstract object a logic."</ref>) that quantum logic is a ''logic'', in the sense of describing a process of reasoning, as opposed to a particularly convenient language to summarize the measurements performed by quantum apparatuses.{{sfn\|Maudlin\|2005\|p=159-161}}{{sfn\|Brody\|1984}} In particular, modern philosophers of science argue that quantum logic attempts to substitute metaphysical difficulties for unsolved problems in physics, rather than properly solving the physics problems.{{sfn\|Brody\|1984\|pp=428-429}} [[Biography:Tim Maudlin\|Tim Maudlin]] writes that quantum "logic 'solves' the [measurement] problem by making the problem impossible to state."{{sfn\|Maudlin\|2005\|p=174}}

	\|Hilary Putnam\|pp. 184-185}}		\|Hilary Putnam\|pp. 184-185}}
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	== Relationship to other logics ==		== 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 ~~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}}		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 ''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>		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>

← Older revision		Revision as of 23:54, 23 May 2026
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	To illustrate why the distributive law fails, consider a particle moving on a line and (using some system of units where the reduced Planck's constant is 1) let<ref group="Note">Due to technical reasons, it is not possible to represent these propositions as quantum-mechanical operators. They are presented here because they are simple enough to enable intuition, and can be considered as limiting cases of operators that ''are'' feasible. See {{Slink\|\|Quantum logic as the logic of observables}} ''et seq.'' for details.</ref>		To illustrate why the distributive law fails, consider a particle moving on a line and (using some system of units where the reduced Planck's constant is 1) let<ref group="Note">Due to technical reasons, it is not possible to represent these propositions as quantum-mechanical operators. They are presented here because they are simple enough to enable intuition, and can be considered as limiting cases of operators that ''are'' feasible. See {{Slink\|\|Quantum logic as the logic of observables}} ''et seq.'' for details.</ref>
	: ''p'' = "the particle has momentum in the interval ~~{{closed-closed\|~~0, +{{frac\|1\|6}}}}"		: ''p'' = "the particle has momentum in the interval [0, +{{frac\|1\|6}}]"
	: ''q'' = "the particle is in the interval "		: ''q'' = "the particle is in the interval "
	: ''r'' = "the particle is in the interval "		: ''r'' = "the particle is in the interval "
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	</ref>		</ref>

	Inspired by [[Biography:Hans Reichenbach\|Hans Reichenbach]]'s recent defence of general relativity, the philosopher [[Biography:Hilary Putnam\|Hilary Putnam]] popularized Mackey's work in two papers in 1968 and 1975,{{sfn\|Maudlin\|2005}} in which he attributed the idea that anomalies associated to quantum measurements originate with a failure of logic itself to his coauthor, physicist [[Biography:David Finkelstein\|David Finkelstein]].{{sfn\|Putnam\|1969}} Putnam hoped to develop a possible alternative to hidden variables or wavefunction collapse in the problem of quantum measurement, but Gleason's theorem presents severe difficulties for this goal.{{sfn\|Maudlin\|2005}}{{sfn\|Wilce}} Later, Putnam retracted his views, albeit with much less fanfare,{{sfn\|Maudlin\|2005}} but the damage had been done. While Birkhoff and von Neumann's original work only attempted to organize the calculations associated with the Copenhagen interpretation of quantum mechanics, a school of researchers had now sprung up, either hoping that quantum logic would provide a viable hidden-variable theory, or obviate the need for one.