Revisiting Connes’ finite spectral distance on noncommutative spaces: Moyal plane and fuzzy sphere

Beginning with a review of the existing literature on the computation of spectral distances on noncommutative spaces like Moyal plane and fuzzy sphere, adaptable to Hilbert–Schmidt operatorial formulation, we carry out a correction, revision and extension of the algorithm provided in [1] i.e. [F. G. Scholtz and B. Chakraborty, J. Phys. A, Math. Theor. 46 (2013) 085204] to compute the finite Connes’ distance between normal states. The revised expression, which we provide here, involves the computation of the infimum of an expression which involves the “transverse” (Δρ⊥) component of the algebra element in addition to the “longitudinal” component Δρ of [1], identified with the difference of density matrices representing the states, whereas the expression given in [1] involves only Δρ and corresponds to the lower bound of the distance. This renders the revised formula less user-friendly, as the determination of the exact transverse component for which the infimum is reached remains a nontrivial task, but und...