Linear Transformations between Multipartite Quantum Systems That Map the Set of Tensor Product of Idempotent Matrices into Idempotent Matrix Set

Abstract

LetHnbe the set ofn×ncomplex Hermitian matrices and𝒫n(resp.,𝒯n) be the set of all idempotent (resp., tripotent) matrices inHn. Inl-partite quantum systemHm1Tml=⊗1lHmi,⊗1l𝒫mi(resp.,⊗1l𝒯mi) denotes the set of all decomposable elements⊗1lAisuch thatAi∈𝒫mi(resp.,Ai∈𝒯mi). In this paper, linear mapsϕfromHm1⋯mltoHnwithn≤m1⋯mlsuch thatϕ⊗1l𝒫mi∈𝒫nare characterized. As its application, the structure of linear mapsϕfromHm1⋯mltoHnwithn≤m1⋯mlsuch thatϕ⊗1l𝒯mi∈𝒯nis also obtained.