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We develop a new $\mathcal{PT}$-symmetric approach for mapping three purequbit states, implement it by the dilation method, and demonstrate it with asuperconducting quantum processor provided by the IBM Quantum Experience. Wederive exact formulas for the population of the post-selected$\mathcal{PT}$-symmetric subspace and show consistency with the Hermitian case,conservation of average projections on reference vectors, and Quantum FisherInformation. When used for discrimination of $N = 2$ pure states, our algorithmgives an equivalent result to the conventional unambiguous quantum statediscrimination. For $N = 3$ states, our approach provides novel propertiesunavailable in the conventional Hermitian case and can transform an arbitraryset of three quantum states into another arbitrary set of three states at thecost of introducing an inconclusive result. For the QKD three-state protocol,our algorithm has the same error rate as the conventional minimum error,maximum confidence, and maximum mutual information strategies. The proposedmethod surpasses its Hermitian counterparts in quantum sensing using non-MSEmetrics, providing an advantage for precise estimations within specific dataspace regions and improved robustness to outliers. Applied to quantum databasesearch, our approach yields a notable decrease in circuit depth in comparisonto traditional Grover's search algorithm while maintaining the same averagenumber of oracle calls, thereby offering significant advantages for NISQcomputers. Additionally, the versatility of our method can be valuable for thediscrimination of highly non-symmetric quantum states, and quantum errorcorrection. Our work unlocks new doors for applying $\mathcal{PT}$-symmetry inquantum communication, computing, and cryptography.

$\mathcal{PT}$-symmetric mapping of three states and its implementation on a cloud quantum processor