The necessary and sufficient condition for the uniqueness of Schmidt decomposition and the necessary condition for the maximal von Neumann entanglement entropy

Abstract

Schmidt decomposition (SD) has been proven to be an important tool in quantum information and computation. Ac{\' i}n et al. proposed the SD for three qubits, but in the meanwhile, they indicated that it is possible to have two different SDs for the same state. The more challenging quest of finding the sufficient and necessary condition for the uniqueness of SD has never been undertaken. In this paper, we propose a necessary and sufficient condition for the uniqueness of SD for three qubits. By examining the condition, one can tell what state has one SD and what state has two SDs without actually performing the Schmidt decomposition. We investigate the relation between the uniqueness of SD and the von Neumann entanglement entropy (vNEE). To this end, we prove that any state having the maximal vNEE $S(\rho _{x})=\ln 2$, $ x=A, B,$ or $C$ must have a unique SD. %Therefore, we should choose a state having a unique SD for its maximal vNEE for quantum information theory. This means if a state has two SDs, then the state does not have the maximal vNEE. Therefore, we should not choose a state having two SDs for its maximal vNEE for quantum information theory. In this paper, we also give all the SD states that have the maximal vNEE and a unique SD, as well as all the SD states that have a unique SD.