Noncommutative Mathematics for Quantum Systems
2 Noncommutative Mathematics for Quantum Systems intends to give an introduction to the theory of quantum stochastic processes w ...
Independence and L ́evy Processes in Quantum Probability 3 physics. The basic theory of these processes is owing to Schurmann, ̈ ...
4 Noncommutative Mathematics for Quantum Systems algebraic probability spaces and review their classifications. The results in t ...
Independence and L ́evy Processes in Quantum Probability 5 This description of randomness is based on the idea that randomness i ...
6 Noncommutative Mathematics for Quantum Systems States of the formφ(X) =〈ψ,Xψ〉are calledpure statesorvector states. Note that u ...
Independence and L ́evy Processes in Quantum Probability 7 on a quantum system in the stateρcan only yield values that belong to ...
8 Noncommutative Mathematics for Quantum Systems withθ ∈ [0,π],φ ∈ [0, 2π), and| 0 〉 = |↑〉,| 1 〉 = |↓〉form an orthonormal basis ...
Independence and L ́evy Processes in Quantum Probability 9 The stateφρ:B(H)→C^2 associated with to a density matrixρis the linea ...
10 Noncommutative Mathematics for Quantum Systems Let us define S(θ,φ) =|u〉〈u|−|u⊥〉〈u⊥| = ( cosθ e−iφsinθ eiφsinθ −cosθ ) =xσx+y ...
Independence and L ́evy Processes in Quantum Probability 11 1.2.1 Distinguishing features of classical and quantum probability T ...
12 Noncommutative Mathematics for Quantum Systems Example 1.2.10 LetH = C^2 ,ψ = | 0 〉,u = cosθ 2 | 0 〉+sinθ 2 | 1 〉, v=cosφ 2 | ...
Independence and L ́evy Processes in Quantum Probability 13 The set of all extreme probability measures on a sample space of n p ...
14 Noncommutative Mathematics for Quantum Systems continued Extreme The set of all probab. The extreme points of the points dist ...
Independence and L ́evy Processes in Quantum Probability 15 randomness does not disappear, except for some special observables ( ...
16 Noncommutative Mathematics for Quantum Systems Note thatψdoes not depend on the direction that we choose for our coordinate s ...
Independence and L ́evy Processes in Quantum Probability 17 for any 0≤θ≤π, 0≤φ< 2 π. This means that Alice and Bob will obser ...
18 Noncommutative Mathematics for Quantum Systems If they measure in directions that are at an angleθ, then joint probabilities ...
Independence and L ́evy Processes in Quantum Probability 19 Aa,Ab,Ac,Ba,Bb,Bc such that P(Ax=−Bx) = 1 for allx∈{a,b,c}and P(AX 1 ...
20 Noncommutative Mathematics for Quantum Systems see for example the discussion in [Gil98]. However, in such a model the random ...
Independence and L ́evy Processes in Quantum Probability 21 1.3.3 The Kochen–Specker theorem We will now review the Kochen–Speck ...
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