State-dependent spectral distribution

Let $H$ be a Hermitian endomorphism of a finite-dimensional complex inner-product space, with distinct eigenvalues $\lambda$ and orthogonal spectral projections $P_{\lambda }$. Let $\psi$ be a unit vector representing a pure state. Its state-dependent spectral distribution is the finite measure on $\sigma (H)$ defined by ${\mu }_{\psi }(\{\lambda \})=\mathrm{\parallel }{P}_{\lambda }\psi {\mathrm{\parallel }}^2=⟨\psi \mathrm{\mid }{P}_{\lambda }\mathrm{\mid }\psi ⟩\mathrm{.}$ The support is restricted to eigenvalues with nonzero weight; degeneracies are represented by one atom per distinct eigenvalue, not one atom per chosen eigenvector. The formula is the definition; its nonnegativity, total mass one, and functional-calculus moment identity are established in QPE output law and conditional state for a spectral input ↗.