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Journal of Function Spaces and Applications
Volume 2013 (2013), Article ID 182569, 7 pages
Linear Transformations between Multipartite Quantum Systems That Map the Set of Tensor Product of Idempotent Matrices into Idempotent Matrix Set
1Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China
2College of Science, Harbin Engineering University, Harbin 150001, China
Received 12 May 2013; Accepted 16 July 2013
Academic Editor: Gen-Qi Xu
Copyright © 2013 Jinli Xu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Let be the set of complex Hermitian matrices and (resp., ) be the set of all idempotent (resp., tripotent) matrices in . In -partite quantum system , (resp., ) denotes the set of all decomposable elements such that (resp., ). In this paper, linear maps ϕ from to with such that are characterized. As its application, the structure of linear maps ϕ from to with such that is also obtained.
Let be the vector space of complex matrices and be the vector space of complex Hermitian matrices. In quantum information theory, a -partite system can be represented as the tensor product space , where is the usual Kronecker product of matrices; a quantum state is represented as a positive semidefinite with trace one in , see . In quantum information science and quantum computing, it is important to understand, characterize, and construct different classes of maps on quantum states . For example, entanglement is one of the main concepts in quantum information theory, and entangled state involves at least bipartite system or multipartite system ; to study entangled states, one should construct entanglement witnesses, which are special types of positive maps, see . 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 , 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.  first characterized linear transformations on preserving idempotent matrices. In , the authors obtained the following result: linear map with satisfies if and only if or where 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 -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, with satisfying where is the set of all idempotent matrices in , that is, and denotes the set of all decomposable elements such that . As application, the forms of linear maps from -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, with satisfying where is the set of all tripotent matrices in , that is, and denotes the set of all decomposable elements such that .
Throughout this paper, we always assume integers and , , . Let be the identity matrix, be the zero matrix which order is omitted in different matrices just for simplicity, and (resp., , ) be the transpose (resp., conjugate transpose, rank) of . () stands for the matrix with 1 at the th entry and 0 otherwise. and , where . For real numbers and with , let be the set of all integers between and . is the usual direct sum of matrices. A linear map is canonical if satisfies , where satisfies or for all .
2. Main Results
Lemma 1 (see ). Suppose such that , . Let . Then, there exists a unitary matrix such that where is the diagonal matrix in which all diagonal entries are zero except those in the st to the th rows.
Lemma 2. Let . Suppose , . Then, there exists such that , where .
Proof. Let be a unitary matrix satisfying . Set where . Then, A straightforward computation shows that ; therefore, the result holds.
Lemma 3. Let and with , . If where , and are all roots of , then, , and is of the form where , .
Proof. It is clear that and . By a direct computation, one can obtain that Replacing by , we have and Choosing , then or , this yields that Thus, and . This, together with the form of and , implies that has the form , where , .
Lemma 4. Let with and . Suppose satisfying and . Then, with and .
Proof. By a direct computation, we have Let and be unitary matrices satisfying and , respectively. Then, By (12), one can assume that where and . This, together with (13), implies that and . Thus, as desired.
Lemma 5. Let , be canonical maps on , and , are invertible matrices of order . If for all , then and for some .
Proof. Let us first point out a simple observation which will be used in our proof. If a matrix commutes with for all real symmetric , then has the form with .
Since are canonical maps on , we assume that where and are the identity maps or the transposition maps for .
Clearly, , , hence .
We prove by induction on . The case of is the well-known fact. We assume that the statement holds true for . We will prove it for . For any real symmetric , since , , it follows that for some matrix . We define linear maps It is easy to see that and are canonical maps on . For any , since we have , thus, by induction hypothesis, . Since is invertible, we have . Thus, and for all .
Theorem 6. Suppose is a linear map from to with . Then, if and only if either or , there exists a unitary matrix and a canonical map on such that
Proof. The sufficiency part is obvious. We prove the necessity part by induction on . When , it is the result in , which has been given in the above introduction. We assume now that the result holds true for and give the proof of the case by the following five steps.
Step 1. Suppose and . Then, where is a linear map from to satisfying .
Proof of Step 1. Set
We prove by induction on that . Assume for a moment that we have already proved this. Then, when is linear, we can complete the proof of Step 1.
Now, we prove the assertion. The case of is just the assumption. Then, we assume that our statement holds true for , and we consider the image of for , . Because we obtain using the property of that
We obtain by induction hypothesis and Lemma 2 that , where , as desired.
Step 2. If . Then, .
Proof of Step 2. Since and , we obtain using the property of that Denote by , . Using Lemma 1 and composing by a similarity transformation, we may obtain that where is the diagonal matrix in which all diagonal entries are zero except those in the st to the th rows.
Since , there exists some . Without loss of generality, we may assume . Thus, . By Step 1, we see that there exists a linear map from to satisfying such that for all . We obtain by induction hypothesis . Thus, For any , and real with , we have , where and are all roots of . We obtain using the property of , (27), and Lemma 3 that and . By using the similar argument, we see that , , . Thus, By the arbitrariness of , we have Therefore, .
We next always assume and .
Step 3. Suppose are of rank one with , . Then, there exists a unitary matrix of order such that where is a canonical map on .
Proof of Step 3. Using the similar approach as in the proof of Step 2, we may show that , . In fact, if for some , then, by Lemma 1, there exists for some . Then, using the similar approach as in the proof of Step 2, we have , which is a contradiction, and there exist a unitary matrix such that . Thus, applying Step 1, we obtain that where linear map satisfying , for any . We obtain by induction hypothesis that there exists unitary matrices and canonical maps on such that Set , we complete the proof of Step 3.
By Step 3, we may assume that where is canonical map on .
Step 4. For , and there exists , with such that
Proof of Step 4. Without loss of generality, we only prove the case of , .
Set , , for (if ). By Step 3, there exists a unitary matrix and canonical maps on such that This, together with (33), implies that Thus, we may set where .
On one hand, noting that we obtain using (33) and (35) that for all ,
On the other hand, for any and real with , we have , where and are all roots of . We obtain using the property of , (33), and Lemma 3 that where . Choosing , we get using (39), (40), and (42) that . Hence, by (39), (40), and (42) again, one can obtain that By Lemma 5, we have and with . Hence, This, together with (41), implies that . Similarly, This completes the proof of Step 4.
Step 5. There exists such that , . If , then , for distinct .
Proof of Step 5. Since we obtain using the property of , (33), and Step 4 that Hence, . It follows from that . Thus, there exists such that .
If , since we obtain using the property of , (33), and Step 4 that It follows from (49) that . This, together with (50), implies that . Similarly, (51) implies that . Thus, all numbers have to be the same for any . This completes the proof of this step.
By Steps , we set , then is unitary such that Thus, set , we have or This completes the proof of Theorem 6.
Remark 7. When , 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
Then and , but
As the application of Theorem 6, we give the following theorem.
Theorem 8. Suppose is a linear map from to with . Then, if and only if either or , there exist a unitary matrix , and a canonical map on such that
Proof. The sufficiency part is clear. We give the proof of the necessity part.
Set It is obvious that and , .
Since , we have . Without loss of generality, we may assume that . We next prove by induction on that where and .
We assume that our statement holds true for and prove it for . For any , choosing , we have by induction hypothesis that with and . Since , and . Using the property of , we have , and . By Lemma 4, we have with and . This implies that (59) holds.
As is linear, we can expend to be a linear map from to and to be a linear map from to , then with and , we obtain using Theorem 6 that(a)if , then , , , thus, , and(b)if , then and or , thus or .
This completes the proof of Theorem 8.
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.
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