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

Abstract

Let 𝐻_𝑛 be the set of 𝑛×𝑛 complex Hermitian matrices and 𝒫_𝑛 (resp., 𝒯_𝑛) be the set of all idempotent (resp., tripotent) matrices in 𝐻_𝑛. In 𝑙-partite quantum system 𝐻_{𝑚1𝑇𝑚𝑙}=⊗^𝑙₁𝐻_𝑚𝑖, ⊗^𝑙₁𝒫_𝑚𝑖 (resp., ⊗^𝑙₁𝒯_𝑚𝑖) denotes the set of all decomposable elements ⊗^𝑙₁𝐴_𝑖 such that 𝐴_𝑖∈𝒫_𝑚𝑖 (resp., 𝐴_𝑖∈𝒯_𝑚𝑖). In this paper, linear maps ϕ from 𝐻_{𝑚1⋯𝑚𝑙} to 𝐻_𝑛 with 𝑛≤𝑚₁⋯𝑚_𝑙 such that 𝜙(⊗^𝑙₁𝒫_𝑚𝑖)∈𝒫_𝑛 are characterized. As its application, the structure of linear maps ϕ from 𝐻_{𝑚1⋯𝑚𝑙} to 𝐻_𝑛 with 𝑛≤𝑚₁⋯𝑚_𝑙 such that 𝜙(⊗^𝑙₁𝒯_𝑚𝑖)∈𝒯_𝑛 is also obtained.

1. Introduction

Let 𝑀_𝑛 be the vector space of 𝑛×𝑛 complex matrices and 𝐻_𝑛⊆𝑀_𝑛 be the vector space of 𝑛×𝑛 complex Hermitian matrices. In quantum information theory, a 𝑙-partite system can be represented as the tensor product space 𝐻_{𝑚1⋯𝑚𝑙}≡⊗^𝑙₁𝐻_𝑚𝑖, where ⊗ is the usual Kronecker product of matrices; a quantum state is represented as a positive semidefinite with trace one in 𝐻_𝑛, see [1]. In quantum information science and quantum computing, it is important to understand, characterize, and construct different classes of maps on quantum states [2]. For example, entanglement is one of the main concepts in quantum information theory, and entangled state involves at least bipartite system or multipartite system [3]; to study entangled states, one should construct entanglement witnesses, which are special types of positive maps, see [4]. On this background, the research on the characterizations of maps leaving invariant, some important subsets or quantum properties attracted more and more researchers’ attention, see ([1, 3, 5]). Especially, in [5], Lim characterized the linear and additive maps on tensor products of spaces of Hermitian matrices that carry the set of tensor product of rank one matrices into itself.

Preserver problem is a hot area in Banach algebra; there are many results about this area, see ([6–11]). Specially, the idempotent preservers and the rank one preservers play an important role; therefore, it is meaningful to study the two preservers. Chan et al. [12] first characterized linear transformations on 𝑀_𝑛 preserving idempotent matrices. In [13], the authors obtained the following result: linear map 𝑇∶𝐻_𝑛→𝐻_𝑚 with 𝑛≤𝑚 satisfies 𝑇(𝒫_𝑛)⊂𝒫_𝑚 if and only if 𝑇=0 or 𝑇(𝑋)=𝑄𝑋𝑄⁻¹or𝑇(𝑋)=𝑄𝑋^𝑇𝑄⁻¹,∀𝑋∈𝐻_𝑛, (1) where 𝑄 is unitary matrix.

Since the study of linear transformations, preserving idempotents is important in many aspects of mathematics and physics, see [14–17], and inspired by the above, the purpose of this paper is to study linear maps from 𝑙-partite system to the space of Hermitian matrices that carry the set of tensor product of idempotent matrices into the set of idempotent matrices in the space of Hermitian matrices, that is, 𝜙∶𝐻_{𝑚1⋯𝑚𝑙}→𝐻_𝑛 with 𝑛≤𝑚₁⋯𝑚_𝑙 satisfying𝜙⊗^𝑙₁𝒫_𝑚𝑖⊂𝒫_𝑛, (2) where 𝒫_𝑛 is the set of all idempotent matrices in 𝐻_𝑛, that is, 𝒫_𝑛={𝐴∈𝐻_𝑛|𝐴²=𝐴} and ⊗^𝑙₁𝒫_𝑚𝑖 denotes the set of all decomposable elements ⊗^𝑙₁𝐴_𝑖 such that 𝐴_𝑖∈𝒫_𝑚𝑖. As application, the forms of linear maps from 𝑙-partite system to the space of Hermitian matrices that carry the set of tensor product of tripotent matrices into the set of tripotent matrices in the space of Hermitian matrices are obtained, that is, 𝜙∶𝐻_{𝑚1⋯𝑚𝑙}→𝐻_𝑛 with 𝑛≤𝑚₁⋯𝑚_𝑙 satisfying 𝜙⊗^𝑙₁𝒯_𝑚𝑖⊂𝒯_𝑛, (3) where 𝒯_𝑛 is the set of all tripotent matrices in 𝐻_𝑛, that is, 𝒯_𝑛={𝐴∈𝐻_𝑛|𝐴³=𝐴} and ⊗^𝑙₁𝒯_𝑚𝑖 denotes the set of all decomposable elements ⊗^𝑙₁𝐴_𝑖 such that 𝐴_𝑖∈𝒯_𝑚𝑖.

Throughout this paper, we always assume integers 𝑙≥1 and 𝑛,𝑚₁, …, 𝑚_𝑙≥2. Let 𝐼_𝑘 be the 𝑘×𝑘 identity matrix, 0 be the zero matrix which order is omitted in different matrices just for simplicity, and 𝑋^𝑇 (resp., 𝑋^∗, rank𝑋) be the transpose (resp., conjugate transpose, rank) of 𝑋. 𝐸^(𝑛)_𝑖𝑗 (1≤𝑖,𝑗≤𝑛) stands for the 𝑛×𝑛 matrix with 1 at the (𝑖,𝑗)th entry and 0 otherwise. 𝐷^(𝑛)_𝑖𝑗=𝐸^(𝑛)_𝑖𝑗+𝐸^(𝑛)_𝑗𝑖 and 𝐾^(𝑛)_𝑖𝑗=√−1𝐸^(𝑛)_𝑖𝑗−√−1𝐸^(𝑛)_𝑗𝑖, where 1≤𝑖≠𝑗≤𝑛. For real numbers 𝑎 and 𝑏 with 𝑎≤𝑏, let [𝑎,𝑏] be the set of all integers between 𝑎 and 𝑏. ⊕ is the usual direct sum of matrices. A linear map 𝜋=⊗^𝑙₁𝜋_𝑖∶𝐻_{𝑚1⋯𝑚𝑙}→𝐻_{𝑚1⋯𝑚𝑙} is canonical if 𝜋 satisfies 𝜋(⊗^𝑙₁𝐴_𝑖)=⊗^𝑙₁𝜋_𝑖(𝐴_𝑖), where 𝜋_𝑖∶𝐻_𝑚𝑖→𝐻_𝑚𝑖 satisfies 𝜋_𝑖(𝐴_𝑖)=𝐴_𝑖 or 𝜋_𝑖(𝐴_𝑖)=𝐴^𝑇_𝑖 for all 𝐴_𝑖∈𝐻_𝑚𝑖.

