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arXiv:2608.14327

Quantum Snapshots Reveal a Compact Conformal Boundary Mode

M. A. Rajabpour

quant-phcond-mat.stat-mechhep-thSubmitted Aug 14, 2026• Announced Aug 17, 2026• 0 citations
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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 $δ_L$. At every finite size, $δ_L$ 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 $2h(ζ)$, where $h(ζ)$ is the rectangle modulus, and hence $\langle e^{\ii qδ_L}\rangle\to e^{-h(ζ)q^2}$. 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.

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Local measurements transform a many body wavefunction into a statistical ensemble of conditional quantum states. In chaotic systems, the higher moments of such ensembles diagnose the emergence of quantum state randomness, but their structure at equilibrium quantum criticality is largely unknown. Here we show that the projected ensemble of a Tomonaga-Luttinger liquid exhibits a universal nonlinear hierarchy of state overlap moments. Remarkably, the leading scaling exponents are independent of the continuously varying Luttinger parameter. Replica boundary conformal field theory reveals a geometric origin: outcome locking combines the active replica swaps into a single collective rotated sector, while intersecting compact replica branes eliminate the leading interaction dependent zero mode contribution. Matrix product state calculations across the interacting XXZ critical phase and direct free fermion calculations independently confirm the interaction independence and nonlinear hierarchy.
Can a probability distribution measured in one basis determine complex quantum geometry? The answer is affirmative for a lattice Laughlin quasihole. The exact occupation law at one generic reference position, together with the known analytic quasihole factor, fixes the complete complex Gram kernel. A finite snapshot ensemble estimates this kernel without preparing another member of the family, and thereby determines finite-distance overlaps, Bargmann phases, the quantum metric, and the Berry curvature. The same analytic structure confines the family to an exact projective subspace whose dimension grows at most linearly with particle number. Exact enumeration of a Nielsen-Cirac-Sierra state demonstrates finite-shot reconstruction of both the metric and a geometric phase. Moreover, a Rényi-2 divergence sets the statistical range of the reconstruction while, at nondegenerate points, occupation readout also attains the local single-copy multiparameter information bound.