Complete Positivity of Two Class of Maps Involving Depolarizing and Transpose-Depolarizing Channels

2021 ◽  
Vol 28 (02) ◽  
Author(s):  
Xiuhong Sun ◽  
Yuan Li
Keyword(s):  
Sufficient Conditions ◽  
Linear Operators ◽  
Complete Positivity ◽  
Completely Positive Map ◽  
Completely Positive ◽  
Bounded Linear ◽  
Positive Map ◽  
Finite Dimensional ◽  
Necessary And Sufficient ◽  
Finite Dimensional Hilbert Space

In this note, we mainly study the necessary and sufficient conditions for the complete positivity of generalizations of depolarizing and transpose-depolarizing channels. Specifically, we define [Formula: see text] and [Formula: see text], where [Formula: see text] (the set of all bounded linear operators on the finite-dimensional Hilbert space [Formula: see text] is given and [Formula: see text] is the transpose of [Formula: see text] in a fixed orthonormal basis of [Formula: see text] First, we show that [Formula: see text] is completely positive if and only if [Formula: see text] is a positive map, which is equivalent to [Formula: see text] Moreover, [Formula: see text] is a completely positive map if and only if [Formula: see text] and [Formula: see text] At last, we also get that [Formula: see text] is a completely positive map if and only if [Formula: see text] with [Formula: see text] for all [Formula: see text] where [Formula: see text] are eigenvalues of [Formula: see text].

scholarly journals Infinite dimensional generalizations of Choi’s Theorem

2019 ◽  
Vol 7 (1) ◽  
pp. 67-77
Author(s):  
Shmuel Friedland
Keyword(s):  
Hilbert Spaces ◽  
Quantum Channel ◽  
Trace Class ◽  
Natural Generalization ◽  
Linear Operators ◽  
Completely Positive Map ◽  
Completely Positive ◽  
Bounded Linear ◽  
Positive Map ◽  
Finite Dimensional

Abstract In this paper we give a simple sequence of necessary and sufficient finite dimensional conditions for a positive map between certain subspaces of bounded linear operators on separable Hilbert spaces to be completely positive. These criterions are natural generalization of Choi’s characterization for completely positive maps between pairs of linear operators on finite dimensional Hilbert spaces. We apply our conditions to a completely positive map between two trace class operators on separable Hilbert spaces. A completely positive map μ is called a quantum channel, if it is trace preserving, and μ is called a quantum subchannel if it decreases the trace of a positive operator.We give simple neccesary and sufficient condtions for μ to be a quantum subchannel.We show that μ is a quantum subchannel if and only if it hasHellwig-Kraus representation. The last result extends the classical results of Kraus and the recent result of Holevo for characterization of a quantum channel.

QUADRATIC EQUATIONS IN HILBERTIAN OPERATORS AND APPLICATIONS

2009 ◽  
Vol 20 (11) ◽  
pp. 1431-1454
Author(s):  
VICTOR J. MIZEL ◽  
M. M. RAO
Keyword(s):  
Random Fields ◽  
Sufficient Conditions ◽  
Linear Operators ◽  
Infinite Dimensions ◽  
Bounded Linear ◽  
Quadratic Equations ◽  
Weak Ordering ◽  
Finite Dimensional ◽  
Necessary And Sufficient ◽  
Adjoint Operators

In this paper bounded linear operators in Hilbert space satisfying general quadratic equations are characterized. Necessary and sufficient conditions for sets of operators satisfying two such equations to compare relative to a weak ordering are presented. In addition, averaging operators in finite dimensional spaces are determined, and in this case it is shown that they are unitary models for all projections. It is pointed out, by an example, that the latter result does not hold in infinite dimensions. A key application to certain second order random fields of Karhunen type is given. The main purpose is to present the structure of bounded non-self adjoint operators solving quadratic equations, and indicate their use.

On the invertibility of length two elementary operators

2015 ◽  
Vol 30 ◽  
pp. 916-913
Author(s):  
Janko Bracic ◽  
Nadia Boudi
Keyword(s):  
Banach Space ◽  
Sufficient Conditions ◽  
Complex Banach Space ◽  
Linear Operators ◽  
Elementary Operator ◽  
Bounded Linear Operators ◽  
Existence Of A Solution ◽  
Bounded Linear ◽  
Elementary Operators ◽  
Necessary And Sufficient

Let X be a complex Banach space and L(X) be the algebra of all bounded linear operators on X. For a given elementary operator P of length 2 on L(X), we determine necessary and sufficient conditions for the existence of a solution of the equation YP=0 in the algebra of all elementary operators on L(X). Our approach allows us to characterize some invertible elementary operators of length 2 whose inverses are elementary operators.

Periodic Solutions of Second Order Degenerate Differential Equations with Delay in Banach Spaces

2018 ◽  
Vol 61 (4) ◽  
pp. 717-737 ◽  
Author(s):  
Shangquan Bu ◽  
Gang Cai
Keyword(s):  
Sufficient Conditions ◽  
Complex Banach Space ◽  
Second Order ◽  
Linear Operators ◽  
Bounded Linear ◽  
Order Degenerate ◽  
Degenerate Differential Equations ◽  
Necessary And Sufficient ◽  
Equations With Delay ◽  
Closed Linear Operators

AbstractWe give necessary and sufficient conditions of the Lp-well-posedness (resp. -wellposedness) for the second order degenerate differential equation with finite delayswith periodic boundary conditions (Mu)(0) = (Mu)(2π), (Mu)′ (0) = (Mu)′ (2π), where A, B, and M are closed linear operators on a complex Banach space X satisfying D(A) ∩ D(B) ⊂ D(M), F and G are bounded linear operators from into X.

