In an attempt to propose more general conditions for decoherence to occur, we study spectral and ergodic properties of unital, completely positive maps on not necessarily unital $C^*$ -algebras, with a particular focus on gapped maps for which the transient portion of the arising dynamical system can be separated from the persistent one. After some general results, we first devote our attention to the abelian case by investigating the unital $*$ -endomorphisms of, in general non-unital, $C^*$ -algebras, and their spectral structure. The finite-dimensional case is also investigated in detail, and examples are provided of unital completely positive maps for which the persistent part of the associated dynamical system is equipped with the new product making it a $C^*$ -algebra, and the map under consideration restricts to a unital $*$ -automorphism for this new $C^*$ -structure, thus generating a conservative dynamics on that persistent partIn an attempt to propose more general conditions for decoherence to occur, we study spectral and ergodic properties of unital, completely positive maps on not necessarily unital $C^*$-algebras, with a particular focus on gapped maps for which the transient portion of the arising dynamical system can be separated from the persistent one. After some general results, we first devote our attention to the abelian case by investigating the unital $*$-endomorphisms of, in general non-unital, $C^*$-algebras, and their spectral structure. The finite-dimensional case is also investigated in detail, and examples are provided of unital completely positive maps for which the persistent part of the associated dynamical system is equipped with the new product making it a $C^*$-algebra, and the map under consideration restricts to a unital $*$-automorphism for this new $C^*$-structure, thus generating a conservative dynamics on that persistent part

Jan 1, 20226 citations

SPECTRAL PROPERTIES OF ENTANGLEMENT WITNESSES AND POSITIVE MAPS

Dariusz Chruściński

Quantum Bio-InformaticsJan 1, 20100 citations

Application of Shemesh theorem to quantum channels

Michał Białończyk, Andrzej Jamiołkowski, Karol Życzkowski

Completely positive maps are useful in modeling the discrete evolution of quantum systems. Spectral properties of operators associated with such maps are relevant for determining the asymptotic dynamics of quantum systems subjected to multiple interactions described by the same quantum channel. We discuss a connection between the properties of the peripheral spectrum of completely positive and trace preserving map and the algebra generated by its Kraus operators A(A1,…,AK). By applying the Shemesh and Amitsur-Levitzki theorems to analyse the structure of the algebra A(A1,…,AK), one can predict the asymptotic dynamics for a class of operations.Completely positive maps are useful in modeling the discrete evolution of quantum systems. Spectral properties of operators associated with such maps are relevant for determining the asymptotic dynamics of quantum systems subjected to multiple interactions described by the same quantum channel. We discuss a connection between the properties of the peripheral spectrum of completely positive and trace preserving map and the algebra generated by its Kraus operators A(A1,…,AK). By applying the Shemesh and Amitsur-Levitzki theorems to analyse the structure of the algebra A(A1,…,AK), one can predict the asymptotic dynamics for a class of operations.

arXiv:1711.02354Journal of Mathematical PhysicsOct 1, 20186 citations

Completely positive maps and entropy inequalities

Göran Lindblad

Communications in Mathematical PhysicsJun 1, 1975664 citations

Metrics on completely positive maps via noncommutative geometry

Are Austad, Erik Bédos, Jonas Eidesen +2 more

We study methods of inducing metrics on unital completely positive maps by employing seminorms arising in noncommutative geometry. Our main approach relies on the development of an infinite-dimensional $C^*$ -algebraic analogue of the Choi-Jamiołkowski isomorphism. Under suitable conditions, we show that the induced metrics satisfy the quantum information theoretic properties of stability and chaining. Moreover, we show how to generate such metrics using constructions native to noncommutative geometry, by for example using external Kasparov products of spectral triples.We study methods of inducing metrics on unital completely positive maps by employing seminorms arising in noncommutative geometry. Our main approach relies on the development of an infinite-dimensional $C^*$-algebraic analogue of the Choi-Jamiołkowski isomorphism. Under suitable conditions, we show that the induced metrics satisfy the quantum information theoretic properties of stability and chaining. Moreover, we show how to generate such metrics using constructions native to noncommutative geometry, by for example using external Kasparov products of spectral triples.

math.OACross-listedarXiv:2512.10842Dec 12, 20250 citations

In an attempt to propose more general conditions for decoherence to occur, we study spectral and ergodic properties of unital, completely positive maps on not necessarily unital

C^*

-algebras, with a particular focus on gapped maps for which the transient portion of the arising dynamical system can be separated from the persistent one. After some general results, we first devote our attention to the abelian case by investigating the unital

*

-endomorphisms of, in general non-unital,

C^*

-algebras, and their spectral structure. The finite-dimensional case is also investigated in detail, and examples are provided of unital completely positive maps for which the persistent part of the associated dynamical system is equipped with the new product making it a

C^*

-algebra, and the map under consideration restricts to a unital

*

-automorphism for this new

C^*

-structure, thus generating a conservative dynamics on that persistent part

C^*

-algebraic analogue of the Choi-Jamiołkowski isomorphism. Under suitable conditions, we show that the induced metrics satisfy the quantum information theoretic properties of stability and chaining. Moreover, we show how to generate such metrics using constructions native to noncommutative geometry, by for example using external Kasparov products of spectral triples.