0.10 Some useful identities

Here is a summary of some particularly useful equalities concerning bras, kets, inner products, outer products, traces, and operators, that we will be using time and time again. In all of these, |a\rangle,|b\rangle\in\mathcal{H} are kets, A,B,C are operators on \mathcal{H}, and \alpha,\beta\in\mathbb{C} are scalars.

Dagger for bras and kets:

|a\rangle^\dagger = \langle a|
\langle a|^\dagger = |a\rangle
(|a\rangle\langle b|)^\dagger = |b\rangle\langle a|
(\alpha|a\rangle+\beta|b\rangle)^\dagger = \alpha^\star\langle a|+\beta^\star\langle b|

Dagger for operators:

(AB)^\dagger = B^\dagger A^\dagger
(A^\dagger)^\dagger = A
(\alpha A+\beta B)^\dagger = \alpha^\star A^\dagger+\beta^\star B^\dagger

Trace:

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