Quantum Information Theory

Abstract

The concepts of classical information theory can be extended to quantum information theory. Since in general measurement yields a result with probability, we may suggest using these probabilities in classical information theory. However the probabilities do not contain phase information, which cannot be neglected. Thus the definitions are given in terms of the density operator. These probabilities depend on the basis used for measurement. A density operator ρ over a n-dimensional Hilbert space H is a positive operator with unit trace. The trace tr(A) is defined as

$$tr(A): = \sum\limits_{j = 1}^n {\left\langle {{\beta _j}|A|{\beta _j}} \right\rangle } $$

where β _j for j = 1,..., n is any orthonormal basis in HThus tr(P)=1. The eigenvalues of a density operator are greater than zero. By the spectral theorem every density operator can be represented as a mixture of pure states

$$\rho = \sum\limits_{j = 1}^n {Pj|} \left\langle {aj} \right.\langle aj|$$

where α _j for j = 1,..., n are the orthonormal eigenvectors of ρ (which form a basis in H and

$$pj \in R,{p_j}0,\sum\limits_{j = 1}^n {{p_j}} = 1$$