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A structure for quantum measurements. (English) Zbl 1098.81014
We are told at the outset that effects are 2-valued measures, “yes/no” experiments or ideally precise measurement outcomes, represented in Hilbert space by projection operators. An effect algebra is generated by a partial binary “orthosum” relation over effects, where \(a\oplus b\) represents “a parallel measurement of \(a\) and \(b\)” (p. 249). A sequential effect algebra has in addition a binary “sequential sum” relation such that \(a\circ b\) represents “a sequential measurement first of \(a\) then of \(b\)”. The author seeks to extend these definitions to “unsharp” measurements “with a finite or even infinite number of real values” (p. 250), represented in Hilbert space by the positive operators bounded above by 1.
In section 2 generalized definitions introduce effect algebras as structures \(({\mathcal E},0,1,\oplus)\) where \(0\), \(1\) are zero and unit elements, and \(\oplus\) is an orthosum. Each effect \(a\) in \({\mathcal E}\) has a unique complement \(a'\). Effects are now “quantum events that may be imprecise or fuzzy” (p. 251). A partial ordering is defined by \(a\leq b\) if there is a \(c\) in \({\mathcal E}\) such that \(a\oplus c= b\), and “sharp” effects are now those where \(a\vee a'= 0\). The \(\sigma\)-effect algebras are effect algebras “where \(a_1\leq a_2\leq\cdots\) implies that the \(\bigvee a_i\) exists.” (Ibid) Sequential effect algebras (SEA) have an binary operator o defined by conditions that for example allow interval \([0,1]\), a subset of reals, to be a \(\sigma\)-SEA with \(a\circ b= ab\) (p. 253).
Measurements are introduced in section 3 as “normalized, effect valued measures on the \(\sigma\)-effect algebra \({\mathcal B}(\mathbb{R})\)” of Borel subsets of the reals (p. 253). So a measurement \(X\) is a morphism \(X: {\mathcal B}(\mathbb{R})\to{\mathcal E}\). The author presents a series of results about the structure \({\mathcal M}({\mathcal E})\), the set of measurements on \({\mathcal E}\). Theorem 3.2 shows \({\mathcal M}({\mathcal E})\) is a “generalized effect algebra” (p. 256) and Theorem 3.4 that it is a “generalized \(\sigma\)-effect algebra” (p. 258). The notions of “globally sharp” and “locally sharp” measurements are introduced, and it is shown that an initial interval of measurements is isomorphic to \({\mathcal E}\) (p. 259). Divisor effect algebras are also discussed. In section 4 “finite measurements” are defined and shown to correspond to orthosums of \((1,0)\)-valued mappings. Lastly in section 5, sequential effect algebras are considered with discussion of conditional measurements and conditional expectations. Various definitions for sequential products are discussed and compared.
The writing in this paper is clear. There is no wider discussion of the generalizations, but the definitions express the required properties of quantum measurements in Hilbert space and in this sense the author provides what he sought, “a mathematical structure for measurements” (p. 250).

81P15 Quantum measurement theory, state operations, state preparations
81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
81P05 General and philosophical questions in quantum theory
Full Text: DOI
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