What would be the less bad synonym of “evidence” : clue, proof, observation, sign (that’s basically my best candidates, translated back in english). I dislike all of them, but that’s the best candidates I found, translated back in english. (substitute evidence in all the test sentences abole, and you will understand my problem. “Clue for a given theory…” is somewhat good, but “conservation of expected clue” less so…)

Of those four, I like "clue" the most. As Lumifer says, the word "proof" in English arguably connotes evidence supporting something; "sign" might have a similar problem; and "observation" feels a bit too vague to me, since an observation may be irrelevant to a hypothesis and hence not evidence at all.

The singular "clue" doesn't read well to me in the phrase "conservation of expected clue", but I think pluralizing it may help ("conservation of expected clues"). It might be feasible to invent a new word meaning something like "clueness", which might align better with the technical meaning of "evidence".

That said, if I examine how "preuve" is actually translated from French to English in official documents, the French "preuve" does sometimes seem to mean "evidence" in pretty much the English sense. So maybe "preuve" doesn't have the potential connotation (of evidence in support of a hypothesis) that Lumifer worries about.

(and don’t get me started on “entangled with”, I think I will lose much hair trying to find an acceptable translation for that one. French sucks.)

Perhaps "intriqué avec"?

(It occurred to me that French quantum physicists must have had to deal with the phrase "entangled with" for a long time, so one could simply borrow whatever French translation those physicists use.

I went to English Wikipedia's "Quantum entanglement" entry to look at the sidebar's list of alternative languages. It links to the French entry "Intrication quantique", though that title isn't the answer, because "intrication" is a noun, not an adjectival phrase. However, the entry's second sentence mentions (in bold, helpfully) "état intriqué", which certainly looks like "entangled state", and when I Google the phrase "entangled with" along with "intrication" & "quantique", I see snippets of French like "intriqués avec" and "états quantiques intriqués". Googling "intriqué avec" confirms that the phrase is used in French discussions of quantum mechanics in contexts where it seems to mean "entangled with".)