Eigenvalues and Eigenspaces of Quantum Dynamical Systems and Their Tensor Products

Łuczak, Andrzej

doi:10.1006/jmaa.1995.4987

Cited by 13 publications

(17 citation statements)

References 5 publications

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“…For a unital channel E 1 ⊗E 2 , the fixed point set Fix E 1 ⊗E 2 is a subalgebra of M d ⊗M d ′ and unlike the multiplicative domain case, this subalgebra does not split nicely. However, using Theorem 2.3, we can provide an exact description of this algebra and characterize when this subalgebra splits and recapture the result of [24]. Our results are specific cases of [32] and [24], but through a vastly different approach.…”

Section: Fixed Points Of Product Channelsmentioning

confidence: 91%

“…However, using Theorem 2.3, we can provide an exact description of this algebra and characterize when this subalgebra splits and recapture the result of [24]. Our results are specific cases of [32] and [24], but through a vastly different approach. The spectrum of the tensor product of two channels is known to be the set product of the two spectra, but this theorem characterizes the eigen operators as only the obvious choices.…”

Section: Fixed Points Of Product Channelsmentioning

confidence: 91%

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Spectral properties of tensor products of channels

Jaques

Rahaman

2018

Journal of Mathematical Analysis and Applications

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We investigate spectral properties of the tensor products of two completely positive and trace preserving linear maps (also known as quantum channels) acting on matrix algebras. This leads to an important question of when an arbitrary subalgebra can split into the tensor product of two subalgebras. We show that for two unital quantum channels the multiplicative domain of their tensor product splits into the tensor product of the individual multiplicative domains. Consequently, we fully describe the fixed points and peripheral eigen operators of the tensor product of channels. Through a structure theorem of maximal unital proper *-subalgebras (MUPSA) of a matrix algebra we provide a non-trivial upper bound of the recentlyintroduced multiplicative index of a unital channel. This bound gives a criteria on when a channel cannot be factored into a product of two different channels. We construct examples of channels which cannot be realized as a tensor product of two channels in any way. With these techniques and results, we found some applications in quantum information theory.

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Section: Fixed Points Of Product Channelsmentioning

confidence: 91%

Section: Fixed Points Of Product Channelsmentioning

confidence: 91%

Spectral properties of tensor products of channels

Jaques

Rahaman

2018

Journal of Mathematical Analysis and Applications

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“…In many interesting situations, the ergodic behavior of dynamical systems is connected with some spectral properties, see e.g. [3,9,11]. It is not possible to extend such results to our situation.…”

Section: Proposition 23 If the C * -Dynamical System (A T ) Is F -mentioning

confidence: 96%

Strict weak mixing of some C∗-dynamical systems based on free shifts

Fidaleo

Mukhamedov

2007

Journal of Mathematical Analysis and Applications

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We define a stronger property than unique ergodicity with respect to the fixed-point subalgebra firstly investigated in [B. Abadie, K. Dykema, Unique ergodicity of free shifts and some other automorphisms of C * -algebras, math.OA/0608227]. Such a property is denoted as F -strict weak mixing (F stands for the unital completely positive projection onto the fixed-point operator system). Then we show that the free shifts on the reduced C * -algebras of RD-groups, including the free group on infinitely many generators, and amalgamated free product C * -algebras, considered in [B. Abadie, K. Dykema, Unique ergodicity of free shifts and some other automorphisms of C * -algebras, math.OA/0608227], are all strictly weak mixing and not only uniquely ergodic.

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“…Let ξ ∈ M a , ξ⊥M a . According to [9,Theorem 4 and Corollary 7], there exists a projection ε a from M onto M a such that for each x ∈ M, we have…”

Section: Weak Mixingmentioning

confidence: 99%

“…Assuming that the α t 's are Schwarz maps we come to deal with von Neumann subalgebras of the initial algebra, and * -automorphisms on them, while assuming only positivity leads us to considering Jordan subalgebras and Jordan morphisms (cf. [1,3,4,7,9,16]). A very similar situation occurs for the ergodic projection, i.e.…”

Section: Introductionmentioning

confidence: 99%

Mixing and asymptotic properties of Markov semigroups on von Neumann algebras

Łuczak¹

2000

Mathematische Zeitschrift

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In the paper, we investigate weak mixing and 'approach to equilibrium' of a Markov semigroup, i.e. a semigroup (α t : t ≥ 0) of linear normal positive unital mappings on a von Neumann algebra. In particular, we show that weak mixing is equivalent to ergodictity and triviality of the (point) spectrum of the semigroup, and give conditions assuring that each normal state tends to an equilibrium state under the action of the semigroup. Classification (1991):46 L 55, 28 D 05. Mathematics Subject

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Eigenvalues and Eigenspaces of Quantum Dynamical Systems and Their Tensor Products

Cited by 13 publications

References 5 publications

Spectral properties of tensor products of channels

Spectral properties of tensor products of channels

Strict weak mixing of some C∗-dynamical systems based on free shifts

Mixing and asymptotic properties of Markov semigroups on von Neumann algebras

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