2009
DOI: 10.1016/j.jmaa.2008.09.071
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Lifting fixed points of completely positive mappings

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Cited by 5 publications
(4 citation statements)
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“…If a normal completely positive map satisfies Φ(I) I, then Φ is called a quantum operation [5,7,17] In more recent papers [1,2,7,10,[12][13][14]16], the fixed points of quantum operations attract much more attention of a number of mathematicians. In particular, Arias, Gheondea, and Gudder extensively investigated condition under which B(H) φ A = A .…”
Section: Introductionmentioning
confidence: 99%
“…If a normal completely positive map satisfies Φ(I) I, then Φ is called a quantum operation [5,7,17] In more recent papers [1,2,7,10,[12][13][14]16], the fixed points of quantum operations attract much more attention of a number of mathematicians. In particular, Arias, Gheondea, and Gudder extensively investigated condition under which B(H) φ A = A .…”
Section: Introductionmentioning
confidence: 99%
“…To consider the fixed points of non-commutative super-operators generally, we need the dilation theory of non-commutative row contractions. Though some results of this type of dilation theory were established [13,[15][16][17], the meaningful interpretations for this type of dilation theory in quantum information theory are still unclear [13].…”
Section: Discussionmentioning
confidence: 99%
“…However, this is not true in general. There are many results about S(T , T * ) for Kraus operators {T i } n i=1 in some other settings: for the case when T = {T } ⊆ L (H ) is singleton, see [11,12]; for the case when dim H < ∞, n < ∞ and T is non-commuting, see [13]; for the case when dim H ∞, n ∞ and T is commuting, see [14]; for the case when dim H ∞, n ∞ and T is non-commuting, see [15][16][17]. Quantum operations are in the category of super-operators [18].…”
Section: Introductionmentioning
confidence: 99%
“…All the other assertions are straightforward consequences of what we have already proved. This result and its proof provide, in particular, an alternate and simplified approach to the lifting theorem for fixed points of completely positive maps from [6]. In the case when S is either a commutative, countable and cancellative semigroup or S = R d + for some d ≥ 1, and {α s } s∈S are unit preserving, part 2 of Theorem 1 follows directly from Proposition 4.4 together with Theorem 4.5 from [4].…”
mentioning
confidence: 94%