2. Main Results

Lemma 1 (see [18]). Suppose 𝑃₁,…,𝑃_𝑘∈𝒫_𝑛 such that 𝑃_𝑖+𝑃_𝑗∈𝒫_𝑛, ∀1≤𝑖<𝑗≤𝑘. Let 𝑟_𝑖=rank𝑃_𝑖. Then, there exists a unitary matrix 𝑈∈𝑀_𝑛 such that 𝑃_𝑖=𝑈diag(0,…,0,1,…,1,0,…,0)𝑈^∗, (4) where diag(0,…,0,1,…,1,0,…,0) is the diagonal matrix in which all diagonal entries are zero except those in the (𝑟₁+⋯+𝑟_𝑖−1+1)st to the (𝑟₁+⋯+𝑟_𝑖)th rows.

Lemma 2. Let 𝑃∈𝒫_𝑟. Suppose 𝑋, 0_𝑠⊕𝑃⊕0_{𝑛−𝑟−𝑠}−X∈𝒫_𝑛. Then, there exists 𝑋_𝑟∈𝒫_𝑟 such that 𝑋=0_𝑠⊕𝑋_𝑟⊕0_{𝑛−𝑟−𝑠}, where 0≤𝑟,𝑠≤𝑛.

Proof. Let 𝑇∈𝑀_𝑟 be a unitary matrix satisfying 𝑃=𝑇(𝐼_𝑡⊕0)𝑇^∗. Set 𝐼_𝑟000𝑇^∗000𝐼_{𝑛−𝑟−𝑠}𝑋𝐼_𝑟000𝑇000𝐼_{𝑛−𝑟−𝑠}=𝑋₁₁𝑋₁₂𝑋₁₃𝑋₂₁𝑋₂₂𝑋₂₃𝑋₃₁𝑋₃₂𝑋₃₃, (5) where 𝑋₁₁∈𝐻_𝑠,𝑋₂₂∈𝐻_𝑡. Then, 𝑋₁₁𝑋₁₂𝑋₁₃𝑋₂₁𝑋₂₂𝑋₂₃𝑋₃₁𝑋₃₂𝑋₃₃∈𝒫_𝑛,0_𝑟000𝐼_𝑡0−𝑋000¹¹𝑋₁₂𝑋₁₃𝑋₂₁𝑋₂₂𝑋₂₃𝑋₃₁𝑋₃₂𝑋₃₃∈𝒫_𝑛. (6) A straightforward computation shows that 𝑋=0_𝑠⊕[𝑇(𝑋₂₂⊕0_𝑟−𝑡)𝑇^∗]⊕0_{𝑛−𝑟−𝑠}; therefore, the result holds.

Lemma 3. Let 𝐴=𝑃_𝑟⊕0_𝑠⊕0_{𝑛−𝑟−𝑠} and 𝐵=0_𝑟⊕𝑃_𝑠⊕0_{𝑛−𝑟−𝑠} with 𝑃_𝑟∈𝒫_𝑟, 𝑃_𝑠∈𝒫_𝑠. If 𝛼𝐴+𝛽𝐵+𝛾𝑋∈𝒫_𝑛𝛾≤1𝑓𝑜𝑟𝑎𝑛𝑦𝑟𝑒𝑎𝑙𝛾𝑤𝑖𝑡ℎ0≠2, (7) where 𝑋∈𝑀_𝑛, 𝛼 and 𝛽 are all roots of 𝑥²−𝑥+𝛾²=0, then, 𝑋²=𝐴𝐵, and 𝑋 is of the form 0_𝑟𝑋₁₂𝑋₂₁0_𝑠⊕0_{𝑛−𝑟−𝑠}, (8) where 𝑋₁₂∈𝑀_𝑟×𝑠, 𝑋₂₁∈𝑀_𝑠×𝑟.

Proof. It is clear that 𝐴,𝐵∈𝒫_𝑛 and 𝐴𝐵=𝐵𝐴=0. By a direct computation, one can obtain that 𝑋=𝛼(𝐴𝑋+𝑋𝐴)+𝛽(𝐵𝑋+𝑋𝐵)+𝛾𝑋²−𝐴−𝐵. (9) Replacing 𝛾 by −𝛾, we have 𝑋²=𝐴+𝐵 and 𝑋=𝛼(𝐴𝑋+𝑋𝐴)+𝛽(𝐵𝑋+𝑋𝐵). (10) Choosing 𝛾=2/5, then 𝛼=1/5,𝛽=4/5 or 𝛼=4/5,𝛽=1/5, this yields that 1𝑋=5()+4𝐴𝑋+𝑋𝐴5(),4𝐵𝑋+𝑋𝐵𝑋=5()+1𝐴𝑋+𝑋𝐴5(𝐵𝑋+𝑋𝐵). (11) Thus, 𝑋=𝐴𝑋+𝑋𝐴 and 𝑋=𝐵𝑋+𝑋𝐵. This, together with the form of 𝐴 and 𝐵, implies that 𝑋 has the form ^0𝑟𝑋12_𝑋210𝑠⊕0_{𝑛−𝑟−𝑠}, where 𝑋₁₂∈𝑀_𝑟×𝑠, 𝑋₂₁∈𝑀_𝑠×𝑟.

Lemma 4. Let 𝐽=𝑃_𝑝⊕−𝑃_𝑞⊕0_{𝑛−𝑝−𝑞} with 𝑃_𝑝∈𝒫_𝑝 and 𝑃_𝑞∈𝒫_𝑞. Suppose 𝑋∈𝒯_𝑛 satisfying 𝐽−𝑋∈𝒯_𝑛 and 𝐽−2𝑋∈𝒯_𝑛. Then, 𝑋=𝑋_𝑝⊕−𝑋_𝑞⊕0_{𝑛−𝑝−𝑞} with 𝑋_𝑝∈𝒫_𝑝 and 𝑋_𝑞∈𝒫_𝑞.

