Quantum neighborhood on the probability space Metricity on the Grassmannian submanifold

Invariance of quantum separation on the probability space is discussed. The idea of quantum separation and metricity on the quantum state space is extended in a scenario wherein metric can be interpreted as chordal distance that is invariant under rotations of sphere and wherein there exist lengths of paths and distances between points on this spherical metric. Thus, the metric ought to admit symmetries following [Formula: see text], and could be represented on the Lagrangian–Grassmannian [Formula: see text], a submanifold of Grassmannian. This metric on the projective Hilbert space is then recast on the probability space and its geometry is depicted graphically with respect to probability variables. Hitherto, this metric was neither explored on the sub-manifold of the quantum state space nor on the corresponding probability space.