A counterexample to the strong spin alignment conjecture

TL;DR

This paper disproves the strong spin alignment conjecture in quantum information theory by providing a specific counterexample involving three qubits, challenging previous assumptions about spectrum majorization.

Contribution

It constructs the first explicit counterexample to the strong spin alignment conjecture for three qubits, demonstrating its invalidity in general.

Findings

Counterexample in three qubits disproves the conjecture

Two-body states incompatible with any three-qubit global state

Challenges the spectrum majorization assumption in quantum channels

Abstract

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.