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arXiv:0811.0539 · International Journal of Theoretical Physics

Embedding Quantum Mechanics Into a Broader Noncontextual Theory: A Conciliatory Result

Claudio Garola, Sandro Sozzo

Published Dec 22, 2009• 27 citations
arXivHTMLPDF
DOI
10.1007/s10773-009-0222-8

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Most scholars concerned with the foundations of quantum mechanics (QM) think that contextuality and nonlocality (hence nonobjectivity of physical properties) are unavoidable features of QM which follow from the mathematical apparatus of QM. Moreover these features are usually considered as basic in quantum information processing. Nevertheless they raise still unsolved problems, as the objectification problem in the quantum theory of measurement. The extended semantic realism (ESR) model offers a possible way out from these difficulties by embedding the mathematical formalism of QM into a broader mathematical formalism and reinterpreting quantum probabilities as conditional on detection rather than absolute. The embedding allows to recover the formal apparatus of QM within the ESR model, and the reinterpretation of QM allows to construct a noncontextual hidden variables theory which justifies the assumptions introduced in the ESR model and proves its objectivity. According to the ESR model both linear and nonlinear time evolution occur, depending on the physical environment, as in QM. In addition, the ESR model, though objective, implies modified Bell's inequalities that do not conflict with QM, supplies different mathematical representations of proper and improper mixtures, provides a general framework in which the local interpretations of the GHZ experiment obtained by other authors are recovered and explained, and supports an interpretation of quantum logic which avoids the introduction of the problematic notion of quantum truth.
We show that the noncontextual inequality proposed by Klyachko et al. [Phys. Rev. Lett. 101, 020403 (2008)] belongs to a broader family of inequalities, one associated to each compatibility structure of a set of events (a graph), and its independence number. These have the surprising property that the maximum quantum violation is given by the Lovasz theta-function of the graph, which was originally proposed as an upper bound on its Shannon capacity. Furthermore, probabilistic theories beyond quantum mechanics may have an even larger violation, which is given by the so-called fractional packing number. We discuss in detail, and compare, the sets of probability distributions attainable by noncontextual, quantum, and generalized models; the latter two are shown to have semidefinite and linear characterizations, respectively. The implications for Bell inequalities, which are examples of noncontextual inequalities, are discussed. In particular, we show that every Bell inequality can be recast as a noncontextual inequality a la Klyachko et al.
Elaborating on a previous work by Simon et al (2000 Phys. Rev. Lett. 85 1783) we propose a realizable quantum optical single-photon experiment using standard present day technology, capable of discriminating maximally between the predictions of quantum mechanics (QM) and noncontextual hidden variable theories (NCHV). Quantum mechanics predicts a gross violation (up to a factor of 2) of the noncontextual Bell-like inequality associated with the proposed experiment. An actual maximal violation of this inequality would demonstrate (modulo fair sampling) an all-or-nothing type contradiction between QM and NCHV.