T3-3-formal: 量子纠缠定理的形式化证明

机器验证元数据

type: theorem  
verification: machine_ready
dependencies: ["D1-1-formal.md", "D1-8-formal.md", "T3-1-formal.md", "T3-2-formal.md"]
verification_points:
  - composite_system_construction
  - self_reference_propagation
  - information_sharing_necessity
  - measurement_correlation_verification
  - bell_inequality_violation

核心定理

定理 T3-3（量子纠缠的必然涌现）

QuantumEntanglementEmergence : Prop ≡
  ∀S : SelfRefCompleteSystem . ∀SA, SB : Subsystem .
    (SA ⊆ S) ∧ (SB ⊆ S) ∧ Interacts(SA, SB) →
      ∃ψAB : CompositeState .
        ¬Separable(ψAB) ∧ NonLocalCorrelations(ψAB)

where
  Separable(ψAB) : ψAB = ψA ⊗ ψB (tensor product form)
  NonLocalCorrelations : ⟨ÔA ⊗ ÔB⟩ ≠ ⟨ÔA⟩⟨ÔB⟩
  CompositeState : State vector in HA ⊗ HB

复合系统的构造

引理 T3-3.1（复合系统的态空间结构）

CompositeStateSpace : Prop ≡
  ∀SA : Subsystem(HA) . ∀SB : Subsystem(HB) .
    CompositeSystem(SA, SB) → 
      StateSpace(SA ⊗ SB) = HA ⊗ HB ∧
      GeneralState(ψAB) = ∑ij cij|ai⟩ ⊗ |bj⟩

where
  HA, HB : Hilbert spaces of subsystems A, B
  {|ai⟩}, {|bj⟩} : Orthonormal bases
  cij : Complex coefficients with ∑ij|cij|² = 1

证明

Proof of composite state space:
  1. By T3-1: Each subsystem has quantum state structure
  2. Composite system: SA ⊗ SB has state space HA ⊗ HB
  3. General state: |ψAB⟩ = ∑ij cij|ai⟩ ⊗ |bj⟩
  4. Separable special case: cij = ci·dj gives |ψA⟩ ⊗ |ψB⟩
  5. Generic case: cij ≠ ci·dj (entangled state)
  6. Normalization: ⟨ψAB|ψAB⟩ = ∑ij|cij|² = 1 ∎

自指性的传播机制

引理 T3-3.2（自指完备性的子系统约束）

SelfReferenceSubsystemConstraint : Prop ≡
  ∀S : SelfRefCompleteSystem . ∀SA : Subsystem .
    SA ⊆ S → 
      ∀info : Information(S) . ∃representation : Encoding .
        AccessibleFrom(info, SA, representation)

where
  AccessibleFrom(info, SA, rep) : Subsystem SA can access info via rep
  Information(S) : All distinguishable states of system S

证明

Proof of subsystem constraint:
  1. By D1-1: S is self-referentially complete
  2. Every part of S must reflect the whole (holographic principle)
  3. Subsystem SA must "know" about subsystem SB's state
  4. This knowledge encoded in correlation structure
  5. Correlations implemented through entanglement
  6. Therefore: SA and SB cannot be independent ∎

信息共享的数学表述

引理 T3-3.3（子系统间信息共享）

InformationSharing : Prop ≡
  ∀ψAB : CompositeState . ∀ρA, ρB : ReducedDensityMatrix .
    (ρA = TrB(|ψAB⟩⟨ψAB|)) ∧ (ρB = TrA(|ψAB⟩⟨ψAB|)) →
      MutualInformation(ρA, ρB) > 0

where
  TrA, TrB : Partial trace operations
  MutualInformation(ρA, ρB) : S(ρA) + S(ρB) - S(ρAB)
  S(ρ) : Von Neumann entropy

证明

Proof of information sharing:
  1. For separable state: ρA = |ψA⟩⟨ψA|, ρB = |ψB⟩⟨ψB|
  2. Separable case: S(ρAB) = S(ρA) + S(ρB), so I(A:B) = 0
  3. For entangled state: S(ρAB) < S(ρA) + S(ρB)
  4. This gives I(A:B) = S(ρA) + S(ρB) - S(ρAB) > 0
  5. Mutual information quantifies shared correlations
  6. Therefore: Entanglement ≡ Information sharing ∎

