Fixed Points Associated to Power of Normal Completely Positive Maps* ()
Received 1 April 2016; accepted 21 May 2016; published 24 May 2016
![](//html.scirp.org/file/8-1720568x19.png)
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).
NOTES
![]()
*Fixed points of completely positive maps.