Necessary and sufficient condition for the existence of Schmidt decomposition in multipartite Hilbert spaces

TL;DR

This paper corrects and generalizes a previous condition for the existence of Schmidt decomposition in multipartite Hilbert spaces, providing a necessary and sufficient criterion applicable to any number of parties.

Contribution

It identifies errors in prior conditions, offers a corrected and generalized criterion, and proves its necessity and sufficiency for multipartite systems.

Findings

Corrected the previous sufficient condition for Schmidt decomposition

Generalized the condition to n-partite Hilbert spaces

Proved the condition is both necessary and sufficient

Abstract

Pati (Physics Letters A, 2000) derived a sufficient condition for the existence of Schmidt decomposition in tripartite Hilbert spaces. In this paper, we show that the condition is erroneous by demonstrating some counter-examples. Moreover, we suitably modify the condition and provide a correctness proof. We also show for the first time how this can be generalized to $n$-partite Hilbert spaces for any $n\geq 2$. Finally, we prove that this condition is also a necessary condition.