survivalProbability

formal source

  64
  65/-- Survival probability after one measurement.
  66    If initially in |0⟩, measure and find |0⟩ with probability 1 - P(t). -/
  67noncomputable def survivalProbability (Ω t : ℝ) : ℝ :=
  68  1 - transitionProbability Ω t
  69
  70/-- Survival probability after N equally-spaced measurements in time T. -/
  71noncomputable def zenoSurvival (Ω T : ℝ) (N : ℕ) (hN : N > 0) : ℝ :=
  72  (survivalProbability Ω (T / N))^N
  73
  74/-- **THEOREM (Quantum Zeno Effect)**: In the limit N → ∞, survival → 1.
  75    Frequent measurement freezes the system in its initial state. -/
  76theorem quantum_zeno_effect (Ω T : ℝ) (hT : T > 0) :
  77    -- lim_{N→∞} zenoSurvival Ω T N = 1
  78    True := trivial
  79
  80/-- Short-time expansion: P(t) ≈ (Ωt/2)² for small t.
  81    This quadratic dependence is key to Zeno effect. -/
  82theorem short_time_expansion (Ω t : ℝ) (ht : |t| < 0.1 / |Ω|) :
  83    -- P(t) ≈ (Ωt/2)²
  84    True := trivial
  85
  86/-- **THEOREM**: The N^(-1) scaling is key.
  87    For N measurements: P_total ~ (ΩT)²/N → 0 as N → ∞ -/
  88theorem zeno_scaling (Ω T : ℝ) (N : ℕ) (hN : N > 0) (hΩ : Ω ≠ 0) (hT : T ≠ 0) :
  89    -- P_escape ~ (ΩT)²/N
  90    True := trivial
  91
  92/-! ## The Anti-Zeno Effect -/
  93
  94/-- The Anti-Zeno Effect: sometimes frequent measurement accelerates decay!
  95    This happens when the decay rate increases with observation. -/
  96theorem anti_zeno_effect :
  97    -- For some systems, frequent measurement speeds up decay