TY - JOUR
T1 - Categorical Equivalence Between Orthomodular Dynamic Algebras and Complete Orthomodular Lattices
AU - Kishida, Kohei
AU - Rafiee Rad, Soroush
AU - Sack, Joshua
AU - Zhong, Shengyang
N1 - Funding Information:
We would like to thank Professor Roberto Giuntini for his suggestion that we consider orthomodular lattices rather than just Hilbert lattices in the equivalence. Kishida’s research has been supported by the grants FA9550-12-1-0136 of the U.S. AFOSR and EP/N018745/1 of EPSRC. Rafiee Rad’s research is funded by the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013)/ERC Grant agreement no. 283963. Zhong’s research is supported by NSSFC Grant 14ZDB015.
Publisher Copyright:
© 2017, Springer Science+Business Media New York.
PY - 2017/12/1
Y1 - 2017/12/1
N2 - This paper provides a categorical equivalence between two types of quantum structures. One is a complete orthomodular lattice, which is used for reasoning about testable properties of a quantum system. The other is an orthomodular dynamic algebra, which is a quantale used for reasoning about quantum actions. The result extends to more restrictive lattices than orthomodular lattices, and includes Hilbert lattices of closed subspaces of a Hilbert space. These other lattice structures have connections to a wide range of different quantum structures; hence our equivalence establishes a categorical connection between quantales and a great variety of quantum structures.
AB - This paper provides a categorical equivalence between two types of quantum structures. One is a complete orthomodular lattice, which is used for reasoning about testable properties of a quantum system. The other is an orthomodular dynamic algebra, which is a quantale used for reasoning about quantum actions. The result extends to more restrictive lattices than orthomodular lattices, and includes Hilbert lattices of closed subspaces of a Hilbert space. These other lattice structures have connections to a wide range of different quantum structures; hence our equivalence establishes a categorical connection between quantales and a great variety of quantum structures.
KW - Complete orthomodular lattice
KW - Orthomodular dynamic algebra
KW - Quantale
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U2 - 10.1007/s10773-017-3433-4
DO - 10.1007/s10773-017-3433-4
M3 - Article
AN - SCOPUS:85023180362
SN - 0020-7748
VL - 56
SP - 4060
EP - 4072
JO - International Journal of Theoretical Physics
JF - International Journal of Theoretical Physics
IS - 12
ER -