Strong nonlocality with more imaginarity and less entanglement

Complex numbers are central to the formulation of quantum mechanics, yet their role as a genuine resource is only beginning to be understood. In this work, we demonstrate that quantum states with intrinsically complex amplitudes provide a fundamental advantage in state discrimination. We construct a set of five orthogonal three qubit pure states and show that the set is strongly nonlocal if and only if it includes imaginary components. Such a set becomes locally indistinguishable not only under local measurements but also against bipartite joint measurements. This enhanced robustness makes imaginarity a valuable resource for quantum cryptography since information encoded in these states remains secure against collaborative group attacks. Our results highlight a new operational role of complex numbers in quantum theory and establish imaginarity as a key enabler of cryptographic security. However, we reconstruct the set by replacing the only product state with a biseparable state whose shared entanglement between two parties nullifies the effect of imaginarity in exhibiting strong nonlocality. In fact, we show how entangling correlations between two distant parties can dilute the effect of imaginarity, and conversely, how imaginarity itself can mimic the role of entanglement. Nevertheless, the set spans a locally indistinguishable subspace, while its complement, in turn, produces distillable genuine entanglement. Notably, this is the smallest possible Unextendible Biseparable Basis (UBB) that resolves the open problem regarding the existence of a UBB of cardinality d²+d-1 in d^{otimes 3}. Our construction yields a highly powerful set, rich in resources from multiple perspectives of quantum information theory, including many-copy discrimination, unambiguous identification, entanglement creation from product state, and non-entangling perturbations.