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T3-4-formal: 量子隐形传态定理的形式化证明

机器验证元数据​

type: theorem  
verification: machine_ready
dependencies: ["D1-4-formal.md", "T3-1-formal.md", "T3-2-formal.md", "T3-3-formal.md"]
verification_points:
  - initial_state_construction
  - bell_measurement_protocol
  - classical_information_transmission
  - unitary_correction_application
  - teleportation_fidelity_verification

核心定理​

定理 T3-4（量子隐形传态的必然实现）​

QuantumTeleportationRealization : Prop ≡
  ∀ψ : StateVector . ∀siteA, siteB, siteC : Location .
    SeparatedSites(siteA, siteC) ∧ EntangledResource(siteB, siteC) →
      ∃protocol : TeleportationProtocol .
        TransferState(ψ, siteA, siteC, protocol) ∧ 
        Fidelity(ψ, protocol) = 1

where
  SeparatedSites : No direct quantum channel between sites
  EntangledResource : Pre-shared entangled state |Φ+⟩BC
  TransferState : Perfect state transfer achieved
  Fidelity : Overlap between original and final states

初始态的张量积构造​

引理 T3-4.1（三粒子复合态构造）​

ThreeParticleCompositeState : Prop ≡
  ∀ψ : StateVector . ∀entangledPair : BellState .
    ψ = α|0⟩ + β|1⟩ ∧ entangledPair = (|00⟩ + |11⟩)/√2 →
      CompositeState(ψ, entangledPair) = 
        (α|0⟩A + β|1⟩A) ⊗ (|00⟩BC + |11⟩BC)/√2

where
  CompositeState : Three-particle state |ψ⟩A ⊗ |Φ+⟩BC
  A, B, C : Particle labels (Alice, Bob, Charlie)

证明​

Proof of composite state construction:
  1. Unknown state: |ψ⟩A = α|0⟩A + β|1⟩A
  2. Entangled resource: |Φ+⟩BC = (|00⟩BC + |11⟩BC)/√2
  3. Tensor product: |Ψ⟩ABC = |ψ⟩A ⊗ |Φ+⟩BC
  4. Expansion: |Ψ⟩ABC = (α|0⟩A + β|1⟩A) ⊗ (|00⟩BC + |11⟩BC)/√2
  5. Full form: |Ψ⟩ABC = (α|000⟩ + α|011⟩ + β|100⟩ + β|111⟩)/√2
  6. This state contains correlations enabling teleportation ∎

Bell基测量的协议​

引理 T3-4.2（Bell基测量分解）​

BellBasisMeasurementDecomposition : Prop ≡
  ∀stateABC : CompositeState .
    BellMeasurement(stateABC, sitesAB) →
      stateABC = ∑k pk|Φk⟩AB ⊗ |ψk⟩C

where
  BellMeasurement : Measurement in Bell basis {|Φ±⟩, |Ψ±⟩}
  |Φk⟩AB : Bell state measured at sites A,B  
  |ψk⟩C : Corresponding state at site C
  pk : Probability of measuring k-th Bell state (pk = 1/4)

证明​

Proof of Bell measurement decomposition:
  1. Initial state: |Ψ⟩ABC = (α|000⟩ + α|011⟩ + β|100⟩ + β|111⟩)/√2
  
  2. Rewrite in Bell basis for AB:
     |00⟩AB = (|Φ+⟩AB + |Φ-⟩AB)/√2
     |01⟩AB = (|Ψ+⟩AB + |Ψ-⟩AB)/√2  
     |10⟩AB = (|Ψ+⟩AB - |Ψ-⟩AB)/√2
     |11⟩AB = (|Φ+⟩AB - |Φ-⟩AB)/√2
  
  3. Substitution yields:
     |Ψ⟩ABC = (1/2)[|Φ+⟩AB ⊗ (α|0⟩C + β|1⟩C) +
                     |Φ-⟩AB ⊗ (α|0⟩C - β|1⟩C) +  
                     |Ψ+⟩AB ⊗ (α|1⟩C + β|0⟩C) +
                     |Ψ-⟩AB ⊗ (α|1⟩C - β|0⟩C)]
  
