C4-2-formal: 波函数坍缩的信息理论推论的形式化规范

机器验证元数据

type: corollary
verification: machine_ready
dependencies: ["A1-formal.md", "C4-1-formal.md", "T3-2-formal.md", "D1-8-formal.md"]
verification_points:
  - information_gain_calculation
  - measurement_operator_optimization
  - collapse_probability_verification
  - post_measurement_state_uniqueness
  - measurement_backaction_necessity
  - information_causality_check

核心推论

推论 C4-2（波函数坍缩的信息理论）

WavefunctionCollapseInformationTheory : Prop ≡
  ∀ψ : QuantumState, M : MeasurementOperator, O : Observer .
    measurement(M, ψ) ↔ information_gain(O, ψ) ∧
    collapse_outcome = argmax_{n} information_content(n) ∧
    optimal_basis(M) = phi_representation_basis

where
  MeasurementOperator : Type = {
    operators : List[Matrix[ℂ]],
    completeness : ∑_n M_n† M_n = I,
    orthogonality : M_i M_j = δ_{ij} M_i
  }
  
  InformationGain : Type = {
    before : ℝ,  // Entropy before measurement
    after : ℝ,   // Entropy after measurement
    gain : ℝ,    // after - before
    positive : gain ≥ 0
  }

形式化组件

1. 测量的信息论表示

MeasurementAsInformation : QuantumState → MeasurementOperator → InformationGain ≡
  λψ, M .
    let ρ_before = |ψ⟩⟨ψ| in
    let outcomes = measure_probabilities(ψ, M) in
    let ρ_after = ∑_n p_n |n⟩_φ⟨n|_φ in
    InformationGain {
      before = von_neumann_entropy(ρ_before),
      after = von_neumann_entropy(ρ_after),
      gain = shannon_entropy(outcomes),
      positive = true  // By construction
    }

MeasureProbabilities : QuantumState → MeasurementOperator → List[ℝ] ≡
  λψ, M .
    [|⟨n|_φ|ψ⟩|² for n in valid_phi_representations]

2. φ-优化测量算子

PhiOptimalMeasurement : Type ≡
  record {
    basis_states : List[PhiBasisState]
    projectors : List[ProjectionOperator]
    completeness : ∑_n P_n = I
    orthogonality : P_i P_j = δ_{ij} P_i
    phi_structured : ∀n . P_n = |n⟩_φ⟨n|_φ
  }

ConstructPhiMeasurement : ℕ → PhiOptimalMeasurement ≡
  λdimension .
    let valid_states = generate_valid_phi_states(dimension) in
    let projectors = [|n⟩_φ⟨n|_φ for n in valid_states] in
    PhiOptimalMeasurement {
      basis_states = valid_states,
      projectors = projectors,
      completeness = verified_by_construction,
      orthogonality = verified_by_construction,
      phi_structured = true
    }

3. 信息增益最大化

InformationMaximization : MeasurementOperator → QuantumState → ℝ ≡
  λM, ψ .
    let probabilities = measure_probabilities(ψ, M) in
    shannon_entropy(probabilities)

OptimalMeasurementBasis : QuantumState → Set[BasisState] ≡
  λψ .
    argmax_{basis} information_gain(measure_in_basis(basis, ψ))
    
ProofOfPhiOptimality : Prop ≡
  ∀ψ : QuantumState .
    optimal_measurement_basis(ψ) ⊆ phi_basis_states
    
// This holds because φ-basis maximizes distinguishability under no-11 constraint

4. 坍缩动力学

CollapseEvolution : QuantumState → MeasurementResult → QuantumState ≡
  λψ, n .
    let M_n = |n⟩_φ⟨n|_φ in
    let ψ_unnormalized = M_n|ψ⟩ in
    let norm = √⟨ψ|M_n†M_n|ψ⟩ in
    ψ_unnormalized / norm

CollapseProbability : QuantumState → ℕ → ℝ ≡
  λψ, n .
    |⟨n|_φ|ψ⟩|²

PostMeasurementState : QuantumState → MeasurementResult → DensityMatrix ≡
  λψ, n .
    |n⟩_φ⟨n|_φ  // Pure state after measurement

5. 测量反作用

MeasurementBackaction : DensityMatrix → MeasurementOperator → DensityMatrix ≡
  λρ, M .
    let outcomes = [M_n ρ M_n† for n in measurement_outcomes] in
    ∑_n outcomes[n]

BackactionMagnitude : DensityMatrix → DensityMatrix → ℝ ≡
  λρ_before, ρ_after .
    trace_distance(ρ_before, ρ_after)
    
NoFreeInformation : Prop ≡
  ∀ρ, M . information_gain(ρ, M) > 0 → 
    backaction_magnitude(ρ, measurement_backaction(ρ, M)) > 0

6. 信息效率度量

InformationEfficiency : MeasurementBasis → ℕ → ℝ ≡
  λbasis, dimension .
    let max_entropy = log(dimension) in
    let actual_entropy = average_measurement_entropy(basis) in
    actual_entropy / max_entropy

