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Phaseless Schrödinger data on a finite connected graph recover every initial state up to global phase for almost every real potential.

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2026-07-30 23:28 UTC pith:NBK3ECXV

load-bearing objection Clean spectral uniqueness criterion for phaseless Schrödinger evolution on finite graphs, with explicit and generic good potentials; solid subfield advance, no load-bearing gaps.

arxiv 2607.26705 v1 pith:NBK3ECXV submitted 2026-07-29 math.CA math-phmath.COmath.MP

Dynamical phase retrieval for Schr{\"o}dinger evolution on finite graphs

Philippe Jaming (IMB) , Azita Mayeli (CUNY) This is my paper
classification math.CA math-phmath.COmath.MP MSC 05C5015A1894A1242C15
keywords dynamical phase retrievalfinite graphsgraph Schrödinger operatorsphaseless reconstructionspectral graph theorydiagonal potentialsgeneric uniquenessB2 sets
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper asks when the time-dependent probability distribution of a quantum particle on a finite graph determines the initial wave function up to a single overall phase. It answers with three spectral conditions on the graph Schrödinger operator: the eigenvalue sums must uniquely label pairs of modes, the matrix of squared eigenvector entries must be invertible, and every pair of eigenvectors must share a common vertex. When those hold, the continuous phaseless trajectory at the vertices pins down every initial state uniquely. The authors then prove the conditions are not rare: every connected graph admits an explicit diagonal potential that works, and the same uniqueness holds for Lebesgue-almost every real potential and every positive observation time. The result turns a classical phase-retrieval question into a generic property of graph Schrödinger dynamics.

Core claim

If a graph Schrödinger operator H_Q = Δ_G + Q satisfies the B₂ spectrum condition, invertibility of the squared-eigenvector matrix, and pairwise support overlap of eigenvectors, then the map sending an initial state u₀ to the family of moduli |e^{-it H_Q} u₀(j)| uniquely determines u₀ up to global phase. These three conditions are realized by an explicit diagonal potential on every finite connected graph and hold for almost every real Q.

What carries the argument

The triple of spectral conditions B₂–(I)–(S): B₂ forces the intensity identities to decouple into modulus and cross-term equations; (I) recovers the individual mode moduli; (S) makes the overlap graph complete so a single unimodular phase propagates to all modes.

Load-bearing premise

Every pair of eigenvectors must share at least one vertex; if two modes live on disjoint sets of vertices, independent phases can be assigned without changing any observed intensity.

What would settle it

On any fixed connected graph, compute the spectrum and eigenvectors of H_Q for a concrete diagonal Q; check whether the pairwise sums λ_j+λ_k are all distinct, whether det(φ_k(j)²) is nonzero, and whether every pair of eigenvectors has a common nonzero coordinate—if any check fails and two non-equivalent initial states still produce identical |e^{-it H_Q}u(j)|, the uniqueness claim is false for that operator.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Continuous-time phaseless measurements can be replaced by finitely many sampling times without losing uniqueness.
  • Every finite connected graph admits an explicit, constructible potential that makes Schrödinger phase retrieval hold for every initial state.
  • Uniqueness is generic: the exceptional potentials form a proper algebraic set of Lebesgue measure zero.
  • Disconnected graphs, multiple eigenvalues, and eigenvectors with disjoint supports each produce concrete non-uniqueness counter-examples.
  • The same spectral criterion applies verbatim to any real symmetric matrix observed in a fixed coordinate basis.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The free Laplacian (Q=0) is systematically excluded by failure of condition (I), so any practical recovery scheme on unweighted graphs will need a designed or random on-site potential.
  • Because the good potentials are Zariski-open, random diagonal disorder of arbitrarily small amplitude already yields uniqueness with probability one, suggesting a simple experimental prescription.
  • The frame-theoretic reformulation indicates that the same three conditions could certify phase retrieval for other continuous frames generated by unitary groups, not only Schrödinger evolution.
  • Algorithmic recovery remains open; existing PhaseLift or alternating-projection methods could be tested directly on the finite sampled frame produced by the paper’s finite-time sampling result.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 5 minor

Summary. The paper studies dynamical phase retrieval for the Schrödinger evolution e^{-it H_Q} on a finite connected graph, with H_Q = Δ_G + Q a real diagonal perturbation of the combinatorial Laplacian. The central result (Theorem 1.5) gives a spectral uniqueness criterion: if the eigenvalues form a B_2-set, the squared-eigenvector matrix (φ_k(j)^2) is invertible (Condition (I)), and every pair of eigenvectors has overlapping support (Condition (S)), then the phaseless data |e^{-it H_Q} u_0(j)| for t ∈ [0,T] and all vertices determine every initial state uniquely up to global phase. The authors then prove that every connected graph admits an explicit diagonal potential realizing the three conditions (Proposition 3.1), and that the conditions hold for Lebesgue-almost every real Q (Theorem 1.6), via non-vanishing of polynomial discriminants/determinants and a cyclic-vector argument for full support. Several sharp obstructions (disconnected graphs, multiple eigenvalues, failure of (S), two-mode swapping) are identified, and intermediate examples show that B_2 and (I) are not always necessary for uniqueness.

