China does not recognise dual nationality
The Nationality Law is short and it is not ambiguous. There is no registration to make and no declaration to sign: naturalising elsewhere ends Chinese nationality by operation of law. That has consequences that surface years later, usually at a border.
What follows from it
Permanent residence, which is a different thing
China does issue a permanent residence card to foreign nationals, sometimes called the Chinese green card. It is residence, not nationality, and it does not conflict with holding a foreign passport. The family of a holder is eligible for a Q visa on the same footing as the family of a Chinese citizen.

What the law actually says
Two articles do the work. One provides that a Chinese national who has settled abroad and voluntarily acquires another nationality loses Chinese nationality.
The other provides that a child born abroad to Chinese parents who had themselves settled abroad, and who acquires a foreign nationality at birth, does not have Chinese nationality.
Everything else in this area follows from those two sentences, including why one American-born child of Chinese parents needs a visa and another needs a travel document.

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