"Lamp-switch set-up number" of $n$ Answer

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"Lamp-switch set-up number" of $n$

Motivation. The following has a real-life (!) inspiration from a discussion about how to connect lamps and switches in an efficient way.

question. let $n\in \mathbb{N}$ be a positive integer and let $\{1,\ldots,n\}$ represent $n$ lamps, each of which is in exactly one of the states OFF or ON. let $E\subseteq {\cal P}(\{1,\ldots,n\})$. For every $e\in E$ we have on "$e$button"such that if that button is pressed, every element of $e$ switches its state (either from OFF to ON, or vice versa). We say that $E$ is state complete if the following condition holds:

If all lamps are OFF and if $k\in \{1,\ldots,n\}$ is given, there is a finite button sequence $e_1, \ldots, e_m \in E$ such that after all the buttons have been pressed, lamp $k$ is ON and all other lamps are OFF.

For example, $\big\{\{k\}: k\in \{1,\ldots,n\}\big\}$ is state-complete, and $\big\{\{1,\ldots,n\}\big\}$ is not state-complete for $n\geq 2$.

Given $n\in \mathbb{N}$what is the least cardinality that a state-complete set $E\subseteq {\cal P}(\{1,\ldots,n\})$ can have?

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