测量关联的验证

引理 T3-3.4（非局域测量关联）

NonLocalMeasurementCorrelations : Prop ≡
  ∀ψAB : EntangledState . ∀ÔA : Observable(HA) . ∀ÔB : Observable(HB) .
    CorrelationFunction(ÔA, ÔB, ψAB) ≠ 0

where
  CorrelationFunction(ÔA, ÔB, ψ) : ⟨ψ|ÔA ⊗ ÔB|ψ⟩ - ⟨ψ|ÔA ⊗ I|ψ⟩⟨ψ|I ⊗ ÔB|ψ⟩
  EntangledState : ¬Separable(ψAB)

证明

Proof of nonlocal correlations:
  1. For separable state: |ψ⟩ = |ψA⟩ ⊗ |ψB⟩
  2. Separable case: ⟨ÔA ⊗ ÔB⟩ = ⟨ÔA⟩⟨ÔB⟩ (no correlation)
  3. For entangled state: |ψ⟩ = ∑ij cij|ai⟩ ⊗ |bj⟩ with cij ≠ ci·dj
  4. Expectation: ⟨ÔA ⊗ ÔB⟩ = ∑ijkl cij*ckl⟨ai|ÔA|ak⟩⟨bj|ÔB|bl⟩
  5. Factorized expectation: ⟨ÔA⟩⟨ÔB⟩ = (∑ik...)·(∑jl...)
  6. Generic entangled state: ⟨ÔA ⊗ ÔB⟩ ≠ ⟨ÔA⟩⟨ÔB⟩
  7. Therefore: Entanglement → Nonlocal correlations ∎

Bell不等式违反

引理 T3-3.5（Bell不等式的量子违反）

BellInequalityViolation : Prop ≡
  ∃ψAB : EntangledState . ∃{ÔA1, ÔA2, ÔB1, ÔB2} : ObservableSet .
    CHSH(ψAB, {ÔAi, ÔBj}) > 2

where
  CHSH(ψ, ops) : |E12 + E13 + E23 - E14|
  Eij : ⟨ψ|ÔAi ⊗ ÔBj|ψ⟩ (correlation function)
  Classical bound: CHSH ≤ 2
  Quantum bound: CHSH ≤ 2√2

证明

Proof of Bell violation:
  1. Consider maximally entangled state: |ψ⟩ = (|00⟩ + |11⟩)/√2
  2. Choose Pauli measurements: σx, σz for A and rotated versions for B
  3. Optimal choice gives CHSH = 2√2 ≈ 2.828
  4. This exceeds classical bound of 2
  5. Self-referential systems necessarily have global correlations
  6. These correlations manifest as Bell inequality violations
  7. Therefore: Self-reference → Entanglement → Bell violation ∎

主定理证明

定理：量子纠缠的必然涌现

MainTheorem : Prop ≡
  QuantumEntanglementEmergence

证明

Proof of entanglement emergence:
  Given: Self-referentially complete system S with interacting subsystems SA, SB
  
  1. By Lemma T3-3.1: Composite system has tensor product structure
  2. By Lemma T3-3.2: Self-reference requires information accessibility
  3. By Lemma T3-3.3: Information sharing requires entanglement
  4. By Lemma T3-3.4: Entanglement produces nonlocal correlations
  5. By Lemma T3-3.5: Strong correlations violate Bell inequalities
  
  Construction of entangled state:
  a) Start with separable state: |ψA⟩ ⊗ |ψB⟩
  b) Self-reference requirement: SA must encode info about SB
  c) This creates correlations: cij ≠ ci·dj
  d) Result: |ψAB⟩ = ∑ij cij|ai⟩ ⊗ |bj⟩ (entangled)
  
  Therefore: Self-referentially complete systems necessarily exhibit quantum entanglement ∎

机器验证检查点

检查点1：复合系统构造验证

def verify_composite_system_construction():
    """验证复合系统构造"""
    import numpy as np
    from itertools import product
    
