Quantum Snapshots Reveal a Compact Conformal Boundary Mode

Abstract

A projective measurement of a many-body state produces a microscopic snapshot, usually viewed as random classical data. We show that partial occupation snapshots of the critical XX chain contain a universal angle with a precise conformal meaning. Dividing the ring into two measured and two unmeasured arcs, we assign geometry-dependent conformal side weights to the observed occupations and obtain a compact variable $\delta_L$. At every finite size, $\delta_L$ is fixed by the measured sites alone; the particular complete-configuration lift $X_L$ used in the proof additionally depends on unobserved particles. This angle is an exact microscopic compact coordinate whose scaling-limit law is that of the relative Dirichlet phase of the associated conformal quadrilateral---the boundary coordinate conjugate to charge in continuum post-measurement descriptions. Exact free-fermion determinants yield all of its Fourier moments. We prove that the lift becomes Gaussian with variance $2h(\zeta)$, where $h(\zeta)$ is the rectangle modulus, and hence $\langle e^{\ii q\delta_L}\rangle\to e^{-h(\zeta)q^2}$. Thus raw quantum snapshots realize the heat kernel on a circle and provide an outcome-level microscopic foundation for the compact zero-mode sector of Born averages over fluctuating conformal boundary conditions.