  4. Each Bell measurement outcome occurs with probability 1/4
  5. Charlie's state depends on Alice's measurement result ∎

经典信息传输约束​

引理 T3-4.3（经典通信时间延迟）​

ClassicalCommunicationDelay : Prop ≡
  ∀measurementResult : ClassicalBits . ∀siteA, siteC : Location .
    Distance(siteA, siteC) = d →
      TransmissionTime(measurementResult, siteA, siteC) ≥ d/c

where
  ClassicalBits : 2-bit measurement outcome encoding Bell state
  Distance : Spatial separation between sites
  c : Speed of light (maximum information transmission speed)
  TransmissionTime : Time required for classical information transfer

证明​

Proof of communication delay constraint:
  1. By D1-4: Time metric requires causal ordering
  2. Alice's measurement yields 2 classical bits: (i,j) ∈ {0,1}²
  3. Information cannot travel faster than light: v ≤ c
  4. Minimum transmission time: Δt = d/c
  5. Charlie cannot apply correction before receiving information
  6. This delay ensures no faster-than-light communication ∎

幺正修正操作的应用​

引理 T3-4.4（状态修正映射）​

UnitaryCorrection : Prop ≡
  ∀measurementOutcome : BellBasisResult . ∀stateC : StateVector .
    CorrectionOperation(measurementOutcome) = 
      match measurementOutcome with
      | |Φ+⟩ → I  (identity)
      | |Φ-⟩ → σz (Pauli-Z)  
      | |Ψ+⟩ → σx (Pauli-X)
      | |Ψ-⟩ → σy (Pauli-Y)

where
  CorrectionOperation : Unitary operation applied to site C
  I, σx, σy, σz : Pauli operators and identity

证明​

Proof of correction operation mapping:
  1. From Bell measurement decomposition:
     - |Φ+⟩AB measurement → |ψ⟩C = α|0⟩ + β|1⟩ (no correction needed)
     - |Φ-⟩AB measurement → |ψ⟩C = α|0⟩ - β|1⟩ (phase flip: σz)
     - |Ψ+⟩AB measurement → |ψ⟩C = α|1⟩ + β|0⟩ (bit flip: σx) 
     - |Ψ-⟩AB measurement → |ψ⟩C = α|1⟩ - β|0⟩ (bit+phase flip: σy)
  
  2. Verification of corrections:
     - σz(α|0⟩ - β|1⟩) = α|0⟩ + β|1⟩ = |ψ⟩original
     - σx(α|1⟩ + β|0⟩) = α|0⟩ + β|1⟩ = |ψ⟩original
     - σy(α|1⟩ - β|0⟩) = -i(α(-i|0⟩) - β(i|1⟩)) = α|0⟩ + β|1⟩
  
  3. All corrections restore original state perfectly ∎

传输保真度验证​

引理 T3-4.5（完美传输保真度）​

PerfectTeleportationFidelity : Prop ≡
  ∀ψoriginal : StateVector . ∀ψfinal : StateVector .
    TeleportationProtocol(ψoriginal) = ψfinal →
      Fidelity(ψoriginal, ψfinal) = |⟨ψoriginal|ψfinal⟩|² = 1

where
  TeleportationProtocol : Complete teleportation process
  Fidelity : Quantum state overlap measure

证明​

Proof of perfect fidelity:
  1. Original state: |ψ⟩original = α|0⟩ + β|1⟩
  2. After teleportation: |ψ⟩final = α|0⟩ + β|1⟩ (after correction)
  3. Inner product: ⟨ψoriginal|ψfinal⟩ = |α|² + |β|² = 1
  4. Fidelity: F = |⟨ψoriginal|ψfinal⟩|² = 1² = 1
  5. Perfect fidelity achieved in ideal conditions
  6. Original state at site A is destroyed (no-cloning) ∎