Proof. By a direct computation, we have 3𝑋=𝐽²𝑋+𝐽𝑋𝐽+𝑋𝐽²,3𝑋=𝐽𝑋²+𝑋𝐽𝑋+𝑋²𝐽. (12)(13) Let 𝑈∈𝑀_𝑝 and 𝑉∈𝑀_𝑞 be unitary matrices satisfying 𝑃_𝑝=𝑈(𝐼_𝑟⊕0_𝑝−𝑟)𝑈^∗ and 𝑃_𝑞=𝑉(𝐼_𝑠⊕0_𝑞−𝑠)𝑉^∗, respectively. Then, 𝑈^∗000𝑉^∗000𝐼_{𝑛−𝑝−𝑞}𝐽𝑈000𝑉000𝐼_{𝑛−𝑝−𝑞}=𝐼_𝑟⊕0_𝑝−𝑟000−𝐼_𝑠⊕0_𝑞−𝑠0000_{𝑛−𝑝−𝑞}. (14) By (12), one can assume that 𝑈^∗000𝑉^∗000𝐼_{𝑛−𝑝−𝑞}𝑋𝑈000𝑉000𝐼_{𝑛−𝑝−𝑞}=𝑋_𝑟⊕0_𝑝−𝑟000−𝑋_𝑠⊕0_𝑞−𝑠0000_{𝑛−𝑝−𝑞}, (15) where 𝑋_𝑟∈𝐻_𝑟 and 𝑋_𝑠∈𝐻_𝑠. This, together with (13), implies that 𝑋_𝑟∈𝒫_𝑟 and 𝑋_𝑠∈𝒫_𝑠. Thus, 𝑋=𝑈𝑋_𝑟⊕0_𝑝−𝑟𝑈^∗⊕−𝑉𝑋_𝑠⊕0_𝑞−𝑠𝑉^∗⊕0_{𝑛−𝑝−𝑞}, (16) as desired.

Lemma 5. Let 𝜓₁, 𝜓₂ be canonical maps on 𝐻_{𝑚1⋯𝑚𝑙}, and 𝑈, 𝑉 are invertible matrices of order 𝑚₁⋯𝑚_𝑙. If 𝜓₁(𝑋)𝑈=𝑉𝜓₂(𝑋) for all 𝑋∈𝐻_{𝑚1⋯𝑚𝑙}, then 𝜓₁=𝜓₂ and 𝑈=𝑉=𝜆𝐼_{𝑚1⋯𝑚𝑙} for some 𝜆≠0.

Proof. Let us first point out a simple observation which will be used in our proof. If a matrix 𝑈∈𝑀_𝑚𝑛 commutes with 𝐼_𝑚⊗𝑆 for all real symmetric 𝑆∈𝑀_𝑛, then 𝑈 has the form 𝑊⊗𝐼_𝑛 with 𝑊∈𝑀_𝑚.
Since 𝜓₁,𝜓₂ are canonical maps on 𝐻_{𝑚1⋯𝑚𝑙}, we assume that 𝜓₁=𝜏₁⊗⋯⊗𝜏_𝑙,𝜓₂=𝜂₁⊗⋯⊗𝜂_𝑙, (17) where 𝜏_𝑖 and 𝜂_𝑖 are the identity maps 𝑋↦𝑋 or the transposition maps 𝑋↦𝑋^𝑇 for 𝑖∈[1,𝑙].
Clearly, 𝜓_𝑖(𝐼_{𝑚1⋯𝑚𝑙})=𝐼_{𝑚1⋯𝑚𝑙}, 𝑖=1,2, hence 𝑈=𝑉.
We prove 𝑈=𝜆𝐼_{𝑚1⋯𝑚𝑙} by induction on 𝑙. The case of 𝑙=1 is the well-known fact. We assume that the statement holds true for 𝑙−1. We will prove it for 𝑙. For any real symmetric 𝑆∈𝑀_𝑚𝑙, since 𝜓_𝑖(𝐼_{𝑚1⋯𝑚𝑙−1}⊗𝑆)=𝐼_{𝑚1⋯𝑚𝑙−1}⊗𝑆, 𝑖=1,2, it follows that 𝑈=𝑊⊗𝐼_𝑚𝑙 for some matrix 𝑊∈𝑀_{𝑚1⋯𝑚𝑙−1}. We define linear maps 𝜑₁=𝜏₁⊗⋯⊗𝜏_𝑙−1,𝜑₂=𝜂₁⊗⋯⊗𝜂_𝑙−1. (18) It is easy to see that 𝜑₁ and 𝜑₂ are canonical maps on 𝐻_{𝑚1⋯𝑚𝑙−1}. For any 𝑌∈𝐻_{𝑚1⋯𝑚𝑙−1}, since 𝜑₁(𝑌)𝑊⊗𝐼_𝑚𝑙=𝜓₁𝑌⊗𝐼_𝑚𝑙𝑈=𝑈𝜓₂𝑌⊗𝐼_𝑚𝑙=𝑊𝜑₂(𝑌)⊗𝐼_𝑚𝑙, (19) we have 𝜑₁(𝑌)𝑊=𝑊𝜑₂(𝑌), thus, by induction hypothesis, 𝑊=𝜆𝐼_{𝑚1⋯𝑚𝑙−1}. Since 𝑈 is invertible, we have 𝜆≠0. Thus, 𝑈=𝜆𝐼_{𝑚1⋯𝑚𝑙} and 𝜓₁(𝑋)=(1/𝜆)𝜓₁(𝑋)𝑈=(1/𝜆)𝑈𝜓₂(𝑋)=𝜓₂(𝑋) for all 𝑋∈𝐻_{𝑚1⋯𝑚𝑙}.

Theorem 6. Suppose 𝜙 is a linear map from 𝐻_{𝑚1⋯𝑚𝑙} to 𝐻_𝑛 with 𝑛≤𝑚₁⋯𝑚_𝑙. Then, 𝜙(⊗^𝑙₁𝒫_𝑚𝑖)⊂𝒫_𝑛 if and only if either 𝜙=0 or 𝑛=𝑚₁⋯𝑚_𝑙, there exists a unitary matrix 𝑈∈𝑀_𝑛 and a canonical map 𝜋 on 𝐻_{𝑚1⋯𝑚𝑙} such that 𝜙(𝑋)=𝑈𝜋(𝑋)𝑈^∗,∀𝑋∈𝐻_{𝑚1⋯𝑚𝑙}. (20)

