C4-1-formal: 量子系统的经典化推论的形式化规范
机器验证元数据
type: corollary
verification: machine_ready
dependencies: ["A1-formal.md", "T3-1-formal.md", "D1-8-formal.md", "T12-1-formal.md"]
verification_points:
- quantum_state_evolution
- decoherence_rate_scaling
- entropy_increase_verification
- classical_limit_convergence
- phi_basis_stability
- irreversibility_proof
核心推论
推论 C4-1（量子系统的经典化）
QuantumClassicalization : Prop ≡
∀ρ : DensityMatrix, system : SelfReferentialSystem .
quantum_state(ρ, system) ∧ complete(system) →
∃t_c : ℝ⁺, ρ_classical : ClassicalState .
lim_{t→∞} evolve(ρ, t) = ρ_classical ∧
S(ρ_classical) > S(ρ) ∧
phi_structured(ρ_classical)
where
DensityMatrix : Type = {
matrix : ℂ^(n×n),
hermitian : matrix† = matrix,
positive : ∀v . ⟨v|matrix|v⟩ ≥ 0,
trace_one : Tr(matrix) = 1
}
ClassicalState : Type = {
state : DensityMatrix,
diagonal : ∀i≠j . state[i,j] = 0,
phi_basis : basis_set ⊆ ValidPhiRepresentations
}
形式化组件
1. 退相干动力学
DecoherenceDynamics : Type ≡
record {
system_dimension : ℕ
decoherence_rates : Matrix[ℝ⁺]
lindblad_operators : List[Matrix[ℂ]]
master_equation : DensityMatrix → DensityMatrix
}
DecoherenceRate : ℕ → ℕ → ℝ⁺ ≡
λi, j .
if i = j then 0 // No decoherence for diagonal elements
else γ₀ × |i - j|^(1/φ)
where γ₀ : ℝ⁺ = environment_coupling_strength
2. 量子态演化
QuantumEvolution : DensityMatrix → ℝ⁺ → DensityMatrix ≡
λρ₀, t .
let coherences = extract_off_diagonal(ρ₀) in
let populations = extract_diagonal(ρ₀) in
let evolved_coherences = [
coherences[i,j] × exp(-DecoherenceRate(i,j) × t)
for all i ≠ j
] in
reconstruct_density_matrix(populations, evolved_coherences)
MasterEquation : DensityMatrix → DensityMatrix ≡
λρ .
-i[H, ρ] + ∑_k γ_k (L_k ρ L_k† - ½{L_k†L_k, ρ})
where
H : Hamiltonian (can be zero for pure decoherence)
L_k : Lindblad operators encoding environment coupling
3. 经典极限
ClassicalLimit : DensityMatrix → ClassicalState ≡
λρ .
let ρ_∞ = lim_{t→∞} QuantumEvolution(ρ, t) in
ClassicalState {
state = diagonalize_in_phi_basis(ρ_∞),
diagonal = true,
phi_basis = extract_phi_basis(ρ_∞)
}
ClassicalEntropy : ClassicalState → ℝ ≡
λρ_c .
let probabilities = diagonal_elements(ρ_c.state) in
-∑_i p_i × log(p_i)
4. φ-基稳定性
PhiBasisStability : Prop ≡
∀n ∈ ValidPhiRepresentations .
let |n⟩_φ = phi_basis_state(n) in
let ρ_n = |n⟩_φ⟨n|_φ in
QuantumEvolution(ρ_n, t) = ρ_n for all t ≥ 0
StabilityProof : Proof[PhiBasisStability] ≡
By construction, φ-basis states are eigenstates of the decoherence
superoperator with eigenvalue 0, hence invariant under evolution. □
5. 退相干时间尺度
DecoherenceTimescale : ℕ → ℝ⁺ ≡
λN . // N is system size
τ₀ × φ^(-log(N))
where τ₀ : ℝ⁺ = microscopic_time_scale
TimescaleScaling : Prop ≡
∀N₁, N₂ : ℕ . N₁ < N₂ →
DecoherenceTimescale(N₁) > DecoherenceTimescale(N₂)
6. 熵增验证
EntropyIncrease : DensityMatrix → ℝ⁺ → Bool ≡
λρ₀, t .
let S₀ = von_neumann_entropy(ρ₀) in
let S_t = von_neumann_entropy(QuantumEvolution(ρ₀, t)) in
S_t ≥ S₀
EntropyMonotonicity : Prop ≡
∀ρ : DensityMatrix, t₁, t₂ : ℝ⁺ .
