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.
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.
We construct a rank-one minimal informationally complete measurement (MIC) with an invertible dual operator and prove that, for arbitrary finite input and output dimensions, a channel universally preserves measurement incompatibility if and only if its input space contains a noiseless code subspace of the full output dimension. Preserving the compatibility threshold of a finite family of binary measurements already guarantees such a code. Furthermore, for every d≥2, there exists a family of binary measurements that is incompatible on the full space but compatible after compression to any proper subspace. These results resolve the conjecture of Chitambar and Zhang and the existence question posed by Uola and coauthors. We also give an explicit construction of exact encodings and show that, for a sufficiently small shift in the compatibility threshold, the transmission error of an approximate encoding is bounded by a constant times the square of the shift. These results establish a connection between measurement compatibility, quantum steering, and quantum error correction.
We prove a spectral majorization law for compatible three-qubit marginals: for every global state, every common single-qubit replacement state, and arbitrary nonnegative weights on all eight supports, the weighted marginal sum is majorized by its perfectly aligned counterpart. For normalized weights, this simultaneously minimizes every Schur-concave output entropy. We characterize equality, establish stability bounds and an incompatibility witness, and determine the single-block Holevo information for correlated unbiased replacement. We also develop input-certification protocols with explicit measurement, noise, calibration, and sample-complexity assumptions, together with rigorously delimited four-qubit results and a channel-compatibility example. The unrestricted noncommuting four-qubit middle-rank problem remains open. Complete proofs, exact certificates, and reproducible source are included.
Quantum measurements can remain incompatible yet become impossible to certify when all probe states first pass through a quantum channel. We formulate this loss of certifiability as operational concealment and identify its exact finite-dimensional threshold. Under tomographically complete input access, a channel conceals some incompatible binary POVM pair if and only if its Heisenberg adjoint has nontrivial kernel. Thus, in this channel restricted single-use setting, rank loss rather than contraction is the universal transition for exact concealment: injective dissipative dynamics may render the effective measurements compatible while leaving exact concealment impossible. We quantify this distinction through a minimum-gain lower bound and show that regular time-local dynamics cannot produce exact concealment at finite time. We then classify all qubit channels whose adjoint fibres admit a positive unital retraction, reducing concealment to ordinary joint measurability of canonical representatives. For orthogonal projected qubit measurements we obtain an exact robustness formula and a strict separation from ordinary incompatibility robustness. Finally, quotient-space duality expresses this robustness as an experimentally accessible additive advantage in positive-payoff channel-output games. We also prove invariance under adjoining a passive finite-dimensional reference system for lifted target measurements, even when arbitrary joint compatible simulators are allowed on the enlarged output space. These results establish rank-sensitive criteria for certifying measurement incompatibility under restricted channel access.
We study the canonical USp(4)-covariant quantum compression problem in which the fifteen-dimensional carrier W = V5 ⊕ V10 is encoded through a four-dimensional covariant memory M = V4 and decoded, and we ask for the optimal entanglement fidelity F*(p) of the composite channel on a trace-one, sector-weighted reference state ρ_p. This problem is the operational face of a structural fact: the same pair of sectors 5 ⊕ 10 that makes USp(4) on C^4 state-homogeneous but not observable-uniform is exactly the modal content a minimal covariant memory is asked to carry. Our contribution is not a resolution of the underlying capacity question but a demonstration that this problem admits an exact reduction and a machine-checkable global bound: the reduction and the final lower- and upper-bound certificates are exact, while only the intermediate relaxation-solving step is numerical. We give an exact representation-theoretic reduction — Wedderburn splitting of the covariant commutant over Q(i), an induced-involution positivity bridge, and an exact trace-preservation reduction to fifteen free rational coordinates — turning F*(p) into a bilinear polynomial optimization over a product of two explicit semialgebraic bodies, each the affine preimage of the product cone (R+ × SOC4 × SOC4). Exploiting correlative sparsity we build an order-two sum-of-squares relaxation, solve it numerically for discovery, reconstruct the resulting SOS upper bounds in exact rational arithmetic, and separately certify exact rational feasible lower bounds from explicit witnesses. At p = 1/2 and p = 7/8: 0.0749 ≤ F*(1/2) ≤ 0.21, and 0.1546 ≤ F*(7/8) ≤ 0.643 (exact rationals in the paper). The upper bounds are Positivstellensatz certificates whose polynomial identity and positive semidefiniteness are verified with no optimizer in the loop by a standalone verifier we provide. Throughout we keep three visibly separate registers — exact theorems, numerical diagnostics, and open or conditional interpretations — and we are explicit that the exact optimum, the exactness of the relaxation, and the width of the certified interval remain open. Package: this record includes the manuscript, a single 30-page combined archival PDF (cover + manuscript + auditable reduction-supplement listing) inside the package, and a verification package with two independent machine-checkable obligations (the exact reduction chain and the certified two-sided enclosure). Manuscript licensed CC BY 4.0; code licensed MIT.
Quantum reference frame (QRF) transformations are usually presented as unitary or gauge-fixing changes of coordinates. That picture is exact for ideal frames but fails for finite clocks, gyroscopes, POVM-defined tokens, observers correlated with the targets they describe, and relativistic frames with quantum boost and clock degrees of freedom; the obstruction is loss of distinguishability. We model a non-ideal frame by a normal covariant perspective channel. A deterministic transformation A → B then exists exactly when the kernel of A's channel is contained in the kernel of B's channel, and is unique on the preparable image. Ancilla-stable transformations are complete-order quotient morphisms, while a laboratory implementation on a chosen output space is a separate Choi-Arveson extension problem. The common invariant content of a family of frames is a Heisenberg operator system, carrying an algebraic product only on its product-compatible core. For finite frames in the regular representation, concrete covariant degradability A → B is decided by a group-convolution linear program with explicit dual witnesses ; for Abelian groups this becomes a Fourier mode-support resource lattice. Our main result is that for every finite group this convolution linear program already equals the full diamond-norm deficiency: coherent, multiplicity-exploiting post-processings never outperform random translations, so the resource theory of frame degradation is entirely classical. The same quotient principle governs process tensors, protected relational sectors, entangled observers, relativistic covariance — the boosted-spin perspective being an explicit U(1) dephasing channel — and observer networks. Reproducible numerical certificates accompany every analytic claim.
Measurement compatibility asks whether a family of quantum measurements can be simulated by one parent measurement followed by classical post-processing. We extend this measurement compatibility to the notion of \emph{d-compatibility} for measurements, channels, and instruments whose parent passes a d-dimensional quantum system and unlimited classical information to the post-processing. We formulate d-compatibility as a common linear factorization of operator spaces and operator systems and show that d-compatible families form a closed set, even with infinitely many settings and a separable infinite-dimensional input. To certify d-incompatibility, we define the compatibility functional as the smallest product of completely bounded norms over common factorizations through a d-dimensional register. Using factorization theory of operator spaces, we show that measure-and-prepare channels defined in terms of mutually unbiased bases and pinching channels defined in terms of Heisenberg-Weyl operators are d-incompatible in certain regimes of d. We also link d-compatibility of multi-instruments to the preparability of Choi assemblages using channel-state duality, and use this connection to show that for bipartite states the (in)compatibility of instruments can be used to certify Schmidt numbers and the implementability of the instrument using one-way LOCC protocols. Finally, our compatibility functional also bounds the generalized robustness of compatibility, which we interpret through a memory-bounded nontransient preparation game.