Quantum state discrimination via generalized bounds on Löwner partial order sets

This paper introduces a novel class of upper and lower bounds for a set of Hermitian operators with $K$ elements. These bounds, derived from the L"owner partial ordering, represent the maximal lower and minimal upper bounds for a pair of Hermitian operators. Additionally, the paper redefines the minimum Bayes cost and minimum error quantum state discrimination problems by identifying the maximal lower and minimal upper bounds for quantum density operators. These proposed bounds are then utilized to establish a tight bound for the error probability of quantum state discrimination problem. Furthermore, the paper provides a geometric interpretation of these bounds, offering valuable insights into the quantum state discrimination problem and aiding in the understanding of the optimal measurement strategy, especially for qubits.