scholarly journals New Results on Markov Moment Problem

2013 ◽  
Vol 2013 ◽  
pp. 1-17 ◽  
Author(s):  
Octav Olteanu
Keyword(s):  
Moment Problem ◽  
Sufficient Conditions ◽  
Linear Operators ◽  
Moment Problems ◽  
Finite Dimensional ◽  
Integrable Functions ◽  
Truncated Moment Problem ◽  
Markov Moment ◽  
The Moment ◽  
Necessary And Sufficient

The present work deals with the existence of the solutions of some Markov moment problems. Necessary conditions, as well as necessary and sufficient conditions, are discussed. One recalls the background containing applications of extension results of linear operators with two constraints to the moment problem and approximation by polynomials on unbounded closed finite-dimensional subsets. Two domain spaces are considered: spaces of absolute integrable functions and spaces of analytic functions. Operator valued moment problems are solved in the latter case. In this paper, there is a section that contains new results, making the connection to some other topics: bang-bang principle, truncated moment problem, weak compactness, and convergence. Finally, a general independent statement with respect to polynomials is discussed.

scholarly journals Solutions of the system of operator equations $BXA=B=AXB$ via the *-order

2017 ◽  
Vol 32 ◽  
pp. 172-183 ◽  
Author(s):  
Mehdi Vosough ◽  
Mohammad Sal Moslehian
Keyword(s):  
Hilbert Space ◽  
General Solution ◽  
Existence Of Solutions ◽  
Sufficient Conditions ◽  
Linear Operators ◽  
Operator Equations ◽  
Bounded Linear ◽  
Mild Conditions ◽  
Necessary And Sufficient ◽  
System Of Operator Equations

In this paper, some necessary and sufficient conditions are established for the existence of solutions to the system of operator equations $BXA=B=AXB$ in the setting of bounded linear operators on a Hilbert space, where the unknown operator $X$ is called the inverse of $A$ along $B$. After that, under some mild conditions, it is proved that an operator $X$ is a solution of $BXA=B=AXB$ if and only if $B \stackrel{*}{ \leq} AXA$, where the $*$-order $C\stackrel{*}{ \leq} D$ means $CC^*=DC^*, C^*C=C^*D$. Moreover, the general solution of the equation above is obtained. Finally, some characterizations of $C \stackrel{*}{ \leq} D$ via other operator equations, are presented.

scholarly journals Cesàro and Abel ergodic theorems for integrated semigroups

2021 ◽  
Vol 8 (1) ◽  
pp. 135-149
Author(s):  
Fatih Barki
Keyword(s):  
Banach Space ◽  
Sufficient Conditions ◽  
Linear Operators ◽  
Ergodic Theorems ◽  
Bounded Linear ◽  
Cesàro Mean ◽  
Integrated Semigroups ◽  
Cesaro Mean ◽  
Necessary And Sufficient ◽  
Resolvent Function

Abstract Let {S(t)}t ≥ 0 be an integrated semigroup of bounded linear operators on the Banach space 𝒳 into itself and let A be their generator. In this paper, we consider some necessary and sufficient conditions for the Cesàro mean and the Abel average of S(t) converge uniformly on ℬ(𝒳). More precisely, we show that the Abel average of S(t) converges uniformly if and only if 𝒳 = ℛ(A) ⊕ 𝒩(A), if and only if ℛ(Ak) is closed for some integer k and ∥ λ 2 R(λ, A) ∥ → 0 as λ→ 0+, where ℛ(A), 𝒩(A) and R(λ, A), be the range, the kernel, the resolvent function of A, respectively. Furthermore, we prove that if S(t)/t 2 → 0 as t → 1, then the Cesàro mean of S(t) converges uniformly if and only if the Abel average of S(t) is also converges uniformly.

Bounded linear operators that preserve the weak supermajorization on $\ell^1(I)^+$

2018 ◽  
Vol 34 ◽  
pp. 407-427 ◽  
Author(s):  
Martin Ljubenović ◽  
Dragan Djordjevic
Keyword(s):  
Linear Operator ◽  
Additional Condition ◽  
Sufficient Conditions ◽  
Linear Operators ◽  
Linear Preserver ◽  
Bounded Linear ◽  
Linear Preservers ◽  
Weak Majorization ◽  
Necessary And Sufficient ◽  
Positive Functions

Linear preservers of weak supermajorization which is defined on positive functions contained in the discrete Lebesgue space $\ell^1(I)$ are characterized. Two different classes of operators that preserve the weak supermajorization are formed. It is shown that every linear preserver may be decomposed as sum of two operators from the above classes, and conversely, the sum of two operators which satisfy an additional condition is a linear preserver. Necessary and sufficient conditions under which a bounded linear operator is a linear preserver of the weak supermajorization are given. It is concluded that positive linear preservers of the weak supermajorization coincide with preservers of weak majorization and standard majorization on $\ell^1(I)$.

scholarly journals Two characterisations of additive *-automorphisms of B(H)

1996 ◽  
Vol 53 (3) ◽  
pp. 391-400 ◽  
Author(s):  
Lajos Molnár
Keyword(s):  
Hilbert Space ◽  
Sufficient Conditions ◽  
Complex Hilbert Space ◽  
Necessary And Sufficient Conditions ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Preserver Problems ◽  
Necessary And Sufficient

Let H be a complex Hilbert space and let B(H) denote the algebra of all bounded linear operators on H. In this paper we give two necessary and sufficient conditions for an additive bijection of B(H) to be a *-automorphism. Both of the results in the paper are related to the so-called preserver problems.

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