A measure theoretic approach to negative probabilities

FEDERICO HOLIK

Vaxjo

Congreso; Quantum Information and Probability: from Foundations to Engineering (QIP22); 2022

Negative probabilities (or quasiprobabilities) have historically given place to deep debates in the foundations of quantum physics [1,2,3]. They find applications in quantum optics [4] and quantum information theory [5]. In this presentation, we discuss a measure theoretic approach to the study of negative probabilities that was recently introduced in [6]. It relies on a definition based in measure theory which does not depend on any prior Hilbert space representation, and incorporates a suitably chosen notion of measurement context that has a direct operational interpretation. We illustrate this definition in several examples of interest in quantum theory, and study its properties in connection to quantifying quantum contextuality in quantum information processing tasks.[1] R. P. Feynman. “Negative Probability”. In Quantum Implications: Essays in Honour of David Bohm; Routledge & Kegan Paul Ltd.: London, UK; New York, NY, USA, 1987, pp. 235-248.[2] R. W. Spekkens. “Negativity and Contextuality are Equivalent Notions of Nonclassicality”, Phys. Rev. Lett. 101, 020401, 2008.[3] S. Abramsky and A. Brandenburger. “The sheaf-theoretic structure of non-locality andcontextuality”, New Journal of Physics, 13, 113036, 2011.[4] K. E. Cahill and R. J. Glauber. “Density Operators and Quasiprobability Distributions”, Physical Review 177 (5), 1882–1902, 1969.[5] C. Ferrie. “Quasi-probability representations of quantum theory with applications to quantum information science”, Reports on Progress in Physics, Volume 74, Number 11, 116001, 2011.[6] J. A. de Barros and F. Holik. “Indistinguishability and Negative Probabilities”, Entropy, 22(8), 829, 2020.