<ref>~~{{wikicite\|~~T. A. Brody, "On Quantum Logic", ''Foundations of Physics'', vol. 14, no. 5, 1984. pp. 409-430.~~\|ref={{harvid\|Brody\|1984}}}}~~</ref> Their work proved fruitless, and now lies in poor repute.{{sfn\|Bacciagaluppi\|2009}}		Inspired by [[Biography:Hans Reichenbach\|Hans Reichenbach]]'s recent defence of general relativity, the philosopher [[Biography:Hilary Putnam\|Hilary Putnam]] popularized Mackey's work in two papers in 1968 and 1975,{{sfn\|Maudlin\|2005}} in which he attributed the idea that anomalies associated to quantum measurements originate with a failure of logic itself to his coauthor, physicist [[Biography:David Finkelstein\|David Finkelstein]].{{sfn\|Putnam\|1969}} Putnam hoped to develop a possible alternative to hidden variables or wavefunction collapse in the problem of quantum measurement, but Gleason's theorem presents severe difficulties for this goal.{{sfn\|Maudlin\|2005}}{{sfn\|Wilce}} Later, Putnam retracted his views, albeit with much less fanfare,{{sfn\|Maudlin\|2005}} but the damage had been done. While Birkhoff and von Neumann's original work only attempted to organize the calculations associated with the Copenhagen interpretation of quantum mechanics, a school of researchers had now sprung up, either hoping that quantum logic would provide a viable hidden-variable theory, or obviate the need for one.<ref>T. A. Brody, "On Quantum Logic", ''Foundations of Physics'', vol. 14, no. 5, 1984. pp. 409-430.</ref> Their work proved fruitless, and now lies in poor repute.{{sfn\|Bacciagaluppi\|2009}}

	Most philosophers find quantum logic an unappealing competitor to classical logic. It is far from evident (albeit true<ref>{{harvnb\|Dalla Chiara\|Giuntini\|2002\|p=94}}: "Quantum logics are, without any doubt, logics. As we have seen, they satisfy all the canonical conditions that the present community of logicians require in order to call a given abstract object a logic."</ref>) that quantum logic is a ''logic'', in the sense of describing a process of reasoning, as opposed to a particularly convenient language to summarize the measurements performed by quantum apparatuses.{{sfn\|Maudlin\|2005\|p=159-161}}{{sfn\|Brody\|1984}} In particular, modern philosophers of science argue that quantum logic attempts to substitute metaphysical difficulties for unsolved problems in physics, rather than properly solving the physics problems.{{sfn\|Brody\|1984\|pp=428-429}} [[Biography:Tim Maudlin\|Tim Maudlin]] writes that quantum "logic 'solves' the [[Physics:Measurement problem\|[measurement] problem]] by making the problem impossible to state."{{sfn\|Maudlin\|2005\|p=174}}		Most philosophers find quantum logic an unappealing competitor to classical logic. It is far from evident (albeit true<ref>{{harvnb\|Dalla Chiara\|Giuntini\|2002\|p=94}}: "Quantum logics are, without any doubt, logics. As we have seen, they satisfy all the canonical conditions that the present community of logicians require in order to call a given abstract object a logic."</ref>) that quantum logic is a ''logic'', in the sense of describing a process of reasoning, as opposed to a particularly convenient language to summarize the measurements performed by quantum apparatuses.{{sfn\|Maudlin\|2005\|p=159-161}}{{sfn\|Brody\|1984}} In particular, modern philosophers of science argue that quantum logic attempts to substitute metaphysical difficulties for unsolved problems in physics, rather than properly solving the physics problems.{{sfn\|Brody\|1984\|pp=428-429}} [[Biography:Tim Maudlin\|Tim Maudlin]] writes that quantum "logic 'solves' the [[Physics:Measurement problem\|[measurement] problem]] by making the problem impossible to state."{{sfn\|Maudlin\|2005\|p=174}}
