Projection Obstruction for Factorized States: Local Correlation Lower Bounds, Additive Diagnostics, and Observer Recovery

Title. Projection Obstruction for Factorized States: Local Correlation Lower Bounds, Additive Diagnostics, and Observer Recovery

Abstract. For a finite quantum system with a time-independent Hamiltonian containing one- and two-site interactions and a product initial state, we bound the departure of each exact two-site reduced state from the product of its own marginals. Define

Fe(t)=ρe(t)−ρa(t)⊗ρb(t),De(t)=∥Fe(t)∥1,F_e(t)=\rho_e(t)-\rho_a(t)\otimes\rho_b(t), \qquad D_e(t)=\|F_e(t)\|_1,

where e={a,b}e=\{a,b\} and ∥⋅∥1\|\cdot\|_1 is the full trace norm. With local interaction norm g=max⁡v∑X∋v∥hX∥∞g=\max_v\sum_{X\ni v}\|h_X\|_\infty and first-order coefficient νe=∥Fe′(0)∥1\nu_e=\|F_e'(0)\|_1, the bound is

De(t)≥νet−24g2t2.D_e(t)\ge\nu_et-24g^2t^2.

More generally, if Fe(j)(0)=0F_e^{(j)}(0)=0 for 0≤j<k0\le j<k, then

De(t)≥∥Fe(k)(0)∥1k!tk−2(k+2)(2g)k+1tk+1.D_e(t)\ge \frac{\|F_e^{(k)}(0)\|_1}{k!}t^k -2(k+2)(2g)^{k+1}t^{k+1}.

The proof uses local support counting and Taylor remainder estimates. Explicit examples exhibit activation orders one, two, and three. Uniform interaction bounds, a positive active-edge fraction, and uniformly positive first-order coefficients give system-size-independent bounds on the edge-averaged defect. The constants are conservative; optimality is not claimed.

Separately, the paper studies detection of compatibility defects by linear observation maps in finite diagrams. A fully covered six-edge example has zero observer outputs and represented defect norm 66; scaling excludes any finite recovery bound based only on those outputs. Injectivity on the realizable defect subspace characterizes raw defect detection. The paper proves minimum additional measurement counts for distinct realizable-defect and repair subspaces, and formulates budget-constrained selection from a finite measurement menu. Disjoint-set additivity and singleton calibration determine an edgewise ℓ1\ell^1 sum relative to specified local gauges.

Relevance. The quantum estimates provide local lower bounds on correlations generated by exact dynamics. A positive bound certifies that the exact reduced state cannot remain factorized. The observation results distinguish coverage from detection and identify the additional measurements required for raw or quotient recovery.

Scope and verification. A positive product defect establishes non-product correlation, but does not alone establish entanglement or the error of a particular numerical solver. The sum over overlapping edges is not identified with total correlation or a global trace norm. The quantum results assume finite-dimensional Hilbert spaces and exact unitary evolution; no thermodynamic-limit theorem is claimed.

Quantum residuals become cochains only after observation maps are specified. Exactness additionally requires a solution to a specified coboundary equation. The quantum derivative estimates and new distinct-subspace completion theorem are proved in prose and are not formalized in Lean. Computational examples and tests provide evidence for their stated instances.

Version and DOI chain.

• Concept DOI: 10.5281/zenodo.18896776.

• Previous published version, v6.3: 10.5281/zenodo.22289548.

• This release, v6.4: 10.5281/zenodo.23174322.