主定理证明​

定理：量子隐形传态实现​

MainTheorem : Prop ≡
  QuantumTeleportationRealization

证明​

Proof of quantum teleportation realization:
  Given: Unknown quantum state |ψ⟩A and entangled resource |Φ+⟩BC
  
  1. By Lemma T3-4.1: Construct composite state |Ψ⟩ABC
  2. By Lemma T3-4.2: Alice measures AB in Bell basis  
  3. By Lemma T3-4.3: Classical result transmitted to Charlie
  4. By Lemma T3-4.4: Charlie applies appropriate correction
  5. By Lemma T3-4.5: Perfect fidelity achieved
  
  Teleportation sequence:
  a) Preparation: |ψ⟩A ⊗ |Φ+⟩BC
  b) Bell measurement: projects onto |Φk⟩AB ⊗ |ψk⟩C  
  c) Classical communication: 2 bits from A to C
  d) Unitary correction: U_k|ψk⟩C = |ψ⟩C
  e) Result: Perfect transfer |ψ⟩A → |ψ⟩C
  
  Therefore: Quantum teleportation is necessarily realizable in self-referential systems ∎

机器验证检查点​

检查点1：初始态构造验证​

def verify_initial_state_construction():
    """验证初始态构造"""
    import numpy as np
    
    # 未知态
    alpha, beta = 0.6, 0.8
    psi_unknown = np.array([alpha, beta])
    
    # 纠缠资源
    bell_state = np.array([1, 0, 0, 1]) / np.sqrt(2)  # |Φ+⟩ = (|00⟩ + |11⟩)/√2
    
    # 构造三粒子复合态
    # |ψ⟩A ⊗ |Φ+⟩BC
    composite_state = np.kron(psi_unknown, bell_state)
    
    # 验证维度
    assert composite_state.shape[0] == 8, "Composite state should be 8-dimensional"
    
    # 验证归一化
    norm = np.vdot(composite_state, composite_state).real
    assert abs(norm - 1.0) < 1e-10, "Composite state should be normalized"
    
    # 验证分量结构
    expected = np.array([
        alpha/np.sqrt(2),  # α|000⟩
        0,                 # |001⟩
        0,                 # |010⟩  
        alpha/np.sqrt(2),  # α|011⟩
        beta/np.sqrt(2),   # β|100⟩
        0,                 # |101⟩
        0,                 # |110⟩
        beta/np.sqrt(2)    # β|111⟩
    ])
    
    assert np.allclose(composite_state, expected), "Composite state components should match expected values"
    
    return True

检查点2：Bell测量协议验证​

def verify_bell_measurement_protocol():
    """验证Bell基测量协议"""
    import numpy as np
    
    def bell_basis():
        """构造Bell基"""
        phi_plus = np.array([1, 0, 0, 1]) / np.sqrt(2)   # |Φ+⟩
        phi_minus = np.array([1, 0, 0, -1]) / np.sqrt(2)  # |Φ-⟩
        psi_plus = np.array([0, 1, 1, 0]) / np.sqrt(2)    # |Ψ+⟩
        psi_minus = np.array([0, 1, -1, 0]) / np.sqrt(2)  # |Ψ-⟩
        return [phi_plus, phi_minus, psi_plus, psi_minus]
    
    def decompose_in_bell_basis(composite_state):
        """将复合态分解到Bell基"""
        bell_states = bell_basis()
        coefficients = []
        charlie_states = []
        
        # 提取AB子系统和C子系统的对应关系
        # 这里简化为直接验证测量概率
        for i, bell_state in enumerate(bell_states):
            # 构造对应的投影算符 (在AB空间)
            projector_AB = np.kron(np.outer(bell_state, bell_state.conj()), np.eye(2))
            
            # 计算投影概率
            prob = np.vdot(composite_state, projector_AB @ composite_state).real
            coefficients.append(prob)
        
        return coefficients
    
    # 测试态
    alpha, beta = 0.6, 0.8
    psi_unknown = np.array([alpha, beta])
    bell_resource = np.array([1, 0, 0, 1]) / np.sqrt(2)
    composite_state = np.kron(psi_unknown, bell_resource)
    
