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{{Short description|Theory of logic to account for observations from quantum theory}}
{{Quantum mechanics|cTopic=[[Interpretation of quantum mechanics|Interpretations]]}}

In the [[Mathematical logic|mathematical study of logic]] and the [[Physics|physical]] analysis of [[quantum foundations]], '''quantum logic''' is a set of rules for manip&shy;ulation of [[proposition]]s inspired by the structure of [[Quantum mechanics|quantum theory]].  The formal system takes as its starting point an obs&shy;ervation of [[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.

A number of other logics have also been proposed to analyze quantum-mechanical phenomena, unfortunately also under the name of "quantum logic(s)".  They are not the subject of this article.  For discussion of the similarities and differences between quantum logic and some of these competitors, see ''{{slink||Relationship to other logics}}''.

Quantum logic has been proposed as the correct logic for propositional inference generally, most notably by the philosopher [[Hilary Putnam]], at least at one point in his career.  This thesis was an important ingredient in Putnam's 1968 paper "[[Is Logic Empirical?]]" in which he analysed the [[epistemology|epistemological]] status of the rules of propositional logic.  Modern philosophers reject quantum logic as a basis for reasoning, because it lacks a [[material conditional]]; a common alternative is the system of [[linear logic]], of which quantum logic is a fragment.

Mathematically, quantum logic is formulated by weakening the [[distributive law]] for a Boolean algebra, resulting in an [[Orthocomplement|ortho&shy;complemented lattice]].   Quantum-mechanical [[observable]]s and [[Quantum state|states]] can be defined in terms of functions on or to the lattice, giving an alternate [[Formalism (mathematics)|formalism]] for quantum computations.

== Introduction ==
The most notable difference between quantum logic and [[classical logic]] is the failure of the [[propositional logic|propositional]] [[distributive law]]:<ref>Peter Forrest, "Quantum logic" in ''[[Routledge Encyclopedia of Philosophy]]'', vol.&nbsp;7, 1998. p.&nbsp;882ff: "[Quantum logic] differs from the standard sentential calculus....The most notable difference is that the distributive laws fail, being replaced by a weaker law known as orthomodularity."</ref>	
: ''p'' and (''q'' or ''r'') = (''p'' and ''q'') or (''p'' and ''r''),
where the symbols ''p'', ''q'' and ''r'' are propositional variables.

To illustrate why the distributive law fails, consider a particle moving on a line and (using some system of units where the [[reduced Planck constant]] is 1) let<ref group="Note">Due to technical reasons, it is not possible to represent these propositions as [[Operator (quantum mechanics)|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}}}}"	
: ''q'' = "the particle is in the interval {{closed-closed|−1, 1}}"
: ''r'' = "the particle is in the interval {{closed-closed|1, 3}}"
We might observe that:
: ''p'' and (''q'' or ''r'') = ''true''	
in other words, that the state of the particle is a weighted [[quantum superposition|superposition]] of momenta between 0 and +1/6 and positions between −1 and +3.

On the other hand, the propositions "''p'' and ''q''" and "''p'' and ''r''" each assert tighter restrictions on simultaneous values of position and momentum than are allowed by the [[uncertainty principle]] (they each have uncertainty 1/3, which is less than the allowed minimum of 1/2). So there are no states that can support either proposition, and
: (''p'' and ''q'') or (''p'' and ''r'') = ''false''

== History and modern criticism ==
In his classic 1932 treatise ''[[Mathematical Foundations of Quantum Mechanics]]'', [[John von&nbsp;Neumann]] noted that [[projection (mathematics)|projection]]s on a [[Hilbert space]] can be viewed as propositions about physical observables; that is, as potential ''yes-or-no questions'' an observer might ask about the state of a physical system, questions that could be settled by some measurement.{{sfn|von&nbsp;Neumann|1932}}  Principles for manipulating these quantum propositions were then called ''quantum logic'' by von&nbsp;Neumann and Birkhoff in a 1936 paper.{{sfn|Birkhoff|von&nbsp;Neumann|1936}}

