Computational Complexity of QSD measurement methods

This study explored Quantum state discrimination (QSD) as a key task in quantum information processing, enabling potential practical application to quantum key distribution, quantum sensing, and quantum communication. The distinction between quantum states with varying probabilities of transmission make it indispensable for certain applications within these fields. The project examines differences between Pretty Good Measurement (PGM) and Helstrom Measurement (HM) at high school level math. PGM offers simplicity and computational efficiency making it more practical for broader applications. HM minimizes error probability, providing the most accurate binary state discrimination, but limited scalability due to complexity. As quantum technologies continue to evolve, the development of hybrid methods bridging the strengths of both strategies will be essential. This research underscores the necessity of optimizing QSD to advance the fields of quantum computation, secure communication, and beyond. The research highlights the importance of Positive Operator-Valued Measures and semidefinite programming as frameworks that enhance the efficiency of QSD methods. Applications of these methods are explored in quantum key distribution, quantum error correction, and communication protocols, emphasizing their role in ensuring security and reliability. By addressing the challenges in QSD, this study contributes to the growing body of knowledge in quantum information science and encourages further exploration in the field.