Schmidt Decomposition of Multipartite States

Quantum states can be written in infinitely many ways depending on the choices of basis. Schmidt decomposition of a quantum state has a lot of properties useful in the study of entanglement. All bipartite states admit Schmidt decomposition, but this does not extend to multipartite systems. We obtain necessary and sufficient conditions for the existence of Schmidt decompositions of multipartite states. Moreover, we provide an efficient algorithm to obtain the decomposition for a Schmidt decomposable multipartite state.

💡 Research Summary

The paper tackles the long‑standing question of whether the elegant Schmidt decomposition, which is guaranteed for any bipartite pure state, can be extended to systems with three or more parties. After reviewing the bipartite case—where the decomposition follows directly from the singular‑value decomposition (SVD) of the coefficient matrix—the authors focus on multipartite states and ask under what precise mathematical conditions a “global” Schmidt form
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