Formulas for Mutually Orthogonal Quantum States in Two-Qubit Systems: Orthogonal Schmidt Decompositions

We present Schmidt decomposition formulas for mutually orthogonal two-qubit pure states and classify orthonormal sets based on their entanglement structure. First, we derive explicit Schmidt decomposition formulas for any pure state and extend them to two orthogonal pure states. For three mutually orthogonal states, we provide formulas for specific cases and discuss the challenges of obtaining analytic expressions for the rest. Additionally, we derive explicit formulas for certain orthonormal bases and analyze those containing one or two maximally entangled states. Finally, we prove that no orthonormal basis can consist of three product states and one entangled state.