Formulas for Mutually Orthogonal Quantum States in Two-Qubit Systems: Orthogonal Schmidt Decompositions
Yonghae Lee, Youngho Min, Sunghyun Bae, Youngrong Lim
TL;DR
This work advances the theory of Schmidt decompositions by incorporating orthogonality constraints for sets of two-qubit pure states, enabling explicit constructions of orthogonal Schmidt decompositions for pairs, triples, and bases. The authors classify orthogonal state sets by their product versus entangled content and derive closed-form decompositions for many configurations, including PPP, PPE, PPPP, and PPEE types, while identifying fundamental obstacles for PEE and EEE types. A key no-go result shows that three product states cannot coexist with a single entangled state in a Schmidt-admitting orthonormal basis, linking to UPB-inspired concepts and local state discrimination. The developed formulas provide a systematic toolkit for designing state-preparation circuits and for constructing spectral decompositions of two-qubit mixed states, with potential impact on entanglement witnesses and channel characterization. Overall, the paper delivers a structured, algebraic framework to map entanglement structure onto Schmidt data in two-qubit systems, with practical implications for quantum information processing.
Abstract
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.