PhiBasisEfficiency : ℕ → ℝ ≡
  λdimension .
    let phi_basis = construct_phi_measurement(dimension) in
    information_efficiency(phi_basis.basis_states, dimension)
    
EfficiencyTheorem : Prop ≡
  ∀dimension, basis .
    information_efficiency(phi_basis, dimension) ≥ 
    information_efficiency(basis, dimension)

算法规范

测量模拟算法

QuantumMeasurementSimulation : Algorithm ≡
  Input: ψ : QuantumState, measurement_basis : MeasurementBasis
  Output: (result : MeasurementResult, ψ_after : QuantumState, info_gain : ℝ)
  
  Process:
    1. # Calculate initial entropy
    2. ρ_initial = |ψ⟩⟨ψ|
    3. S_initial = von_neumann_entropy(ρ_initial)  # = 0 for pure state
    
    4. # Calculate measurement probabilities
    5. probabilities = []
    6. for basis_state in measurement_basis:
         p = |⟨basis_state|ψ⟩|²
         probabilities.append(p)
    
    7. # Sample measurement outcome
    8. result = sample_from_distribution(probabilities)
    
    9. # Calculate collapsed state
    10. ψ_after = measurement_basis[result]
    
    11. # Calculate information gain
    12. info_gain = -∑_i p_i log(p_i)  # Shannon entropy of outcome distribution
    
    13. return (result, ψ_after, info_gain)
  
  Invariants:
    - ∑ probabilities = 1
    - |ψ_after⟩ is normalized
    - info_gain ≥ 0

φ-最优测量构造算法

ConstructOptimalPhiMeasurement : Algorithm ≡
  Input: dimension : ℕ
  Output: measurement : PhiOptimalMeasurement
  
  Process:
    1. φ = (1 + √5) / 2
    2. valid_indices = []
    3. 
    4. # Generate valid φ-representation indices (no consecutive 1s)
    5. for i in range(dimension):
         if is_valid_phi_representation(i):
           valid_indices.append(i)
    
    6. # Construct basis states
    7. basis_states = []
    8. for idx in valid_indices:
         state = zeros(dimension)
         state[idx] = 1
         basis_states.append(state)
    
    9. # Construct projectors
    10. projectors = []
    11. for state in basis_states:
          P = outer_product(state, state)
          projectors.append(P)
    
    12. # Verify completeness (may need padding for non-square dimensions)
    13. if len(projectors) < dimension:
          # Add projectors onto subspace orthogonal to φ-basis
          add_completion_projectors(projectors, dimension)
    
    14. return PhiOptimalMeasurement(basis_states, projectors)

信息因果性验证算法

VerifyInformationCausality : Algorithm ≡
  Input: measurement_sequence : List[(time, location, result)]
  Output: causality_preserved : Bool
  
  Process:
    1. # Sort measurements by time
    2. sorted_measurements = sort_by_time(measurement_sequence)
    3. 
    4. # Check light-cone constraints
    5. for i in range(len(sorted_measurements) - 1):
         m1 = sorted_measurements[i]
         m2 = sorted_measurements[i + 1]
         
         Δt = m2.time - m1.time
         Δx = distance(m1.location, m2.location)
         
         # Information cannot propagate faster than light
         if Δx > c * Δt:
           # Check if measurements are correlated
           if are_correlated(m1.result, m2.result):
             return False  # Causality violation
    
    6. return True  # All measurements respect causality

数学性质验证

性质1：Born规则的信息论推导

BornRuleDerivation : Prop ≡
  ∀ψ : QuantumState, n : MeasurementOutcome .
    P(n|ψ) = |⟨n|ψ⟩|² ↔ 
    P(n|ψ) = argmax_{probability_distribution} expected_information_gain

性质2：量子Zeno效应

QuantumZenoEffect : Prop ≡
  ∀ψ : QuantumState, H : Hamiltonian, τ : ℝ⁺ .
    lim_{τ→0} evolution_under_repeated_measurement(ψ, H, τ) = ψ
    
where repeated measurement happens at intervals τ

性质3：测量不可克隆

MeasurementNoCloning : Prop ≡
  ¬∃M : MeasurementOperator .
    ∀ψ . measurement(M, ψ) yields complete_information(ψ) ∧
         post_measurement_state = ψ

验证检查点

1. 信息增益计算验证

def verify_information_gain_calculation(quantum_state, measurement_basis):
    """验证信息增益计算的正确性"""
    # Calculate probabilities
    probabilities = []
    for basis_state in measurement_basis:
        p = abs(inner_product(basis_state, quantum_state))**2
        probabilities.append(p)
    
    # Verify normalization
    assert abs(sum(probabilities) - 1.0) < 1e-10
    
    # Calculate Shannon entropy (information gain)
    info_gain = 0
    for p in probabilities:
        if p > 0:
            info_gain -= p * np.log2(p)
    
