scholarly journals Weak Log-majorization of Unital Trace-preserving Completely Positive Maps

2019 ◽  
Vol 35 ◽  
pp. 524-532
Author(s):  
Pan Shun Lau ◽  
Tin-Yau Tam
Keyword(s):  
Positive Definite Matrix ◽  
Affirmative Answer ◽  
Completely Positive Map ◽  
Matrix Inequalities ◽  
Completely Positive Maps ◽  
Completely Positive ◽  
Positive Map ◽  
Weak Majorization ◽  
Fischer’S Inequality ◽  
Determinantal Inequalities

Let Φ : Mn → Mn be a unital trace preserving completely positive map and A ∈ Mn be a positive definite matrix. Weak log-majorization and weak majorization between Φ(A) and A are studied. Determinantal inequalities between Φ(A) and A are obtained as a consequence. By considering special classes of unital trace preserving completely positive map, some known matrix inequalities such as Fischer’s inequality are rediscovered. An affirmative answer to a question of Tam and Zhang in 2019 is given.

scholarly journals Asymmetric invariant sets for completely positive maps on C*-algebras

1986 ◽  
Vol 33 (3) ◽  
pp. 471-473 ◽  
Author(s):  
A. Guyan Robertson
Keyword(s):  
Invariant Sets ◽  
Completely Positive Map ◽  
Completely Positive Maps ◽  
Completely Positive ◽  
Positive Map ◽  
C Algebras ◽  
Positive Maps

Let A be a noncommutative C*-algebra other than M2(I). We show that there exists a completely positive map φ of norm one on A and an element a ɛ A such that φ(a) = a, φ(a*a) = a*a, but φ(aa*) ≠ aa*.

scholarly journals CUNTZ–KRIEGER ALGEBRAS AND ONE-SIDED CONJUGACY OF SHIFTS OF FINITE TYPE AND THEIR GROUPOIDS

2019 ◽  
Vol 109 (3) ◽  
pp. 289-298
Author(s):  
KEVIN AGUYAR BRIX ◽  
TOKE MEIER CARLSEN
Keyword(s):  
Conjugacy Class ◽  
Finite Type ◽  
Negative Answer ◽  
The Other ◽  
Completely Positive Map ◽  
Completely Positive ◽  
Positive Map ◽  
Shift Of Finite Type ◽  
Abelian Subalgebra ◽  
The One

AbstractA one-sided shift of finite type $(\mathsf{X}_{A},\unicode[STIX]{x1D70E}_{A})$ determines on the one hand a Cuntz–Krieger algebra ${\mathcal{O}}_{A}$ with a distinguished abelian subalgebra ${\mathcal{D}}_{A}$ and a certain completely positive map $\unicode[STIX]{x1D70F}_{A}$ on ${\mathcal{O}}_{A}$. On the other hand, $(\mathsf{X}_{A},\unicode[STIX]{x1D70E}_{A})$ determines a groupoid ${\mathcal{G}}_{A}$ together with a certain homomorphism $\unicode[STIX]{x1D716}_{A}$ on ${\mathcal{G}}_{A}$. We show that each of these two sets of data completely characterizes the one-sided conjugacy class of $\mathsf{X}_{A}$. This strengthens a result of Cuntz and Krieger. We also exhibit an example of two irreducible shifts of finite type which are eventually conjugate but not conjugate. This provides a negative answer to a question of Matsumoto of whether eventual conjugacy implies conjugacy.

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.

Quantum collections

2017 ◽  
Vol 28 (12) ◽  
pp. 1750085 ◽  
Author(s):  
Andre Kornell
Keyword(s):  
Normal State ◽  
Completely Positive Map ◽  
Completely Positive ◽  
Positive Map ◽  
Exponential Formula

We develop the viewpoint that [Formula: see text], the opposite of the category of [Formula: see text]-algebras and unital normal ∗-homomorphisms, is analogous to the category of sets and functions. For each pair of [Formula: see text]-algebras [Formula: see text] and [Formula: see text], we construct their free exponential [Formula: see text], which in the context of this analogy corresponds to the collection of functions from [Formula: see text] to [Formula: see text]. We also show that every unital normal completely positive map [Formula: see text] arises naturally from a normal state on [Formula: see text].

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 ASYMPTOTIC STABILITY I: COMPLETELY POSITIVE MAPS

2004 ◽  
Vol 15 (03) ◽  
pp. 289-312 ◽  
Author(s):  
WILLIAM ARVESON
Keyword(s):  
Asymptotic Behavior ◽  
Asymptotic Stability ◽  
Von Neumann Algebras ◽  
Completely Positive Map ◽  
Completely Positive Maps ◽  
Completely Positive ◽  
Locally Finite ◽  
Von Neumann ◽  
C Algebra ◽  
Positive Maps

We show that for every "locally finite" unit-preserving completely positive map P acting on a C*, there is a corresponding *-automorphism α of another unital C*-algebra such that the two sequences P, P2, P3, … and α, α2, α3, … have the same asymptotic behavior. The automorphism α is uniquely determined by P up to conjugacy. Similar results hold for normal completely positive maps on von Neumann algebras, as well as for one-parameter semigroups. These results are operator algebraic counterparts of the classical theory of Perron and Frobenius on the structure of square matrices with nonnegative entries.

QUANTUM DYNAMICAL ENTROPY FOR COMPLETELY POSITIVE MAP

1999 ◽  
Vol 02 (02) ◽  
pp. 267-282 ◽  
Author(s):  
A. KOSSAKOWSKI ◽  
M. OHYA ◽  
N. WATANABE
Keyword(s):  
Markov Chain ◽  
Completely Positive Map ◽  
Completely Positive ◽  
Positive Map ◽  
Quantum Markov Chain ◽  
Quantum Dynamical ◽  
Dynamical Entropy

A dynamical entropy for not only shift but also completely positive (CP) map is defined by generalizing the AOW entropy1 defined through quantum Markov chain and AF entropy defined by a finite operational partition. Our dynamical entropy is numerically computed for several models.

scholarly journals Topological entropy for the canonical completely positive maps on graph C*-Algebras

2004 ◽  
Vol 70 (1) ◽  
pp. 101-116 ◽  
Author(s):  
Ja A. Jeong ◽  
Gi Hyun Park
Keyword(s):  
Topological Entropy ◽  
Finite Graph ◽  
Infinite Graph ◽  
Completely Positive Map ◽  
Completely Positive Maps ◽  
Completely Positive ◽  
Locally Finite ◽  
Finite Graphs ◽  
C Algebras ◽  
Positive Maps

Let C*(E) = C*(se, pv) be the graph C*-algebra of a directed graph E = (E0, E1) with the vertices E0 and the edges E1. We prove that if E is a finite graph (possibly with sinks) and φE: C*(E) → C*(E) is the canonical completely positive map defined by then Voiculescu's topological entropy ht(φE) of φE is log r(AE), where r(AE) is the spectral radius of the edge matrix AE of E. This extends the same result known for finite graphs with no sinks. We also consider the map φE when E is a locally finite irreducible infinite graph and prove that , where the supremum is taken over the set of all finite subgraphs of E.

scholarly journals Depolarizing channel as a completely positive map with memory

2004 ◽  
Vol 70 (1) ◽  
Author(s):  
Sonja Daffer ◽  
Krzysztof Wódkiewicz ◽  
James D. Cresser ◽  
John K. McIver
Keyword(s):  
Completely Positive Map ◽  
Completely Positive ◽  
Positive Map ◽  
With Memory ◽  
Depolarizing Channel
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