Quantum Coding Theorems

Abstract

We restrict ourselves to finite level systems. Elementary classical probability theory is woven around the notions of a finite sample space or an alphabet A, events which are subsets of A, random variables which are real or complex-valued functions on A and probability distributions which are, once again, functions on A with values in the unit interval adding to unity. An event E ⊂ A can be equivalently replaced by the two-valued indicator function 1 _E which is one on the set E and zero on the complement E′ of E in A. Thus all the basic objects, namely, events, random variables and distributions are functions on A. Complex-valued functions on A constitute a commutative algebra $\mathcal{A}$ with complex conjugation as an involution. Thus an event can be identified as an element f ∈ $\mathcal{A}$ satisfying f = f² = f̄ where the bar indicates complex conjugation. A real-valued random variable is an element f ∈ $\mathcal{A}$ satisfying f = f̄ whereas a probability distribution is an element p ∈ $\mathcal{A}$ satisfying p(x) ≥ 0 ∀ x ∈ A and $\sum\limits_{x \in A} {p\left( x \right) = 1} .$. Note that every nonnegative function on A has the form f f̄ for some f ∈ $\mathcal{A}$.