PhysicsLogic / Local Quantum Dynamics

Local Operations and Measured Influence

A small commutator controls a readout change; it does not make that change exactly zero.

Definitions and assumptions

Scope and physical limitations

The operational comparison applies a unitary U to the initial vector, keeps the Hamiltonian and later observable fixed, and compares their actual expectations. A normalized vector supplies positivity and a contracting readout functional. An independent two-qubit model has H = |11><11|, initial state |0,+>, a first-site bit flip, and a second-site Y readout. Its expectation changes from zero to -sin(t). With unit separation and time in inverse interaction-energy units, every positive candidate velocity and tolerance admit a positive time before that candidate front arrives with a nonzero but smaller-than-tolerance readout change and commutator norm. These are vector-state expectations, not guaranteed individual outcomes or detectability at a fixed sample budget. This is not an infinite-lattice Lieb-Robinson theorem, a relativistic spacelike claim, or a protocol exploiting postselection.

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Examples and distinctions

Constructed examples of this model; each answer includes the Lean statement.

Exact root statement

‖initial‖ = 1 ∧
  IsSelfAdjoint readout ∧
    PhysicsLogic.Framework.QuantumCommutator.commutatorNorm dynamics readout (↑operation) 0 = 0 ∧
      ∀ (epsilon velocity : ℝ),
        0 < epsilon →
          0 < velocity →
            ∃ time,
              0 < time ∧
                velocity * time < 1 ∧
                  PhysicsLogic.Framework.QuantumCommutator.commutatorNorm dynamics readout (↑operation) time < epsilon ∧
                    after time ≠ before time ∧ ‖after time - before time‖ < epsilon
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