Editing Quantum logic (section)

== 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>