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Compatibility Restores Spin Alignment for Three Qubits

Dongming Zhang, Qihang Wang

Published Aug 30, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.5281/zenodo.22168800

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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.
The strong spin-alignment conjecture fails for three qubits when the subset states are chosen independently. We prove that it holds when they are marginals of one global state: for every three-qubit state, every positive semidefinite qubit operator Q, and every nonnegative set of subset weights, the aligned output majorizes the corresponding marginal mixture. The proof classifies the extreme rays of cones on which a fixed set of aligned eigenvalues is largest. This reduces arbitrary weights to two-term sums and the complete one- and two-body layers. For arbitrary numbers of subsystems, we also characterize when one unital quantum channel simultaneously realizes all marginal replacements. This criterion proves alignment for fully PPT states. A separate argument proves alignment for locally rotated stabilizer states in every prime local dimension, at maximally mixed background. These results determine the minimum output Rényi entropies of the associated three-qubit replacement channels. Under either of the two input restrictions above, they also determine the many-use classical capacity and the exact strong-converse exponent for correlated replacement channels of arbitrary size.
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.
The spin alignment conjecture was originally formulated in connection with the additivity of coherent information for a class of quantum channels known as platypus channels. Recently, a stronger majorization-based version was proposed by M. A. Alhejji and E. Knill [Commun. Math. Phys. 405, 119, 2024], asserting that the spectrum of the alignment operator is always majorized by that of the perfectly aligned configuration. In this letter, we show that this strong spin alignment conjecture is false in general by constructing an explicit counterexample in the smallest unresolved case, namely three qubits. The example uses two-body states that are not jointly compatible with any single three-qubit global state, which naturally leads to a compatibility-constrained variant of the conjecture.
Majorization and von Neumann entropy provide related but inequivalent descriptions of spectral disorder in quantum optical states. We investigate the channels preserving these descriptions and propose a three-mode photonic test of their difference. For channels with equal input and output dimensions, we show that constant output purity and third spectral moment on the unitary orbit of one simple-spectrum state determine the majorization-preserving channel forms, without assuming unitality.For qubits, output purity alone is sufficient. Preservation of entropy equality is substantially more restrictive: in dimension d⩾3d\geqslant3d⩾3, even a local condition near the maximally mixed state permits only replacement and unitary channels. Qubits admit an additional depolarizing family, but this exception disappears under extension by an idle auxiliary qubit. We construct exactly entropy-matched qutrit inputs whose output entropies split at cubic order under nontrivial depolarizing noise. A proposed single-photon implementation uses three optical modes, randomized Weyl transformations and population or spectral-moment measurements.Explicit expressions are also obtained for preparation mismatch,readout uncertainty and input-independent loss. The results connect quadratic and cubic spectral information with experimentally interpretable tests of quantum-channel structure.
When absolute maximal entanglement is unavailable in a state family, different objectives can select different distributions of residual entanglement. We determine exact trade-offs for qutrit stabilizer states and binary--ternary product states. For eight qutrits, evaluation of the complete published classification gives three nondominated balanced-cut entanglement profiles, characterized by the probabilities of obtaining three or four maximally entangled qutrit pairs. We relate the squared entropy-deficit objective to code weight enumerators and use a uniformity-forcing bound to prove that its eleven-qutrit minimum is six. The unique minimizing Clifford class is an already classified code, whose entropy profile also attains a sharp bound on average reference--erasure mutual information among four-uniform eleven-qutrit stabilizer states. In local dimension six, explicit sector alignments attain 56 maximally mixed four-party reductions, the maximum for arbitrary binary--ternary product states. Within the stabilizer product family, the exact cost--count Pareto boundary consists of (44,48)(44,48)(44,48) and (48,56)(48,56)(48,56). These results identify when aggregate optimization determines operational entanglement properties and when it does not. Published classifications supply coverage; analytical bounds, explicit witnesses, and independent arithmetic checks establish the stated consequences.
Mutually unbiased bases capture perfect complementarity between two quantum measurements. Extensions beyond pairwise unbiasedness have been proposed, but essentially no non-trivial higher-order constructions are known. We introduce kkk-fold unbiased measurements (kkk-UMs), extending the kkk-fold unbiased bases notion of [arXiv:1706.04446] from rank-one basis measurements to arbitrary-rank projective measurements, and show that this higher-rank setting supports a much richer theory. We develop the notions of algebraic and spectral kkk-UMs and prove that they coincide for rank-one measurements and triples of measurements (3-UMs). We establish strong no-go results for higher-order rank-one constructions and three-outcome 3-UMs, but obtain infinitely many higher-rank triples using Hadamard matrices and Clifford algebras. We then give these structures an exact operational interpretation in terms of measurement incompatibility. For 3-UMs with any number of outcomes, we determine their generalised incompatibility robustness exactly and construct an explicit joint measurement for their noisy versions at the compatibility threshold. Finally, we implement the symmetry reduction of the sum-of-squares hierarchy recently introduced in [New J. Phys. 28, 064509 (2026)] and give numerical evidence that the noise thresholds arising from the kkk-UM analysis may characterise the asymptotic behaviour of this hierarchy. In particular, with very high precision, we numerically show that our constructed four-outcome 3-UMs are among the most incompatible triples of four-outcome measurements.