{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/probability-vector-representation-of-the","title":"Probability vector representation of the Schrödinger equation and Leggett-Garg-type experiments","arxiv_id":"2312.16281","date":"2023-12-26","proceeding":null,"authors":["Masahiro Hotta","Sebastian Murk"],"abstract":"Leggett-Garg inequalities place bounds on the temporal correlations of a system based on the principles of macroscopic realism $\\textit{per se}$ 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 $\\textit{a priori}$.","url_abs":"https://arxiv.org/abs/2312.16281v3","url_pdf":"https://arxiv.org/pdf/2312.16281v3.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"probability-vector-representation-of-the","repo_url":"https://github.com/s-murk/2312.16281","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}