math.OA — Operator Algebras

Bures Geometry of Projection States

Contributed by Zhenyu Sun, Jilin Wang

We study the Bures metric on states obtained by conditioning a faithful normal state on nonzero projections of a von Neumann algebra. The metric determines the reference state, projection probabilities, orthogonality, and order. For algebras with no type $I_2$ direct summand, surjective isometries are exactly those induced by reference-state-preserving normal Jordan $^*$-isomorphisms; the same conclusion holds for $M_2(\mathbb C)$. A semifinite tracial version determines the trace up to a single positive factor. In finite tracial algebras, we characterize optimal finite projective measurements and express their fidelity deficit as a sum of squared $L^2$ errors. Spectral measurements with at most $n+1$ outcomes approximate fidelity with error $O(n^{-2})$; the interval grid is minimax optimal, and the order is sharp within spectral measurements. We compare the fidelity defect of a common commutative context with approximation by commuting projections, obtain sharp spectral rounding constants, and solve both optimization problems in the algebra generated by two projections. The space of normalized finite-trace projection states is complete; in a finite tracial algebra its rectifiable components are unitary orbits, and within each orbit the intrinsic Bures distance is the root mean square of the principal angles. Finally, a fidelity bound on local unitary orbits characterizes tracial reference states.

Author comments: 38 pages.