On properties of Schmidt Decomposition
Mithilesh Kumar
TL;DR
This work interrogates which properties of Schmidt decomposition extend from bipartite to multipartite quantum states, highlighting both structural and computational aspects. It shows that, while Schmidt coefficients govern entanglement in bipartite systems, multipartite decomposability imposes stricter constraints, with examples like the $|W\rangle$ state illustrating non-decomposability in general. The paper proves key results: Schmidt coefficients determine local eigenvalue spectra for decomposable multipartite states and are preserved under local unitaries, establishes a rank-based characterization of Schmidt number, and demonstrates that the problem SCHMIDT-PARTITION (seeking the maximal Schmidt number across partitions) is NP-complete through a reduction from SUBSET SUM. It also analyzes purification, proving that purifications related by local unitaries maintain Schmidt-decomposability properties and can be either decomposable or not. Overall, the work clarifies the entanglement-structure landscape of multipartite systems and underscores computational hardness and purification considerations relevant to quantum information tasks, while outlining directions for extending Schmidt-number concepts to mixed states.
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.