[[George Mackey]], in his 1963 book (also called ''Mathematical Foundations of Quantum Mechanics''), attempted to axiomatize quantum logic as the structure of an [[orthocomplemented lattice|ortho&shy;complemented lattice]], and recognized that a physical observable could be ''defined'' in terms of quantum propositions.  Although Mackey's presentation still assumed that the ortho&shy;complemented lattice is the [[Lattice (order)|lattice]] of [[closed set|closed]] [[linear subspace]]s of a [[separable space|separable]] Hilbert space,{{sfn|Mackey|1963}} [[Constantin Piron]], Günther Ludwig and others later developed axiomatizations that do not assume an underlying Hilbert space.<ref>Piron:
* C. Piron, "Axiomatique quantique" (in French), ''Helvetica Physica Acta'' vol.&nbsp;37, 1964.  DOI:&nbsp;[http://doi.org/10.5169/seals-113494 10.5169/seals-113494].
* {{harvnb|Piron|1976}}.
Ludwig:
* Günther Ludwig, "[https://link.springer.com/content/pdf/10.1007/BF01653647.pdf Attempt of an Axiomatic Foundation of Quantum Mechanics and More General Theories]"&nbsp;II, ''Commun. Math. Phys.'', vol.&nbsp;4, 1967.  pp.&nbsp;331-348.  
* {{harvnb|Ludwig|1954}}
</ref>

Inspired by [[Hans Reichenbach]]'s then-recent defence of [[general relativity]], the philosopher [[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 [[David Finkelstein]].{{sfn|Putnam|1969}}  Putnam hoped to develop a possible alternative to [[Hidden-variable theory|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&nbsp;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.&nbsp;A. Brody, "On Quantum Logic", ''Foundations of Physics'', vol.&nbsp;14, no.&nbsp;5, 1984.  pp.&nbsp;409-430.|ref={{harvid|Brody|1984}}}}</ref>  Their work proved fruitless, and now lies in poor repute.{{sfn|Bacciagaluppi|2009}}

Most philosophers would agree that quantum logic is not a competitor to [[classical logic]]. It is far from evident (albeit true<ref>{{harvnb|Dalla&nbsp;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, some modern [[philosopher of science|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}}  [[Tim Maudlin]] writes that quantum "logic 'solves' the [[measurement problem|[measurement] problem]] by making the problem impossible to state."{{sfn|Maudlin|2005|p=174}}

Quantum logic remains in use among logicians {{sfn|Dalla&nbsp;Chiara|Giuntini|2002}} and interests are expanding through the recent development of [[quantum computing]], which has engendered a proliferation of new logics for formal analysis of quantum protocols and algorithms (see also {{slink||Relationship to other logics}}).{{sfn|Dalla&nbsp;Chiara|Giuntini|Leporini|2003}}  The logic may also find application in (computational) linguistics.

== Algebraic structure ==
Quantum logic can be axiomatized as the theory of propositions modulo the following identities:{{sfn|Megill|2019}}
* ''a'' {{=}} &not;&not;''a''
* &or; is [[commutative]] and [[associative]].
* There is a maximal element ⊤, and ⊤ {{=}} ''b''&or;&not;''b'' for any ''b''.
* ''a''&or;&not;(&not;''a''&or;''b'') {{=}} ''a''.

("&not;" is the traditional notation for "[[negation (logic)|not]]", "&or;" the notation for "[[or (logic)|or]]", and "&and;" the notation for "[[and (logic)|and]]".)

Some authors restrict to [[orthomodular lattice]]s, which additionally satisfy the orthomodular law:<ref>{{harvnb|Kalmbach|1974}} and {{harvnb|Kalmbach|1983}}</ref> 
* If ⊤ {{=}} &not;(&not;''a''&or;&not;''b'')&or;&not;(''a''&or;''b'') then ''a'' {{=}} ''b''.

("⊤" is the traditional notation for [[truth]] and ""⊥" the traditional notation for [[Deception|falsity]].)

