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 $δ_L$. 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 $\langle e^{\ii qδ_L}\rangle\to e<sup>{-h(ζ)q<sup>2}$. 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.

• The paper establishes that a compact angle computed from partial occupation snapshots is exactly record-measurable and converges to a heat-kernel distribution with integer harmonics exp[-h(ζ)q²].
• The paper combines determinantal-process identities, Toeplitz determinant asymptotics, and a finite-size variance formula to prove a Gaussian limit whose variance equals the conformal modulus h(ζ).
• The paper provides an experimentally accessible bridge between microscopic Born probabilities and continuum boundary ensembles, while leaving extensions to interacting systems, merging endpoints, and multiple intervals open.

Overview

This Letter by Rajabpour establishes an exact bridge between projective measurement records of a critical quantum many-body state and the compact boundary zero mode of conformal field theory. Working in the half-filled periodic spin-$1/2$ XX chain, the author shows that a specific angle constructed from partial occupation snapshots—bits measured on two arcs of a ring divided into four consecutive arcs $A$, $C_1$, $B$, $C_2$—is exactly determined by the observed record alone, and that its Born distribution converges in the thermodynamic limit to the heat kernel on a circle at time equal to the conformal modulus of the associated quadrilateral. The headline result is

$$\lim_{L\to\infty}\mathbb{E}\left[e^{iq\delta_L}\right]=e^{-h(\zeta)q^2},\qquad q\in\mathbb{Z},$$

where $\zeta$ is the cross ratio of the four marked boundary points and $h(\zeta)=\pi K(\zeta)/K(1-\zeta)$ is the rectangle width under the Schwarz–Christoffel map. This provides an outcome-level microscopic foundation for the Born average over fluctuating Dirichlet boundary conditions anticipated in continuum treatments of measurement-induced entanglement (Khanna et al., 15 Dec 2025), without replicas or entanglement observables.

Microscopic setting and the record-only decoder

The Jordan–Wigner transformation maps the XX ground state to a half-filled Fermi sea of consecutive momenta, so the Born probability of a complete occupation configuration is a circular Vandermonde determinant—a discrete $\beta=2$ log gas and a projection determinantal process. Four ordered endpoints on the circle define a conformal quadrilateral; the map to a rectangle sends the unmeasured arcs to horizontal sides (where the coordinate $f=2\,\mathrm{Im}\,W$ equals $A$0 and $A$1) and the measured arcs to vertical sides along which $A$2 winds once. The central finite-size observation is algebraic: because the unmeasured contribution to the centered linear statistic $A$3 is $A$4, its class modulo $A$5,

$A$6

is fixed by the measured record alone, even though the real lift depends on the unknown particle number $A$7 in arc $A$8. Consequently every physical character $A$9 is accessible from incomplete data—an exact information-theoretic statement holding at every finite size, not a scaling-limit identification. Experimentally this requires only deterministic, tabulable weights and classical post-processing of the observed bits.

Determinant representations and Gaussian asymptotics

The Fourier moments admit four equivalent exact representations: determinants on the full lattice, on the occupied one-particle subspace, on the measured sites only, and a discrete Toeplitz determinant. The measured-site reduction for integer $C_1$0 follows because $C_1$1 vanishes identically on the unmeasured arcs, making the full determinant block triangular; it is an exact algebraic consequence of compactification, not a thermodynamic approximation.

The asymptotic analysis must handle the square-root corner singularities of $C_1$2, which give Fourier coefficients decaying as $C_1$3—sufficient for finite homogeneous $C_1$4 energy but beyond naive strong Szegő applicability. The proof combines three ingredients: midpoint aliasing estimates showing discrete Toeplitz determinants converge exponentially fast to continuous ones at fixed Fourier cutoff; the complex strong Szegő theorem for the truncated analytic symbol; and an exact finite-size variance identity,

$C_1$5

whose weight counts particle–hole pairs across the Fermi points and provides uniform control when the cutoff is removed. The order of limits ($C_1$6 first, then cutoff) yields the real Gaussian limit $C_1$7.

Geometry of the variance

The quadratic form $C_1$8 is evaluated exactly by conformal invariance: the Poisson extension of $C_1$9 is $B$0, Dirichlet energy is conformally invariant in two dimensions, and on the rectangle the potential $B$1 has energy $B$2. Hence $B$3 exactly—the abstract variance of a microscopic fermion statistic equals the conformal modulus of the four measurement regions. This also explains why uniform angular weights fail: they produce a different, explicitly computable energy $B$4, confirmed numerically to relative error $B$5 at $B$6, while the conformal weights reproduce $B$7 to relative deviations below $B$8 for the first harmonic. The wrapped-Gaussian limit interpolates between localization near a definite phase as $B$9 ($C_2$0) and a uniform distribution as $C_2$1 ($C_2$2).

Relation to continuum boundary ensembles

The limiting density matches the normalized zero-mode factor of a twisted Dirac/compact-boson partition function on the doubled cylinder, $C_2$3, whose dual expansion carries integer charge labels $C_2$4. The relative phase acts as a twist conjugate to conserved $C_2$5 charge, mapping to an effective chemical potential $C_2$6 in the post-measurement entanglement Hamiltonian (Eisler et al., 6 Aug 2026). The paper's contribution relative to prior continuum work (Khanna et al., 15 Dec 2025) is precisely the microscopic random variable, the proof of partial-record measurability, and the derivation of the law from exact Born probabilities rather than an assumed effective ensemble.

Limitations and open questions

The exact theorem is confined to the free-fermion point. For interacting Tomonaga–Luttinger liquids, where continuum theory predicts a stiffness-rescaled heat kernel, no outcome-level decoder is established: the projection determinant disappears, irrelevant operators can contaminate height variables, and the compactification radius varies continuously. The degenerate regime where endpoints merge on an $C_2$7-dependent scale is excluded; there the finite-energy Szegő problem should cross over to Fisher–Hartwig counting statistics, and the universal double-scaling kernels remain undetermined. The corner-regularity threshold used in the proof may not be optimal. Multicomponent extensions to several measured intervals are formulated—the covariance matrix is identified as a Dirichlet capacitance matrix—but the connection to period matrices of multiply connected domains is conjectural. Whether the compact heat kernel controls portions of measurement-induced entanglement distributions, rather than only their averaged partition functions, is left open.

Conclusion

The paper identifies a compact collective coordinate of critical XX-chain snapshots that is exactly measurable from partial records, derives its full Born law via exact determinant asymptotics, and evaluates its variance as a conformal modulus. The result demonstrates that universal conformal geometry can be decoded directly from individual measurement outcomes through classical post-processing, providing a concrete experimental target via the harmonics $C_2$8 and a structural template—record-only compactification, Gaussian determinant asymptotics, conformal Dirichlet energy—for analogous constructions in broader critical systems.