- 능동 변환과 수동 변환

1. 기저 변환 ↔ 벡터 및 연산자(행렬) 변환 #

$${|base\rangle} \to {U^\dagger|base\rangle}$$ $$\Leftrightarrow \quad |\psi\rangle \to U|\psi\rangle, A \to UAU^\dagger (|a\rangle \to U|a\rangle)$$

2. 기댓값 $\langle A \rangle$ 변환 #

$$\langle\psi|A|\psi\rangle \to \langle\psi’|A|\psi’\rangle \quad (|\psi’\rangle = U|\psi\rangle)$$

또는

$$\langle\psi|A|\psi\rangle \to \langle\psi|U^\dagger AU|\psi\rangle$$

따라서,

능동 변환: $|\psi\rangle \to U|\psi\rangle, \quad A \to A$

$\Leftrightarrow$

수동 변환: $|\psi\rangle \to |\psi\rangle, \quad A \to U^\dagger AU$