Main Results – Finite Dimensions Sample Clauses

Main Results – Finite Dimensions. L  Consider the quantum degraded broadcast channel A→BE1E2 with layered secrecy. We obtain a regularized formula for the capacity region in this setting. Define RLS(L) = [ A (R0, R1, R2) : R0 ≤ I(X0; E2)ρ R1 ≤ [I(X1; E1|X0)ρ − I(X1; E2|X0)ρ]+  , (83) pX ,X ,X , ϕx0,x1,x2  R2 ≤ [I(X2; B|X0, X1)ρ − I(X2; E1E2|X0, X1)ρ]+  x0,x ,x1 2 where the union is over the distribution of the auxiliary random variables X0, X1, X2 and the collections of quantum states {ϕA }, with 2 0 1 2 2 A ρX X X BE E = Σ pX ,X ,X (x0, x1, x2)LA→BE E (ϕx0,x1,x2 ) . (84) x0,x1,x2 L Theorem 9. The layered-secrecy capacity region of the quantum degraded broadcast channel A→BE1E2 in finite dimensions is given by [ ∞ CLS(L) = n=1 1 RLS(L⊗n) . (85) n The proof of Theorem 9 is given in Appendix D. Remark 14. Given a classical degraded broadcast channel WY Z1Z2|X , the bound on the top-secret rate R2 in (83) can be simplified. In particular, we can choose X2 to be the channel input, i.e., X2 = X. Furthermore, the auxiliary random variables can be chosen such that X0 X1 X Y Z1 Z2 form a Markov chain, given a physically degraded broadcast channel. That is, the joint distribution of those variables can be expressed as pX0X1XY Z1Z2 (x0, x1, x, y, z1, z2) = pX0 (x0)pX1|X0 (x1|x0)pX|X1 (x|x1)WY |X (y|x)D1(z1|y)D2(z2|z1) , (86) for some input distributions pX0 , pX1|X0 , and pX|X1 , and degrading channels Dj, for j = 1, 2. In this case, we have