Proof. The sufficiency part is obvious. We prove the necessity part by induction on 𝑙. When 𝑙=1, it is the result in [13], which has been given in the above introduction. We assume now that the result holds true for 𝑙−1 and give the proof of the case 𝑙 by the following five steps.
Step 1. Suppose 𝑃∈𝒫_𝑚1 and 𝜙(𝑃⊗𝐼_{𝑚2⋯𝑚𝑙})=0_𝑠⊕𝐼_𝑟⊕0_{𝑛−𝑟−𝑠}. Then, 𝜙(𝑃⊗𝑋)=0_𝑠⊕𝜓(𝑋)⊕0_{𝑛−𝑟−𝑠},∀𝑋∈𝐻_{𝑚2⋯𝑚𝑙}, (21) where 𝜓 is a linear map from 𝐻_{𝑚2⋯𝑚𝑙} to 𝐻_𝑟 satisfying 𝜓(⊗^𝑙₂𝒫_𝑚𝑖)⊂𝒫_𝑟.
Proof of Step 1. Set Γ₀=𝐼_𝑚2⊗⋯⊗𝐼_𝑚𝑙,Γ₁=𝑃₂⊗𝐼_{𝑚3⋯𝑚𝑙}∶𝑃₂∈𝒫_𝑚2,Γ₂=𝑃₂⊗𝑃₃⊗𝐼_{𝑚4⋯𝑚𝑙}∶𝑃_𝑖∈𝒫_𝑚𝑖[,⋯Γ,𝑖∈2,3]_𝑙−1=𝑃₂⊗⋯⊗𝑃_𝑚𝑙∶𝑃_𝑖∈𝒫_𝑚𝑖,𝑖∈[2,𝑙]. (22)
We prove by induction on 𝑘 that 𝜙(𝑃⊗𝑋)=0_𝑠⊕𝜓(𝑋)⊕0_{𝑛−𝑟−𝑠},∀𝑋∈Γ_𝑘,𝑘∈[0,𝑙−1]. Assume for a moment that we have already proved this. Then, when 𝜙 is linear, we can complete the proof of Step 1.
Now, we prove the assertion. The case of 𝑘=0 is just the assumption. Then, we assume that our statement holds true for 𝑘−1, and we consider the image of 𝑃⊗𝑃₂⊗⋯⊗𝑃_𝑘+1⊗𝐼_{𝑚𝑘+2⋯𝑚𝑙} for 𝑃_𝑖∈𝒫_𝑚𝑖, 𝑖∈[2,𝑘+1]. Because 𝑃⊗𝑃₂⊗⋯⊗𝑃_𝑘⊗𝐼_𝑚𝑘+1⊗𝐼_{𝑚𝑘+2⋯𝑚𝑙}∈⊗^𝑙₁𝒫_𝑚𝑖,𝑃⊗𝑃₂⊗⋯⊗𝑃_𝑘⊗𝑃_𝑘+1⊗𝐼_{𝑚𝑘+2⋯𝑚𝑙}∈⊗^𝑙₁𝒫_𝑚𝑖,𝑃⊗𝑃₂⊗⋯⊗𝑃_𝑘⊗𝐼_𝑚𝑘+1−𝑃_𝑘+1⊗𝐼_{𝑚𝑘+2⋯𝑚𝑙}∈⊗^𝑙₁𝒫_𝑚𝑖, (23) we obtain using the property of 𝜙 that 𝜙𝑃⊗𝑃₂⊗⋯⊗𝑃_𝑘⊗𝐼_𝑚𝑘+1⊗𝐼_{𝑚𝑘+2⋯𝑚𝑙}∈𝒫_𝑛,𝜙𝑃⊗𝑃₂⊗⋯⊗𝑃_𝑘⊗𝑃_𝑘+1⊗𝐼_{𝑚𝑘+2⋯𝑚𝑙}∈𝒫_𝑛,𝜙𝑃⊗𝑃₂⊗⋯⊗𝑃_𝑘⊗𝐼_𝑚𝑘+1⊗𝐼_{𝑚𝑘+2⋯𝑚𝑙},−𝜙𝑃⊗𝑃₂⊗⋯⊗𝑃_𝑘+1⊗𝐼_{𝑚𝑘+2⋯𝑚𝑙}∈𝒫_𝑛. (24)
We obtain by induction hypothesis and Lemma 2 that 𝜙(𝑃⊗𝑃₂⊗⋯⊗𝑃_𝑘+1⊗𝐼_{𝑚𝑘+2⋯𝑚𝑙})=0_𝑠⊕𝑋_𝑟⊕0_{𝑛−𝑟−𝑠}, where 𝑋_𝑟∈𝒫_𝑟, as desired.
Step 2. If 𝑚₁⋯𝑚_𝑙>𝑛. Then, 𝜙=0.
Proof of Step 2. Since 𝐸^(𝑚1)_𝑖𝑖⊗𝐼_{𝑚2⋯𝑚𝑙}∈⊗^𝑙₁𝒫_𝑚𝑖,∀𝑖∈[1,𝑚₁] and (𝐸^(𝑚1)_𝑖𝑖+𝐸^(𝑚1)_𝑗𝑗)⊗𝐼_{𝑚2⋯𝑚𝑙}∈⊗^𝑙₁𝒫_𝑚𝑖,∀𝑖≠𝑗∈[1,𝑚₁], we obtain using the property of 𝜙 that 𝜙𝐸^(𝑚1)_𝑖𝑖⊗𝐼_{𝑚2⋯𝑚𝑙}∈𝒫_𝑛,∀𝑖∈1,𝑚₁,𝜙𝐸^(𝑚1)_𝑖𝑖⊗𝐼_{𝑚2⋯𝑚𝑙}+𝜙𝐸^(𝑚1)_𝑗𝑗⊗𝐼_{𝑚2⋯𝑚𝑙}∈𝒫_𝑛,∀𝑖≠𝑗∈1,𝑚₁. (25) Denote by 𝑟_𝑖=rank𝜙(𝐸^(𝑚1)_𝑖𝑖⊗𝐼_{𝑚2⋯𝑚𝑙}), ∀𝑖∈[1,𝑚₁]. Using Lemma 1 and composing 𝜙 by a similarity transformation, we may obtain that 𝜙𝐸^(𝑚1)_𝑖𝑖⊗𝐼_{𝑚2⋯𝑚𝑙}=diag(0,…,0,1,…,1,0,…,0), (26) where diag(0,…,0,1,…,1,0,…,0) is the diagonal matrix in which all diagonal entries are zero except those in the (𝑟₁+⋯+𝑟_𝑖−1+1)st to the (𝑟₁+⋯+𝑟_𝑖)th rows.
Since 𝑛<𝑚₁⋯𝑚_𝑙, there exists some 𝑟_𝑖<𝑚₂⋯𝑚_𝑙. Without loss of generality, we may assume 𝑟₁<𝑚₂⋯𝑚_𝑙. Thus, 𝜙(𝐸^(𝑚1)₁₁⊗𝐼_{𝑚2⋯𝑚𝑙})=𝐼_𝑟1⊕0_𝑛−𝑟1. By Step 1, we see that there exists a linear map 𝜓 from 𝐻_{𝑚2⋯𝑚𝑙} to 𝐻_𝑟1 satisfying 𝜓(⊗^𝑙₂𝒫_𝑚𝑖)⊂𝒫_𝑟1 such that 𝜙(𝐸^(𝑚1)₁₁⊗𝑋)=𝜓(𝑋)⊕0_𝑛−𝑟1 for all 𝑋∈𝐻_{𝑚2⋯𝑚𝑙}. We obtain by induction hypothesis 𝜓=0. Thus, 𝜙𝐸^(𝑚1)₁₁⊗𝑋=0,∀𝑋∈𝐻_{𝑚2⋯𝑚𝑙}. (27) For any 𝑗∈[2,𝑚₁], 𝑃∈⊗^𝑙₂𝒫_𝑚𝑖 and real 𝛾 with 0≠|𝛾|≤1/2, we have (𝛼𝐸^(𝑚1)₁₁+𝛽𝐸^(𝑚1)_𝑗𝑗+𝛾𝐷^(𝑚1)_1𝑗)⊗𝑃∈⊗^𝑙₁𝒫_𝑚𝑖, where 𝛼 and 𝛽 are all roots of 𝑥²−𝑥+𝛾²=0. We obtain using the property of 𝜙, (27), and Lemma 3 that 𝜙(𝐸^(𝑚1)_𝑗𝑗⊗𝑃)=0 and 𝜙(𝐷^(𝑚1)_1𝑗⊗𝑃)=0. By using the similar argument, we see that 𝜙(𝐷^(𝑚1)_𝑖𝑗⊗𝑃)=0, 𝜙(𝐾^(𝑚1)_𝑖𝑗⊗𝑃)=0, ∀𝑖≠𝑗∈[1,𝑚₁]. Thus, 𝜙(𝐴⊗𝑃)=0,∀𝐴∈𝐻_𝑚1. (28) By the arbitrariness of 𝑃∈⊗^𝑙₂𝒫_𝑚𝑖, we have 𝜙(𝐴⊗𝑋)=0,∀𝐴∈𝐻_𝑚1,𝑋∈𝐻_{𝑚2⋯𝑚𝑙}. (29) Therefore, 𝜙=0.