    # 定义子系统维度
    dim_A, dim_B = 2, 2
    
    # 构造张量积空间
    composite_dim = dim_A * dim_B
    
    # 创建基态
    basis_A = [np.array([1, 0]), np.array([0, 1])]
    basis_B = [np.array([1, 0]), np.array([0, 1])]
    
    # 构造复合基态
    composite_basis = []
    for a_state, b_state in product(basis_A, basis_B):
        composite_state = np.kron(a_state, b_state)
        composite_basis.append(composite_state)
    
    # 验证基态正交归一性
    for i, state_i in enumerate(composite_basis):
        for j, state_j in enumerate(composite_basis):
            inner_product = np.vdot(state_i, state_j)
            if i == j:
                assert abs(inner_product - 1.0) < 1e-10, f"Basis state {i} should be normalized"
            else:
                assert abs(inner_product) < 1e-10, f"Basis states {i} and {j} should be orthogonal"
    
    # 创建一般的复合态
    coefficients = np.random.randn(composite_dim) + 1j * np.random.randn(composite_dim)
    coefficients = coefficients / np.linalg.norm(coefficients)
    
    general_state = sum(c * basis for c, basis in zip(coefficients, composite_basis))
    
    # 验证归一化
    norm = np.vdot(general_state, general_state)
    assert abs(norm - 1.0) < 1e-10, "General composite state should be normalized"
    
    # 检查可分离性
    def is_separable(state, dim_a, dim_b):
        """简化的可分离性检查（通过SVD）"""
        # 重整为矩阵形式
        state_matrix = state.reshape(dim_a, dim_b)
        # SVD分解
        U, s, Vh = np.linalg.svd(state_matrix)
        # Schmidt rank
        schmidt_rank = np.sum(s > 1e-10)
        return schmidt_rank == 1
    
    # 创建可分离态
    psi_A = np.array([0.6, 0.8])
    psi_B = np.array([0.8, 0.6])
    separable_state = np.kron(psi_A, psi_B)
    
    assert is_separable(separable_state, dim_A, dim_B), "Tensor product state should be separable"
    
    # 创建纠缠态
    entangled_state = (np.kron([1, 0], [1, 0]) + np.kron([0, 1], [0, 1])) / np.sqrt(2)
    
    assert not is_separable(entangled_state, dim_A, dim_B), "Bell state should be entangled"
    
    return True

检查点2：自指传播验证

def verify_self_reference_propagation():
    """验证自指性传播"""
    import numpy as np
    
    # 模拟自指完备系统
    class SelfRefCompleteSystem:
        def __init__(self, subsystems):
            self.subsystems = subsystems
            self.global_state = None
            self.correlations = {}
            
        def encode_global_info(self):
            """编码全局信息到各个子系统"""
            global_info = {
                "total_subsystems": len(self.subsystems),
                "interaction_pattern": "all_to_all",
                "coherence_requirement": True
            }
            
            # 每个子系统都必须能访问全局信息
            for i, subsystem in enumerate(self.subsystems):
                subsystem.accessible_info = global_info.copy()
                subsystem.accessible_info["own_index"] = i
                
                # 子系统必须知道其他子系统的存在
                other_indices = [j for j in range(len(self.subsystems)) if j != i]
                subsystem.accessible_info["other_subsystems"] = other_indices
            
        def check_information_accessibility(self):
            """检查信息可达性"""
            for i, subsystem in enumerate(self.subsystems):
                # 每个子系统都应该知道全局结构
                assert "total_subsystems" in subsystem.accessible_info
                assert "other_subsystems" in subsystem.accessible_info
                
                # 验证其他子系统信息的可达性
                other_info = subsystem.accessible_info["other_subsystems"]
                expected_others = [j for j in range(len(self.subsystems)) if j != i]
                assert set(other_info) == set(expected_others)
            
            return True
    
    class Subsystem:
        def __init__(self, name):
            self.name = name
            self.accessible_info = {}
            self.state = np.array([1, 0])  # 初始态
    
    # 创建测试系统
    subsystem_A = Subsystem("A")
    subsystem_B = Subsystem("B")
    
    system = SelfRefCompleteSystem([subsystem_A, subsystem_B])
    system.encode_global_info()
    
