Two finite-order quantum controls may each be compatible with a qubit while their joint algebra is not. We study a cyclic control Q and an independent involution R with target algebra C_n x C_2. For even n >= 4, no faithful projective qubit realization exists: in PU(2) ≃ SO(3), a nontrivial involution commuting with an order-n rotation collapses to Q^(n/2). An explicit qutrit construction realizes the full algebra, so the minimum projective dimension is three; n = 4 is the first obstructed even member and has an eight-ray regular orbit. The representation-theory obstruction itself lies in a mature mathematical area. The operational contribution developed here is a calibrated qubit-consistency test: for exact quarter-turn and half-turn targets, the commutator angle gives a tight state-independent fidelity bound and a tight optimized binary-contrast cap. We derive a sufficient angular-error version and extend the test to general CPTP gate deviations through diamond distance. The explicit qutrit construction reaches the algebraic maximum binary contrast of one using two control sequences and a common binary measurement. The result is a calibrated falsification test for a qubit control model, not a device-independent dimension witness, and it remains entirely within standard finite-dimensional quantum mechanics.
Two finite-order quantum controls may each be compatible with a qubit while their joint algebra is not. We study a cyclic control Q and an independent involution R with target algebra C_n x C_2. For even n >= 4, no faithful projective qubit realization exists: in PU(2) ≃ SO(3), a nontrivial involution commuting with an order-n rotation collapses to Q^(n/2). An explicit qutrit construction realizes the full algebra, so the minimum projective dimension is three; n = 4 is the first obstructed even member and has an eight-ray regular orbit. The representation-theory obstruction itself lies in a mature mathematical area. The operational contribution developed here is a calibrated qubit-consistency test: for exact quarter-turn and half-turn targets, the commutator angle gives a tight state-independent fidelity bound and a tight optimized binary-contrast cap. We derive a sufficient angular-error version and extend the test to general CPTP gate deviations through diamond distance. The explicit qutrit construction reaches the algebraic maximum binary contrast of one using two control sequences and a common binary measurement. The result is a calibrated falsification test for a qubit control model, not a device-independent dimension witness, and it remains entirely within standard finite-dimensional quantum mechanics.
Can an encoded quantum state be controlled continuously without repeated measurements to prevent leakage? This study gives a constructive answer for a four-spin-2 AKLT plaquette, where a three-dimensional singlet code sits inside a 625-dimensional physical Hilbert space. The result combines an exact restriction with a coherent control scheme: strictly leakage-free two-site Hamiltonians cannot provide universal control of the full encoded qutrit, yet two explicitly constructed interactions enable universal single-qubit control without intermediate measurements or postselection within a two-dimensional subspace. The classification begins with the complete physical leakage map for all six spin pairs and exchange polynomials through degree four. Exact rational calculations establish rank L = 6 and dim ker L = 18 for the 255 × 24 matrix. Despite this large family of leakage-cancelling combinations, every accessible logical Hamiltonian obeys [A, R] = 0, where R is a fixed geometric involution. Its two-dimensional and one-dimensional eigenspaces therefore remain separate, obstructing universal SU(3) control. The restriction extends to arbitrary anisotropic two-site Hamiltonians that preserve the code. Within the rotationally invariant pair-polynomial model, interactions confined to the four plaquette edges act only as logical scalars; noncommuting protected controls require at least three pair supports and polynomial degree three. The constructive solution uses quadratic and cubic exchange Hamiltonians acting simultaneously on three pairs, including a diagonal link. Their coefficients cancel physical leakage exactly, so QH₂P = QH₃P = 0 throughout the full system, with P projecting onto the code and Q onto its complement. Their logical generators satisfy Lie(iA, iB) ≅ u(2) ⊕ u(1), allowing arbitrary coherent qubit gates while preserving the code at every intermediate time. The manuscript supplies exact certificates and verification procedures, together with operator-norm and calibration-error bounds that make the engineering requirements explicit. The practical opportunity is to remove intermediate code-check measurements from this control scheme; demonstrating a hardware advantage requires calibrated simultaneous interactions, including the diagonal channel, and measured gate fidelity and latency.
Consider exact scalar gate-sequence data consisting of one preparation, one binary effect, and the probability of that effect after every word over a fixed two-gate alphabet. This preprint constructs a single word law of exact noncommutative Hankel rank four that admits a control-irreducible unitary quantum realization on C^d for every finite d >= 2. In each realization the state is full rank, the binary effect is strict, and the two controls generate the full matrix algebra M_d(C), excluding direct-sum padding and invariant spectator sectors. The associated four-dimensional angular-momentum operator systems have the same first-level Lorentz cone in every dimension, while a level-2 matrix pencil is positive exactly when t >= (d+1)/(d-1). Thus matrix-level-two order distinguishes all members of the constructed family. The result is a scoped non-identifiability theorem for the Hilbert-space dimension of an arbitrary compatible irreducible realization. It does not claim that the minimum compatible dimension is unidentifiable, does not contradict informationally complete gate set tomography within a declared model dimension, and does not reconstruct a tensor-product structure. The record includes the preprint PDF, LaTeX source, bibliography, numerical verification script and output, citation metadata, and a targeted priority audit.