We next always assume 𝑚₁⋯𝑚_𝑙=𝑛 and 𝜙≠0.
Step 3. Suppose 𝐹₁,…,𝐹_𝑚1∈𝒫_𝑚1 are of rank one with 𝐹_𝑖+𝐹_𝑗∈𝒫_𝑚1, ∀𝑖≠𝑗∈[1,𝑚₁]. Then, there exists a unitary matrix 𝑈 of order 𝑚₁⋯𝑚_𝑙 such that 𝜙𝐹_𝑘⊗𝑋=𝑈𝐸^(𝑚1)_𝑘𝑘⊗𝜋_𝑘(𝑋)𝑈^∗,∀𝑋∈𝐻_{𝑚2⋯𝑚𝑙},𝑘∈1,𝑚₁, (30) where 𝜋_𝑘 is a canonical map on 𝐻_{𝑚2⋯𝑚𝑙},∀𝑘∈[1,𝑚₁].
Proof of Step 3. Using the similar approach as in the proof of Step 2, we may show that rank𝜙(𝐹_𝑘⊗𝐼_{𝑚2⋯𝑚𝑙})=𝑚₂⋯𝑚_𝑙, ∀𝑘∈[1,𝑚₁]. In fact, if rank𝜙(𝐹_𝑘⊗𝐼_{𝑚2⋯𝑚𝑙})>𝑚₂⋯𝑚_𝑙 for some 𝑘, then, by Lemma 1, there exists rank𝜙(𝐹_𝑖⊗𝐼_{𝑚2⋯𝑚𝑙})<𝑚₂⋯𝑚_𝑙 for some 𝑖. Then, using the similar approach as in the proof of Step 2, we have 𝜙=0, which is a contradiction, and there exist a unitary matrix 𝑄 such that 𝜙(𝐹_𝑘⊗𝐼_{𝑚2⋯𝑚𝑙})=𝑄[𝐸^(𝑚1)_𝑘𝑘⊗𝐼_{𝑚2⋯𝑚𝑙}]𝑄^∗. Thus, applying Step 1, we obtain that 𝜙𝐹_𝑘⊗𝑋=𝑄𝐸^(𝑚1)_𝑘𝑘⊗𝜓_𝑘(𝑋)𝑄^∗,∀𝑋∈𝐻_{𝑚2⋯𝑚𝑙}, (31) where linear map 𝜓_𝑘∶𝐻_{𝑚2⋯𝑚𝑙}→𝐻_{𝑚2⋯𝑚𝑙} satisfying 𝜓_𝑘(⊗^𝑙₂𝒫_𝑚𝑖)⊂𝒫_{𝑚2⋯𝑚𝑙}, for any 𝑘∈[1,𝑚₁]. We obtain by induction hypothesis that there exists unitary matrices 𝑈_𝑘∈ℳ_{𝑚2⋯𝑚𝑙} and canonical maps 𝜋_𝑘 on 𝐻_{𝑚2⋯𝑚𝑙} such that 𝜓_𝑘(𝑋)=𝑈_𝑘𝜋_𝑘(𝑋)𝑈^∗_𝑘,∀𝑋∈𝐻_{𝑚2⋯𝑚𝑙},𝑘∈1,𝑚₁. (32) Set 𝑈=𝑄diag(𝑈₁,…,𝑈_𝑚1), we complete the proof of Step 3.
By Step 3, we may assume that 𝜙𝐸^(𝑚1)_𝑘𝑘⊗𝑋=𝐸^(𝑚1)_𝑘𝑘⊗𝜋_𝑘(𝑋),∀𝑋∈𝐻_{𝑚2⋯𝑚𝑙},𝑘∈1,𝑚₁, (33) where 𝜋_𝑘 is canonical map on 𝐻_{𝑚2⋯𝑚𝑙},∀𝑘∈[1,𝑚₁].
Step 4. For ∀𝑖≠𝑗∈[1,𝑚₁], 𝜋_𝑖=𝜋_𝑗 and there exists 𝜆_𝑖𝑗, 𝜇_𝑖𝑗 with |𝜆_𝑖𝑗|=|𝜇_𝑖𝑗|=1 such that 𝜙𝐷^(𝑚1)_𝑖𝑗⊗𝑋=𝜆_𝑖𝑗𝐸^(𝑚1)_𝑖𝑗+𝜆_𝑖𝑗𝐸^(𝑚1)_𝑗𝑖⊗𝜋₁(𝑋),∀𝑋∈𝐻_{𝑚2⋯𝑚𝑙},𝜙𝐾^(𝑚1)_𝑖𝑗⊗𝑋=𝜇_𝑖𝑗𝐸^(𝑚1)_𝑖𝑗+𝜇_𝑖𝑗𝐸^(𝑚1)_𝑗𝑖⊗𝜋₁(𝑋),∀𝑋∈𝐻_{𝑚2⋯𝑚𝑙}. (34)
Proof of Step 4. Without loss of generality, we only prove the case of 𝑖=1, 𝑗=2.
Set 𝐹₁=(1/2)𝐸^(𝑚1)₁₁+(1/2)𝐸^(𝑚1)₂₂+(1/2)𝐷^(𝑚1)₁₂, 𝐹₂=(1/2)𝐸^(𝑚1)₁₁+(1/2)𝐸^(𝑚1)₂₂−(1/2)𝐷^(𝑚1)₁₂, 𝐹_𝑘=𝐸^(𝑚1)_𝑘𝑘 for 𝑘∈[3,𝑚₁] (if 𝑚₁≥3). By Step 3, there exists a unitary matrix 𝑉 and canonical maps 𝜂_𝑘 on 𝐻_{𝑚2⋯𝑚𝑙} such that 𝜙𝐹_𝑘⊗𝑋=𝑉𝐸^(𝑚1)_𝑘𝑘⊗𝜂_𝑘(𝑋)𝑉^∗,∀𝑋∈𝐻_{𝑚2⋯𝑚𝑙},𝑘∈1,𝑚₁. (35) This, together with (33), implies that 𝐸^(𝑚1)_𝑘𝑘⊗𝐼_{𝑚2⋯𝑚𝑙}=𝑉𝐸^(𝑚1)_𝑘𝑘⊗𝐼_{𝑚2⋯𝑚𝑙}𝑉^∗,∀𝑘∈3,𝑚₁. (36) Thus, we may set 𝑉𝑉=¹¹𝑉₁₂𝑉₂₁𝑉₂₂⊕diag𝑉₃₃,…,𝑉_𝑚1𝑚1, (37) where 𝑉_𝑘𝑘∈𝑀_{𝑚2⋯𝑚𝑙},𝑘∈[1,𝑚₁].
On one hand, noting that 𝐷^(𝑚1)₁₂=2𝐹₁−𝐸^(𝑚1)₁₁−𝐸^(𝑚1)₂₂,𝐷^(𝑚1)₁₂=𝐸^(𝑚1)₁₁+𝐸^(𝑚1)₂₂−2𝐹₂, (38) we obtain using (33) and (35) that for all 𝑋∈𝐻_{𝑚2⋯𝑚𝑙}, 𝜙𝐷^(𝑚1)₁₂=⊗𝑋2𝑉₁₁𝜂₁(𝑋)𝑉^∗₁₁−𝜋₁(𝑋)2𝑉₁₁𝜂₁(𝑋)𝑉^∗₂₁2𝑉₂₁𝜂₁(𝑋)𝑉^∗₁₁2𝑉₂₁𝜂₁(𝑋)𝑉^∗₂₁−𝜋₂(𝑋)𝜙𝐷⊕0,_(𝑚1)12=𝜋⊗𝑋₁(𝑋)−2𝑉₁₂𝜂₂(𝑋)𝑉^∗₁₂−2𝑉₁₂𝜂₂(𝑋)𝑉^∗₂₂−2𝑉₂₂𝜂₂(𝑋)𝑉^∗₁₂𝜋₂(𝑋)−2V₂₂𝜂₂(𝑋)𝑉^∗₂₂⊕0. (39)(40)
