<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">P</mml:mi><mml:mi class="MJX-tex-caligraphic" mathvariant="script">T</mml:mi></mml:mrow></mml:math>-sy

<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">P</mml:mi><mml:mi class="MJX-tex-caligraphic" mathvariant="script">T</mml:mi></mml:mrow></mml:math>-symmetric systems have garnered significant attention due to their unconventional properties. Despite the growing interest, there remains an ongoing debate about whether these systems outperform their Hermitian counterparts in practical applications, and if so, by what metrics this performance should be measured. We developed <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">P</mml:mi><mml:mi class="MJX-tex-caligraphic" mathvariant="script">T</mml:mi></mml:mrow></mml:math>-symmetric approach for mapping <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>N</mml:mi><mml:mo>=</mml:mo><mml:mn>3</mml:mn></mml:math> pure qubit states to address this, implemented it using the dilation method, and demonstrated it on a superconducting quantum processor from the IBM Quantum Experience. For the first time, we derived exact expressions for the population of the post-selected <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">P</mml:mi><mml:mi class="MJX-tex-caligraphic" mathvariant="script">T</mml:mi></mml:mrow></mml:math>-symmetric subspace for both <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>N</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>N</mml:mi><mml:mo>=</mml:mo><mml:mn>3</mml:mn></mml:math> states. When applied to the discrimination of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>N</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:math> pure states, our algorithm provides an equivalent result to the conventional unambiguous quantum state discrimination. For <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>N</mml:mi><mml:mo>=</mml:mo><mml:mn>3</mml:mn></mml:math> states, our approach introduces novel capabilities not available in traditional Hermitian systems, enabling the transformation of an arbitrary set of three pure quantum states into another, at the cost of introducing an inconclusive outcome. Our algorithm has the same error rate for the attack on the three-state QKD protocol as the conventional minimum error, maximum confidence, and maximum mutual information strategies. For post-selected quantum metrology, our results provide precise conditions where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">P</mml:mi><mml:mi class="MJX-tex-caligraphic" mathvariant="script">T</mml:mi></mml:mrow></mml:math>-symmetric quantum sensors outperform their Hermitian counterparts in terms of information-cost rate. Combined with punctuated unstructured quantum database search, our method significantly reduces the qubit readout requirements at the cost of adding an ancilla, while maintaining the same average number of oracle calls as the original punctuated Grover's algorithm. Our work opens new pathways for applying <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">P</mml:mi><mml:mi class="MJX-tex-caligraphic" mathvariant="script">T</mml:mi></mml:mrow></mml:math> symmetry in quantum communications, computing, and cryptography.