MeasurementIrreversibility : Prop ≡
∀M : MeasurementProcess .
¬∃M⁻¹ : Process . M⁻¹ ∘ M = id_S
where
MeasurementProcess : Type ≡ (S × O) → (S' × O' × R)
id_S : S → S is identity function
Proof of information creation:
Let M be a measurement process, (s,o) initial state.
1. Before measurement:
- System in state s
- Observer doesn't know exact state
2. After measurement:
- Produces result r
- r contains information about s
- r must be recorded (self-ref completeness)
3. New information:
- r ∉ s (didn't exist before)
- s' = s ∪ {r, Desc(r)}
- Added at least 2 elements
Therefore measurement creates information ∎
Proof by contradiction:
Assume ∃M⁻¹ such that M⁻¹(s',o',r) = (s,o).
1. Information conservation requirement:
- M⁻¹ must "delete" record r
- In self-ref complete system, info cannot be deleted
- Deletion itself needs recording
2. Entropy monotonicity:
- By axiom: H(S_{t+1}) > H(S_t)
- Reverse requires: H(s) < H(s')
- But M⁻¹ would need H(s) = H(s')
- Violates entropy axiom
3. Causal paradox:
- Record r may have influenced evolution
- Other parts may have "seen" r
- Complete elimination needs time reversal
4. Self-reference paradox:
- M⁻¹ is system operation
- Executing M⁻¹ creates new record r'
- Need (M⁻¹)⁻¹ to eliminate r'
- Infinite regress
Contradiction. Therefore measurement irreversible ∎
Analysis:
1. Reversible parts:
- Some changes to measured object
- Unitary part of quantum operations
2. Irreversible parts:
- Existence of record r
- Observer state change
- Total entropy increase
3. Essential difference:
- Local reversibility ≠ Global reversibility
- Operation reversibility ≠ Information reversibility
def verify_information_creation(measurement, system, observer):
s_prime, o_prime, record = measurement.apply(system, observer)
# 验证记录是新创建的
assert record not in system.states
# 验证系统包含新信息
assert record in s_prime.states
assert f"desc_{record}" in s_prime.states
# 验证状态数增加
assert len(s_prime.states) > len(system.states)
return True