Alternative formulations include propositions derivable via a [[natural deduction]],{{sfn|Dalla&nbsp;Chiara|Giuntini|2002}} [[sequent calculus]]<ref>{{cite journal | jstor  = 44084050 | author1=N.J. Cutland |author2= P.F. Gibbins | title=A regular sequent calculus for Quantum Logic in which ∨ and ∧ are dual | journal=Logique et Analyse |series=Nouvelle Série | volume=25 | number=99 | pages=221–248 | date=Sep 1982 }}</ref><ref>
* {{cite journal | author=Hirokazu Nishimura | title=Proof theory for minimal quantum logic I | journal=International Journal of Theoretical Physics | volume=33 | number=1 | pages=103–113 | date=Jan 1994 |bibcode = 1994IJTP...33..103N |doi = 10.1007/BF00671616 | s2cid=123183879 |ref=none}}
* {{cite journal | author=Hirokazu Nishimura | title=Proof theory for minimal quantum logic II | journal=International Journal of Theoretical Physics | volume=33 | number=7 | pages=1427–1443 | date=Jul 1994 | doi=10.1007/bf00670687| bibcode=1994IJTP...33.1427N | s2cid=189850106 |ref=none}}</ref> or [[method of analytic tableaux|tableaux]] system.<ref>{{cite conference|url=http://www.kr.tuwien.ac.at/staff/tompits/papers/tableaux-99.pdf |author1=Uwe Egly |author2=Hans Tompits |title=Gentzen-like Methods in Quantum Logic |conference=8th Int. Conf. on Automated Reasoning with Analytic Tableaux and Related Methods (TABLEAUX) |publisher=[[SUNY Albany]] | year=1999 |citeseerx=10.1.1.88.9045 }}</ref>  Despite the relatively developed [[proof&nbsp;theory]], quantum logic is not known to be [[decidability (logic)|decidable]].{{sfn|Megill|2019}}

== Quantum logic as the logic of observables ==
The remainder of this article assumes the reader is familiar with the [[spectral theory]] of [[self-adjoint operator]]s on a Hilbert space. However, the main ideas can be under&shy;stood in the [[Dimension (vector space)|finite-dimensional]] case.

=== Logic of classical mechanics ===
The [[Hamiltonian mechanics|Hamiltonian]] formulations of [[classical mechanics]] have three ingredients: [[Classical mechanics|states]], [[observable]]s and [[Dynamics (mechanics)|dynamics]].  In the simplest case of a single particle moving in '''R'''<sup>3</sup>, the state space is the position–momentum space '''R'''<sup>6</sup>. An observable is some [[real-valued function]] ''f'' on the state space.  Examples of observables are position, momentum or energy of a particle.   For  classical systems, the value ''f''(''x''), that is the value of ''f'' for some particular system state ''x'', is obtained by a process of measurement of ''f''.

The [[proposition]]s concerning a classical system are generated from basic statements of the form
: "Measurement of ''f'' yields a value in the interval [''a'', ''b''] for some real numbers ''a'', ''b''."

through the conventional arithmetic operations and [[Limit (mathematics)|pointwise limits]].  It follows easily from this characterization of propositions in classical systems that the corresponding logic is identical to the [[Boolean algebra (structure)|Boolean algebra]] of [[Borel subset]]s of the state space.  They thus obey the laws of [[Classical logic|classical]] [[propositional logic]] (such as [[de&nbsp;Morgan's laws]]) with the set operations of union and intersection corresponding to the [[Boolean operator (Boolean algebra)|Boolean conjunctives]] and subset inclusion corresponding to [[Material implication (rule of inference)|material implication]].

In fact, a stronger claim is true: they must obey the [[infinitary logic]] {{Math|''L''<sub>&omega;<sub>1</sub>,&omega;</sub>}}.

We summarize these remarks as follows: The proposition system of a classical system is a lattice with a distinguished ''orthocomplementation'' operation:  The lattice operations of ''meet'' and ''join'' are respectively set intersection and set union.  The orthocomplementation  operation is set complement.  Moreover, this lattice is ''sequentially complete'', in the sense that any sequence {''E''<sub>''i''</sub>}<sub>''i''∈'''N'''</sub> of elements of the lattice has a [[least upper bound]], specifically the set-theoretic union: <math display="block"> \operatorname{LUB}(\{E_i\}) = \bigcup_{i=1}^\infty E_i\text{.} </math>

=== Propositional lattice of a quantum mechanical system ===
In the [[Hilbert space]] formulation of quantum mechanics as presented by von&nbsp;Neumann, a physical observable is represented by some (possibly [[bounded operator|unbounded]]) densely defined [[self-adjoint operator]] ''A'' on a Hilbert space ''H''.  ''A'' has a [[Spectral theorem|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 eigenvector]]s of ''A'' with eigenvalue in {{closed-closed|''a'',''b''}}.  That subspace can be interpreted as the quantum analogue of the classical proposition
* Measurement of ''A'' yields a value in the interval [''a'', ''b''].

