Cardinality Bounds Sample Clauses

Cardinality Bounds. To bound the alphabet size of the random variables T and X, we use the Xxxxxxx-Xxxxxxxxx-Carathe´odory lemma [108] and arguments similar to [109]. Let N0 =|HA|4 + 3 (115) N1 =|T |(|HA|4 + 2) . (116) First, fix pX|T (x|t), and consider the ensemble {pT (t)pX|T (x|t) , ϕt,x}. Every mixed state ϕA has a unique parametric representation u(ϕA) of dimension |HA|4 − 1, since the corresponding density matrix has |HA|2 complex entries and the constraint Tr(ϕA) = 1. Then, define a map f0 : X → RN0 by f0(t) = .u(ρt ) , H(B|T = t)ρ , H(E|T = t)ρ , H(B|X, T = t)ρ , H(E|X, T = t)ρΣ (117) where ρt = Σx pX|T (x|t)θt,x. The map f0 can be extended to probability distributions as follows, Σ F0 : pT ›→ pT (t)f0(t) = (u(ρA) , H(B|T )ρ , H(E|T )ρ , H(B|X, T )ρ , H(E|X, T )ρ) (118) where ρA = Σt pT (t)ρt . According to the Xxxxxxx-Xxxxxxxxx-Carathe´odory lemma [108], any point in the convex closure of a connected compact set within Rd belongs to the convex hull of d points in the set. Since the map F0 is linear, it maps the set of distributions on T to a connected compact set in RN0 . Thus, for every pT , there exists a probability distribution pT¯ on a subset T ⊆ T of size N0, such that F0(pT¯) = F0(pT ). We deduce that the alphabet size can be restricted to |T | ≤ N0, while preserving ρA and ρBE ≡ LA→BE(ρA); I(T ; B)ρ = H(B)ρ − H(B|T )ρ, I(X; B|T )ρ = H(B|T )ρ − H(B|X, T )ρ; and similarly, I(T ; E)ρ and I(X; E|T )ρ. f (t)(x) = .pT (t) , u(ϕt,x) , H(B|X = x, T = t)ρ , H(E|X = x, T = t)ρΣ (119) We move to the alphabet size of X. We keep T and pT fixed. Then, define the map f (t) : X → RN1 by 1 A X T =t for t ∈ T . Now, the extended map is F (t) : p | | (x|t)f (t)(x) = .pT (t) , u(ρt ) , H(B|X, T = t)ρ , H(E|X, T = t)ρ}t∈T Σ . (120) By the Xxxxxxx-Xxxxxxxxx-Carathe´odory lemma [108], for every pX|T , there exists pX¯ |T on a subset X ⊆ X of size N1, such that F (t)(p ¯ ) = F (t)(pX|T =t) for all t ∈ T . Thus, we can restrict the alphabet size to |X | ≤ N1, while preserving ρt ,