scholarly journals Strict Positivity of Operators and Inflated Schur Products

2018 ◽  
Vol 10 (6) ◽  
pp. 30
Author(s):  
Ching-Yun Suen
Keyword(s):  
Main Diagonal ◽  
Completely Positive ◽  
Strict Positivity ◽  
Linear Maps ◽  
Positive Matrices ◽  
Positive Linear Maps ◽  
Invertible Elements ◽  
Equivalent Conditions

In this paper we provide a characterization of strictly positive matrices of operators and a factorization of their inverses. Consequently, we provide a test of strict positivity of matrices in . We give equivalent conditions for the inequality . We prove a theorem involving inflated Schur products [4, P. 153] of positive matrices of operators with invertible elements in the main diagonal which extends the results [3, P. 479, Theorem 7.5.3 (b), (c)]. We also discuss strictly completely positive linear maps in the paper.

Positive Linear Maps on C*-Algebras

1972 ◽  
Vol 24 (3) ◽  
pp. 520-529 ◽  
Author(s):  
Man-Duen Choi
Keyword(s):  
Vector Spaces ◽  
Complex Numbers ◽  
Complex Matrices ◽  
Completely Positive ◽  
C Algebras ◽  
Linear Maps ◽  
Positive Linear Maps

The objective of this paper is to give some concrete distinctions between positive linear maps and completely positive linear maps on C*-algebras of operators.Herein, C*-algebras possess an identity and are written in German type . Capital letters A, B, C stand for operators, script letters for vector spaces, small letters x, y, z for vectors. Capital Greek letters Φ, Ψ stand for linear maps on C*-algebras, small Greek letters α, β, γ for complex numbers.We denote by the collection of all n × n complex matrices. () = ⊗ is the C*-algebra of n × n matrices over .

scholarly journals FACIAL STRUCTURES FOR VARIOUS NOTIONS OF POSITIVITY AND APPLICATIONS TO THE THEORY OF ENTANGLEMENT

2013 ◽  
Vol 25 (02) ◽  
pp. 1330002 ◽  
Author(s):  
SEUNG-HYEOK KYE
Keyword(s):  
Convex Cones ◽  
Edge States ◽  
Completely Positive ◽  
Matrix Algebras ◽  
Linear Maps ◽  
Entanglement Witnesses ◽  
Positive Linear Maps

In this expository note, we explain facial structures for the convex cones consisting of positive linear maps, completely positive linear maps, and decomposable positive linear maps between matrix algebras, respectively. These will be applied to study the notions of entangled edge states with positive partial transposes and optimality of entanglement witnesses.

scholarly journals Diagonal unitary and orthogonal symmetries in quantum theory

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 519
Author(s):  
Satvik Singh ◽  
Ion Nechita
Keyword(s):  
Sufficient Conditions ◽  
Quantum Channels ◽  
Completely Positive ◽  
Matrix Algebras ◽  
Completely Positive Matrices ◽  
Linear Maps ◽  
Positive Matrices ◽  
Invariant States ◽  
Class Of Matrices ◽  
Necessary And Sufficient

We analyze bipartite matrices and linear maps between matrix algebras, which are respectively, invariant and covariant, under the diagonal unitary and orthogonal groups' actions. By presenting an expansive list of examples from the literature, which includes notable entries like the Diagonal Symmetric states and the Choi-type maps, we show that this class of matrices (and maps) encompasses a wide variety of scenarios, thereby unifying their study. We examine their linear algebraic structure and investigate different notions of positivity through their convex conic manifestations. In particular, we generalize the well-known cone of completely positive matrices to that of triplewise completely positive matrices and connect it to the separability of the relevant invariant states (or the entanglement breaking property of the corresponding quantum channels). For linear maps, we provide explicit characterizations of the stated covariance in terms of their Kraus, Stinespring, and Choi representations, and systematically analyze the usual properties of positivity, decomposability, complete positivity, and the like. We also describe the invariant subspaces of these maps and use their structure to provide necessary and sufficient conditions for separability of the associated invariant bipartite states.

scholarly journals Extremal problems for completely positive maps

1994 ◽  
Vol 17 (3) ◽  
pp. 607-608
Author(s):  
Mingze Yang
Keyword(s):  
Extremal Problems ◽  
Completely Positive Maps ◽  
Completely Positive ◽  
Positive Maps ◽  
Linear Maps ◽  
Positive Linear Maps ◽  
Convex Subsets

In this note, we study the faces of some convex subsets ofCPc(A,B(ℋ))(the continuous completely positive linear maps from pro-C*-algebraAtoB(ℋ)).

scholarly journals On the convergence of a sequence of completely positive maps to the identity

2000 ◽  
Vol 68 (3) ◽  
pp. 340-348 ◽  
Author(s):  
George A. Elliott
Keyword(s):  
Completely Positive Maps ◽  
Completely Positive ◽  
Positive Maps ◽  
Linear Maps ◽  
Positive Linear Maps

AbstractIt is shown that a sequence of completely positive linear maps on a W*-algebra that converges pointwise in norm to the identity converges uniformly.

scholarly journals On Partially Trace Distance Preserving Maps and Reversible Quantum Channels

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Long Jian ◽  
Kan He ◽  
Qing Yuan ◽  
Fei Wang
Keyword(s):  
Quantum States ◽  
Quantum Channels ◽  
Trace Distance ◽  
Linear Maps ◽  
Unitary Transformations ◽  
Pure States ◽  
Positive Linear Maps

We give a characterization of trace-preserving and positive linear maps preserving trace distance partially, that is, preservers of trace distance of quantum states or pure states rather than all matrices. Applying such results, we give a characterization of quantum channels leaving Helstrom's measure of distinguishability of quantum states or pure states invariant and show that such quantum channels are fully reversible, which are unitary transformations.

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