Fixed Points Associated to Power of Normal Completely Positive Maps*

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http://dx.doi.org/10.4236/jamp.2016.45101

How to cite this paper:Zhang, H.Y. and Si, H.Y. (2016) Fixed Points Associated to Power of Normal Completely Positive Maps. Journal of Applied Mathematics and Physics, 4, 925-929. http://dx.doi.org/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

Received 1 April 2016; accepted 21 May 2016; published 24 May 2016

This work is licensed under the Creative Commons Attribution International License (CC BY).

http://creativecommons.org/licenses/by/4.0/

Abstract

Let ϕ_Α be a normal completely positive map with Kraus operators

{ }

n k k

E

1

=

Α = . An operator X is said to be a fixed point of ϕ_Α_,if ϕ_Α

( )

X = X. Let B H

( )

ϕΑ be the fixed points set of ϕ_Α. In this

paper, fixed points of ϕj

Α are considered for 1< < +∞j , where ϕΑj means j-power of ϕΑ. We

obtain that B H

( )

j B H

( )

B H

( )

2

ϕ_Α ϕ_Α −ϕ_Α

= + and B H

( )

j B H

( )

2 1

ϕ − ϕ

Α

Α ₌ _{for integral} _j_>₁ _{when A is}

self-adjoint and commutable. Moreover,

( )

j

( )

B H ϕΑ =B H ϕΑ 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. Es-pecially recent years, it has a great development since a quantum channel can be represented by a trace preserv-ing completely positive map. Fixed points of completely positive map are useful in theory of quantum error cor-rection 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 B H

( )

be the set of all bounded linear operators on H. Let Φ on B H

( )

be a contractive

( )

Φ map. As we know, every contrac-tive and normal (or weak * continuous) completely positive map Φ on B H

( )

is determined by a row con-traction on H in the sense that

*

(2)

( )

*

( )

1 ,

n k k k

X ₌ E XE X B H

Φ =

∑

∀ ∈

where if n= +∞, the convergence is in the weak * topology (see [13] and [14]) and then denoting ϕΑ = Φ, we

call ϕΑ a completely positive map associated with A.

Let Α =

{ }

Ek nk=₁ be an at most countable subset of B H

( )

with

* 1 n

k k k= E E ≤I

∑

, where the series is con-vergent in the strong operator topology. In this case, A is called a row contraction. Then A

( )

1 *

n k k k

X E XE

ϕ =

∑

₌

is well defined on B H

( )

and also a normal completely positive map. Moreover, we denote j-power for

1< < +∞j by ϕj

Α, that is A A A A

j

ϕ =ϕ ϕ_ _ϕ . In addition, For a row contraction Α =

{ }

Ek n_k₌₁, we say

that the operator sequence A is unital if

∑

nk=₁E Ek k*=I is commutative, if E Ek j=E Ej k for all 1≤k j, ≤n

is normal, if each Ek is normal and positive, and if every Ek is positive. If A is unital (resp. commutative) then we say that ϕ_Α is unital (resp. commutative). Moreover, ϕ_Α or A is called trace preserving if

* 1 n

k k k= E E ≤I

∑

. For a subset S⊂B H

( )

, we denote the commutant of S in B H

( )

by S′. We say that an

( )

X∈B H is a fixed point of ϕΑ or a fixed point associated to the row contraction A if ϕΑ

( )

X =X . Let

( )

B H ϕΑ be the set of fixed points of ϕΑ. Some authors compared the commutant

{

E Ek, k*,1 k n

}

′ ≤ ≤ of

*

Α ∪ Α , where *

{ }

*

1

n k _k

E =

Α = , and some conditions for which

( )

{

*

}

, ,1

k k

B H ϕΑ = E E ≤ ≤k n′ are given (as in

[1], [10]).

For a trace preserving quantum operation ϕΑ, it was proved that B H

( )

{

E Ek, k*,1 k n

}

ϕΑ ₌ _{≤ ≤} ′

if dimH < ∞

in [1]. And B H

( )

ϕΑ = Α′, 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 B H

( )

ϕΑ = Α′

in those cases. Moreover, the fixed points set B H

( )

ϕΑ of ϕΑ 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 ϕj

Α for

1

j≥ . It is obtained that

( )

2j

B H ϕΑ =B H ϕΑ +B H −ϕΑ and

( )

2j1

B H ϕ B H ϕ

−

Α Α ₌

when A is self-adjoint

and commutable. Furthermore,

( )

j

B H ϕΑ =B H ϕΑ 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

no-tations and lemmas. Let . . lim j

( )

j

Q_Α s o ϕ_Α I

→∞

= − be the strong operator topology limit of

{

( )

}

1 j

j

I

ϕΑ ∞₌ .