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	===Historical works===		===Historical works===
	:''Organized chronologically''		:''Organized chronologically''
	* ~~{{wikicite\|~~J. von Neumann, ''Mathematical Foundations of Quantum Mechanics'', trans. Robert T. Beyer, ed. Nicholas A. Wheeler; Princeton University Press, 2018 (original 1932). pp. 160-164. [[Organization:JSTOR\|JSTOR]] [https://www.jstor.org/stable/j.ctt1wq8zhp j.ctt1wq8zhp]. [https://archive.org/details/mathematicalfoun0000vonn 1955 edition] available at the [[Internet Archive]].\|ref={{harvid\|von Neumann\|1932}}}}		* J. von Neumann, ''Mathematical Foundations of Quantum Mechanics'', trans. Robert T. Beyer, ed. Nicholas A. Wheeler; Princeton University Press, 2018 (original 1932). pp. 160-164. [[Organization:JSTOR
	* ~~{{wikicite\|~~[[Biography:Garrett Birkhoff\|G. Birkhoff]] and [[Biography:John von Neumann\|J. von Neumann]], "[http://www.fulviofrisone.com/attachments/article/451/the%20logic%20of%20quantum%20mechanics%201936.pdf The Logic of Quantum Mechanics]," ''Annals of Mathematics'', series II, vol. 37, issue 4, pp. 823–843, 1936. JSTOR [https://www.jstor.org/stable/pdf/1968621.pdf 1968621]. DOI [http://dx.doi.org/10.2307/1968621 10.2307/1968621]. \|ref={{harvid\|Birkhoff\|von Neumann\|1936}}}}		* [[Biography:Garrett Birkhoff
	* ~~{{wikicite\|~~[[Biography:George Mackey\|G. Mackey]], ''[https://archive.org/details/mathematicalfoun0000unse_t7f9 Mathematical Foundations of Quantum Mechanics]'', W. A. Benjamin, 1963. HathiTrust [https://hdl.handle.net/2027/mdp.39015001329567 2027/mdp.39015001329567].\|ref={{harvid\|Mackey\|1963}}}}		* [[Biography:George Mackey
	* ~~{{wikicite\|~~[[Biography:Hilary Putnam~~\|H. Putnam]], ''Is Logic Empirical?'', Boston Studies in the Philosophy of Science V, ed. Robert S. Cohen and Marx W. Wartofsky, 1969.\|ref={{harvid\|Putnam\|1969}}}}~~		* [[Biography:Hilary Putnam
	* ~~{{wikicite\|~~G. Kalmbach ''Orthomodular Logic'', Z. Logik und Grundl. Math., vol. 20, 1974, pp. 395-406.~~\|ref={{harvid\|Kalmbach\|1974}}}}~~		* G. Kalmbach ''Orthomodular Logic'', Z. Logik und Grundl. Math., vol. 20, 1974, pp. 395-406.
	* ~~{{wikicite\|~~G. Kalmbach ''Orthomodular Logic as a Hilbert Type Calculus'', in Current Issues in Quantum Logic, Plenum Press, New York, ed. E. Beltrametti et al., 1981, pp. 333-340~~\|ref={{harvid\|Kalmbach\|1981}} }}~~ <!--{{harvid\|Ka2}}}}-->		* G. Kalmbach ''Orthomodular Logic as a Hilbert Type Calculus'', in Current Issues in Quantum Logic, Plenum Press, New York, ed. E. Beltrametti et al., 1981, pp. 333-340 <!--{{harvid\|Ka2}}}}-->
	* ~~{{wikicite\|~~G. Kalmbach ''Orthomodular Lattices'', Academic Press, London, 1983~~\|ref={{harvid\|Kalmbach\|1983}}}}~~		* G. Kalmbach ''Orthomodular Lattices'', Academic Press, London, 1983

	===Modern philosophical perspectives===		===Modern philosophical perspectives===
	* ~~{{wikicite\|~~Guido Bacciagaluppi, "[http://perso.ens-lyon.fr/jacques.jayez/Cours/Logique_Classique/Handbook_of_Quantum_Logic_2009.pdf Is Logic Empirical?]", in ''Handbook of Quantum Logic and Quantum Structures: Quantum Logic'', ed. K. Engesser, D. M. Gabbay, and D. Lehmann; Elsevier, 2009. pp. 49-78.~~\|ref={{harvid\|Bacciagaluppi\|2009}}}}~~		* Guido Bacciagaluppi, "[http://perso.ens-lyon.fr/jacques.jayez/Cours/Logique_Classique/Handbook_of_Quantum_Logic_2009.pdf Is Logic Empirical?]", in ''Handbook of Quantum Logic and Quantum Structures: Quantum Logic'', ed. K. Engesser, D. M. Gabbay, and D. Lehmann; Elsevier, 2009. pp. 49-78.