Significance. The work cleanly transplants the Schrödinger–Pauli / dynamical phase-retrieval question to finite graphs and supplies a usable spectral criterion together with both an explicit construction and a genericity theorem. The reduction via exponential independence, the phase-propagation argument on the eigenvector-overlap graph Γ_Q, and the algebraic-genericity proofs are standard but carefully executed; the separation between uniqueness for every state (completeness of Γ_Q) and almost-everywhere-in-state uniqueness (connectedness of Γ_Q) is particularly clear. The finite-time sampling observation and the frame-measurement variant broaden the scope. The paper is a solid contribution to spectral graph theory and structured phase retrieval.

minor comments (5)
  1. [Title / §4.2] Title page / running head: “DYNAMICAL PHASE RETRIEV AL” and “F ailure of Condition (S)” contain stray spaces; fix throughout.
  2. [Example 2.6] In Example 2.6 the difference table and the four positive frequencies are asserted to be pairwise distinct; a one-line verification that 0 < 2β < α−β < α+β < 2α for the concrete α,β would help the reader.
  3. [§3.1] Proposition 3.1: the constant R = R(G) is existential. A brief remark on how large ρ must be in terms of n and M = ∥A∥ would make the construction more immediately usable.
  4. [§2.5 / Introduction] Section 2.5 reformulates Condition (I) via the matrix B_Q of diagonal entries of powers of H_Q. Cross-referencing this reformulation earlier (e.g., when Condition (I) is first introduced) would improve readability.
  5. [§1.3, §3.2.3] A few references to continuous-frame phase retrieval and to controllability of graph Laplacians could be expanded slightly for readers coming from either community; the existing citations are adequate but sparse.

Circularity Check

0 steps flagged

No significant circularity: uniqueness and genericity are derived from spectral expansion, exponential independence, and non-vanishing of explicit polynomials.

full rationale

The central criterion (Theorem 1.5) is obtained directly from the spectral theorem: equality of intensities expands into a finite linear combination of distinct exponentials e^{-i(λ_k-λ_ℓ)t}; Condition B₂ plus Lemma 2.1 forces the algebraic system (2.1)–(2.2); Condition (I) collapses the diagonal block to |a_k|=|b_k|; Condition (S) makes the overlap graph complete so phases propagate to a single global factor (Proposition 2.3). None of B₂, (I), or (S) is defined in terms of the retrieval conclusion. Genericity (Theorem 1.6) is proved by exhibiting one explicit large-diagonal potential (Proposition 3.1 / scalings tC) at which the relevant discriminants and controllability determinants are nonzero, hence the exceptional sets are proper algebraic varieties of measure zero—standard and non-circular. There are no fitted constants, no data-driven “predictions,” and no load-bearing uniqueness theorem imported from the authors’ prior work; self-citations (e.g. Jaming’s free Schrödinger–Pauli paper) appear only as motivation. Obstructions (Section 4) are likewise derived, not assumed. The derivation chain is self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 6 axioms · 0 invented entities

Load-bearing background is standard finite-dimensional spectral theory and elementary algebraic geometry (nonzero polynomials vanish on measure-zero sets). The three named conditions are hypotheses of the uniqueness theorem, not hidden axioms; connectedness is a stated domain assumption needed for genericity of (S) and for necessity of phase retrieval. No free parameters are fitted. No new physical entities are postulated.

axioms (6)
  • standard math Real symmetric matrices are orthogonally diagonalizable with real eigenvalues and a real orthonormal eigenbasis (spectral theorem).
    Used throughout §2 to expand e^{-it H_Q} and reduce intensity equality to algebraic equations on coefficients.
  • standard math Distinct complex exponentials t ↦ e^{i μ_r t} are linearly independent on every nontrivial interval (Vandermonde).
    Lemma 2.1; converts continuous-time intensity equality into the system (2.1)–(2.2) under B₂.
  • standard math The zero set of a non-identically-vanishing real polynomial on R^n has Lebesgue measure zero.
    Used in Propositions 3.3, 3.4, 3.6 to upgrade existence of one good potential into almost-everywhere statements.
  • standard math Weyl eigenvalue perturbation and Levy–Desplanques strict diagonal dominance imply invertibility for large diagonal potentials.
    Proposition 3.1 construction of Q_ρ.
  • domain assumption The underlying graph G is finite, undirected, simple, and connected; potentials Q are real and diagonal so that H_Q is self-adjoint and the evolution is unitary.
    Stated in §1.2; connectedness is necessary (disconnected components allow independent phases) and is used for genericity of full support (S).
  • domain assumption Observation times form a nondegenerate interval [0,T] (or a finite sampling set separating the relevant frequencies).
    Problem 1.3 and Proposition 2.9; T itself drops out once B₂ holds (Remark 2.2).

pith-pipeline@v1.2.0-daily-grok45 · 30409 in / 3139 out tokens · 63659 ms · 2026-07-30T23:28:17.684329+00:00 · methodology

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We study dynamical phase retrieval for Schr\''odinger evolutions on finite connected graphs. Let \[ H\_Q=\Delta\_G+Q \] be a graph Schr\''odinger operator with a real diagonal potential. We investigate when phaseless data obtained from the associated Schr\''odinger evolution \[ |e^{-itH\_Q}u\_0(j)|, \qquad 0\leq t\leq T,\ j\in V, \] determines the initial state $u\_0\in\C^V$ up to a global phase. We give a uniqueness criterion in terms of the eigenvalues and eigenvectors of $H\_Q$. The assumptions are a $B\_2$ condition on the spectrum, meaning that the sums $\lambda\_j+\lambda\_k$ determine the unordered pair $\{j,k\}$, invertibility of the squared-eigenvector matrix $\bigl(\phi\_k(j)^2\bigr)\_{j,k}$ and an overlap condition on the supports of pairs of eigenvectors. Under these hypotheses, the phaseless Schr\''odinger data determine every initial state uniquely, modulo global phase. We then show that the criterion is both realized and generic. Every finite connected graph admits an explicit real diagonal potential for which the criterion holds. Moreover, for every finite connected graph, dynamical phase retrieval holds for Lebesgue-almost every real potential $Q\in\R^V$ and every $T>0$. We also give several obstructions to uniqueness.

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