Lemma 1 ([10]) Let Α =

{ }

Ek nk=₁ be a unital and normal commuting row contractions. Then B H

( )

ϕΑ _{= Α}_′

.

Lemma 2 ([10]) Let Α =

{ }

Ek nk=₁⊂B H

( )

be a commuting row contraction. If QΑ≠0, then there exists a

triple

{

{ }

}

1

, , k n_k

K Γ U ₌ where K is a Hilbert space, Γ is a bounded operator from K to H and

{ }

Uk nk=₁ is a

spherical unitary on K satisfying the following properties: 1) *

Q_Α

ΓΓ = ;

2) EkΓ = ΓUk for all k;

3) K is the smallest reducing subspace for

{ }

Uk nk=₁ containing

*_H

Γ ;

4) The mapping

{ }

₁

( )

: Uk n_k B H

ϕ

ρ Α

=′ →

defined by

( )

*

{ }

1

, k nk

Y Y Y U

ρ = Γ Γ ∈ ₌

is a complete isometry from the commutant of

{ }

Uk n_k₌₁ onto the space B H

( )

ϕΑ

; 5) There exists a *-homomorphism *

{

( )

}

{ }

1

: _H, _k n

k

C I B H ϕ U

π Α

=

→ such that

( )

(

)

,

{ }

1

n k k

Y Y Y U

(3)

Lemma 3 ([16]) (Fuglede-Putnam Theorem) Let A B C, , ∈B H

( )

, if A and B are normal, then AC=CB

implies A C* =CB*.

In general, there is no concrete relation between B H

( )

ϕΑk and

( )

j

B H ϕΑ for different positive integers k and

j.

Example 4 Let

1 3 0 2 2 1 3 0 2 2 i E i   +     =   −    

and Α =

{ }

E , then ϕΑ

( )

X =EXE* is well defined and

( )

11

11 22 22 0 : , 0 a

B H a a C

a

ϕΑ ₌  _∈ 

  

  

 . However, by a direct computation,

( )

3

X X

ϕΑ = and

( )

3

2

B H ϕΑ =M C .

Hence, B H

( )

ϕΑ3 ≠B H

( )

ϕΑ.

But if A is self-adjoint and commutable, the following result holds.

Theorem 5 If A is unital, self-adjoint and commutable, then

( )

2j1

B H ϕ B H ϕ

−

Α Α ₌

and

( )

2j

( )

B H ϕΑ =B H ϕΑ +B H −ϕΑ for any j≥1.

Proof. For any j≥1, we first prove

( )

2j1

B H ϕ B H ϕ

−

Α Α ₌

. For any X∈B H

( )

ϕΑ, then ϕ_Α

( )

X = X . So

( )

1

( )

j j

X X X

ϕ ϕ −

Α = Α == for any j. It is only to prove

( )

2j1

B H ϕ B H ϕ

−

Α

Α _⊂

. According to A is unital, self-

adjoint and commutable, then

{

₁ ₂ : for 1

}

j

k k k i

E E E k i j

Α = _ ∈ Α ≤ ≤ is so. For any operator

( )

2j1

A B H ϕ

− Α

∈ ,

then A and Ek2j1

−

are commutable for any k by lemma 1. By the function calculus, A and Ek are commutable since Ek is self-adjoint, and so A B H

( )

ϕΑ

∈ . Therefore,

( )

2j1

( )

B H ϕ B H ϕ

−

Α Α ₌

.

Next, we prove B H

( )

ϕΑ2j =B H

( )

ϕΑ +B H

( )

−ϕΑ. For any A∈B H

( )

ϕ2Αj, then

1 1 1 1

2 2 2 2

j j

k k k l k k k l

AE E E ₋E =E E E ₋E A, for any k k k, 1, 2,,kj−1,l∈

{

1, 2,,n

}

.

