Papers › Probability vector representation of the Schrödinger equation and Leggett-Garg-type experiments
Probability vector representation of the Schrödinger equation and Leggett-Garg-type experiments
Masahiro Hotta, Sebastian Murk
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Leggett-Garg inequalities place bounds on the temporal correlations of a system based on the principles of macroscopic realism per se and noninvasive measurability. Their conventional formulation relies on the ensemble-averaged products of observables measured at different instants of time. However, a complete description that enables a precise understanding and captures all physically relevant features requires the study of probability distributions associated with noncommuting observables. In this article, we propose a scheme to describe the dynamics of generic N-level quantum systems ("qudits") via a probability vector representation of the Schr\"odinger equation and define a precise notion of no-signaling in time (NSIT) for the probability distributions of noncommuting observables. This provides a systematic way of identifying the interferences responsible for nonclassical behavior. In addition, we introduce an interference witness measure to quantify violations of NSIT for arbitrary general probabilistic states. For single-qubit systems, we pinpoint the pivotal relation that establishes a connection between the disturbance of observables incurred during a measurement and the resulting NSIT violation. For large-N systems where a manual determination is infeasible, the classification of states as either NSIT-conforming or NSIT-violating may be performed by a machine learning algorithm. We present a proof-of-principle implementation of such an algorithm in which the classifier function is prepared via supervised learning using pseudorandomly generated training data sets composed of states whose corresponding classifications are known a priori.
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