Entanglement-breaking channels with general outcome operator algebras
| Autor: | Yui Kuramochi |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2018 |
| Předmět: |
Pure mathematics
Quantum Physics 010102 general mathematics Mathematics - Operator Algebras FOS: Physical sciences Statistical and Nonlinear Physics Quantum entanglement 01 natural sciences Infimum and supremum Injective function Linear map symbols.namesake Separable state Operator algebra Von Neumann algebra 0103 physical sciences symbols FOS: Mathematics Coherent states 0101 mathematics 010306 general physics Quantum Physics (quant-ph) Operator Algebras (math.OA) Mathematical Physics Mathematics |
| Popis: | A unit-preserving and completely positive linear map, or a channel, $\Lambda \colon \mathcal{A} \to \mathcal{A}_{\mathrm{in}}$ between $C^\ast$-algebras $\mathcal{A}$ and $\mathcal{A}_{\mathrm{in}}$ is called entanglement-breaking (EB) if $\omega \circ( \Lambda \otimes \mathrm{id}_{\mathcal{B}} ) $ is a separable state for any $C^\ast$-algebra $\mathcal{B}$ and any state $\omega$ on the injective $C^\ast$-tensor product $\mathcal{A}_{\mathrm{in}} \otimes \mathcal{B} .$ In this paper, we establish the equivalence of the following conditions for a channel $\Lambda$ with a quantum input space and with a general outcome $C^\ast$-algebra, generalizing known results in finite dimensions: (i) $\Lambda$ is EB; (ii) $\Lambda$ has a measurement-prepare form (Holevo form); (iii) $n$ copies of $\Lambda$ are compatible for all $2 \leq n < \infty ;$ (iv) countably infinite copies of $\Lambda$ are compatible. By using this equivalence, we also show that the set of randomization-equivalence classes of normal EB channels with a fixed input von Neumann algebra is upper and lower Dedekind-closed, i.e. the supremum or infimum of any randomization-increasing or decreasing net of EB channels is also EB. As an example, we construct an injective normal EB channel with an arbitrary outcome operator algebra $\mathcal{M}$ acting on an infinite-dimensional separable Hilbert space by using the coherent states and the Bargmann measure. Comment: 25 pages. New sections have been added |
| Databáze: | OpenAIRE |
| Externí odkaz: |
|