TY - JOUR

T1 - Entanglement-breaking channels with general outcome operator algebras

AU - Kuramochi, Yui

N1 - Funding Information:
This work was supported by the National Natural Science Foundation of China (Grant Nos. 11374375 and 11574405). The author would like to thank Erkka Haapasalo for helpful comments on the manuscript.
Publisher Copyright:
© 2018 Author(s).

PY - 2018/10/1

Y1 - 2018/10/1

N2 - A unit-preserving and completely positive linear map, or a channel, Λ:A→Ain between C∗-algebras A and Ain is called entanglement-breaking (EB) if ω' - ' (Λ ' - idB) is a separable state for any C∗-algebra B and any state ω on the injective C∗-tensor product Ain' - B. In this paper, we establish the equivalence of the following conditions for a channel Λ with a quantum input space and with a general outcome C∗-algebra, generalizing the known results in finite dimensions: (i) Λ is EB; (ii) Λ has a measurement-prepare form (Holevo form); (iii) n copies of Λ are compatible for all 2 ≤ n < ∞; (iv) countably infinite copies of Λ 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 M acting on an infinite-dimensional separable Hilbert space by using the coherent states and the Bargmann measure.

AB - A unit-preserving and completely positive linear map, or a channel, Λ:A→Ain between C∗-algebras A and Ain is called entanglement-breaking (EB) if ω' - ' (Λ ' - idB) is a separable state for any C∗-algebra B and any state ω on the injective C∗-tensor product Ain' - B. In this paper, we establish the equivalence of the following conditions for a channel Λ with a quantum input space and with a general outcome C∗-algebra, generalizing the known results in finite dimensions: (i) Λ is EB; (ii) Λ has a measurement-prepare form (Holevo form); (iii) n copies of Λ are compatible for all 2 ≤ n < ∞; (iv) countably infinite copies of Λ 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 M acting on an infinite-dimensional separable Hilbert space by using the coherent states and the Bargmann measure.

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U2 - 10.1063/1.5044700

DO - 10.1063/1.5044700

M3 - Article

AN - SCOPUS:85055699341

SN - 0022-2488

VL - 59

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

IS - 10

M1 - 102206

ER -