We study an inverse problem for quantum cellular automata (QCA) and strictly locality-preserving discrete quantum dynamics: what can an observer infer about a finite local update from physical data, and which inferences remain impossible even with exact observations? We separate general interacting locality-preserving automorphisms from homogeneous finite-dimensional linear excitation sectors. In the latter, finite range implies a matrix Laurent-polynomial update and algebraic quasienergy branches. We derive a reciprocal-lattice monodromy obstruction: if repeated Brillouin windings generate infinitely many distinct analytic germs of a target quasienergy phase, the target cannot be an exact band of any finite-Laurent update; moreover, M distinct continuations require internal block dimension q ≥ M. Massive relativistic dispersion violates this criterion, and generic transverse slicing extends the obstruction to isotropic massive and massless dispersion in d ≥ 2. Independently, positive minimum site separation, exact finite-speed causality, and arbitrarily divisible microscopic time imply ultralocal dynamics. We then formulate the inverse direction. In an independently specified operational locality basis, finite full-matrix Fourier support plus pointwise unitarity is constructively equivalent to a homogeneous finite-range update, whereas spectra alone are insufficient. Exact reconstruction generally fixes only a blocking/refinement equivalence class. Conditional on such reconstruction, reachability is a compact-group control problem. Counterexamples accompany each nontrivial converse, and a complexity-controlled evidence hierarchy separates weak spectral consistency from process-level reconstruction.
We study an inverse problem for quantum cellular automata (QCA) and strictly locality-preserving discrete quantum dynamics: what can an observer infer about a finite local update from physical data, and which inferences remain impossible even with exact observations? We separate general interacting locality-preserving automorphisms from homogeneous finite-dimensional linear excitation sectors. In the latter, finite range implies a matrix Laurent-polynomial update and algebraic quasienergy branches. We derive a reciprocal-lattice monodromy obstruction: if repeated Brillouin windings generate infinitely many distinct analytic germs of a target quasienergy phase, the target cannot be an exact band of any finite-Laurent update; moreover, M distinct continuations require internal block dimension q ≥ M. Massive relativistic dispersion violates this criterion, and generic transverse slicing extends the obstruction to isotropic massive and massless dispersion in d ≥ 2. Independently, positive minimum site separation, exact finite-speed causality, and arbitrarily divisible microscopic time imply ultralocal dynamics. We then formulate the inverse direction. In an independently specified operational locality basis, finite full-matrix Fourier support plus pointwise unitarity is constructively equivalent to a homogeneous finite-range update, whereas spectra alone are insufficient. Exact reconstruction generally fixes only a blocking/refinement equivalence class. Conditional on such reconstruction, reachability is a compact-group control problem. Counterexamples accompany each nontrivial converse, and a complexity-controlled evidence hierarchy separates weak spectral consistency from process-level reconstruction.
The formulation of quantum measurements compatible with relativistic causality is obstructed by the tension between the locality of observable algebras and the dynamical state transformations induced by measurement processes. We develop a local measurement theory for quantum cellular automata, lattices of finite-dimensional quantum systems with a causal reversible dynamics evolving in discrete time steps. Measurement processes are described by localized completely positive maps and organized into operational measurement protocols, finite non-adaptive sequences of local instruments distributed across the causal cone induced by the automaton local dynamics. Each protocol induces an effective measurement on the initial time slice, whose support is constrained by the causal structure generated by the automaton. Non-local measurement effects therefore arise through causal propagation and composition of local operations, rather than from primitive non-local interventions or instantaneous collapse prescriptions. We establish the corresponding spatial support bounds and characterize the resulting measurement architecture. As an explicit example, we construct a two-step protocol for a one-dimensional nearest-neighbour qubit automaton which, for suitable automaton parameters, realizes a Bell-basis projective measurement on two neighbouring sites after coarse-graining of the local outcomes. Away from this point, the same protocol generates a continuous family of deformed Bell measurements whose entanglement and stabilizer Rényi non-classicality can be quantified explicitly. The framework thus provides a causally consistent operator-algebraic mechanism for synthesizing spatially extended quantum measurements from strictly local operations.