	* ~~{{wikicite\|~~[[Biography:Tim Maudlin\|Tim Maudlin]], "The Tale of Quantum Logic" in ''[https://archive.org/details/HilaryPutnam-ContemporaryPhilosophyInFocus Hilary Putnam]''; {{wipe\|Cambridge University Press}} "Contemporary Philosophy in Focus" series, 2005. DOI: [http://dx.doi.org/10.1017/CBO9780511614187.006 10.1017/CBO9780511614187.006] {{ISBN\|9780521012546}}.\|ref={{harvid\|Maudlin\|2005}}}}		* [[Biography:Tim Maudlin
	*		*
	* {{cite SEP \|url-id=qt-quantlog \|title=Quantum Logic and Probability Theory \|last=Wilce \|first=Alexander}}		* {{cite SEP \|url-id=qt-quantlog \|title=Quantum Logic and Probability Theory \|last=Wilce \|first=Alexander}}

	===Mathematical study and computational applications===		===Mathematical study and computational applications===
	* ~~{{wikicite\|~~A. Baltag and S. Smets, "[https://www.cambridge.org/core/services/aop-cambridge-core/content/view/0AF9C4FCB2681EECC692859F031C80C2/S0960129506005299a.pdf/lqp_the_dynamic_logic_of_quantum_information.pdf LQP: The Dynamic Logic of Quantum Information]", ''Mathematical Structures in Computer Science'', vol. 16, issue 3, pp. 491-525, 2006. DOI [https://dx.doi.org/10.1017/S0960129506005299 10.1017/S0960129506005299] arXiv [https://arxiv.org/abs/2110.01361 2110.01361]~~\|ref={{harvid\|Baltag\|Smets\|2006}}}}~~		* A. Baltag and S. Smets, "[https://www.cambridge.org/core/services/aop-cambridge-core/content/view/0AF9C4FCB2681EECC692859F031C80C2/S0960129506005299a.pdf/lqp_the_dynamic_logic_of_quantum_information.pdf LQP: The Dynamic Logic of Quantum Information]", ''Mathematical Structures in Computer Science'', vol. 16, issue 3, pp. 491-525, 2006. DOI [https://dx.doi.org/10.1017/S0960129506005299 10.1017/S0960129506005299] arXiv [https://arxiv.org/abs/2110.01361 2110.01361]
	* ~~{{wikicite\|~~A. Baltag, J. Bergfeld, K. Kishida, J. Sack, S. Smets and S. Zhong, "[https://link.springer.com/article/10.1007/s10773-013-1987-3 PLQP & Company: Decidable Logics for Quantum Algorithms]", ''International Journal of Theoretical Physics'', vol. 53, issue 10, pp. 3628-3647, 2014.~~\|ref={{harvid\|Baltag\|Bergfeld\|Kishida\|Sack\|2014}}}}~~		* A. Baltag, J. Bergfeld, K. Kishida, J. Sack, S. Smets and S. Zhong, "[https://link.springer.com/article/10.1007/s10773-013-1987-3 PLQP & Company: Decidable Logics for Quantum Algorithms]", ''International Journal of Theoretical Physics'', vol. 53, issue 10, pp. 3628-3647, 2014.
	* ~~{{wikicite\|~~[[Biography:Maria Luisa Dalla Chiara\|M. L. Dalla Chiara]] and R. Giuntini, "[https://core.ac.uk/download/pdf/25303784.pdf Quantum Logics]", in ''Handbook of Philosophical Logic'', vol. 6, D. Gabbay and F. Guenthner (eds.), Kluwer, 2002. arXiv [https://arxiv.org/abs/quant-ph/0101028 quant-ph/0101028]\|ref={{harvid\|Dalla Chiara\|Giuntini\|2002}}}}		* [[Biography:Maria Luisa Dalla Chiara
	* ~~{{wikicite\|~~M. L. Dalla Chiara, R. Giuntini, and R. Leporini, "[https://link.springer.com/chapter/10.1007/978-94-017-3598-8_9 Quantum Computational Logics: A Survey]", in ''Trends in Logic'', vol. 21, V. F. Hendricks and J. Malinowski (eds.), Springer, 2003. arXiv [https://arxiv.org/abs/quant-ph/0305029 quant-ph/0305029]~~\|ref={{harvid\|Dalla Chiara\|Giuntini\|Leporini\|2003}}}}~~		* M. L. Dalla Chiara, R. Giuntini, and R. Leporini, "[https://link.springer.com/chapter/10.1007/978-94-017-3598-8_9 Quantum Computational Logics: A Survey]", in ''Trends in Logic'', vol. 21, V. F. Hendricks and J. Malinowski (eds.), Springer, 2003. arXiv [https://arxiv.org/abs/quant-ph/0305029 quant-ph/0305029]
	* ~~{{wikicite\|~~Norman Megill, [http://us.metamath.org/qleuni/mmql.html Quantum Logic Explorer] at [[Software:Metamath~~\|Metamath]], 2019.\|ref={{harvid\|Megill\|2019}}}}~~		* Norman Megill, [http://us.metamath.org/qleuni/mmql.html Quantum Logic Explorer] at [[Software:Metamath
	* N. Papanikolaou, "[http://wrap.warwick.ac.uk/61398/7/WRAP_cs-rr-416.pdf Reasoning Formally About Quantum Systems: An Overview]", ''ACM SIGACT News'', 36(3), 2005. pp. 51–66. arXiv [https://arxiv.org/abs/cs/0508005 cs/0508005].		* N. Papanikolaou, "[http://wrap.warwick.ac.uk/61398/7/WRAP_cs-rr-416.pdf Reasoning Formally About Quantum Systems: An Overview]", ''ACM SIGACT News'', 36(3), 2005. pp. 51–66. arXiv [https://arxiv.org/abs/cs/0508005 cs/0508005].

