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arXiv:0805.3917 · Journal of Functional Analysis

Covariant quantum instruments

Claudio Carmeli, Teiko Heinosaari, Alessandro Toigo

Published Sep 16, 2009• 10 citations
arXivHTMLPDF
DOI
10.1016/j.jfa.2009.08.013

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We study completely positive (CP) [Formula: see text]-sesquilinear-form-valued maps on a unital [Formula: see text]-algebra [Formula: see text], where the sesquilinear forms operate on a module over a [Formula: see text]-algebra [Formula: see text]. We also study the cases when either one or both of the algebras are von Neumann algebras. Moreover, we assume that the CP maps are covariant with respect to actions of a symmetry group. This allows us to view these maps as generalizations of covariant quantum instruments. We determine minimal covariant dilations (KSGNS constructions) for covariant CP maps to find necessary and sufficient conditions for a CP map to be extreme in convex subsets of normalized covariant CP maps. As a special case, we study covariant quantum observables and instruments whose value space is a transitive space of a unimodular type-I group. Finally, we discuss the case of instruments that are covariant with respect to a square-integrable representation.
A convenient representation for Radon–Nikodym derivatives of completely positive (c.p.) instruments on B(H) with respect to a scalar measure is suggested, similar to the Stinespring–Kraus representation for c.p. maps, but involving possibly nonclosable unbounded operators. The structure of covariant c.p. instruments is studied in detail. In particular, an exhaustive description is given to instruments covariant with respect to shifts or rotations, corresponding to “position” or “angle” measurements.
In this thesis, I reflect on quantum instruments that measure the state of pure finite dimensional quantum systems. As the Heisenberg principle dictates, there exists a joint restriction to the information gain and distortion by measurement of a quantum system. I minimize the distortion of measurement and maximize the quality of the outcome by finding the optimal (covariant) instrument. In order to optimize this joint restriction, the family of covariant instruments is classified. This is an important side result.