Abstract
In this chapter we review the basic elements and structures of Hilbert space quantum mechanics. We build on the idea of a statistical duality arising from the analysis of an experiment as a preparation-measurement-registration scheme, as sketched in Sect. 1.2. The description of a physical system \(\mathcal {S}\) is thus based on the notions of states as equivalence classes of preparations, observables as equivalence classes of measurements, and on the probability measures for the possible measurement outcomes.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
The notion of superposition of states goes back to Dirac [2]. Ever since, this notion has been considered as one of the basic principles of quantum mechanics. The superposition of states is generally formulated in terms of the linear combination of state vectors, or wave functions. Here we adopt a slightly more abstract point of view, formulating it in terms of the pure states, i.e., one-dimensional projections. This perspective is needed in the axiomatic context of Chap. 23.
- 2.
The need to form statistical mixtures of (pure) states was equally evident in the early stages of quantum mechanics. The explicit formula \(\varrho =\sum _nw_nP[\varphi _n]\) for a statistical mixture of pure states \(P[\varphi _n]\) with the weights \(w_n\) (Gemisch von Zuständen) was already developed and investigated in detail by von Neumann [3].
- 3.
As discussed in Sect. 1.2, this assumption is built into the very idea of a mixed state.
- 4.
This argument is essentially the same as that used by von Neumann in [1] to show that the observables of a quantum system are to be represented as hypermaximal symmetric operators (which are now called selfadjoint operators). The sole difference between von Neumann’s and the present approach is that here we do not adopt his assumption that the measurement outcome statistics of an observable are to be represented in terms of a single (selfadjoint) operator.
References
von Neumann, J.: Wahrscheinlichkeitstheoretischer Aufbau der Quantenmechanik. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse 1927, 245–272 (1927)
Dirac, P.: The Principles of Quantum Mechanics, 4th edn. Oxford, Clarendon Press (1981)
von Neumann, J.: Mathematische Grundlagen der Quantenmechanik. Die Grundlehren der mathematischen Wissenschaften, Band 38. Springer-Verlag, Berlin (1968). (Reprint of the 1932 original)
Hughston, L.P., Jozsa, R., Wootters, W.K.: A complete classification of quantum ensembles having a given density matrix. Phys. Lett. A 183(1), 14–18 (1993)
Hadjisavvas, N.: Properties of mixtures on non-orthogonal states. Lett. Math. Phys. 5, 327–332 (1981). July
Cassinelli, G., De Vito, E., Levrero, A.: On the decompositions of a quantum state. J. Math. Anal. Appl. 210(2), 472–483 (1997)
Davies, E.B.: Quantum Theory of Open Systems. Academic Press, New York (1976)
Lahti, P., Mączyński, M.: Partial order of quantum effects. J. Math. Phys. 36(4), 1673–1680 (1995)
Gleason, A.M.: Measures on the closed subspaces of a Hilbert space. J. Math. Mech. 6, 885–893 (1957)
R. S. Ingarden. Information theory and thermodynamics of light. I. Foundations of information theory. Fortschr. Physik, 12, 567–594 (1964)
Holevo, A.S.: Statistical Structure of Quantum Theory. Lecture Notes in Physics, vol. 67. Monographs. Springer, Berlin (2001)
Heinosaari, T., Pellonpää, J.-P.: Generalized coherent states and extremal positive operator valued measures. J. Phys. A 45(24), 244019 (2012)
Cassinelli, G., De Vito, E., Lahti, P., Levrero, A.: The Theory of Symmetry Actions in Quantum Mechanics—With an application to the Galilei group. Lecture Notes in Physics, vol. 654. Springer-Verlag, Berlin (2004)
Molnár, L.: Selected Preserver Problems on Algebraic Structures of Linear Operators and on Function Spaces, Lecture Notes in Mathematics, vol. 1895. Springer, Heidelberg (2007)
Varadarajan, V.S.: Geometry of Quantum Theory, 2nd edn. Springer-Verlag, New York (1985)
Mackey, G.W.: Unitary Group Representations in Physics, Probability, and Number Theory, 2nd edn. Advanced Book Classics. Addison-Wesley Publishing Company Advanced Book Program, Redwood City (1989)
Mackey, G.W.: Axiomatics of particle interactions. In Proceedings of the International Quantum Structures Association, Part I (Castiglioncello, 1992), vol. 32, pp 1643–1659 (1993)
Cassinelli, G., Zanghì, N.: Conditional probabilities in quantum mechanics. I. Conditioning with respect to a single event. Nuovo Cim. B (11), 73(2), 237–245 (1983)
Cassinelli, G., Zanghì, N.: Conditional probabilities in quantum mechanics. II. Additive conditional probabilities. Nuovo Cim. B (11), 79(2), 141–154 (1984)
Aerts, D., Daubechies, I.: Physical justification for using the tensor product to describe two quantum systems as one joint system. Helv. Phys. Acta, 51(5-6), 661–675 (1979), (1978)
Pulmannová, S.: Tensor products of Hilbert space effect algebras. Rep. Math. Phys. 53(2), 301–316 (2004)
Werner, R.: Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model. Phys. Rev. A 40, 4277–4281 (1989)
Peres, A.: Separability criterion for density matrices. Phys. Rev. Lett. 77, 1413–1415 (1996)
Cirel\(^{\prime }\)son, B.S.: Quantum generalizations of Bell’s inequality. Lett. Math. Phys. 4(2), 93–100 (1980)
Khalfin, L.A., Tsirelson, B.S.: Quantum and quasiclassical analogs of Bell inequalities. In: Symposium on the Foundations of Modern Physics (Joensuu, 1985), pp. 441–460. World Scientific Publishing, Singapore (1985)
Scholz, V.B., Werner, R.F.: Tsirelson’s problem (2008). arXiv:0812.4305
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Busch, P., Lahti, P., Pellonpää, JP., Ylinen, K. (2016). States, Effects and Observables. In: Quantum Measurement. Theoretical and Mathematical Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-43389-9_9
Download citation
DOI: https://doi.org/10.1007/978-3-319-43389-9_9
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-43387-5
Online ISBN: 978-3-319-43389-9
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)