Quantum Computing: Bra and ket (p. 187)

Community PragProg Customers
1

On page 187 in section “Can the Quantum Gate Matrix Be Anything?”, psi-dagger is defined as <psi^dagger| = (w0* w1*).

As per my understanding, the conjugate transpose of a bra is actually the ket, meaning <psi^dagger| = |psi> with |psi> being a column vector with entries w0 and w1 (not w0* and w1*). Am I missing something here?

(Also quoting Wikipedia here, section “Bras and kets as row and column vectors” in https://en.wikipedia.org/wiki/Bra–ket_notation: “The conjugate transpose (also called Hermitian conjugate) of a bra is the corresponding ket and vice versa”)

Thank you very much for any clarification on this topic!

2

You are right that the bra and kets are conjugate transpose (or Hermitian conjugates) of each other. So, if |psi> is a column vector whose elements are w0 and w1, then <psi^dagger| is the row vector whose elements are w0* and w1* as shown on the bottom of page 187. That is, |psi>^dagger = <psi^dagger|.
The math notation can sometimes seem confusing. So, it’s best to understand the intent of what’s going on and then the notation will follow. Also, using actual numbers instead
of letters may help to clarify.