On the other hand, for any 𝑃∈⊗^𝑙₂𝒫_𝑚𝑖 and real 𝛾 with 0≠|𝛾|≤1/2, we have (𝛼𝐸^(𝑚1)₁₁+𝛽𝐸^(𝑚1)₂₂+𝛾𝐷^(𝑚1)₁₂)⊗𝑃∈⊗^𝑙₁𝒫_𝑚𝑖, where 𝛼 and 𝛽 are all roots of 𝑥²−𝑥+𝛾²=0. We obtain using the property of 𝜙, (33), and Lemma 3 that 𝜙𝐷^(𝑚1)₁₂⊗𝐼_{𝑚2⋯𝑚𝑙}²=𝜙𝐸^(𝑚1)₁₁⊗𝐼_{𝑚2⋯𝑚𝑙}+𝜙𝐸^(𝑚1)₂₂⊗𝐼_{𝑚2⋯𝑚𝑙},𝜙𝐷^(𝑚1)₁₂=0⊗𝑋^𝑚1𝑌₁₂𝑌^∗₁₂0_𝑚1⊕0,∀𝑋∈𝐻_{𝑚2⋯𝑚𝑙}, (41)(42) where 𝑌₁₂∈𝑀_𝑚1. Choosing 𝑋=𝐼_{𝑚2⋯𝑚𝑙}, we get using (39), (40), and (42) that 2𝑉₁₁𝑉^∗₁₁=2𝑉₂₂𝑉^∗₂₂=2𝑉₂₁𝑉^∗₂₁=2𝑉₁₂𝑉^∗₁₂=𝐼_{𝑚2⋯𝑚𝑙}. Hence, by (39), (40), and (42) again, one can obtain that 𝜋₁(𝑋)𝑉^∗₁₁⁻¹𝑉^∗₂₁=2𝑉₁₁𝜂₁(𝑋)𝑉^∗₂₁=−2𝑉₁₂𝜂₂(𝑋)𝑉^∗₂₂=−𝑉₁₂𝑉⁻¹₂₂𝜋₂(𝑋). (43) By Lemma 5, we have 𝜋₁=𝜋₂ and (𝑉^∗₁₁)⁻¹𝑉^∗₂₁=𝜆₁₂𝐼_{𝑚2⋯𝑚𝑙} with 𝜆₁₂≠0. Hence, 𝜙𝐷^(𝑚1)₁₂⊗𝑋=𝜆₁₂𝐸^(𝑚1)₁₂+𝜆₁₂𝐸^(𝑚1)₂₁⊗𝜋₁(𝑋),∀𝑋∈𝐻_{𝑚2⋯𝑚𝑙},𝜆₁₂≠0. (44) This, together with (41), implies that |𝜆₁₂|=1. Similarly, 𝜙𝐾^(𝑚1)₁₂⊗𝑋=𝜇₁₂𝐸^(𝑚1)₁₂+𝜇₁₂𝐸^(𝑚1)₂₁⊗𝜋₁(𝑋),∀𝑋∈𝐻_{𝑚2⋯𝑚𝑙},𝜇₁₂=1. (45) This completes the proof of Step 4.
Step 5. There exists 𝜀∈{−1,1} such that 𝜇_𝑖𝑗=𝜀√−1𝜆_𝑖𝑗, ∀𝑖≠𝑗∈[1,𝑚₁]. If 𝑚₁≥3, then 𝜆_𝑖𝑗𝜆_𝑗𝑘=𝜆_𝑖𝑘, for distinct 𝑖,𝑗,𝑘∈[1,𝑚₁].
Proof of Step 5. Since 12𝐸^(𝑚1)_𝑖𝑖+12𝐸^(𝑚1)_𝑗𝑗+√24𝐷^(𝑚1)_𝑖𝑗+√24𝐾^(𝑚1)_𝑖𝑗⊗𝐼_{𝑚2⋯𝑚𝑙}∈⊗^𝑙₁𝒫_𝑚𝑖, (46) we obtain using the property of 𝜙, (33), and Step 4 that 12√24𝜆_𝑖𝑗+𝜇_𝑖𝑗√24𝜆_𝑖𝑗+𝜇_𝑖𝑗12∈𝒫₂. (47) Hence, (𝜆_𝑖𝑗+𝜇_𝑖𝑗)(𝜆_𝑖𝑗+𝜇_𝑖𝑗)=2. It follows from |𝜆_𝑖𝑗|=|𝜇_𝑖𝑗|=1 that 𝜆²_𝑖𝑗=−𝜇²_𝑖𝑗. Thus, there exists 𝜀_𝑖𝑗∈{−1,1} such that 𝜇_𝑖𝑗=𝜀_𝑖𝑗√−1𝜆_𝑖𝑗.
If 𝑚₁≥3, since 13𝐸^(𝑚1)_𝑖𝑖+𝐸^(𝑚1)_𝑗𝑗+𝐸^(𝑚1)_𝑘𝑘+𝐷^(𝑚1)_𝑖𝑗+𝐷^(𝑚1)_𝑖𝑘+𝐷^(𝑚1)_𝑗𝑘⊗𝐼_{𝑚2⋯𝑚𝑙}∈⊗^𝑙₁𝒫_𝑚𝑖,13𝐸^(𝑚1)_𝑖𝑖+𝐸^(𝑚1)_𝑗𝑗+𝐸^(𝑚1)_𝑘𝑘+𝐾^(𝑚1)_𝑖𝑗+𝐾^(𝑚1)_𝑖𝑘+𝐷^(𝑚1)_𝑗𝑘⊗𝐼_{𝑚2⋯𝑚𝑙}∈⊗^𝑙₁𝒫_𝑚𝑖,13𝐸^(𝑚1)_𝑖𝑖+𝐸^(𝑚1)_𝑗𝑗+𝐸^(𝑚1)_𝑘𝑘+𝐷^(𝑚1)_𝑖𝑗+𝐾^(𝑚1)_𝑖𝑘+𝐾^(𝑚1)_𝑗𝑘⊗𝐼_{𝑚2⋯𝑚𝑙}∈⊗^𝑙₁𝒫_𝑚𝑖, (48) we obtain using the property of 𝜙, (33), and Step 4 that 131𝜆_𝑖𝑗𝜆_𝑖𝑘𝜆_𝑖𝑗1𝜆_𝑗𝑘𝜆_𝑖𝑘𝜆_𝑗𝑘1∈𝒫₃1,31𝜀_𝑖𝑗𝜆_𝑖𝑗√−1𝜀_𝑖𝑘𝜆_𝑖𝑘√−1−𝜀_𝑖𝑗𝜆_𝑖𝑗√−11𝜆_𝑗𝑘−𝜀_𝑖𝑘𝜆_𝑖𝑘√−1𝜆_𝑗𝑘1∈𝒫₃1,31𝜆_𝑖𝑗𝜀_𝑖𝑘𝜆_𝑖𝑘√−1𝜆_𝑖𝑗1𝜀_𝑗𝑘𝜆_𝑗𝑘√−1−𝜀_𝑖𝑘𝜆_𝑖𝑘√−1−𝜀_𝑗𝑘𝜆_𝑗𝑘√−11∈𝒫₃. (49)(50)(51) It follows from (49) that 𝜆_𝑖𝑗𝜆_𝑗𝑘=𝜆_𝑖𝑘. This, together with (50), implies that 𝜀_𝑖𝑗=𝜀_𝑖𝑘. Similarly, (51) implies that 𝜀_𝑖𝑘=𝜀_𝑗𝑘. Thus, all numbers 𝜀_𝑖𝑗 have to be the same for any 𝑖≠𝑗∈[1,𝑚₁]. This completes the proof of this step.
By Steps 3--5, we set 𝑄=diag(1,𝜆₁₂,…,𝜆_1𝑚1), then 𝑄 is unitary such that 𝜙𝐸^(𝑚1)_𝑘𝑘⊗𝑋=𝑄𝐸^(𝑚1)_𝑘𝑘𝑄^∗⊗𝜋₁(𝑋),∀𝑋∈𝐻_{𝑚2⋯𝑚𝑙},𝑘∈1,𝑚₁,𝜙𝐷^(𝑚1)_𝑖𝑗⊗𝑋=𝑄𝐷^(𝑚1)_𝑖𝑗𝑄^∗⊗𝜋₁(𝑋),∀𝑋∈𝐻_{𝑚2⋯𝑚𝑙},𝑖≠𝑗∈1,𝑚₁,𝜙𝐾^(𝑚1)_𝑖𝑗⊗𝑋=𝜀𝑄𝐾^(𝑚1)_𝑖𝑗𝑄^∗⊗𝜋₁(𝑋),∀𝑋∈𝐻_{𝑚2⋯𝑚𝑙},𝑖≠𝑗∈1,𝑚₁. (52) Thus, set 𝑈=𝑄⊗𝐼_{𝑚2⋯𝑚𝑙}, we have 𝜙(𝐴⊗𝑋)=𝑈𝐴⊗𝜋₁(𝑋)𝑈^∗,∀𝐴∈𝐻_𝑚1,𝑋∈𝐻_{𝑚2⋯𝑚𝑙} (53) or 𝜙(𝐴⊗𝑋)=𝑈𝐴^𝑇⊗𝜋₁(𝑋)𝑈^∗,∀𝐴∈𝐻_𝑚1,𝑋∈𝐻_{𝑚2⋯𝑚𝑙}. (54) This completes the proof of Theorem 6.

