Logical independence in quantum logic |
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Authors: | Miklós Rédei |
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Affiliation: | (1) Faculty of Natural Sciences, Loránd Eötvös University, Rákóczi út 5., H-1088 Budapest, Hungary |
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Abstract: | The projection latticesP( 1),P( 2) of two von Neumann subalgebras 1, 2 of the von Neumann algebra are defined to be logically independent if A B 0 for any 0 A P( 1), 0 BP( 2). After motivating this notion in independence, it is shown thatP( 1),P( 2) are logically independent if 1 is a subfactor in a finite factor andP( 1),P( 2 commute. Also, logical independence is related to the statistical independence conditions called C*-independence W*-independence, and strict locality. Logical independence ofP( 1,P( 2 turns out to be equivalent to the C*-independence of ( 1, 2) for mutually commuting 1, 2 and it is shown that if ( 1, 2) is a pair of (not necessarily commuting) von Neumann subalgebras, thenP( 1,P( 2 are logically independent in the following cases: is a finite-dimensional full-matrix algebra and 1, 2 are C*-independent; ( 1, 2) is a W*-independent pair; 1, 2 have the property of strict locality. |
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