Appl of Coherent Communication to Quantum Information Theory by A. Harrow

By A. Harrow

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Extra resources for Appl of Coherent Communication to Quantum Information Theory [thesis]

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30. For equivalence classes in R: 1. if α1 ≥ α2 and β1 ≥ β2 then α1 + β1 ≥ α2 + β2 2. 27 has essentially allowed us to replace resources with their equivalence classes and ≡ with =. Henceforth we shall equate the two, and drop the ∼ superscript. The one exception to this rule is when writing relative resources as (β : γ) where β is a proper dynamic resource and γ is a proper static resource; in this case replacing (β : γ) with its equivalence class is well-defined, but replacing β and γ with their equivalence classes wouldn’t make sense.

We shall use the following convention: if β = (Nn )n , where all Nn are proper dynamic resources and γ = (ωn )n , where all ωn are proper static resources, then (β : γ) := (Nn : ωn )n . Note that typically ωn is product state, so the resource γ reduces to the null resource ∅; however this is no problem as long as we are interested in γ only as a test state for β. Our next goal is to define what it means to simulate one (asymptotic) resource by another. 16 (Asymptotic resource inequalities). e. k sequential uses of α⌊n/k⌋ ) for which P(n) [(α⌊n/k⌋ )×k ] − S[β⌊(1−δ)n⌋ ] ≤ ǫ.

Recall that in the QQ formalism classical systems are unchanged under the copying operation ∆. This means we can consider an equivalent protocol in which the systems associated with the classical resource (N : ω)ℓ are copied into a composite classical system Z, which includes all the copies of all the random variables involved. Let P′ be the modified version of P which ′ retains Z in the final state. Now P ′ := P′ [((N : ω)ℓ , (N : ω)ℓ )] ⊇ P takes a particular extension RA′ A∗ B ∗ A∗ B ∗ ZRABA∗ B ∗ Υ ⊇Ω to some state σ .

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