This suggests the following quantum mechanical replacement for the orthocomplemented lattice of propositions in classical mechanics, essentially Mackey's ''Axiom VII'':
* The propositions of a quantum mechanical system correspond to the lattice of closed subspaces of ''H''; the negation of a proposition ''V'' is the orthogonal complement ''V''<sup>⊥</sup>.

The space ''Q'' of quantum propositions is also sequentially complete: any pairwise-disjoint sequence {''V''<sub>''i''</sub>}<sub>''i''</sub> of elements of ''Q'' has a least upper bound.  Here disjointness of ''W''<sub>1</sub> and ''W''<sub>2</sub> means ''W''<sub>2</sub> is a subspace of ''W''<sub>1</sub><sup>⊥</sup>. The least upper bound of {''V''<sub>''i''</sub>}<sub>''i''</sub> is the closed internal [[direct sum]].

=== Standard semantics ===
The standard semantics of quantum logic is that quantum logic is the logic of [[projection operator]]s in a [[separable space|separable]] [[Hilbert space|Hilbert]] or [[pre-Hilbert space]], where an observable ''p'' is associated with the [[eigenspace|set of quantum states]] for which ''p'' (when measured) has [[eigenvalue]] 1.  From there,
* ''¬p'' is the [[orthogonal complement]] of ''p'' (since for those states, the probability of observing ''p'', P(''p'') = 0),
* ''p''∧''q'' is the intersection of ''p'' and ''q'', and
* ''p''∨''q'' = ¬(¬''p''∧¬''q'') refers to states that [[quantum superposition|superpose]] ''p'' and ''q''.

This semantics has the nice property that the pre-Hilbert space is complete (i.e., Hilbert) if and only if the propositions satisfy the orthomodular law, a result known as the [[Solèr theorem]].<ref>{{harvnb|Dalla Chiara|Giuntini|2002}} and {{harvnb|de&nbsp;Ronde|Domenech|Freytes}}.  Despite suggestions otherwise in Josef Jauch, ''Foundations of Quantum Mechanics'', Addison-Wesley Series in Advanced Physics; Addison-Wesley, 1968, this property cannot be used to deduce a vector space structure, because it is not peculiar to (pre-)Hilbert spaces.  An analogous claim holds in most [[Category (math)|categories]]; see John Harding, "[https://www.ams.org/journals/tran/1996-348-05/S0002-9947-96-01548-6/S0002-9947-96-01548-6.pdf Decompositions in Quantum Logic]," ''Transactions of the AMS'', vol.&nbsp;348, no.&nbsp;5, 1996.  pp.&nbsp;1839-1862.</ref> Although much of the development of quantum logic has been motivated by the standard semantics, it is not characterized by the latter; there are additional properties satisfied by that lattice that need not hold in quantum logic.{{sfn|Dalla&nbsp;Chiara|Giuntini|2002}}

== Differences with classical logic ==
The structure of ''Q'' immediately points to a difference with the partial order structure of a classical proposition system.  In the classical case, given a proposition ''p'', the equations
: ⊤ = ''p''&or;''q'' and
: ⊥ = ''p''&and;''q''
have exactly one solution, namely the set-theoretic complement of ''p''.  In the case of the lattice of projections there are infinitely many solutions to the above equations (any closed, algebraic complement of ''p'' solves it; it need not be the orthocomplement).

More generally, [[Valuation (logic)|propositional valuation]] has unusual properties in quantum logic.  An orthocomplemented lattice admitting a [[Total function|total]] [[Lattice Homomorphism|lattice homomorphism]] to {{mset|⊥,⊤}} must be Boolean.  A standard workaround is to study maximal partial homomorphisms ''q'' with a filtering property:
: if ''a''≤''b'' and ''q''(''a'') = ⊤, then ''q''(''b'') = ⊤.{{sfn|Bacciagaluppi|2009}}

=== Failure of distributivity ===
Expressions in quantum logic describe observables using a syntax that resembles classical logic.  However, unlike classical logic, the distributive law ''a'' ∧ (''b'' ∨ ''c'') = (''a'' ∧ ''b'') ∨ (''a'' ∧ ''c'') fails when dealing with [[Observable#Incompatibility of observables in quantum mechanics|noncommuting observables]], such as position and momentum.  This occurs because measurement affects the system, and measurement of whether a disjunction holds does not measure which of the disjuncts is true.

