Decoherence-controlled collective criticality in a two-dimensional quantum Stag Hunt

Physical decoherence can preserve the microscopic strategic neutrality condition of a quantum game while changing the thermodynamic regime of the corresponding interacting population. We demonstrate this for an Eisert–Wilkens–Lewenstein ($\mathit{\text{EWL}}$) Stag Hunt embedded as independent nearest neighbor encounters on a square lattice. For the restricted strategies $\mathsf{Q}=i\mathbb{Z}$ and $\mathsf{D}=i\mathbb{Y}$, noisy two-player payoff matrices are determined for phase damping, depolarization, and amplitude damping and mapped exactly to channel dependent Ising parameters $\mathfrak{J}(\mathrm{\Gamma },p)$ and $\mathfrak{H}(\mathrm{\Gamma },p)$. Phase damping and depolarization show the clearest contrast: they share the same microscopic neutrality branch $\mathfrak{H}=0$, while only depolarization suppresses the interaction as $(1-p{)}^2$. At $\beta =1$, this produces an exact depolarization-driven square-lattice critical point at $p_{\ast }\approx 0.233460\dots$, whereas phase damping remains in the ordered coexistence regime along the same neutrality branch. Monte Carlo finite size scaling is consistent with two-dimensional Ising criticality and distinguishes field driven coexistence below $p_{\ast }$ from a smooth crossover above it. Amplitude damping additionally reveals a strong dependence on channel placement: the post-strategy neutrality branch reaches $\mathrm{\Gamma }=0$ at $p=1\mathrm{/}3$ and then disappears. Resource negativity further shows that microscopic two-qubit entanglement and collective interaction strength are distinct quantities. The resulting extended lattice remains an ordinary classical Ising system.