On properties of Schmidt Decomposition

TL;DR

This paper reviews properties of bipartite Schmidt decomposition, explores their extension to multipartite states, and demonstrates computational complexity and preservation properties related to Schmidt number.

Contribution

It analyzes which Schmidt decomposition properties extend to multipartite states and proves NP-completeness for partitioning states with maximal Schmidt number.

Findings

Schmidt number defines an equivalence class under separable unitaries

Partitioning multipartite states with highest Schmidt number is NP-complete

Purifications preserve Schmidt decomposability

Abstract

Schmidt decomposition is a powerful tool in quantum information. While Schmidt decomposition is universal for bipartite states, its not for multipartite states. In this article, we review properties of bipartite Schmidt decompositions and study which of them extend to multipartite states. In particular, Schmidt number (the number of non-zero terms in Schmidt decomposition) define an equivalence class using separable unitary transforms. We show that it is NP-complete to partition a multipartite state that attains the highest Schmidt number. In addition, we observe that purifications of a density matrix of a composite system preserves Schmidt decomposability.