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.
We develop a device-independent framework for testing quantum channels. That is, we falsify a hypothesis about a quantum channel based only on an observed set of input-output correlations. Formally, the problem consists of characterizing the set of input-output correlations compatible with any arbitrary given quantum channel. For binary (i.e. two input symbols, two output symbols) correlations, we show that extremal correlations are always achieved by orthogonal encodings and measurements, irrespective of whether or not the channel preserves commutativity. We further provide a full, closed-form characterization of the sets of binary correlations in the case of: (i) any dihedrally covariant qubit channel (such as any Pauli and amplitude-damping channels) and (ii) any universally-covariant commutativity-preserving channel in an arbitrary dimension (such as any erasure, depolarizing, universal cloning and universal transposition channels).
Quantum measurements on a two-level system can have more than two independent outcomes, and in this case, the measurement cannot be projective. Measurements of this general type are essential to an operational approach to quantum theory, but so far, the nonprojective character of a measurement can only be verified experimentally by already assuming a specific quantum model of parts of the experimental setup. Here, we overcome this restriction by using a device-independent approach. In an experiment on pairs of polarization-entangled photonic qubits we violate by more than 8 standard deviations a Bell-like correlation inequality that is valid for all sets of two-outcome measurements in any dimension. We combine this with a device-independent verification that the system is best described by two qubits, which therefore constitutes the first device-independent certification of a nonprojective quantum measurement.
The next frontier in device-independent quantum information lies in the certification of scalable and parallel quantum resources, which underpin advanced quantum technologies. We put forth a simultaneous self-testing framework for maximally entangled two-qudit state of local dimension
m∗=2⌊n/2⌋ (equivalently
⌊n/2⌋ copies of maximally entangled two-qubit pairs), together with
n numbers of anti-commuting observables on one side. To this end, we employ an
n-settings Bell inequality comprising two space-like separated observers, Alice and Bob, having
2n−1 and
n number of measurement settings, respectively. We derive the local ontic bound of this inequality and, crucially, employ the Sum-of-Squares decomposition to determine the optimal quantum bound without presupposing the dimension of the state or observables. We then establish that any physical realisation achieving the maximal quantum violation must, up to local isometries and complex conjugation, correspond to a reference strategy consisting of a maximally entangled state of local dimension of at least
2⌊n/2⌋ and local observables forming an irreducible representation of the Clifford algebra. This construction thereby demonstrates that the minimal dimension compatible with
n mutually anticommuting observables is naturally self-tested by the maximal violation of the proposed Bell functional. Finally, we analyse the robustness of the protocol by establishing quantitative bounds relating deviations in the observed Bell value to the fidelity between the realised and the ideal strategies. Our results thus provide a scalable, dimension-independent route for the certification of high-dimensional entanglement and Clifford measurements in a fully device-independent framework.
This work determines the sharp finite-dimensional onset of incompatibility for finite families of U(1)-covariant phase observables. In dimension two, every covariant phase observable is shown to be a convolutional postprocessing of a single canonical qubit phase observable. Consequently, every finite family of qubit covariant phase observables, including arbitrary finite coarse-grainings, is jointly measurable. In dimension three this universal compatibility breaks down. Four rank-two qutrit covariant phase observables are constructed whose binary interval coarse-grainings form a rigorously certified four-Specker family. Triple compatibility is established using explicit parent POVMs with exact-rational positivity tests and interval-bounded affine corrections, while four-way incompatibility is certified by independent dual semidefinite witnesses with exact rational normalization and interval evaluation. The work therefore establishes the sharp threshold d_min = 3 for finite-family incompatibility within this phase-covariant class. It also characterizes the qutrit canonical-smearing sector, proves robustness on a full-dimensional 28-parameter neighborhood, and provides a common-commutant certificate excluding a hidden qubit reduction. Supplementary material contains detailed analytic derivations and certificates. The accompanying reproducibility package provides verification code, exact/interval certificates, numerical summaries, and machine-readable outputs.
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.