By Luc Bouten
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E. its eigenvalues are positive. Let us denote them by E+ and E− such that E+ ≥ E− ≥ 0. Define α := exp(−t|z|2 ) and β := exp(− 2t ). Then for all t ≥ 0 α ≥ 0, 1 − β ≥ 0, 0 ≤ 1 − α ≤ |z|2 t, 0 ≤ 1 − αβ 2 ≤ (|z|2 + 1)t. Therefore, for all t ≥ 0 E t Ω[−t, 0) × Ω[−t, 0)\{(∅, ∅)} (I) ≤ E+ I ≤ (E+ + E− )I = Tr E t Ω[−t, 0) × Ω[−t, 0)\{(∅, ∅)} (I) = (1 − αβ 2 ) + (1 − α) − 4α|z|2 |κf |2 (β − 1)2 I ≤ (1 − αβ 2 ) + (1 − α) I ≤ (2|z|2 + 1)tI. 5 Quantum trajectories In the seventies Davies studied the structure of what we now call Davies processes [33].
The change in the state is given by the sum of two terms: a deterministic one proportional with dt and a stochastic one proportional to the number of detected photons dNt in the interval dt. For other schemes such as homodyne detection the corresponding stochastic differential equation is obtained as the diffusive limit of photon counting where the jumps in the state space decrease in size but become increasingly frequent [9], [27], [92]. In this limit the stochastic term in the differential equation is replaced by a process with continuous paths.
The distribution functions FXi (x) := Pρ [Xi ≤ x] are equal and for i, j = 1, 2, . . the joint probability distribution function FXi ,Xj (x, y) := Pρ [Xi ≤ x ∧ Xj ≤ y] factorizes: FXi ,Xj (x, y) = FXi (x)FXj (y). Let us first introduce some convenient notation. 6), we have for all E ∈ Σt : Pρ [E] = Ptρ [E] = Tr ρ Tr ρ E WZ,Js (ω)dµ(ω)(I) = E−t Zω1 Js Zω2 −ω1 Js . . Js Zt−ωk (I)dµ(ω) . We will denote: x1 := ω1 , x2 := ω2 − ω1 , . . , xk+1 := t − ωk , then we can write: Pρ [E] = E Let P denote the matrix Tr ρZx1 Js Zx2 Js .