Multi-copy quantum state discrimination provides a minimal setting in which the nonlocality of measurements can be revealed even when the input states themselves are product states. In this work, we investigate the two-copy discrimination of three qubit pure states with equal prior probabilities, and compare the optimal performances of global and separable measurements in two representative discrimination tasks: minimum-error discrimination (ME) and unambiguous state discrimination (USD). A general three-state qubit ensemble is parametrized by three Bloch-sphere parameters, which determine the pairwise overlaps and relative phase of the states. After the two-copy embedding, the three signal states lie in the symmetric subspace of two qubits. The optimal success probabilities under global measurements are formulated as semidefinite programs. For separable measurements, we use the positive-partial-transpose condition, which exactly characterizes separability in the 2 ⊗ 2 system, and thus obtain the exact optimal separable performance.
Using the double-trine ensemble as an analytical benchmark, we first recover its known measurement hierarchy: in ME, separable measurements can attain the global optimum, whereas in USD, global measurements strictly outperform all separable measurements. We then extend the analysis to general three-state geometries by scanning the parameter space and refining the regions where a nonzero globalover-separable gap appears. Our results show that the equality between global and separable measurements in ME is not generic. For certain asymmetric, nearly degenerate configurations, in which two states are very close while the third state has a moderate separation from them and a nontrivial relative phase, global measurements can strictly outperform all separable measurements; a representative gap is about 8.1×10^-3. In contrast, the largest USD advantage is concentrated near the uniformly symmetric doubletrine geometry, where the gap reaches 0.25, and perturbations away from this symmetry generally reduce the advantage.
These results demonstrate that measurement nonlocality in two-copy three-state discrimination is strongly task dependent. The global-measurement advantage in ME is favored by nonuniform, nearly degenerate state geometries, whereas that in USD is closely tied to the symmetric double-trine-type geometry. This distinction clarifies that the advantage of global measurements in multi-copy state discrimination cannot be inferred solely from pairwise overlaps, but must be understood jointly from the discrimination criterion and the overall geometry of the state set.