    # Information gain should be non-negative
    assert info_gain >= 0
    
    # For uniform superposition, should be maximal
    if is_uniform_superposition(quantum_state):
        expected_max = np.log2(len([p for p in probabilities if p > 0]))
        assert abs(info_gain - expected_max) < 1e-6
    
    return info_gain

2. 测量算子优化验证

def verify_measurement_operator_optimization(dimension):
    """验证φ-基测量的最优性"""
    # Construct φ-basis measurement
    phi_measurement = construct_optimal_phi_measurement(dimension)
    
    # Construct alternative measurement (e.g., computational basis)
    comp_measurement = construct_computational_measurement(dimension)
    
    # Test on various quantum states
    test_states = generate_test_states(dimension, num_states=100)
    
    phi_efficiency_sum = 0
    comp_efficiency_sum = 0
    
    for state in test_states:
        phi_info = calculate_information_gain(state, phi_measurement)
        comp_info = calculate_information_gain(state, comp_measurement)
        
        phi_efficiency_sum += phi_info
        comp_efficiency_sum += comp_info
    
    # φ-basis should have higher average information efficiency
    assert phi_efficiency_sum >= comp_efficiency_sum
    
    return True

3. 坍缩概率验证

def verify_collapse_probability_verification(quantum_state, measurement_basis):
    """验证坍缩概率的正确性"""
    probabilities = []
    
    for i, basis_state in enumerate(measurement_basis):
        # Calculate probability using Born rule
        amplitude = inner_product(basis_state, quantum_state)
        probability = abs(amplitude)**2
        probabilities.append(probability)
        
        # Verify probability bounds
        assert 0 <= probability <= 1
        
        # Verify collapse state
        if probability > 0:
            collapsed_state = collapse_evolution(quantum_state, i)
            # Collapsed state should be the basis state
            assert np.allclose(collapsed_state, basis_state)
    
    # Verify probability normalization
    total_probability = sum(probabilities)
    assert abs(total_probability - 1.0) < 1e-10
    
    return probabilities

4. 后测量态唯一性验证

def verify_post_measurement_state_uniqueness(quantum_state, measurement_result):
    """验证测量后状态的唯一性"""
    # Apply measurement
    post_state_1 = collapse_evolution(quantum_state, measurement_result)
    
    # Apply same measurement again
    post_state_2 = collapse_evolution(post_state_1, measurement_result)
    
    # Should get same state (eigenstate property)
    assert np.allclose(post_state_1, post_state_2)
    
    # Verify it's a pure state
    density_matrix = np.outer(post_state_1, np.conj(post_state_1))
    purity = np.trace(density_matrix @ density_matrix)
    assert abs(purity - 1.0) < 1e-10
    
    return True

5. 测量反作用必然性验证

def verify_measurement_backaction_necessity(quantum_state, measurement):
    """验证测量反作用的必然性"""
    # Initial state
    rho_initial = np.outer(quantum_state, np.conj(quantum_state))
    
    # Apply measurement
    rho_after = measurement_backaction(rho_initial, measurement)
    
    # Calculate information gain
    info_gain = calculate_information_gain(quantum_state, measurement)
    
    # Calculate backaction magnitude
    backaction = trace_distance(rho_initial, rho_after)
    
    # If information was gained, there must be backaction
    if info_gain > 1e-10:
        assert backaction > 1e-10, "No free information: backaction required"
    
    # Verify backaction is bounded
    assert backaction <= 2.0  # Maximum trace distance
    
    return True

实用函数

def simulate_quantum_measurement(quantum_state, measurement_type='phi'):
    """模拟量子测量过程"""
    if measurement_type == 'phi':
        measurement = construct_optimal_phi_measurement(len(quantum_state))
    else:
        measurement = construct_measurement_basis(len(quantum_state), measurement_type)
    
    # Calculate probabilities and sample outcome
    probabilities = calculate_probabilities(quantum_state, measurement)
    outcome = np.random.choice(len(probabilities), p=probabilities)
    
    # Get collapsed state
    collapsed_state = measurement.basis_states[outcome]
    
    # Calculate information gain
    info_gain = shannon_entropy(probabilities)
    
    return {
        'outcome': outcome,
        'collapsed_state': collapsed_state,
        'information_gain': info_gain,
        'probabilities': probabilities
    }

def compare_measurement_bases(quantum_state, bases_list):
    """比较不同测量基的信息效率"""
    results = {}
    
    for basis_name, basis in bases_list:
        info_gain = calculate_information_gain(quantum_state, basis)
        distinguishability = calculate_distinguishability(basis)
        
        results[basis_name] = {
            'information_gain': info_gain,
            'distinguishability': distinguishability,
            'efficiency': info_gain / np.log2(len(basis))
        }
    
    return results

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）
• C4-1: 量子经典化（退相干基础）
• T3-2: 量子测量定理（测量形式）
• D1-8: φ-表示定义（最优基）

支撑的理论

• 为C4-3（测量装置涌现）提供信息论基础
• 为量子信息论提供新诠释
• 为量子密码学提供理论支撑

$$\boxed{\text{形式化规范：波函数坍缩等价于最优信息提取过程}}$$