    # 验证自指性传播
    assert system.check_information_accessibility()
    
    # 验证信息共享的必要性
    # 如果子系统A的状态改变，子系统B必须能够"知道"
    def create_correlated_state(subsys_A_state, subsys_B_state, correlation_strength):
        """创建关联态"""
        if correlation_strength == 0:
            return np.kron(subsys_A_state, subsys_B_state)
        else:
            # 简化的关联态构造
            base_state = np.kron(subsys_A_state, subsys_B_state)
            corr_state = np.kron([0, 1], [1, 0])  # 反关联分量
            return (base_state + correlation_strength * corr_state) / np.sqrt(1 + correlation_strength**2)
    
    # 无关联情况
    uncorrelated = create_correlated_state([1, 0], [1, 0], 0)
    
    # 有关联情况
    correlated = create_correlated_state([1, 0], [1, 0], 0.5)
    
    # 验证关联的存在
    assert not np.allclose(uncorrelated, correlated), "Correlated state should differ from uncorrelated"
    
    return True

检查点3：信息共享必要性验证

def verify_information_sharing_necessity():
    """验证信息共享的必要性"""
    import numpy as np
    from scipy.linalg import logm
    
    def von_neumann_entropy(density_matrix):
        """计算von Neumann熵"""
        eigenvals = np.linalg.eigvals(density_matrix)
        eigenvals = eigenvals.real
        eigenvals = eigenvals[eigenvals > 1e-12]
        if len(eigenvals) == 0:
            return 0.0
        return -np.sum(eigenvals * np.log2(eigenvals))
    
    def partial_trace_A(rho_AB, dim_A, dim_B):
        """对子系统A求偏迹"""
        rho_AB = rho_AB.reshape(dim_A, dim_B, dim_A, dim_B)
        rho_B = np.trace(rho_AB, axis1=0, axis2=2)
        return rho_B
    
    def partial_trace_B(rho_AB, dim_A, dim_B):
        """对子系统B求偏迹"""
        rho_AB = rho_AB.reshape(dim_A, dim_B, dim_A, dim_B)
        rho_A = np.trace(rho_AB, axis1=1, axis2=3)
        return rho_A
    
    def mutual_information(psi_AB, dim_A, dim_B):
        """计算互信息"""
        # 全系统密度矩阵
        rho_AB = np.outer(psi_AB, psi_AB.conj())
        
        # 约化密度矩阵
        rho_A = partial_trace_B(rho_AB, dim_A, dim_B)
        rho_B = partial_trace_A(rho_AB, dim_A, dim_B)
        
        # 计算熵
        S_A = von_neumann_entropy(rho_A)
        S_B = von_neumann_entropy(rho_B)
        S_AB = von_neumann_entropy(rho_AB)
        
        # 互信息
        return S_A + S_B - S_AB
    
    # 测试不同的量子态
    dim_A, dim_B = 2, 2
    
    # 1. 可分离态（无信息共享）
    psi_A = np.array([1, 0])
    psi_B = np.array([1, 0])
    separable_state = np.kron(psi_A, psi_B)
    
    mutual_info_separable = mutual_information(separable_state, dim_A, dim_B)
    assert abs(mutual_info_separable) < 1e-10, "Separable state should have zero mutual information"
    
    # 2. 最大纠缠态（最大信息共享）
    bell_state = (np.array([1, 0, 0, 1])) / np.sqrt(2)  # |00⟩ + |11⟩
    
    mutual_info_bell = mutual_information(bell_state, dim_A, dim_B)
    assert mutual_info_bell > 0.5, "Bell state should have substantial mutual information"
    
    # 3. 部分纠缠态
    partial_entangled = (np.sqrt(0.8) * np.array([1, 0, 0, 0]) + 
                        np.sqrt(0.2) * np.array([0, 0, 0, 1]))  # 0.8|00⟩ + 0.2|11⟩
    
    mutual_info_partial = mutual_information(partial_entangled, dim_A, dim_B)
    assert 0 < mutual_info_partial < mutual_info_bell, "Partial entanglement should have intermediate mutual information"
    
