Journal of Applied Mathematics and Physics
Vol.04 No.05(2016), Article ID:66721,5 pages
10.4236/jamp.2016.45101
Fixed Points Associated to Power of Normal Completely Positive Maps*
Haiyan Zhang, Hongying Si
College of Mathematics and Information Science, Shangqiu Normal University, Shangqiu, China
Copyright © 2016 by authors and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY).
http://creativecommons.org/licenses/by/4.0/
Received 1 April 2016; accepted 21 May 2016; published 24 May 2016
ABSTRACT
Let be a normal completely positive map with Kraus operators
. An operator X is said to be a fixed point of
, if
. Let
be the fixed points set of
. In this paper, fixed points of
are considered for
, where
means j-power of
. We obtain that
and
for integral
when A is self-adjoint and commutable. Moreover,
holds under certain condition.
Keywords:
Fixed Point, Power, Completely Positive Map
1. Introduction
Completely positive maps are founded to be very important in operator algebras and quantum information. Especially recent years, it has a great development since a quantum channel can be represented by a trace preserving completely positive map. Fixed points of completely positive map are useful in theory of quantum error correction and quantum measurement theory and have been studied in several papers from different aspects, many interesting results have been obtained (see [1] - [12] ).
For the convenience of description, let H be a separable complex Hilbert space and be the set of all bounded linear operators on H. Let
on
be a contractive
map. As we know, every contractive and normal (or weak
continuous) completely positive map
on
is determined by a row contraction on H in the sense that
where if, the convergence is in the weak * topology (see [13] and [14] ) and then denoting
, we call
a completely positive map associated with A.
Let be an at most countable subset of
with
, where the series is convergent in the strong operator topology. In this case, A is called a row contraction. Then
is well defined on
and also a normal completely positive map. Moreover, we denote j-power for
by
, that is
. In addition, For a row contraction
, we say that the operator sequence A is unital if
is commutative, if
for all
is normal, if each
is normal and positive, and if every
is positive. If A is unital (resp. commutative) then we say that
is unital (resp. commutative). Moreover,
or A is called trace preserving if
. For a subset
, we denote the commutant of S in
by
. We say that an
is a fixed point of
or a fixed point associated to the row contraction A if
. Let
be the set of fixed points of
. Some authors compared the commutant
of
, where
, and some conditions for which
are given (as in [1] , [10] ).
For a trace preserving quantum operation, it was proved that
if
in [1] . And, if Kraus operators A is a spherical unitary [10] . On the other hand, the authors [12] consider some conditions for a unital and commuting row contraction A to be normal and therefore
in those cases. Moreover, the fixed points set
of
is represented when A is a commuting and trace preserving row contraction [15] .
The purpose of this paper is to investigate fixed points of j-power of the completely positive map for
. It is obtained that
and
when A is self-adjoint and commutable. Furthermore,
holds under certain condition.
2. Main Results
In this section, let A be a normal and commuting row contraction. To give main results, we begin with some notations and lemmas. Let be the strong operator topology limit of
.
Lemma 1 ( [10] ) Let be a unital and normal commuting row contractions. Then
.
Lemma 2 ( [10] ) Let be a commuting row contraction. If
, then there exists a triple
where K is a Hilbert space,
is a bounded operator from K to H and
is a spherical unitary on K satisfying the following properties:
1);
2) for all k;
3) K is the smallest reducing subspace for containing
;
4) The mapping
defined by
is a complete isometry from the commutant of onto the space
;
5) There exists a *-homomorphism such that
.
Lemma 3 ( [16] ) (Fuglede-Putnam Theorem) Let, if A and B are normal, then
implies
.
In general, there is no concrete relation between and
for different positive integers k and j.
Example 4 Let and
, then
is well defined and
. However, by a direct computation,
and
. Hence,
.
But if A is self-adjoint and commutable, the following result holds.
Theorem 5 If A is unital, self-adjoint and commutable, then and
for any
.
Proof. For any, we first prove
. For any
, then
. So
for any j. It is only to prove
. According to A is unital, self- adjoint and commutable, then
is so. For any operator
, then A and
are commutable for any k by lemma 1. By the function calculus, A and
are commutable
since is self-adjoint, and so
. Therefore,
.
Next, we prove. For any
, then
, for any
.
So since
, thus
. It follows that
and. So
. Conversely, for any
,
, then
. So
for any j. Therefore
. So
and
. Therefore, the result holds. This completes the proof.
Corollary 6 Let be unital, self-adjoint and commutable, then
,
where
Proof. From Theorem 5 and Lemma 1, it is only to prove that. Let
and
, for any operator
, then
.
From Lemma 1, we have
.
It follows that for any k and then
. This completes the proof.
Theorem 7 Let be unital and commutable. Supposing that there is an
such that
is positive and invertible, then
, where
.