Remark 7. When 𝑙≥2, the linear transformation that maps the set of tensor product of idempotent matrices into idempotent matrix set does not necessarily preserve idempotent matrices. For example, let 𝜙𝐴₁⊗𝐴₂⊗⋯⊗𝐴_𝑙=𝐴^𝑇₁⊗𝐴₂⊗⋯⊗𝐴_𝑙,∀𝐴_𝑖∈ℋ_𝑚𝑖[],1,𝑖∈1,𝑙𝐶=2𝐸^(𝑚1)₁₁⊗𝐸^(𝑚2)₁₁+𝐸^(𝑚1)₂₂⊗𝐸^(𝑚2)₂₂+𝐸^(𝑚1)₁₂⊗𝐸^(𝑚2)₁₂+𝐸^(𝑚1)₂₁⊗𝐸^(𝑚2)₂₁⊗𝐼_{𝑚3⋯𝑚𝑙}. (55) Then 𝜙(⊗^𝑙₁𝒫_𝑚𝑖)⊂𝒫_{𝑚1⋯𝑚𝑙} and 𝐶∈𝒫_{𝑚1⋯𝑚𝑙}, but 𝜙(𝐶)=12𝐸^(𝑚1)₁₁⊗𝐸^(𝑚2)₁₁+𝐸^(𝑚1)₂₂⊗𝐸^(𝑚2)₂₂+𝐸^(𝑚1)₂₁⊗𝐸^(𝑚2)₁₂+𝐸^(𝑚1)₁₂⊗𝐸^(𝑚2)₂₁⊗𝐼_{𝑚3⋯𝑚𝑙}∉𝒫_{𝑚1⋯𝑚𝑙}. (56)
As the application of Theorem 6, we give the following theorem.