We develop a finite operational response theory for controlled quantum systems in which observer closure, hidden microscopic structure, and higher-order response are treated within a single algebraic framework. The central distinction is between generic higher-order observability and Closure–Residue (CR)-constrained generative response. Standard finite-dimensional quantum theory does not, by itself, impose the specific CR accessibility condition introduced here; physically realizable controls are ordinarily determined by the microscopic Hamiltonian, hardware connectivity, conservation laws, and environmental couplings of the system under study. We therefore formulate First-Residue Accessibility as an additional finite selection principle for the coherent-control sector studied here: the infinitesimal deformation of the observer closure algebra generated by any admissible tangent in the declared target control manifold must lie in the first-residue derivation space inherited from an underlying microscopic CR sector. The algebraic consequences are rigid. Within this coherent-control sector, the maximal abelianness of the population algebra implies that the infinitesimal action of a control on the closure algebra determines its anti-Hermitian generator uniquely modulo a diagonal population gauge. First-Residue Accessibility therefore forces the declared coherent control class, up to this gauge, to lie in a canonical graph-internal subtype extracted from the microscopic CR sector. For diagonal prepared states, the visible first-order population response is silent, while the second variation factorizes through a graph-divergence map of a microscopic bilinear edge flow. A three-site single-excitation benchmark realizes the factorization exactly and yields an even response law of the form Sigma(epsilon X) = epsilon^2 Q_CR(X,X) + O(epsilon^4). We also separate exact discrete support transport from continuous-time quasilocality. A hard one-step support cone has a singular continuous-time limit whenever the limiting generator genuinely spreads local observables; finite propagation in the continuous limit must therefore be formulated by soft tails rather than exact support. We introduce a normalized-trace visible-leakage profile and derive a soft causal readout bound. In the coherent single-excitation sector this profile is uniformly equivalent, up to fixed observer-volume constants, to intrinsic operator-norm quasilocality. For finite-range exchange with simultaneous site dephasing we prove an N-uniform normalized-trace quasilocality estimate without using rank preservation. This genuinely nonunitary extension establishes a common finite-time exponential locality class with standard open-system quasilocality, while the stronger two-sided intrinsic equivalence is claimed only for the coherent unitary subclass. We further show that divergence-image membership is non-discriminating for connected full-population readout, construct an explicit standard CPTP control family that is first-order silent but violates the CR component constraint on a disconnected graph, and isolate experimentally measurable null tests. We distinguish a declared target-control manifold from the larger independently calibrated manifold of physically available coherent controls, thereby separating a conditional declared-manifold accessibility hypothesis from the stronger Physical First-Residue Accessibility statement. Under star-compatible oriented provenance we also derive an observable cross-prediction bound that links independently fixed static CR data to a measurable quadratic response coefficient. A framework-discriminating test requires both that the CR structure be fixed without using the target Hessian data and that the resulting constraint be stronger than what follows from the same calibrated microscopic information in a standard quantum model. We further exhibit a connected three-site collective-mode candidate in which CR Accessibility predicts an exact left–right quadratic null, while a same-support standard comparator constrained only by independently fixed aggregate tangent norms permits a nonzero response envelope. This yields a strict prediction-set separation under an explicit Dataset-A information condition, while also making clear that the separation disappears if standard microscopic calibration independently fixes the same collective tangent relation. We map this strict comparator onto a single-electron linear triple quantum dot, construct a four-point mixed-Hessian estimator, and propagate swap-odd instrumental asymmetry, finite-difference bias, and shot noise into a preregistered experimental envelope. The deeper derivation of the accessibility principle from the full Generative Map is left to the subsequent axiomatic stage of the program.
Abstract This paper develops a discrete information-theoretic framework in which physical systems are modeled on a graph or lattice of locally interacting space cells. Its principal contribution is an integrated architecture separating exact consequences of locality and unitarity, continuum-limit derivations, model-family results, toy-model mechanisms, and unresolved physical mappings. Matter-like identity is treated as propagated relational information rather than persistence of the same occupied cells, and candidate mechanisms are audited for causal locality, conservation or ledger closure, and false reconstruction. The quantum branch assumes finite-dimensional local Hilbert spaces and local unitary or completely positive trace-preserving (CPTP) updates. Under those assumptions, norm preservation, finite causal cones, and reduced-state no-signaling follow directly; spinor quantum walks admit a Dirac-like long-wavelength limit and a secondary nonrelativistic Schrödinger envelope. A representative reciprocal q–χ lattice action shows how compact local dynamics can support continuity, spectral response, action-derived scattering, and internal symmetry/charge, while later extensions supply a provisional local gauge/Gauss/holonomy structure. Two geometric realizations remain deliberately distinct: distortable cells and scalar extrinsic-χ compression. Toy models provide mechanism and failure-boundary evidence for relational identity, reconstruction, scattering, photon-like bookkeeping, and topology/charge-current response. The framework does not derive the Standard Model, quantum field theory, or general relativity; it defines a falsifiable program for testing whether compact local information-processing laws can recover required low-energy structures without hidden nonlocal repair or post-hoc tuning.
