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**Read or Download Probability, Statistics, and Random Processes For Electrical Engineering (3rd Edition) PDF**

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**Extra resources for Probability, Statistics, and Random Processes For Electrical Engineering (3rd Edition)**

**Example text**

Thus for all m ≤ n, ||T∗,tn (rm ) ([ρ]h )|| ≤ ||T∗,tn (rn ) ([ρ]h )|| ≤ ||T∗,tn (min) ([ρ]h )|| − (min) 0. 46 Quantum dynamical semigroups Keeping m fixed and letting n tend to ∞, we obtain that ||T∗,t0 (rm ) ([ρ]h )|| ≤ ||T∗,t0 (min) ([ρ]h )|| − 0; and then by letting m tend to ∞, ||T∗,t0 (min) ([ρ]h )|| ≤ ||T∗,t0 (min) ([ρ]h )|| − 0, ✷ which is clearly a contradiction. 13 The generator of (T∗,t J ; and we have the following minimality property. Whenever (T∗,t )t≥0 is a positive, C0 contraction semigroup on Ah,∗ whose (min) generator (say A ) extends Z + J , we must have T∗,t ≥ T∗,t for all t ≥ 0.

But by a simple computation it can be verified that T (X ) is the 4 × 4 matrix ⎛ ⎞ 2 0 −i 0 ⎜ 0 0 −i 0 ⎟ ⎜ ⎟, ⎝i i 1 1⎠ 0 0 1 1 which is not positive since its determinant = −2 < 0. We say that T is n-positive if Tk is positive for all k ≤ n, but not necessarily for k = n + 1. T is said to be completely positive (CP for short) if it is n-positive for each n. The role of positivity in classical probability is played by complete positivity in the quantum theory. Let us now formulate the notion of complete positivity in a slightly different but convenient language, namely that of positive definite kernels.

Take ρ1 = (1 − G/n)−1 σn+ (1 − G ∗ /n)−1 , ρ2 = (1 − G/n)−1 σn− (1 − G ∗ /n)−1 ; and observe that ||[ρ1 ]h || + ||[ρ2 ]h || = ||(1 − G/n)−1 σn+ (1 − G ∗ /n)−1 ||1 + ||(1 − G/n)−1 σn− (1 − G ∗ /n)−1 ||1 (as ρ1 , ρ2 are positive) ≤ ||σn+ ||1 + ||σn− ||1 = ||[σn+ ]h || + ||[σn− ]h || = ||[σn ]h || ≤ ||[ρ]h || + . Now, [ρ1 ]h − [ρ2 ]h = [(1 − G/n)−1 (σn+ − σn− )(1 − G ∗ /n)−1 ]h = [(1 − G/n)−1 σn (1 − G ∗ /n)−1 ]h = [ρ]h , because (σn+ − σn− ) ∼ σn implies (1 − G/n)−1 (σn+ − σn− )(1 − G ∗ /n)−1 ∼ (1 − G/n)−1 σn (1 − G ∗ /n)−1 .