    # 分解到Bell基
    probabilities = decompose_in_bell_basis(composite_state)
    
    # 验证概率和为1
    total_prob = sum(probabilities)
    assert abs(total_prob - 1.0) < 1e-10, "Bell measurement probabilities should sum to 1"
    
    # 验证各概率相等（对称性）
    for prob in probabilities:
        assert abs(prob - 0.25) < 1e-10, "Each Bell measurement outcome should have probability 1/4"
    
    return True

检查点3：经典信息传输验证​

def verify_classical_information_transmission():
    """验证经典信息传输"""
    import numpy as np
    
    def encode_measurement_result(bell_state_index):
        """编码测量结果为经典比特"""
        # Bell态索引到经典比特的映射
        encoding_map = {
            0: (0, 0),  # |Φ+⟩ → 00
            1: (0, 1),  # |Φ-⟩ → 01  
            2: (1, 0),  # |Ψ+⟩ → 10
            3: (1, 1)   # |Ψ-⟩ → 11
        }
        return encoding_map[bell_state_index]
    
    def transmission_delay(distance, speed_of_light=1.0):
        """计算传输延迟"""
        return distance / speed_of_light
    
    # 测试所有可能的测量结果
    for bell_index in range(4):
        classical_bits = encode_measurement_result(bell_index)
        
        # 验证编码
        assert len(classical_bits) == 2, "Should encode to 2 classical bits"
        assert all(bit in [0, 1] for bit in classical_bits), "Bits should be 0 or 1"
        
    # 验证传输延迟
    distances = [1.0, 10.0, 100.0]
    for d in distances:
        delay = transmission_delay(d)
        assert delay >= 0, "Transmission delay should be non-negative"
        assert delay >= d, "Delay should respect speed limit"
    
    # 验证信息容量
    total_outcomes = 4  # 4个可能的Bell态
    information_content = np.log2(total_outcomes)
    assert abs(information_content - 2.0) < 1e-10, "Should carry exactly 2 bits of information"
    
    return True

检查点4：幺正修正操作验证​

def verify_unitary_correction_application():
    """验证幺正修正操作"""
    import numpy as np
    
    # Pauli算符
    I = np.eye(2)
    sigma_x = np.array([[0, 1], [1, 0]])
    sigma_y = np.array([[0, -1j], [1j, 0]])  
    sigma_z = np.array([[1, 0], [0, -1]])
    
    def get_correction_operator(bell_outcome):
        """根据Bell测量结果获取修正算符"""
        corrections = {
            0: I,        # |Φ+⟩ → I
            1: sigma_z,  # |Φ-⟩ → σz
            2: sigma_x,  # |Ψ+⟩ → σx
            3: sigma_y   # |Ψ-⟩ → σy
        }
        return corrections[bell_outcome]
    
    def get_charlie_state_after_measurement(bell_outcome, alpha, beta):
        """获取测量后Charlie的状态"""
        states = {
            0: np.array([alpha, beta]),      # α|0⟩ + β|1⟩
            1: np.array([alpha, -beta]),     # α|0⟩ - β|1⟩  
            2: np.array([beta, alpha]),      # α|1⟩ + β|0⟩ → β|0⟩ + α|1⟩
            3: np.array([beta, -alpha])      # α|1⟩ - β|0⟩ → β|0⟩ - α|1⟩
        }
        return states[bell_outcome]
    
    # 测试所有修正操作
    alpha, beta = 0.6, 0.8
    target_state = np.array([alpha, beta])
    
    for bell_outcome in range(4):
        # 获取测量后的状态
        charlie_state = get_charlie_state_after_measurement(bell_outcome, alpha, beta)
        
        # 应用修正操作
        correction = get_correction_operator(bell_outcome)
        corrected_state = correction @ charlie_state
        
        # 验证修正后状态等于目标状态
        assert np.allclose(corrected_state, target_state), \
            f"Correction for outcome {bell_outcome} should restore target state"
        