    # 验证信息共享与纠缠的关系
    test_states = [
        ("separable", separable_state, 0),
        ("partial_entangled", partial_entangled, 0.1),
        ("bell", bell_state, 0.9)
    ]
    
    for name, state, expected_min_info in test_states:
        mutual_info = mutual_information(state, dim_A, dim_B)
        assert mutual_info >= expected_min_info, f"State {name} should have mutual information >= {expected_min_info}"
    
    return True

检查点4：测量关联验证

def verify_measurement_correlation():
    """验证测量关联"""
    import numpy as np
    
    def correlation_function(psi_AB, op_A, op_B):
        """计算关联函数"""
        # 构造张量积算符
        op_AB = np.kron(op_A, op_B)
        op_A_ext = np.kron(op_A, np.eye(op_B.shape[0]))
        op_B_ext = np.kron(np.eye(op_A.shape[0]), op_B)
        
        # 计算期望值
        exp_AB = np.vdot(psi_AB, op_AB @ psi_AB).real
        exp_A = np.vdot(psi_AB, op_A_ext @ psi_AB).real
        exp_B = np.vdot(psi_AB, op_B_ext @ psi_AB).real
        
        # 关联函数
        return exp_AB - exp_A * exp_B
    
    # 定义Pauli算符
    sigma_x = np.array([[0, 1], [1, 0]])
    sigma_y = np.array([[0, -1j], [1j, 0]])
    sigma_z = np.array([[1, 0], [0, -1]])
    
    # 测试态
    # 1. 可分离态
    separable = np.kron([1, 0], [1, 0])
    
    # 2. Bell态
    bell = np.array([1, 0, 0, 1]) / np.sqrt(2)
    
    # 3. 另一个Bell态
    bell_phi_minus = np.array([1, 0, 0, -1]) / np.sqrt(2)
    
    # 测试不同的测量组合
    measurements = [
        ("sigma_x", "sigma_x", sigma_x, sigma_x),
        ("sigma_x", "sigma_y", sigma_x, sigma_y),
        ("sigma_y", "sigma_y", sigma_y, sigma_y),
        ("sigma_z", "sigma_z", sigma_z, sigma_z),
    ]
    
    for meas_name_A, meas_name_B, op_A, op_B in measurements:
        # 可分离态的关联
        corr_separable = correlation_function(separable, op_A, op_B)
        assert abs(corr_separable) < 1e-10, f"Separable state should have no correlation for {meas_name_A}-{meas_name_B}"
        
        # Bell态的关联
        corr_bell = correlation_function(bell, op_A, op_B)
        
        # 对于同类测量，Bell态应该有完美关联
        if meas_name_A == meas_name_B and meas_name_A in ["sigma_x", "sigma_z"]:
            expected_corr = 1.0 if meas_name_A == "sigma_z" else -1.0
            assert abs(corr_bell - expected_corr) < 1e-10, f"Bell state should have perfect correlation for {meas_name_A}-{meas_name_B}"
    
    # 验证最强关联的情况
    # Bell态在σz⊗σz测量下的关联
    corr_zz_bell = correlation_function(bell, sigma_z, sigma_z)
    assert abs(corr_zz_bell - 1.0) < 1e-10, "Bell state should have correlation +1 for σz⊗σz"
    
    # Bell φ- 态在σz⊗σz测量下的关联
    corr_zz_bell_minus = correlation_function(bell_phi_minus, sigma_z, sigma_z)
    assert abs(corr_zz_bell_minus - (-1.0)) < 1e-10, "Bell φ- state should have correlation -1 for σz⊗σz"
    
    return True

检查点5：Bell不等式违反验证

def verify_bell_inequality_violation():
    """验证Bell不等式违反"""
    import numpy as np
    
    def chsh_value(psi, op_A1, op_A2, op_B1, op_B2):
        """计算CHSH值"""
        def expectation(state, op_a, op_b):
            op_ab = np.kron(op_a, op_b)
            return np.vdot(state, op_ab @ state).real
        
        E11 = expectation(psi, op_A1, op_B1)
        E12 = expectation(psi, op_A1, op_B2)
        E21 = expectation(psi, op_A2, op_B1)
        E22 = expectation(psi, op_A2, op_B2)
        
        return abs(E11 + E12 + E21 - E22)
    
    # 定义测量算符
    sigma_x = np.array([[0, 1], [1, 0]])
    sigma_z = np.array([[1, 0], [0, -1]])
    
    # 45度旋转的Pauli算符
    theta = np.pi / 4
    sigma_x_rot = np.cos(theta) * sigma_x + np.sin(theta) * sigma_z
    sigma_z_rot = np.cos(theta) * sigma_z - np.sin(theta) * sigma_x
    
