0.9 Some useful identities

  • |a\rangle^\dagger = \langle a|
  • \langle a|^\dagger = |a\rangle
  • (\alpha|a\rangle+\beta|b\rangle)^\dagger = \alpha^\star\langle a|+\beta^\star\langle b|
  • (|a\rangle\langle b|)^\dagger = |b\rangle\langle a|
  • (AB)^\dagger=B^\dagger A^\dagger
  • (\alpha A+\beta B)^\dagger=\alpha^\star A^\dagger+\beta^\star B^\dagger
  • (A^\dagger)^\dagger=A
  • \operatorname{tr}(\alpha A+ \beta B) = \alpha \operatorname{tr}(A)+\beta\operatorname{tr}(B)
  • \operatorname{tr}|a\rangle\langle b| = \langle b|a\rangle
  • \operatorname{tr}(ABC) = \operatorname{tr}(CAB) = \operatorname{tr}(BCA)