2025/02/25 by Fouad Naderi, Naderi, Fouad
Physics and Astronomy · Computer Science · #Quantum Mechanics and Applications #Quantum Information and Cryptography
paper · pdf · doi:10.48550/arxiv.2506.12018
A quantum expectation is a positive linear functional of norm one on a non-commutative probability space (i.e., a C*-algebra). For a given pair of quantum expectations μ and λ on a non-commutative probability space A, we propose a definition for weak* continuity and weak* singularity of μ with respect to λ. Then, using the theory of von Neumann algebras, we obtain the natural weak* continuous and weak* singular parts of μ with respect to λ. If λ satisfies a weak tracial property known as the KMS condition, we show that our weak* decomposition coincides with the Arveson-Gheondea-Kavruk Lebesgue (AGKL) decomposition. This equivalence allows us to compute the Radon-Nikodym derivative of μ with respect to λ. We also discuss the possibility of extending our results to the positive linear functionals defined on the Cuntz-Toeplitz operator system.