Gudder, Stanley Contexts in quantum measurement theory. (English) Zbl 1431.81022 Found. Phys. 49, No. 6, 647-662 (2019). Summary: State transformations in quantum mechanics are described by completely positive maps which are constructed from quantum channels. We call a finest sharp quantum channel a context. The result of a measurement depends on the context under which it is performed. Each context provides a viewpoint of the quantum system being measured. This gives only a partial picture of the system which may be distorted and in order to obtain a total accurate picture, various contexts need to be employed. We first discuss some basic definitions and results concerning quantum channels. We briefly describe the relationship between this work and ontological models that form the basis for contextuality studies. We then consider properties of channels and contexts. For example, we show that the set of sharp channels can be given a natural partial order in which contexts are the smallest elements. We also study properties of channel maps. The last section considers mutually unbiased contexts. These are related to mutually unbiased bases which have a large current literature. Finally, we connect them to completely random channel maps. Cited in 4 Documents MSC: 81P15 Quantum measurement theory, state operations, state preparations 46L07 Operator spaces and completely bounded maps 81P16 Quantum state spaces, operational and probabilistic concepts 47B65 Positive linear operators and order-bounded operators 81P47 Quantum channels, fidelity 81P13 Contextuality in quantum theory Keywords:contexts; quantum measurements; observables PDFBibTeX XMLCite \textit{S. Gudder}, Found. Phys. 49, No. 6, 647--662 (2019; Zbl 1431.81022) Full Text: DOI arXiv References: [1] Bandyopadhyay, S., Boykin, P., Roychowdhury, V., Vatan, F.: A new proof for the existence of mutually unbiased bases, vol. 3. arXiv:quant-ph/0103162 (2001) · Zbl 1012.68069 [2] Busch, P., Cassinelli, G., Lahti, P.: Probability structures for quantum state spaces. Rev. Math. Phys. 7, 1105-1121 (1995) · Zbl 0839.46071 [3] Busch, P., Lahti, P., Mittlestaedt, P.: The Quantum Theory of Measurement. Springer, Berlin (1996) [4] Busch, P., Grabowski, M., Lahti, P.: Operational Quantum Physics. Springer, Berlin (1997) · Zbl 0863.60106 [5] Carmeli, C., Heinonen, T., Toigo, A.: Intrinsic unsharpness and approximate repeatability of quantum measurements. J. Phys. A 40, 1303-1323 (2007) · Zbl 1109.81012 [6] Choi, M.: Completely positive linear maps on complex matrices. Linear Algebra Appl. 10, 285-290 (1975) · Zbl 0327.15018 [7] Durt, T., Englert, B.-G., Bengtsson, I., Zyczkowski, K.: On mutually unbiased bases. Int. J. Quant. Inf. 8, 535-640 (2010) · Zbl 1208.81052 [8] Gudder, S.: Fuzzy probability theory. Demonstr. Math. 31, 235-254 (1998) · Zbl 0952.60002 [9] Gudder, S.: Sharp and unsharp quantum effects. Adv. Appl. Math. 20, 169-187 (1998) · Zbl 0913.46062 [10] Gudder, S.: Convex and sequential effect algebras. arXiv:1802.01265 [phys-gen] (2018) · Zbl 1486.81007 [11] Gudder, S.: Contexts in convex and sequential effect algebras (to appear) · Zbl 1486.81007 [12] Heinosaari, T., Ziman, M.: The Mathematical Language of Quantum Theory. Cambridge University Press, Cambridge (2012) · Zbl 1243.81008 [13] Kraus, K.: States, effects and operations: fundamental notions of quantum theory. In: Lecture Notes in Physics, vol. 190, Springer, Berlin (1983) · Zbl 0545.46049 [14] Lillystone, P., Wallman, J., Emerson, J.: Contextuality and the single-qubit stabilizer subtheory, vol. 1. arXiv:1802.06121 [quant-phys] (2018) [15] Nielson, M., Chuang, I.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000) [16] Wootters, W., Fields, B.: Optimal state-determination by mutually unbiased measurements. Ann. Phys. 191, 363-381 (1989) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.