        # 验证修正算符的幺正性
        assert np.allclose(correction @ correction.conj().T, I), \
            f"Correction operator {bell_outcome} should be unitary"
        
        # 验证修正算符的Hermitian性（对于Pauli算符）
        assert np.allclose(correction, correction.conj().T), \
            f"Pauli operator {bell_outcome} should be Hermitian"
    
    return True

检查点5：隐形传态保真度验证​

def verify_teleportation_fidelity():
    """验证隐形传态保真度"""
    import numpy as np
    
    def teleportation_fidelity(original_state, final_state):
        """计算传输保真度"""
        overlap = np.vdot(original_state, final_state)
        return abs(overlap)**2
    
    def simulate_teleportation(original_state):
        """模拟完整的隐形传态过程"""
        # 假设所有测量结果等概率出现
        bell_outcomes = [0, 1, 2, 3]
        corrections = [
            np.eye(2),                                    # I
            np.array([[1, 0], [0, -1]]),                 # σz
            np.array([[0, 1], [1, 0]]),                  # σx  
            np.array([[0, -1j], [1j, 0]])                # σy
        ]
        
        # 模拟每种可能的测量结果
        fidelities = []
        for outcome in bell_outcomes:
            # 根据Bell测量结果确定Charlie的初始状态
            alpha, beta = original_state[0], original_state[1]
            if outcome == 0:
                charlie_state = np.array([alpha, beta])
            elif outcome == 1:  
                charlie_state = np.array([alpha, -beta])
            elif outcome == 2:
                charlie_state = np.array([beta, alpha])
            else:  # outcome == 3
                charlie_state = np.array([beta, -alpha])
            
            # 应用修正操作
            final_state = corrections[outcome] @ charlie_state
            
            # 计算保真度
            fidelity = teleportation_fidelity(original_state, final_state)
            fidelities.append(fidelity)
        
        return fidelities
    
    # 测试不同的输入态
    test_states = [
        np.array([1, 0]),           # |0⟩
        np.array([0, 1]),           # |1⟩  
        np.array([1, 1])/np.sqrt(2), # |+⟩
        np.array([1, -1])/np.sqrt(2), # |-⟩
        np.array([0.6, 0.8])        # 一般态
    ]
    
    for original_state in test_states:
        fidelities = simulate_teleportation(original_state)
        
        # 验证每种情况都达到完美保真度
        for i, fidelity in enumerate(fidelities):
            assert abs(fidelity - 1.0) < 1e-10, \
                f"Teleportation fidelity should be 1.0 for outcome {i}, got {fidelity}"
        
        # 验证平均保真度
        avg_fidelity = np.mean(fidelities)
        assert abs(avg_fidelity - 1.0) < 1e-10, \
            "Average teleportation fidelity should be 1.0"
    
    # 验证no-cloning：原态被破坏
    # 这通过Bell测量的投影性质自动保证
    
    return True

自指完备性的体现​

推论 T3-4.1（信息传输的自指约束）​

SelfReferentialTeleportationConstraint : Prop ≡
  QuantumTeleportationRealization →
    ∀ψ : StateVector .
      InformationContent(ψ) = InformationContent(TeleportedState(ψ)) ∧
      NoCloning(ψ) ∧ ClassicalChannelRequired(ψ)

where
  InformationContent : Quantum information measure
  NoCloning : Original state destroyed during process  
  ClassicalChannelRequired : 2 classical bits must be transmitted

证明​

Proof of self-referential constraints:
  1. Information conservation: Quantum information perfectly preserved
  2. No-cloning enforcement: Bell measurement destroys original
  3. Classical communication: Required to complete protocol
  4. Causal structure: Respects relativistic constraints
  5. Self-reference maintained: System describes its own information transfer ∎

形式化验证状态​

  • 定理语法正确
  • 初始态构造完整
  • Bell测量协议形式化
  • 经典信息传输约束
  • 幺正修正操作验证
  • 传输保真度证明
  • 最小完备