Quantum cellular automata (QCAs) describe locality-preserving quantum dynamics and link quantum information, many-body physics, and topological quantum field theory (TQFT). Constructing a QCA from a TQFT, however, remains challenging: although a topological action yields a Hamiltonian with commuting terms that realizes the desired ground state, it does not by itself specify an automorphism of the full operator algebra. We develop a unified algebraic construction that extends such Hamiltonians to complete separator-flipper algebras on the full tensor-product Hilbert space, thereby defining the corresponding QCAs microscopically. In 3 spatial dimensions, our framework unifies all previously known QCA constructions associated with the Z8×Z2 subgroup of the Witt group, including the U(1)2 and U(1)4 QCAs. The same algebraic structure yields new infinite families of generalized U(1)2 and U(1)4 non-Clifford QCAs in spatial dimensions d=4k−1. We also reformulate the 4-dimensional w2w3 QCA and use it to construct QCAs from TQFTs associated with arbitrary products of Stiefel--Whitney classes. For example, we explicitly construct two infinite families: w2nw3m QCAs in dimension d=2n+3m−1 and w2w4k−1 QCAs in dimension d=4k. For the 5-dimensional w32 and w23 QCAs, our construction gives finite-depth quantum circuits, consistent with the cobordism classification. More generally, we show that our construction yields a finite-depth quantum circuit whenever the corresponding Stiefel-Whitney number vanishes on every closed oriented manifold. Overall, these results convert invertible TQFTs into microscopic QCAs, provide higher-dimensional constructions beyond the Clifford setting, and open a systematic route to characterizing their stable structures and boundary anomalies.
This paper examines a deceptively simple question: can a localized physical controller acquire complete control over an entire physical system? Rather than treating “global control” as a single property, the paper decomposes the question into a hierarchy of distinct notions: unitary controllability, exact state controllability, approximate and infinite-time controllability, operational realizability, causal reach, and macroscopic domination. The analysis begins by rejecting locality alone as a no-go principle. Established results in finite quantum networks, together with a reproducible three-spin Heisenberg calculation, show that a single local actuator can generate global unitary controllability under suitable assumptions. The paper then identifies narrower and genuine obstructions. A finite-time causal proposition shows that a compactly generated intervention cannot exactly prepare a fixed global target whose local statistics differ from the initial state at arbitrarily large causal distances. The Ball–Marsden–Slemrod obstruction and its (L^1_{\mathrm{loc}}) extension provide a distinct topological limitation for broad classes of infinite-dimensional bilinear control systems. These negative results are deliberately constrained by positive counterexamples. Approximate controllability and exact infinite-time controllability remain possible in important infinite-dimensional systems, while Reeh–Schlieder density in quantum field theory does not by itself imply deterministic operational control. The central conclusion is therefore not a universal “no-global-control” law. It is a hierarchy of claims, assumptions, counterexamples, and inferential limits. In particular, failure of exact microscopic controllability does not imply protection against catastrophic macroscopic domination. The accompanying reproducibility package includes the Lie-closure verification code, tolerance analysis, source files, and claim-audit materials.
Quantum cellular automata (QCAs) are automorphisms of tensor product algebras that preserve locality, with local quantum circuits as a simple example. We study approximate QCAs, where the locality condition is only satisfied up to a small error, as occurs for local quantum dynamics on the lattice. A priori, approximate QCAs could exhibit genuinely new behavior, failing to be well-approximated by any exact QCA. We show this does not occur in one dimension: every approximate QCA on a finite circle can be rounded to a strict QCA with approximately the same action on local operators, so these systems are classified by the same index as in the exact case. Previous work considered the case of the infinite line, by using global methods not amenable to finite systems. Our new approach proceeds locally and now applies to finite systems, including circles or homomorphisms from sub-intervals. We extract exact local boundary algebras from the approximate QCA restricted to local patches, then glue these to form a strict QCA. The key technical ingredient is a robust notion of the intersection of two subalgebras: when the projections onto two subalgebras approximately commute, we construct an exact subalgebra that serves as a stable proxy for their intersection. This construction uses a recent theorem of Kitaev on the rigidity of approximate C∗-algebras.