	===Quantum foundations===		===Quantum foundations===
	* D. Cohen, ''An Introduction to Hilbert Space and Quantum Logic'', Springer-Verlag, 1989. Elementary and well-illustrated; suitable for advanced undergraduates.		* D. Cohen, ''An Introduction to Hilbert Space and Quantum Logic'', Springer-Verlag, 1989. Elementary and well-illustrated; suitable for advanced undergraduates.
	* ~~{{wikicite\|~~Günther Ludwig, ''Der Grundlagen der Quantenmechanik'' (in German), Springer, 1954. The definitive work. Released in English as: <ul><li>Günther Ludwig, ''[https://archive.org/details/foundations-of-quantum-mechanics-i-g.-ludwig-c.-a.-hein Foundations of Quantum Mechanics]'', vol. 1, trans. Carl A. Hein; Springer-Verlag, 1983.</li><li>Günther Ludwig, ''An Axiomatic Basis for Quantum Mechanics'', vol. 1: "Derivation of Hilbert Space Structure", trans. Leo F. Boron, ed. Karl Just; Springer, 1985. DOI: [http://dx.doi.org/10.1007/978-3-642-70029-3 10.1007/978-3-642-70029-3]. {{ISBN\|978-3-642-70029-3}}.</li></ul>~~\|ref={{harvid\|Ludwig\|1954}}}}~~		* Günther Ludwig, ''Der Grundlagen der Quantenmechanik'' (in German), Springer, 1954. The definitive work. Released in English as: <ul><li>Günther Ludwig, ''[https://archive.org/details/foundations-of-quantum-mechanics-i-g.-ludwig-c.-a.-hein Foundations of Quantum Mechanics]'', vol. 1, trans. Carl A. Hein; Springer-Verlag, 1983.</li><li>Günther Ludwig, ''An Axiomatic Basis for Quantum Mechanics'', vol. 1: "Derivation of Hilbert Space Structure", trans. Leo F. Boron, ed. Karl Just; Springer, 1985. DOI: [http://dx.doi.org/10.1007/978-3-642-70029-3 10.1007/978-3-642-70029-3]. {{ISBN\|978-3-642-70029-3}}.</li></ul>
	* ~~{{wikicite\|~~{{nlab\|id=quantum+logic\|title=Quantum Logic~~}}\|ref={{harvid\|nLab}}~~}}		* {{nlab\|id=quantum+logic\|title=Quantum Logic}}
	* ~~{{wikicite\|~~C. Piron, ''Foundations of Quantum Physics'', W. A. Benjamin, 1976.~~\|ref={{harvid\|Piron\|1976}}}}~~		* C. Piron, ''Foundations of Quantum Physics'', W. A. Benjamin, 1976.
	{{DEFAULTSORT:Quantum Logic}}		{{DEFAULTSORT:Quantum Logic}}

	{{Sourceattribution\|Quantum logic}}		{{Sourceattribution\|Quantum logic}}