For example, consider a simple one-dimensional particle with position denoted by ''x'' and momentum by ''p'', and define observables:
* ''a'' — |''p''| ≤ 1 (in some units)
* ''b'' — x ≤ 0
* ''c'' — x ≥ 0

Now, position and momentum are Fourier transforms of each other, and the [[Fourier transform]] of a [[square-integrable]] nonzero function with a [[compact support]] is [[Entire function|entire]] and hence does not have non-isolated zeroes.  Therefore, there is no wave function that is both [[Normalizable wave function|normalizable]] in momentum space and vanishes on precisely ''x'' ≥ 0.  Thus, ''a'' ∧ ''b'' and similarly ''a'' ∧ ''c'' are false, so (''a'' ∧ ''b'') ∨ (''a'' ∧ ''c'') is false.  However, ''a'' ∧ (''b'' ∨ ''c'') equals ''a'', which is certainly not false (there are states for which it is a viable [[quantum measurement|measurement outcome]]).  Moreover: if the relevant Hilbert space for the particle's dynamics only admits momenta no greater than 1, then ''a'' is true.

To understand more, let ''p''<sub>1</sub> and ''p''<sub>2</sub> be the momentum functions (Fourier transforms) for the projections of the particle wave function to ''x'' ≤ 0 and ''x'' ≥ 0 respectively.  Let |''p''<sub>i</sub>|↾<sub>≥1</sub> be the restriction of ''p''<sub>i</sub> to momenta that are (in absolute value) ≥1.

(''a'' ∧ ''b'') ∨ (''a'' ∧ ''c'') corresponds to states with |''p''<sub>1</sub>|↾<sub>≥1</sub> = |''p''<sub>2</sub>|↾<sub>≥1</sub> = 0 (this holds even if we defined ''p'' differently so as to make such states possible; also, ''a'' ∧ ''b'' corresponds to |''p''<sub>1</sub>|↾<sub>≥1</sub>=0 and ''p''<sub>2</sub>=0). Meanwhile, ''a'' corresponds to states with |''p''|↾<sub>≥1</sub> = 0. As an operator, ''p'' = ''p''<sub>1</sub> + ''p''<sub>2</sub>, and nonzero |''p''<sub>1</sub>|↾<sub>≥1</sub> and |''p''<sub>2</sub>|↾<sub>≥1</sub> might interfere to produce zero |''p''|↾<sub>≥1</sub>.  Such interference is key to the richness of quantum logic and quantum mechanics.

== Relationship to quantum measurement ==

=== Mackey observables ===
Given a [[Orthocomplement|orthocomplemented lattice]] ''Q'', a Mackey observable φ is a [[Countably additive measure|countably additive homomorphism]] from the orthocomplemented lattice of Borel subsets of '''R''' to ''Q''.  In symbols, this means that for any sequence {''S''<sub>''i''</sub>}<sub>''i''</sub> of pairwise-disjoint Borel subsets of '''R''', {φ(''S''<sub>''i''</sub>)}<sub>''i''</sub> are pairwise-orthogonal propositions (elements of ''Q'') and
: <math> \varphi\left(\bigcup_{i=1}^\infty S_i\right) = \sum_{i=1}^\infty \varphi(S_i). </math>

Equivalently, a Mackey observable is a [[projection-valued measure]] on '''R'''.

'''Theorem''' ([[Spectral theorem]]). If ''Q'' is the lattice of closed subspaces of Hilbert ''H'', then there is a bijective correspondence between Mackey observables and densely-defined self-adjoint operators on ''H''.

=== Quantum probability measures ===
{{Main|Gleason's theorem|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
: <math> \operatorname{P}\!\left(\bigvee_{i=1}^\infty E_i\right) = \sum_{i=1}^\infty \operatorname{P}(E_i). </math>