Theorem 8. Suppose 𝜙 is a linear map from 𝐻_{𝑚1⋯𝑚𝑙} to 𝐻_𝑛 with 𝑛≤𝑚₁⋯𝑚_𝑙. Then, 𝜙(⊗^𝑙₁𝒯_𝑚𝑖)⊂𝒯_𝑛 if and only if either 𝜙=0 or 𝑛=𝑚₁⋯𝑚_𝑙, there exist a unitary matrix 𝑈∈𝑀_𝑛, 𝜀∈{−1,1} and a canonical map 𝜋 on 𝐻_{𝑚1⋯𝑚𝑙} such that 𝜙(𝑋)=𝜀𝑈𝜋(𝑋)𝑈^∗,∀𝑋∈𝐻_{𝑚1⋯𝑚𝑙}. (57)

Proof. The sufficiency part is clear. We give the proof of the necessity part.
Set Γ₀=𝐼_𝑚1⊗⋯⊗𝐼_𝑚𝑙,Γ₁=𝑃₁⊗𝐼_{𝑚2⋯𝑚𝑙}∶𝑃₁∈𝒫_𝑚1,Γ₂=𝑃₁⊗𝑃₂⊗𝐼_{𝑚3⋯𝑚𝑙}∶𝑃_𝑖∈𝒫_𝑚𝑖[,⋯Γ,𝑖∈1,2]_𝑙=𝑃₁⊗⋯⊗𝑃_𝑚𝑙∶𝑃_𝑖∈𝒫_𝑚𝑖,𝑖∈[1,𝑙]. (58) It is obvious that Γ_𝑙=⊗^𝑙₁𝒫_𝑚𝑖 and Γ_𝑘⊂⊗^𝑙₁𝒯_𝑚𝑖, ∀𝑘∈[0,𝑙].
Since 𝐼_{𝑚1⋯𝑚𝑙}∈⊗^𝑙₁𝒯_𝑚𝑖, we have 𝜙(𝐼_{𝑚1⋯𝑚𝑙})∈𝒯_𝑛. Without loss of generality, we may assume that 𝜙(𝐼_{𝑚1⋯𝑚𝑙})=𝐼_𝑝⊕−𝐼_𝑞⊕0_{𝑛−𝑝−𝑞}. We next prove by induction on 𝑘 that 𝜙(𝑋)=𝜓_𝑝(𝑋)⊕−𝜓_𝑞(𝑋)⊕0_{𝑛−𝑝−𝑞},∀𝑋∈Γ_𝑘, (59) where 𝜓_𝑝(𝑋)∈𝒫_𝑝 and 𝜓_𝑞(𝑋)∈𝒫_𝑞.
We assume that our statement holds true for 𝑘−1 and prove it for 𝑘. For any 𝑋=𝑃₁⊗⋯⊗𝑃_𝑘⊗𝐼_{𝑚𝑘+1⋯𝑚𝑙}∈Γ_𝑘, choosing 𝑌=𝑃₁⊗⋯⊗𝑃_𝑘−1⊗𝐼_𝑚𝑘⊗𝐼_{𝑚𝑘+1⋯𝑚𝑙}∈Γ_𝑘−1, we have by induction hypothesis that 𝜙(𝑌)=𝜓_𝑝(𝑌)⊕−𝜓_𝑞(𝑌)⊕0_{𝑛−𝑝−𝑞}, (60) with 𝜓_𝑝(𝑌)∈𝒫_𝑝 and 𝜓_𝑞(𝑌)∈𝒫_𝑞. Since 𝑋∈⊗^𝑙₁𝒯_𝑚𝑖, 𝑌−𝑋∈⊗^𝑙₁𝒯_𝑚𝑖 and 𝑌−2𝑋∈⊗^𝑙₁𝒯_𝑚𝑖. Using the property of 𝜙, we have 𝜙(𝑋)∈𝒯_𝑛, 𝜙(𝑌)−𝜙(𝑋)∈𝒯_𝑛 and 𝜙(𝑌)−2𝜙(𝑋)∈𝒯_𝑛. By Lemma 4, we have 𝜙(𝑋)=𝜓_𝑝(𝑋)⊕−𝜓_𝑞(𝑋)⊕0_{𝑛−𝑝−𝑞}, (61) with 𝜓_𝑝(𝑋)∈𝒫_𝑝 and 𝜓_𝑞(𝑋)∈𝒫_𝑞. This implies that (59) holds.
As 𝜙 is linear, we can expend 𝜓_𝑝 to be a linear map from 𝐻_{𝑚1⋯𝑚𝑙} to 𝐻_𝑝 and 𝜓_𝑞 to be a linear map from 𝐻_{𝑚1⋯𝑚𝑙} to 𝐻_𝑞, then 𝜙(𝑋)=𝜓_𝑝(𝑋)⊕−𝜓_𝑞(𝑋)⊕0_{𝑛−𝑝−𝑞},∀𝑋∈𝐻_{𝑚1⋯𝑚𝑙}, (62) with 𝜓_𝑝(⊗^𝑙₁𝒫_𝑚𝑖)⊂𝒫_𝑝 and 𝜓_𝑞(⊗^𝑙₁𝒫_𝑚𝑖)⊂𝒫_𝑞, we obtain using Theorem 6 that(a)if 𝑛<𝑚₁⋯𝑚_𝑙, then 𝑝,𝑞<𝑚₁⋯𝑚_𝑙, 𝜓_𝑝=0, 𝜓_𝑞=0, thus, 𝜙=0, and(b)if 𝜙≠0, then 𝑛=𝑚₁⋯𝑚_𝑙 and 𝑝=𝑛 or 𝑞=𝑛, thus 𝜙=𝜓_𝑝 or 𝜙=−𝜓_𝑞.
This completes the proof of Theorem 8.

Acknowledgments

The authors show great thanks to the referee for his/her careful reading of the paper and valuable comments which greatly improved the readability of the paper. Jinli Xu is supported in part by NSFC (11171294), Natural Science Foundation of Heilongjiang Province of China (Grant no. A201013). Baodong Zheng is supported by the National Natural Science Foundation Grants of China (Grant no. 10871056). Hongmei Yao is supported by the Fundamental Research Funds for the Central Universities.