Achievability proof Sample Clauses

Achievability proof. A ∼ L We modify and extend Devetak and Xxxxxx’s methods. When Xxxxx performs the measurement Λ on each of her systems Ai, she obtains a classical memoryless source sequence Xn, which is i.i.d. pX (x) = Tr(Λx ωA). Roughly speaking, one may encode this source using indices, such that this source sequence “appears” to Bob and Xxx as a codeword for the classical- quantum-quantum broadcast channel X→BE that corresponds to ωXBE. Our key-agreement coding schemes is specified below in further details. Fix the POVM ΛA and the conditional distribution pT0,T1|X . Denote the joint distribution of X, T0, and T1 by 0 A 0 1

Achievability proof. L ∈ X X We show that for every ζ0, ε0, δ0 > 0, there exists a (2n(R0−ζ0), 2n(R1−ζ0), 2n(R2−ζ0), n, ε0, δ0) layered-secrecy code for the quantum broadcast channel A→BE1E2 , provided that (R0, R1, R2) LS( ). To prove achievability, we extend the classical combination of super-position coding with random binning, and then apply the quantum packing lemma and the quantum covering lemma. We use the gentle measurement lemma [110], which guarantees that multiple decoding measurements can be performed without “destroying” the output state. Useful lemmas We make heavy use of the quantum packing lemma, quantum covering lemma, and gentle-measurement lemma. Those lemmas are given below. We begin with the quantum packing lemma, which is a useful tool in proofs of channel coding theorems. Σ Lemma 11 (Quantum Packing Lemma [111][58, Corollary 16.5.1]). Let ρ = pX (x)ρx (175) x∈X where {pX (x), ρx}x∈X is a given ensemble. Furthermore, suppose that there is a code projector Π and codeword projectors Πxn , xn ∈ Aδ (pX ), that satisfy for every α > 0 and sufficiently large n, Tr(Πσxn ) ≥ 1 − α (176) ≥ − Tr(Πxn σxn ) 1 α (177) ne0 Tr(Πxn ) ≤ 2

Achievability proof. Similar to the proof of converse part, we begin with the GS model and then argue the CS model.