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Springer eBooks

The quantum mechanical Hilbert space formalism and the quantum mechanical probability space of the outcomes of measurements

Mioara Mugur-Schächter

Published Apr 6, 2008• 2 citations
Publisher
DOI
10.1007/3-540-06725-6_23

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Quantum mechanics describes seemingly paradoxical relations between the outcomes of measurements that cannot be performed jointly. In Hilbert space, the outcomes of such incompatible measurements are represented by nonorthogonal states. In this paper we investigate how the relation between outcomes represented by nonorthogonal quantum states differs from the relations suggested by a joint assignment of measurement outcomes that do not depend on the actual measurement context. The analysis is based on a well-known scenario where three statements about the impossibilities of certain outcomes would seem to make a specific fourth outcome impossible as well, yet quantum theory allows the observation of that outcome with a nonvanishing probability. We show that the Hilbert space formalism modifies the relation between the four measurement outcomes by defining a lower bound of the fourth probability that increases as the total probability of the first three outcomes drops to zero. Quantum theory thus not only makes the violation of noncontextual consistency between the measurement outcomes possible, but actually requires it as a necessary consequence of the Hilbert space inner products that describe the contextual relation between the outcomes of different measurements.
Recently, [{arXiv:0810.3134}] is accepted and published. We present ultimate version of no-hidden-variables theorem. We derive a proposition concerning the quantum theory under the existence of the Bloch sphere in a single spin-1/2 system. The existence of a single classical probability space for measurement outcome within the formalism of von Neumann's projective measurement does not coexist with the proposition concerning the quantum theory. We have to give up the existence of such a classical probability space for measurement outcome in the two-dimensional Hilbert space formalism of the quantum theory. The quantum theory does not accept a hidden-variable interpretation in the two-dimensional space.
Beginning with an intuitive consideration of sequences of measurements, we define a time-ordered event space representing the collection of all imaginable outcomes for measurement sequences. We then postulate the generalized distributive relation on the event space and examine the class of measurements for which this relation can be experimentally validated. The generalized distributive relation is shown to lead to a σ-additive conditional probability on the event space and to a predictive and retrodictive formalism for stochastic processes. We then show that this formalism has a predictive and a retrodictive representation in a separable Hilbert space H, which has no counterpart in unitary quantum dynamics.