Formula for a single unit gate acting on a lexicographically represented state

Question

I've been trying to find in textbooks a discussion on the action of arbitrary single qubit gates on a lexicographic state. That is, given an operator $G_{l}= 1\otimes 1...\otimes G \otimes ... \otimes 1$ which acts on the $l$-th qubit and a state

$|\psi\rangle = \sum \alpha_{J} |J\rangle$

where $ 0 \leq J < 2^N $, $N$ is the number of qubits, $J \in \mathbb{Z}$, $|J\rangle$ is standard product state representing $J$ in binary, and the sum runs through the range of J, what can be said about:

$G_{l} |\psi \rangle$

in terms of the single qubit operator $G$ and how do the original $\alpha_{J}$ map.

It shouldn't be too difficult to derive some formulas (there will likely be some transitions between $J$ and it's bit flipped int) but I figured since this seems like a basic thing it would have been discussed somewhere.

Edit: Using the notation of the answer below it's can be shown that if:

$G = g_{00} |0\rangle \langle 0| + g_{01} |0\rangle \langle 1| + g_{10} |1\rangle \langle 0| + g_{11} |1\rangle \langle 1|$

then the coefficent mappings are:

$\alpha'_{J} = \alpha_J g_{00} + \alpha_{f_l(J)} g_{01} $ if $J_{l} = 0$

$\alpha'_{J} = \alpha_J g_{11} + \alpha_{f_l(J)} g_{10} $ if $J_{l} = 1$

where $f_l(J)$ flips the $l$-th bit of $J$.

Answer 1

That $G_{l}$ action is linear, so you can write $$ G_{l} |\psi \rangle = G_{l} \sum \alpha_{J} |J\rangle = \sum \alpha_{J} G_{l} |J\rangle. $$

To calculate $G_{l} |J\rangle$ you can use tensor mixed-product property:

$$ G_{l} |J\rangle = \big(1\otimes ...\otimes G \otimes ... \otimes 1\big) ~\cdot~ \big(|J_1\rangle \otimes ... \otimes |J_l\rangle \otimes ... \otimes |J_N\rangle\big) = $$ $$ = |J_1\rangle \otimes ... \otimes G|J_l\rangle \otimes ... \otimes |J_N\rangle, $$ here $J_i$ is the $i$-th bit of $J$.

You can simplify the total sum by grouping terms in two groups based on the value of $J_l$: $$ G_{l} |\psi \rangle = \sum_{J_l=0} \alpha_{J} |J_1... J_{l-1}\rangle \otimes G|0\rangle \otimes |J_{l+1} ...J_n\rangle + $$ $$ + \sum_{J_l=1} \alpha_{J} |J_1... J_{l-1}\rangle \otimes G|1\rangle \otimes |J_{l+1} ...J_n\rangle $$