Fix two qubits and one interaction, the controlled-NOT with control 𝑄2 and target 𝑄1. Give an experimenter local capabilities on one qubit only — resets, local unitaries, projective measurements with adaptive classical processing — and no operation on the other qubit except through the fixed gate. What can be detected about the two-qubit input? The accessible effects are derived from the declared operations and compared only afterwards with an algebra. Three results. (1) With capabilities on the target qubit, the span of accessible effects is exactly 𝒜1 = 𝑀2 ⊗ span{𝐼, 𝑍}, and every effect of 𝒜1 is the effect of one finite adaptive protocol: a non-classical algebra is accessible while the second qubit’s phase is not. (2) Restricting the local unitaries to {𝐼, 𝐻} leaves the span unchanged but strictly shrinks the set of implementable effects: two resource sets can distinguish exactly the same input pairs while differing in which exact measurements they can perform. (3) Relocating the capabilities to the control qubit gives span{𝐼, 𝑋} ⊗ 𝑀2; the two sites are exchanged by a symmetry of the gate, their access is incomparable, their common effects are those of an abelian algebra, and four distinctions are accessible from neither. The upper bounds are conserved-quantity arguments of the kind standard in quantum observability and indirect control; the exact effect-level statements are worked examples whose novelty is not claimed. Every identity and example is checked exactly in the accompanying scripts.
Fix two qubits and one interaction, the controlled-NOT with control 𝑄2 and target 𝑄1. Give an experimenter local capabilities on one qubit only — resets, local unitaries, projective measurements with adaptive classical processing — and no operation on the other qubit except through the fixed gate. What can be detected about the two-qubit input? The accessible effects are derived from the declared operations and compared only afterwards with an algebra. Three results. (1) With capabilities on the target qubit, the span of accessible effects is exactly 𝒜1 = 𝑀2 ⊗ span{𝐼, 𝑍}, and every effect of 𝒜1 is the effect of one finite adaptive protocol: a non-classical algebra is accessible while the second qubit’s phase is not. (2) Restricting the local unitaries to {𝐼, 𝐻} leaves the span unchanged but strictly shrinks the set of implementable effects: two resource sets can distinguish exactly the same input pairs while differing in which exact measurements they can perform. (3) Relocating the capabilities to the control qubit gives span{𝐼, 𝑋} ⊗ 𝑀2; the two sites are exchanged by a symmetry of the gate, their access is incomparable, their common effects are those of an abelian algebra, and four distinctions are accessible from neither. The upper bounds are conserved-quantity arguments of the kind standard in quantum observability and indirect control; the exact effect-level statements are worked examples whose novelty is not claimed. Every identity and example is checked exactly in the accompanying scripts.
An earlier note derived, for two qubits coupled by a fixed controlled-NOT and an experimenterwith local capabilities on the target qubit only, that the accessible effects are exactly those of thealgebra A1 = M2 ⊗ span{I, Z}: a non-classical algebra is accessible while the control qubit’s phaseis not. That boundary was derived for a remainder with no dynamics of its own. This note askswhether it is maintained when the remainder evolves freely under a time-independent HamiltonianH = h1 X + h2 Y + h3 Z that is fixed and known, and the experimenter may insert waits ofany chosen duration, adaptively, anywhere in a protocol. The answer is an exact dichotomy. Theaccessible effect set, and its span, remain unchanged under every finite adaptive protocol if andonly if [H, Z] = 0, i.e. h1 = h2 = 0: the remainder may precess about the axis the interaction reads.For every other member of the family the failure is demonstrated operationally — an explicit pairof states that no protocol without waits can distinguish, and the existing readout protocol with asingle wait of duration π/(4|h|) whose acceptance probabilities differ by 1 − n3^4 > 0, n = h/|h|.For the control H = X the accessible span becomes all of M4, by thirteen named protocols; whetherevery effect of M4 is then implementable is left open. Invariance of a protocol’s statistics underchanges of its wait durations is defined and reported separately from preservation of the accessibleset; in this family the two coincide, for a reason that is stated. The closed form of the wait, thecommutation criterion and the span bound are standard material; the contribution is the worked,exactly-verified criterion at both the effect and the span level. Nothing about robustness or genericHamiltonians is claimed. Technical note, not peer-reviewed. The deposit contains the note and areproducibility archive (one exact-arithmetic script with 34 checks, a failure-path harness, the runrecord and a checksum manifest).
