Schmidt Decomposition of Multipartite States
Mithilesh Kumar
TL;DR
This work addresses the challenge of Schmidt decomposition in multipartite quantum systems by introducing a matrix-set framework that yields a necessary-and-sufficient condition for decomposability and by providing constructive, polynomial-time algorithms for tripartite, quadripartite, and general multipartite cases. Central to the approach is the notion of positive commutativity among matrix sets and centrality of their diagonalizing pairs, which together enable a Schmidt form $| ext{state}\rangle = ∑_ℓ λ_ℓ |ℓ_{A_1}⟩|ℓ_{A_2}⟩⋯|ℓ_{A_n}⟩$ with nonnegative coefficients and orthonormal local bases. The paper also proves that the SCHMIDT-PARTITION problem is NP-complete and discusses purification and state-equivalence under local unitaries, highlighting practical tools for entanglement analysis and quantum information tasks that rely on structured multipartite decompositions. Overall, the results provide a rigorous, algorithmic pathway to identify and construct multipartite Schmidt decompositions, with broad implications for entanglement quantification and state transformation. $| ext{state}⟩$ forms and coefficient tensors are treated in a way that generalizes familiar bipartite Schmidt theory to N parties, preserving key properties while exposing clear computational procedures.
Abstract
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.