Kaushlendra Kumar
Scalar-anchored Connes distance and Helstrom geometry of the qubit Bloch ball
Trace distance gives the operational distinguishability of two qubit states in equal-prior Helstrom discrimination. This talk describes a finite scalar-qubit-scalar spectral model in which this trace geometry is obtained from Connes distance, up to a single calibration factor. The algebra is ${\mathbb C}_L\oplus M_2({\mathbb C})\oplus{\mathbb C}_R$, with scalar reference sectors coupled to the qubit block by isotropic identity links. On the middle qubit sector, the spectral distance is the usual trace distance multiplied by a scale fixed by the two Dirac couplings. Hence the full Bloch ball, including mixed states, inherits its standard chordal trace-distance geometry from the finite spectral metric. Once the scale is known, the same distance gives the Helstrom success and error probabilities. The scalar sectors play a separate role. Their distances are calibration lengths for the finite geometry, not additional qubit discrimination probabilities, and they provide a scalar-scalar consistency check for the two links.