t₁ < t₂ →
von_neumann_entropy(QuantumEvolution(ρ, t₁)) ≤
von_neumann_entropy(QuantumEvolution(ρ, t₂))
算法规范
经典化过程模拟算法
ClassicalizationSimulation : Algorithm ≡
Input: ρ₀ : DensityMatrix, t_max : ℝ⁺, dt : ℝ⁺
Output: trajectory : List[(time, density_matrix, entropy)]
Process:
1. trajectory = [(0, ρ₀, von_neumann_entropy(ρ₀))]
2. ρ_current = ρ₀
3. for t in range(dt, t_max + dt, dt):
# Apply decoherence for time step dt
ρ_next = apply_decoherence(ρ_current, dt)
# Calculate entropy
S = von_neumann_entropy(ρ_next)
# Store in trajectory
trajectory.append((t, ρ_next, S))
# Update current state
ρ_current = ρ_next
# Check for classical limit convergence
if is_diagonal(ρ_current, tolerance=1e-10):
break
4. return trajectory
Invariants:
- Tr(ρ) = 1 at all times
- ρ† = ρ at all times
- S(t₂) ≥ S(t₁) for t₂ > t₁
退相干率计算算法
CalculateDecoherenceRates : Algorithm ≡
Input: dimension : ℕ, coupling_strength : ℝ⁺
Output: Γ : Matrix[ℝ⁺]
Process:
1. φ = (1 + √5) / 2
2. Γ = zeros(dimension, dimension)
3. for i in range(dimension):
for j in range(dimension):
if i ≠ j:
Γ[i,j] = coupling_strength × |i - j|^(1/φ)
4. return Γ
φ-基稳定性验证算法
VerifyPhiBasisStability : Algorithm ≡
Input: phi_states : List[PhiBasisState], evolution_time : ℝ⁺
Output: all_stable : Bool
Process:
1. all_stable = True
2. for |n⟩_φ in phi_states:
# Create density matrix
ρ_n = |n⟩_φ⟨n|_φ
# Evolve under decoherence
ρ_evolved = QuantumEvolution(ρ_n, evolution_time)
# Check if unchanged
if ||ρ_evolved - ρ_n||_F > tolerance:
all_stable = False
break
3. return all_stable
数学性质验证
性质1：严格熵增
StrictEntropyIncrease : Prop ≡
∀ρ : DensityMatrix . ¬is_diagonal(ρ) →
dS/dt|_{t=0} > 0
where dS/dt = -Tr(𝓛[ρ] ln ρ)
性质2：经典极限唯一性
UniquenessOfClassicalLimit : Prop ≡
∀ρ : DensityMatrix .
∃! ρ_c : ClassicalState .
lim_{t→∞} QuantumEvolution(ρ, t) = ρ_c.state
性质3：退相干的φ-普适性
PhiUniversality : Prop ≡
∀ system : QuantumSystem .
optimal_decoherence_basis(system) = phi_basis
where optimal means fastest approach to classical limit
验证检查点
1. 量子态演化验证
def verify_quantum_state_evolution(initial_state, time_points):
"""验证量子态演化的正确性"""
trajectory = []
for t in time_points:
evolved_state = quantum_evolution(initial_state, t)
# Check density matrix properties
assert is_hermitian(evolved_state)
assert is_positive_semidefinite(evolved_state)
assert abs(trace(evolved_state) - 1.0) < 1e-10
# Check entropy increase
if len(trajectory) > 0:
assert von_neumann_entropy(evolved_state) >= trajectory[-1]['entropy']
trajectory.append({
'time': t,
'state': evolved_state,
'entropy': von_neumann_entropy(evolved_state)
})
return trajectory
2. 退相干率标度验证
def verify_decoherence_rate_scaling(dimensions):
"""验证退相干率的φ-标度关系"""
φ = (1 + np.sqrt(5)) / 2
rates = []
for dim in dimensions:
Γ = calculate_decoherence_rates(dim)
# Check scaling for maximum separation
max_rate = Γ[0, dim-1]
expected_rate = (dim - 1)**(1/φ)
relative_error = abs(max_rate / expected_rate - 1)
assert relative_error < 0.01, f"Scaling error: {relative_error}"
rates.append(max_rate)
# Verify scaling between dimensions
for i in range(len(dimensions) - 1):
ratio = rates[i+1] / rates[i]
expected_ratio = ((dimensions[i+1] - 1) / (dimensions[i] - 1))**(1/φ)
assert abs(ratio / expected_ratio - 1) < 0.01
return True
3. 经典极限收敛验证
def verify_classical_limit_convergence(quantum_state, max_time, tolerance):
"""验证向经典极限的收敛"""
t = 0
dt = 0.1
while t < max_time:
state = quantum_evolution(quantum_state, t)
# Check if diagonal
off_diagonal_norm = 0
n = state.shape[0]
for i in range(n):
for j in range(n):
if i != j:
off_diagonal_norm += abs(state[i,j])**2
if np.sqrt(off_diagonal_norm) < tolerance:
# Reached classical limit
classical_state = np.diag(np.diag(state))
# Verify it's a fixed point
evolved = quantum_evolution(classical_state, dt)
assert np.allclose(evolved, classical_state)
return True, t, classical_state
t += dt
return False, max_time, None
实用函数
def create_superposition_state(coefficients, phi_basis_states):
"""创建φ-基的量子叠加态"""
state = sum(c * |n⟩ for c, |n⟩ in zip(coefficients, phi_basis_states))
return normalize(state)
def measure_classicality(density_matrix):
"""测量态的经典性（0=纯量子，1=完全经典）"""
# Ratio of diagonal to total Frobenius norm
diag_norm = np.sum(np.abs(np.diag(density_matrix))**2)
total_norm = np.sum(np.abs(density_matrix)**2)
return diag_norm / total_norm
def estimate_decoherence_time(system_size, environment_coupling):
"""估计系统的退相干时间"""
φ = (1 + np.sqrt(5)) / 2
τ₀ = 1.0 # Microscopic time scale
return τ₀ * φ**(-np.log(system_size)) / environment_coupling
与其他理论的联系
依赖关系
- A1: 自指完备系统必然熵增（基础公理）
- T3-1: 量子态涌现（量子基础）
- D1-8: φ-表示定义（编码基础）
- T12-1: 量子-经典过渡（宏观理论）
支撑的理论
- 为C4-2（波函数坍缩）提供动力学基础
- 为C4-3（测量装置涌现）提供理论支撑
- 为C12-1（意识涌现）提供物理前提