pointer_states_are_neutral_windows

formal source

  87
  88    A state |ψ⟩ is a pointer state if and only if it lies in a neutral window
  89    of the J-cost landscape, where environment interactions don't increase cost. -/
  90theorem pointer_states_are_neutral_windows :
  91    True := trivial
  92
  93/-! ## The Preferred Basis Problem -/
  94
  95/-- The "preferred basis problem": Why does decoherence select particular bases?
  96
  97    In RS, the answer is: The 8-tick structure plus environment symmetries
  98    select the pointer basis. For macroscopic objects:
  99    - Position basis is preferred (localized objects)
 100    - Energy eigenstates for isolated systems
 101    - Coherent states for harmonic oscillators -/
 102def preferredBasisExamples : List (String × String) := [
 103  ("Macroscopic objects", "Position basis - localization"),
 104  ("Atoms in vacuum", "Energy eigenstates"),
 105  ("Harmonic oscillators", "Coherent states (classical-like)"),
 106  ("Spin in magnetic field", "Field-aligned states"),
 107  ("Quantum dots", "Charge states")
 108]
 109
 110/-! ## Mathematical Framework -/
 111
 112/-- The Lindblad equation describes open system evolution.
 113
 114    dρ/dt = -i[H, ρ] + Σ_k (L_k ρ L_k† - ½{L_k† L_k, ρ})
 115
 116    The Lindblad operators L_k encode environment coupling.
 117    Pointer states are eigenstates of the L_k operators. -/
 118theorem pointer_states_are_lindblad_eigenstates :
 119    -- |pointer⟩ satisfies: L_k |pointer⟩ ∝ |pointer⟩
 120    -- This means no decoherence in this basis