Song–Chen Conjecture 2 at n = 4 — the uniform case

A complete proof for uniform weights; the general case reduced to a finite linear program.

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Background

The spin-alignment conjecture of Alhejji and Knill [2] would, in its strong form, single-letterise the quantum capacity of platypus-type families — capacities otherwise out of reach, since regularisation is genuinely needed [4] and non-additivity is generic [3].

Song and Chen [1] refuted the strong form with an explicit n = 3 counterexample, and in the same work isolated a weaker statement — their Conjecture 2, a compatible-marginal majorization inequality — which survives that counterexample and would still serve several of the intended applications. They prove it at n = 3 for the maximally mixed reference with weights on 2-subsets. The next case is n = 4.

The problem

For a state ρ on four qubits and a weight μ on subsets, let H be the Hamiltonian assembled from the compatible marginals of ρ and let T be the reference spectrum. Conjecture 2 asserts that for every r,

Sr(λ(H)) ≤ Sr(T),

where Sr is the sum of the r largest entries. The inequality is not vacuous: fed the n = 3 counterexample of [1], the same quantity reaches 0.8344… > 5/6.

Uniform weights

For uniform μ the inequality holds at n = 4, for every r.

The obstruction was one tail inequality previously argued on a grid. Because the slack is exactly zero — attained at the all-zeros vertex — a grid cannot certify it, and the step has to be made finite and exact.

Writing the spectrum as d(x) = 1 + ⟨ε(x), u⟩ with ε ∈ {±1}⁴ and u ∈ [0, ½]⁴, the complementary pairing d(x) + d(x̄) = 2 holds exactly, so the sixteen values sort into eight pair-minima below eight pair-maxima. Each tail sum then becomes a minimum over which pair to drop — piecewise linear on an arrangement with exactly seventeen vertices, small enough to enumerate in exact rationals. All six tail inequalities hold, each meeting its threshold with slack exactly zero.

Clipping is essential: the naive bound fails, so the vertex enumeration is necessary rather than a convenience.

General weights

The linear-programming bound of [2, Thm. 3.2] applies and its validity was confirmed to machine precision. But the reduction to the target holds only after taking a minimum over the three matching-groupings — any fixed grouping overshoots. That makes the remaining obligation sharper than it first looked.

Verification

The search ran on an independently built embedding agreeing with the original to machine zero, with the detector validated by firing correctly on the n = 3 counterexample. Roughly 120k structured non-uniform weight configurations, a differential-evolution attack on the joint problem in (ρ, μ), and 60-digit arithmetic on the delicate cases returned no violation.

What remains open

  1. General weights at n = 4. The LP route is a verified computational lead, not a proof; the min-over-groupings requirement is the specific obstacle.
  2. General n. Nothing here is tied to n = 4 except the size of the arrangement, which grows fast. The complementary pairing is the part worth trying to preserve.
  3. What the conjecture would buy. Even granting Conjecture 2 in full, the route to a single-letter capacity runs through the bridge lemma of [2] — which is false. What survives of the original programme deserves restating carefully.

References

  1. Z. Song and L. Chen, A counterexample to the strong spin alignment conjecture, arXiv:2603.25410.
  2. M. A. Alhejji and E. Knill, Towards a resolution of the spin alignment problem, arXiv:2307.06894.
  3. F. Leditzky, D. Leung, V. Siddhu, G. Smith and J. A. Smolin, Generic nonadditivity of quantum capacity in simple channels, arXiv:2202.08377.
  4. T. Cubitt, D. Elkouss, W. Matthews, M. Ozols, D. Pérez-García and S. Strelchuk, Unbounded number of channel uses are required to see quantum capacity, arXiv:1408.5115.