So AE Ek l=E E Ak l since

2 1 n

k k=E =I

∑

, thus A∈B H

( )

ϕΑ2. It follows that 1

(

( )

)

( )

2 A A B H

ϕ

ϕ Α

Α

+ ∈

and 1

(

( )

)

( )

2 A A B H

ϕ

ϕ −Α

Α

− ∈ . So

( )

2j

B H ϕΑ ⊂B H ϕΑ +B H −ϕΑ . Conversely, for any X∈B H

( )

ϕΑ

,

( )

Y∈B H −ϕΑ , then ϕ_Α

( )

Y = −Y . So ϕΑ2j

( )

Y =ϕΑ2j−1

( )

−Y =ϕΑ2j−2

( )

Y ==Y for any j. Therefore

(

)

(

)

(

)

2j 2j1 2j2

X Y X Y X Y Y

ϕ ϕ − ϕ −

Α + = Α − = Α + == . So

( )

2j

B H ϕΑ +B H −ϕΑ ⊂B H ϕΑ and

( )

2j

B H ϕΑ +B H −ϕΑ =B H ϕΑ . Therefore, the result holds. This completes the proof.

Corollary 6 Let Α =

{ }

Ek nk=₁ be unital, self-adjoint and commutable, then

(

,

)

Iw

′

Α ⋅ Α = Α + Α −Α ,

where Iw

(

Α −Α =,

)

{

X∈B H

( )

:E Xk +XEk =0, for anyk

}

.

Proof. From Theorem 5 and Lemma 1, it is only to prove that B H

( )

−ϕΑ =Iw

(

Α −Α,

)

. Let 0

0 k k k E F E   =  ₋   

and Β =

{ }

Fk nk=₁, for any operator X B H

( )

ϕ_Α −

∈ , then 0

(

)

0 0

X

B H H ϕΒ

 

∈ ⊕

 

  .

From Lemma 1, we have

0 0

0 0 0 0

(4)

It follows that E X_k +XE_k =0 for any k and then X∈Iw

(

Α −Α,

)

. This completes the proof.

Theorem 7 Let Α =

{ }

Ek n_k₌₁ be unital and commutable. Supposing that there is an k0 such that Ek0∈ Α

is positive and invertible, then B H

( )

ϕΑj =B H

( )

ϕΑ, where 1≤ < ∞j .

Proof. From Lemma 2, there exists a triple

{

{ }

}

1

, , _k n

k

K Γ U ₌ where K is a Hilbert space, Γ:K→H is

ab-ounded operator and

{ }

Uk nk=₁ is a normal unital and commuting operator sequence on K having the properties 1)

* _I

ΓΓ = ; 2) EkΓ = ΓUk; 3) K is the smallest reducing subspace for

{ }

1 n k k

U ₌ containing _Γ*_H

; 4) The mapping

{ }

₁

( )

: _k n

k

U B H ϕ

ρ Α

=′ → defined by

( )

*

Y Y

ρ = Γ Γ and

{ }

1

n k k

Y∈ U ₌′ is a complete isometry from the

commu-tant of

{ }

Uk nk=₁ onto the space B H

( )

ϕΑ

; also it is obtained that U_k =π

( )

E_k for any k, where

( )

{

}

( )

*

:C I_H,B H ϕ B K

π Α _→

is a unital *-homomorphism. Then

( )

0 0

k k

U =π E is positive and invertible since

0

k

E is positive and invertible. Next, we write Λ =

{

k k1, 2,,kj

}

:1≤k k1, 2,,kj ≤n

}

and Uα =U Uk1 k2Ukj

for any α =

{

k k1, 2,,kj

}

∈ Λ. In fact, U Aα =AUα if and only if U Ak =AUk for any 1≤ ≤k n. On one hand, U A_α =AU_α if U Ak =AUk ; on the other hand, if U Aα =AUα , then 0 0

j j

k k

U A=AU and so

0 0

k k

U A=AU by the function calculus. Moreover, AUkj₀1Uk Ukj₀1U A Uk kj₀1AUk

− ₌ − ₌ −

, thus U Ak =AUk since

0

1 j k

U − is invertible. That is to say,

{ }

U_{α α∈Λ} and

{ }

Uk nk=₁ have the same reducing subspace. It follows that K is

also the smallest reducing subspace for the unital, normal and commuting operator sequence

{ }

U_{α α∈Λ}

contain-ing _Γ*_H

. Thus

( )

*

{ }

,

Y Y Y U_{α α}

ρ = Γ Γ ∈ _∈Λ′ is also a complete isometry from the commutant of

{ }

U_{α α∈Λ}

onto the space

( )

j

H ϕΑ. Combining with

( )

j

B H ϕΑ ⊂B H ϕΑ, it is easy to get

( )

j

B H ϕΑ =B H ϕΑ. 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).

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

(5)

[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