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Mathematics for industry

Measurements of Negative Joint Probabilities in Optical Quantum System

M. Iinuma, Yutaro Suzuki, Holger Friedrich Hofmann

Published Jul 31, 2015• 0 citations
Publisher
DOI
10.1007/978-4-431-55429-5_4

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Quantum paradoxes show that quantum statistics can exceed the limits of positive joint probabilities for physical properties that cannot be measured jointly. It is therefore impossible to describe the relations between the different physical properties of a quantum system by assigning joint realities to their observable values. Instead, recent experimental results obtained by weak measurements suggest that nonclassical correlations could be expressed by complex valued quasiprobabilities, where the phases of the complex probabilities express the action of transformations between the noncommuting properties [H. F. Hofmann, New J. Phys. 13, 103009 (2011)]. In these relations, negative probabilities necessarily emerge whenever the physical properties involved are related to each other by half-periodic transformations, since such transformations are characterized by action phases of $\ensuremath{\pi}$ in their complex probabilities. It is therefore possible to trace the failure of realist assumptions back to a fundamental and universally valid relation between statistics and dynamics that associates half-periodic transformations with negative probabilities.
General relativity required the abandonment of Euclidean geometry. Here we show that quantum theory requires the abandonment of classical logic. We show that the Hilbert space representation of quantum theory is logically inevitable. There are no fundamental principles of quantum theory. We find an inevitable generalization of classical logic. The conjunction can take negative values, and reduces to the classical conjunction for order invariant measurements. The expectation of the conjunction is a generalized joint probability that can take negative values. A commutative conjunction leads to the Hilbert space formalism of quantum theory. Quantum theory applies both to microscopic and macroscopic systems. Classicality is represented by non-negativity of the generalized joint probability. We illustrate this by applying the logic to explain puzzling results in quantum cognition.
Quantum mechanics violates Leggett–Garg inequalities because the operator formalism predicts correlations between different spin components that would correspond to negative joint probabilities for the outcomes of joint measurements. However, the uncertainty principle ensures that such joint measurements cannot be implemented without errors. In a sequential measurement of the spin components, the resolution and back-action errors of the intermediate measurement can be described by random spin flips acting on an intrinsic joint probability. If the error rates are known, the intrinsic joint probability can be reconstructed from the noisy statistics of the actual measurement outcomes. In this paper, we use the spin-flip model of measurement errors to analyze experimental data on photon polarization obtained with an interferometric setup that allows us to vary the measurement strength and hence the balance between resolution and back-action errors. We confirm that the intrinsic joint probability obtained from the experimental data is independent of measurement strength, and show that the same violation of the Leggett–Garg inequality can be obtained for any combination of measurement resolution and back-action.