Every quantum probability measure on the closed subspaces of a Hilbert space is induced by a [[density matrix]]&nbsp;&mdash; a [[Positive operator|nonnegative operator]] of [[trace (linear algebra)#Generalizations|trace]] 1.  Formally,
: '''Theorem'''.<ref>[[Andrew Gleason|A. Gleason]], "Measures on the Closed Subspaces of a Hilbert Space", ''Indiana University Mathematics Journal'', vol.&nbsp;6, no.&nbsp;4, 1957.  pp.&nbsp;885-893.  DOI:&nbsp;[http://dx.doi.org/10.1512/iumj.1957.6.56050 10.1512/iumj.1957.6.56050].  Reprinted in ''The Logico-Algebraic Approach to Quantum Mechanics'', University of Western Ontario Series in Philosophy of Science 5a, ed.&nbsp;C.&nbsp;A. Hooker; D. Riedel, c.&nbsp;1975-1979.  pp.&nbsp;123-133.</ref> Suppose ''Q'' is the lattice of closed subspaces of a separable Hilbert space of complex dimension at least 3. Then for any quantum probability measure ''P'' on ''Q'' there exists a unique [[trace class]] operator ''S'' such that <math display=block>\operatorname{P}(E) = \operatorname{Tr}(S E)</math> for any self-adjoint projection ''E'' in ''Q''.

== 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 ''Work&shy;shop on Physics and Computation (PhysComp '92)'' proceedings.  See also the dis&shy;cuss&shy;ion 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.&nbsp;15 issue 1-2: pp.&nbsp;11–25, 1975.</ref> and the [[modal logic]] ''B''.{{sfn|Dalla&nbsp;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.&nbsp;Robert&nbsp;S. Cohen and Marx&nbsp;W. Wartofsky; D.&nbsp;Riedel, 1974.  pp.&nbsp;92-122.  DOI:&nbsp;[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>

== Limitations ==
Although many treatments of quantum logic assume that the underlying lattice must be orthomodular, such logics cannot handle multiple interacting quantum systems.  In an example due to Foulis and Randall, there are orthomodular propositions with finite-dimensional Hilbert models whose pairing admits no orthomodular model.{{sfn|Wilce}}  Likewise, quantum logic with the orthomodular law falsifies the [[deduction theorem]].{{sfn|Kalmbach|1981}}

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&nbsp;T. Ivanov, Prakash Panangaden, Lutz Straß&shy;burger, "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.&nbsp;90-107.  DOI:&nbsp;[http://dx.doi.org/10.1007/978-3-642-54789-8_6 10.1007/978-3-642-54789-8_6].  HAL&nbsp;[https://hal.inria.fr/hal-01092279/ 01092279].</ref>

== See also ==
{{cols|colwidth=26em}}
* [[Fuzzy logic]]
* [[HPO formalism]] (An approach to temporal quantum logic)
* [[Linear logic]]
* [[Mathematical formulation of quantum mechanics]]
* [[Multi-valued logic]]
* [[Quantum Bayesianism]]
* [[Quantum cognition]]
* [[Quantum contextuality]]
* [[Quantum field theory]]
* [[Quantum probability]]
* [[Quasi-set theory]]
* [[Solèr's theorem]]
* [[Vector logic]]
{{colend}}

== Notes ==
{{reflist|group=Note}}

=== Citations ===
{{reflist|2}}

== Sources ==
=== Historical works ===
: ''Arranged chronologically''
* {{wikicite|J. von&nbsp;Neumann, ''Mathematical Foundations of Quantum Mechanics'', trans.&nbsp;Robert&nbsp;T. Beyer, ed.&nbsp;Nicholas&nbsp;A. Wheeler; Princeton University Press, 2018 (original&nbsp;1932).  pp.&nbsp;160-164.  [[JSTOR]]&nbsp;[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}}}}
* {{wikicite|[[Garrett Birkhoff|G. Birkhoff]] and [[John von Neumann|J. von&nbsp;Neumann]], "[http://www.fulviofrisone.com/attachments/article/451/the%20logic%20of%20quantum%20mechanics%201936.pdf The Logic of Quantum Mechanics]," ''Annals of Mathematics'', series&nbsp;II, vol.&nbsp;37, issue&nbsp;4, pp.&nbsp;823–843, 1936.  JSTOR&nbsp;[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}}}}
* {{wikicite|[[George Mackey|G. Mackey]], ''[https://archive.org/details/mathematicalfoun0000unse_t7f9 Mathematical Foundations of Quantum Mechanics]'', W.&nbsp;A.&nbsp;Benjamin, 1963.  HathiTrust&nbsp;[https://hdl.handle.net/2027/mdp.39015001329567 2027/mdp.39015001329567].|ref={{harvid|Mackey|1963}}}}
* {{wikicite|[[Hilary Putnam|H. Putnam]], ''Is Logic Empirical?'', Boston Studies in the Philosophy of Science V, ed.&nbsp;Robert&nbsp;S. Cohen and Marx&nbsp;W. Wartofsky, 1969.|ref={{harvid|Putnam|1969}}}}
* {{wikicite|[[Gudrun Kalmbach|G. Kalmbach]] ''Orthomodular Logic'', Z. Logik und Grundl. Math., vol. 20, 1974, pp.&nbsp;395-406.|ref={{harvid|Kalmbach|1974}}}}
* {{wikicite|[[Gudrun Kalmbach|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.&nbsp;333-340|ref={{harvid|Kalmbach|1981}} }} <!--{{harvid|Ka2}}}}-->
* {{wikicite|[[Gudrun Kalmbach|G. Kalmbach]] ''Orthomodular Lattices'', Academic Press, London, 1983|ref={{harvid|Kalmbach|1983}}}}