We introduce the unitary-accessible coherence-erasure distance, which measures the minimum unitary-control cost required to transform a quantum state into an incoherent state without changing its spectrum. The target states are restricted to incoherent states on the same unitary orbit, leading to a geometry different from conventional state-space proximity. We establish inequalities relating orbit-restricted quantum transport, Bures geometry, and the unitary-control distance. These quantities coincide for pure states but can differ for mixed states. For qubits, we derive an exact expression and show that, in the nondegenerate case, the coherence-erasure cost depends on the eigenbasis orientation rather than on the eigenvalues. For nondegenerate qutrits, we reduce the problem to an optimization over the monomial unitary group. We derive the tight universal bound arccos[(22−1)/4] and construct an explicit eigenbasis that attains it. The resulting distance also determines the minimum coherence-erasure time under bounded unitary driving, giving it a direct operational meaning.
We study the reconstruction of a qubit factor from global operator relations without presupposing a tensor-product structure. Given two Hermitian operators (A,B) on an unpartitioned finite-dimensional Hilbert space, we quantify how approximate involution and anticommutation relations,[|A^2-I|,\qquad |B^2-I|,\qquad |{A,B}|,]control the existence of a nearby exact same-space qubit representation. The construction is explicit and yields quantitative upper and lower bounds relating observable relation defects to distance from an exact marked qubit factor. A central obstruction is then exhibited: arbitrarily small perturbations of an exact qubit-factor pair can generate the full ambient matrix algebra, causing the exact commutant of the raw operators to collapse discontinuously from the complementary matrix factor to the scalars. Thus exact-algebra or exact-commutant reconstruction from noisy operators is intrinsically unstable. Despite this discontinuity, the complementary factor remains quantitatively recoverable from a soft commutant. For the commutator map associated with an exact rounded pair, we determine the complete singular-value structure and use its fixed spectral separation to control the low-singular-value subspace of the original unrounded operators. If[\rho=2\sqrt{p^2+q^2}<1,]where (p) and (q) bound the same-space rounding errors, the reconstructed complementary subspace satisfies[\left|P_{\widehat{\mathcal C}}-P_{\mathcal A’}\right|_{2\to2}\le\frac{\rho}{2-\rho},]while the observed spectral gap is at least (2-2\rho). Composing the rounding bounds with the relation defects gives an explicit relation-defect-to-complement certificate. The results separate three notions that coincide only in the noiseless idealization: exact generated algebra, exact commutant, and robustly recoverable complementary factor structure. The paper also clarifies the distinction between factor existence and tensor-product-structure uniqueness and positions the result relative to stability of approximate qubit relations, almost-commuting operator algebras, finite-dimensional (C^*)-algebra stability, and numerical commutant reconstruction.
This preprint studies finite-shot reconstruction of hidden quantum factor structure from global calibration-free sequence statistics without assuming an internal tensor-product cut in advance. A probability-only complex-projective 3-design self-test is used as an imported anchor for the active Hilbert/Jordan geometry. From this anchor, global control maps are reconstructed with explicit confidence bounds, including a weighted estimator for unequal shot counts and missing reference settings whenever the surviving frame remains full rank. The main certification result is a whole-commutant-energy lower bound. For reconstructed controls, the sum of the lowest commutator singular energies yields a basis-free lower bound on the aggregate projective Hilbert–Schmidt distance to every common hidden matrix factor of a specified multiplicity. Combined with exact Choi-to-projective-unitary conversion and finite-shot confidence balls, this produces a one-sided lower bound on distance to the complete common-factor model class. An explicit exact factor witness gives the complementary PROXIMAL-RECOVER certificate; otherwise the procedure returns ABSTAIN. The paper also proves that exact generated-algebra / tensor-product-structure type is not uniformly identifiable from finite samples without a structural separation margin. For the fixed two-qubit 60-state stabilizer benchmark, the paper provides an anchor-inclusive raw-count validation and an exact-rational / outward-rounded one-sided trust path from integer counts through anchor and control confidence radii to factor RECOVER/REJECT. The scope is deliberately operational: the result certifies active factor structure supported by the augmented data and does not claim identification of arbitrary inactive spectator degrees of freedom or a uniquely true microscopic decomposition.