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 ∈ \( \mathcal{A} \).
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© 2013 Hindustan Book Agency
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Parthasarathy, K.R. (2013). Quantum Coding Theorems. In: Coding Theorems of Classical and Quantum Information Theory. Texts and Readings in Mathematics, vol 45. Hindustan Book Agency, Gurgaon. https://doi.org/10.1007/978-93-86279-59-0_4
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DOI: https://doi.org/10.1007/978-93-86279-59-0_4
Publisher Name: Hindustan Book Agency, Gurgaon
Print ISBN: 978-93-80250-41-0
Online ISBN: 978-93-86279-59-0
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