Structural Perspectives from Quantum States and Measurements in Optimal State Discrimination

Abstract: Quantum state discrimination is a fundamental concept in quantum information theory, which refers to a class of techniques to identify a specific quantum state through a positive operator-valued measure. In this work, we investigate how structural information about either the quantum states or the measurement operators can influence our ability to determine or bound the optimal discrimination probability. First, we observe that for single-qubit states, pairwise fidelities are sufficient to completely characterize the optimal discrimination. In contrast, for multi-qubit states, this correspondence breaks down. Motivated by this, we analytically derive the optimal discrimination probability for three equiprobable single-qubit states with equal pairwise fidelities in terms of fidelity. Secondly, we consider partial information about the optimal measurement, specifically the measurement operators that vanish in the optimal solution. We show that such information can be leveraged to tighten existing upper bounds on the optimal discrimination probability. Lastly, we show that in some cases, subsets and supersets of nonvanishing operators can be identified without semidefinite programming.