Strategic Non-Shareability of Quantum Correlations

Correlations distributed by a mediator can be useful for coordination but vulnerable to inheritance by a colluder. We formalize the obstruction to such inheritance as a source-certified resource theory of strategic non-shareability. The free objects are symmetrically extendible sources, the free operations are shareability-preserving maps, and the trace distance to the free set is a faithful convex monotone. For Werner and isotropic sources in arbitrary local dimension, the resource has the exact form $D_m=c(d)(p-p_m^{\ast }{)}_{+}$, with $p_m^{\ast }$ the Johnson–Viola shareability threshold. For qubit Werner sources, tomographically complete Pauli measurements yield the exact one-colluder capacity$$C_1^{\mathrm{t}\mathrm{o}\mathrm{m}\mathrm{o}}(p)=\frac{1}{12}[(3p-1)-\sqrt{(3p+1)(1-p)}{]}_{+}\mathrm{.}$$ We prove that this anti-collusion resource is independent of Bellnonlocality: the Bell and shareability orderings cross, so some Bell-nonlocal states are strictly less collusion-resistant than Bell-local ones. Finally, we give an aligned Pauli coordination game whose observed behaviour has a local hidden-variable model for every visibility, making device-independent certification empty, while source-certified quantum anti-collusion is positive exactly above the extendibility threshold. These results identify symmetric non-extendibility, rather than Bell nonlocality, as the boundary of source-certified collusion resistance.