I Solving Dirac Algebra Arithmetic

[43arcsec]

I have a question regarding the Dirac notation arithmetic. Below is a measurement of a general 2 qubit state with the measurement operator M=|0><0| ⊗ I , where I is the identity operator. To go from equation (2) to equation (3), I've been converting all the Dirac notation to matrices and column vectors and then carrying out the arithmetic, but this is very cumbersome. Is there a more simple way to go through the arithmetic with the Dirac notation to directly solve (2) for (3) instead of first converting? Thanks for looking.

Prob = <ψ|M† M |ψ > (1)
= (α ∗ <00| + β ∗ <01| + γ ∗ <10| + δ ∗ <11|) (|0><0| ⊗ I) (α |00>+ β |01>+ γ |10>+ δ |11>) (2)
= (α ∗ <00| + β ∗ <01|) (α |00>+ β |01>) (3)
= |α|^2 + |β|^2 (4)

 

[Orodruin]

$M$ is clearly a projection onto the space spanned by states on the form $|0x\rangle = |0\rangle \otimes |x\rangle$. As such, it should be clear that
$$
|\phi\rangle = M|\psi\rangle = \alpha |00\rangle + \beta |01\rangle
$$
and therefore $\langle \phi| = \langle \psi| M^\dagger = \alpha^* \langle 00| + \beta^* \langle 01|$, leading directly to your end result.