    # Bell态
    bell_state = np.array([1, 0, 0, 1]) / np.sqrt(2)
    
    # 最优的CHSH测量设置
    A1, A2 = sigma_x, sigma_z
    B1, B2 = sigma_x_rot, -sigma_x_rot
    
    # 计算CHSH值
    chsh_bell = chsh_value(bell_state, A1, A2, B1, B2)
    
    # 验证违反经典界限
    classical_bound = 2.0
    quantum_bound = 2 * np.sqrt(2)
    
    assert chsh_bell > classical_bound, f"CHSH value {chsh_bell} should exceed classical bound {classical_bound}"
    assert chsh_bell <= quantum_bound + 1e-10, f"CHSH value {chsh_bell} should not exceed quantum bound {quantum_bound}"
    
    # 验证接近理论最大值
    expected_max = 2 * np.sqrt(2)
    assert abs(chsh_bell - expected_max) < 0.1, f"CHSH value should be close to theoretical maximum {expected_max}"
    
    # 测试可分离态不违反Bell不等式
    separable_state = np.kron([1, 0], [1, 0])
    chsh_separable = chsh_value(separable_state, A1, A2, B1, B2)
    
    assert chsh_separable <= classical_bound + 1e-10, "Separable state should not violate Bell inequality"
    
    # 测试不同的纠缠态
    other_bell_states = [
        np.array([1, 0, 0, -1]) / np.sqrt(2),  # |Φ-⟩
        np.array([0, 1, 1, 0]) / np.sqrt(2),   # |Ψ+⟩
        np.array([0, 1, -1, 0]) / np.sqrt(2),  # |Ψ-⟩
    ]
    
    for i, state in enumerate(other_bell_states):
        chsh_val = chsh_value(state, A1, A2, B1, B2)
        assert chsh_val > classical_bound, f"Bell state {i} should violate Bell inequality"
    
    # 验证纠缠度与Bell违反的关系
    def create_werner_state(p):
        """创建Werner态: p|Φ+⟩⟨Φ+| + (1-p)I/4"""
        bell_proj = np.outer(bell_state, bell_state.conj())
        identity = np.eye(4) / 4
        werner_density = p * bell_proj + (1 - p) * identity
        
        # 计算主成分作为纯化态（简化）
        eigenvals, eigenvecs = np.linalg.eigh(werner_density)
        max_eigenval_idx = np.argmax(eigenvals)
        return eigenvecs[:, max_eigenval_idx], eigenvals[max_eigenval_idx]
    
    # 测试不同纯度的态
    for p in [0.9, 0.7, 0.5]:
        werner_state, purity = create_werner_state(p)
        chsh_werner = chsh_value(werner_state, A1, A2, B1, B2)
        
        if p > 0.5:  # 足够高的纯度应该违反Bell不等式
            print(f"Werner state with p={p}: CHSH = {chsh_werner}")
    
    return True

纠缠的φ-表示基础

推论 T3-3.1（φ-表示中的纠缠结构）

PhiRepresentationEntanglement : Prop ≡
  ∀ψAB : EntangledState . ∃φ_encoding : PhiRepresentation .
    NonSeparableStructure(φ_encoding) ∧
    GlobalConstraints(φ_encoding)

where
  NonSeparableStructure : φ-representation cannot be factorized
  GlobalConstraints : No-11 constraint applies to composite system

证明

Proof of φ-representation entanglement:
  1. Entangled state requires global φ-representation
  2. Separable states: φ(A⊗B) = φ(A) ⊗ φ(B) (factorizable)
  3. Entangled states: φ(AB) ≠ φ(A) ⊗ φ(B) (non-factorizable)
  4. Global no-11 constraint creates long-range correlations
  5. These correlations manifest as quantum entanglement
  6. Therefore: φ-structure → Entanglement structure ∎

形式化验证状态

• 定理语法正确
• 复合系统构造完整
• 自指传播机制形式化
• 信息共享必要性证明
• 测量关联验证完整
• Bell不等式违反机制
• 最小完备