=== 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.&nbsp;K. Engesser, D.&nbsp;M. Gabbay, and D. Lehmann; Elsevier, 2009.  pp.&nbsp;49-78.|ref={{harvid|Bacciagaluppi|2009}}}}
* {{wikicite|[[Tim Maudlin]], "The Tale of Quantum Logic" in ''[https://archive.org/details/HilaryPutnam-ContemporaryPhilosophyInFocus Hilary Putnam]''; [[Cambridge University Press]] "Contemporary Philosophy in Focus" series, 2005.  DOI:&nbsp;[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&nbsp;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}}

=== 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.&nbsp;16, issue&nbsp;3, pp.&nbsp;491-525, 2006.  DOI&nbsp;[https://dx.doi.org/10.1017/S0960129506005299 10.1017/S0960129506005299] arXiv&nbsp;[https://arxiv.org/abs/2110.01361 2110.01361]|ref={{harvid|Baltag|Smets|2006}}}}
* {{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.&nbsp;53, issue&nbsp;10, pp.&nbsp;3628-3647, 2014.|ref={{harvid|Baltag|Bergfeld|Kishida|Sack|2014}}}}
* {{wikicite|[[Maria Luisa Dalla Chiara|M.&nbsp;L. Dalla&nbsp;Chiara]] and R.&nbsp;Giuntini, "[https://core.ac.uk/download/pdf/25303784.pdf Quantum Logics]", in ''Handbook of Philosophical Logic'', vol.&nbsp;6, D.&nbsp;Gabbay and F.&nbsp;Guenthner (eds.), Kluwer, 2002.  arXiv&nbsp;[https://arxiv.org/abs/quant-ph/0101028 quant-ph/0101028]|ref={{harvid|Dalla Chiara|Giuntini|2002}}}}
* {{wikicite|M.&nbsp;L. Dalla&nbsp;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.&nbsp;21, V.&nbsp;F. Hendricks and J. Malinowski (eds.), Springer, 2003.  arXiv&nbsp;[https://arxiv.org/abs/quant-ph/0305029 quant-ph/0305029]|ref={{harvid|Dalla Chiara|Giuntini|Leporini|2003}}}}
* {{wikicite|Norman Megill, [http://us.metamath.org/qleuni/mmql.html Quantum Logic Explorer] at [[Metamath]], 2019.|ref={{harvid|Megill|2019}}}}
* 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.&nbsp;51–66.  arXiv [https://arxiv.org/abs/cs/0508005 cs/0508005].

=== Quantum foundations ===
* 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.&nbsp;1, trans.&nbsp;Carl&nbsp;A. Hein; Springer-Verlag, 1983.</li><li>Günther Ludwig, ''An Axiomatic Basis for Quantum Mechanics'', vol.&nbsp;1: "Derivation of Hilbert Space Structure", trans.&nbsp;Leo&nbsp;F. Boron, ed.&nbsp;Karl Just; Springer, 1985.  DOI:&nbsp;[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}}}}  
* {{wikicite|{{nlab|id=quantum+logic|title=Quantum Logic}}|ref={{harvid|nLab}}}}
* {{wikicite|[[Constantin Piron|C. Piron]], ''Foundations of Quantum Physics'',  W.&nbsp;A.&nbsp;Benjamin, 1976.|ref={{harvid|Piron|1976}}}}

{{wiktionary|quantum logic}}
{{Quantum mechanics topics}}
{{emerging technologies|quantum=yes|other=yes}}

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