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.
A small quantum arrangement — a reader qubit Q₁ coupled to a remainder qubit Q₂ through one fixed gate, with declared operations on Q₁ only — has access to some distinctions of the remainder and not to others. This manuscript follows that arrangement through fourteen research units — thirteen with a model, exact proofs and an executable checker (with certified interval enclosures wherever a propagation is read), and one, the maintenance criterion, a definition with checked decision logic — and organizes what they establish around four distinctions. First, protecting an access restriction is different from preserving the distinctions a reader can use: a universal leakage bound and an exact one-gate value are proved; a reset-based maintenance pulse prolongs limited leakage at the price of previously accessible information, while a reversible pulse prolongs it and preserves the full declared input class. Second, meeting protection criteria does not identify a mechanism: a prepared finite clock-controller meets four declared criteria at four sizes, and so does a static field with the clock decoupled, and so does the clock's free mean field at the largest size. Third, restoring a supporting condition does not necessarily recover the capability: coupling alignment is shown to be a supporting condition of the capability, and a restoring process that improves alignment in six matched realizations meets the full restoration standard in one, misses it narrowly in one and fails in four, while useful access is preserved in none. Fourth, retaining a relation, delivering it to a reader and using it internally are different achievements, and a supplied design's advantage can reverse when its context changes without the comparison identifying the mechanism: a sector record restores access at three of four endpoints under an added permission; an internal coupling improves the primary endpoint without reaching the reference; a two-by-two of axis term and gating cell reverses both contrasts. Each result is stated with what it does not establish. No conclusion about self-maintenance, ownership or experience is drawn. The deposit is a technical manuscript, not peer-reviewed, that consolidates fourteen closed research notes and adds no theorem; the supporting proofs are in the notes. The package contains the manuscript (PDF and Markdown), the fourteen notes with two errata, every checker and its run record, the failure-path harness with its recorded outputs and a release adapter, the two earlier baseline notes, a claim-to-proof map, a correction ledger, a source lock and a manifest with the SHA-256 of every file. Every number printed in the manuscript was read by script from a packaged run record. Running checks/run_all.sh reproduces every checker's recorded summary line with the declared dependencies only (Python 3.11 or later; sympy 1.14, mpmath 1.3, numpy); the harness coverage, including the checkers it does not cover, is stated in §10 and Appendix A. Drafting and computation were AI-assisted under the author's direction. Archive SHA-256: 119d835b2d2e19463f3eb6478ea34ddbdf4deea220a5d6d6fe528dcf7061bedb.
An essential element of classical computation is the ``if-then'' construct, that accepts a control bit and an arbitrary gate, and provides conditional execution of the gate depending on the value of the controlling bit. On the other hand, quantum theory prevents the existence of an analogous universal construct accepting a control qubit and an arbitrary quantum gate as its input. Nevertheless, there are controllable sets of quantum gates for which such a construct exists. Here we provide a necessary and sufficient condition for a set of unitary transformations to be controllable, and we give a complete characterization of controllable sets in the two-dimensional case. This result reveals an interesting connection between the problem of controllability and the problem of extracting information from an unknown quantum gate while using it.
Alice and Bob each have a qubit, and they would like to perform a two-qubit gate on them with the assistance of Charlie. The action of this gate depends on a parameter, which either Charlie knows, will receive later as classical information, or is encoded in a program state. We discuss both the case in which Alice and Bob send Charlie their qubits, which he entangles with his state, and the case where Alice and Bob use teleportation to entangle their states with Charlie's state. All of these procedures depend on the ability of Alice and Bob to apply local correction operations that depend on the results of measurements that Charlie makes, and we discuss under what conditions such operations are possible.
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.