Proof. From Lemma 2, there exists a triple where K is a Hilbert space,
is abounded operator and
is a normal unital and commuting operator sequence on K having the properties 1)
; 2)
; 3) K is the smallest reducing subspace for
containing
; 4) The mapping
defined by
and
is a complete isometry from the commutant of
onto the space
; also it is obtained that
for any k, where
is a unital *-homomorphism. Then
is positive and invertible since
is positive and invertible. Next, we write
and
for any
. In fact,
if and only if
for any
. On one hand,
if
; on the other hand, if
, then
and so
by the function calculus. Moreover,
, thus
since
is invertible. That is to say,
and
have the same reducing subspace. It follows that K is also the smallest reducing subspace for the unital, normal and commuting operator sequence
containing
. Thus
is also a complete isometry from the commutant of
onto the space
. Combining with
, it is easy to get
. The proof is completed.
Acknowledgements
This research was supported by the Natural Science Basic Research Plan of Henan Province (No. 14 B110010 and No. 1523000410221).
Cite this paper
Haiyan Zhang,Hongying Si, (2016) Fixed Points Associated to Power of Normal Completely Positive Maps*. Journal of Applied Mathematics and Physics,04,925-929. doi: 10.4236/jamp.2016.45101
References
- 1. Arias, A., Gheondea, A. and Gudder, S. (2002) Fixed Points of Quantum Operations. Journal of Mathematical Physics, 43, 5872-5881.
http://dx.doi.org/10.1063/1.1519669 - 2. Li, Y. (2011) Characterizations of Fixed Points of Quantum Operations. Journal of Mathematical Physics, 52, Article ID: 052103.
http://dx.doi.org/10.1063/1.3583541 - 3. Long, L. and Zhang, S.F. (2011) Fixed Points of Commutative Super-Operators. Journal of Physics A: Mathematical and Theoretical, 44, Article ID: 09521.
http://dx.doi.org/10.1088/1751-8113/44/9/095201 - 4. Long, L. (2011) A Class of General Quantum Operations. International Journal of Theoretical Physics, 50, 1319-1324.
http://dx.doi.org/10.1007/s10773-010-0627-4 - 5. Liu, W.H. and Wu, J.D. (2010) Fixed Points of Commutative Lüders Operations. Journal of Physics A: Mathematical and Theoretical, 43, Article ID: 395206.
http://dx.doi.org/10.1088/1751-8113/43/39/395206 - 6. Magajna, B. (2012) Fixed Points of Normal Completely Positive Maps on B(H). Journal of Mathematical Analysis and Applications, 389, 1291-1302.
http://dx.doi.org/10.1016/j.jmaa.2012.01.007 - 7. Nagy, G. (2008) On Spectra of Lüders Operations. Journal of Mathematical Physics, 49, Article ID: 022110.
http://dx.doi.org/10.1063/1.2840472 - 8. Popescu, G. (2003) Similarity and Ergodic Theory of Positive Linear maps. Journal für die reine und angewandte Mathematik, 561, 87-129.
http://dx.doi.org/10.1515/crll.2003.069 - 9. Prunaru, B. (2008) Toeplitz Operators Associated to Commuting Row Contractions. Journal of Functional Analysis, 254, 1626-1641.
http://dx.doi.org/10.1016/j.jfa.2007.11.001 - 10. Prunaru, B. (2011) Fixed Points for Lüders Operations and Commutators. Journal of Physics A: Mathematical and Theoretical, 44, Article ID: 185203.
http://dx.doi.org/10.1088/1751-8113/44/18/185203 - 11. Zhang, H.Y. and Ji, G.X. (2012) A Note on Fixed Points of General Quantum Operations. Reports on Mathematical Physics, 70, 111-117.
http://dx.doi.org/10.1016/S0034-4877(13)60016-6 - 12. Zhang, H.Y. and Ji, G.X. (2013) Normality and Fixed Points Associated to Commutative Row Contractions. Journal of Mathematical Analysis and Applications, 400, 247-253.
http://dx.doi.org/10.1016/j.jmaa.2012.10.042 - 13. Choi, M.D. (1975) Completely Positive Linear Maps on Complex Matrices. Linear Algebra and Its Applications, 10, 285-290.
http://dx.doi.org/10.1016/0024-3795(75)90075-0 - 14. Kraus, K. (1971) General State Changes in Quantum Theory. Annals of Physics, 64, 311-355.
http://dx.doi.org/10.1016/0003-4916(71)90108-4 - 15. Zhang, H.Y. and Xue, M.Z. (2016) Fixed Points of Trace Preserving Completely Positive maps. Linear Multilinear A, 64, 404-411.
http://dx.doi.org/10.1080/03081087.2015.1043718 - 16. Hou, J.C. (1985) On Putnam-Fuglede Theorem of Non-normal Operators. Acta Mathematica Sinica, 28, 333-340.
NOTES
*Fixed points of completely positive maps.