My daughter has one Chinese parent, and one parent from an other country (A). She was born outside of China. At birth she acquired citizenship of country A, and she has an A passport. How can she travel with us to China (P.R.C)?
China will not issue a visa to a child with a Chinese national parent, as the child is considered to be a Chinese national. China does not allow dual citizenship, and will not issue a full Chinese passport either.
You should apply for a Chinese Travel Document, which looks similar to a passport and will allow the child to enter and exit China. It is valid for two years. After that you may be able to renew it, or China may demand that one of the nationalities is renounced. You can apply for it in the same places where Chinese nationals abroad apply to renew their passports. She will still need her other passport to enter countries other than China.
If the parents (one Chinese, one not) settle permanently outside China then the child does not acquire Chinese nationality via 'jus sanguinis', so long as they acquire country A's nationality at birth (see this page). So, in this case the child can obtain a visa in the usual way. (However, the situation may be different if the parents have not settled permanently outside China.)
My personal experience: my wife is Chinese, I'm British and we live in the U.K. Our son was born in the U.K., acquired British nationality, and was able to apply for a Chinese visa prior to his first trip to China.
The first issue is whether the child has Chinese (PRC) citizenship. According to Article 5 of the PRC Nationality Law, a child born abroad to at least one Chinese-citizen parent will automatically have Chinese nationality at birth, unless at least one parent was a Chinese citizen who has "settled abroad", and the child has a foreign nationality at birth. In this case, we know that one parent was a Chinese citizen at the time of the child's birth, and the child has a foreign nationality at birth. So it hinges on whether the Chinese-citizen parent has "settled abroad" at the time of the child's birth. The Chinese government interprets "settled abroad" to mean having permanent residency abroad (e.g. a green card). If the Chinese-citizen parent had foreign permanent residency at the time of the child's birth, then the child does not have Chinese citizenship, and would need to get a Chinese visa to visit China.
If the Chinese-citizen parent did not have foreign permanent residency at the time of the child's birth, then the child is a Chinese citizen, and needs to use a Chinese-issued travel document, not a foreign passport, to enter and leave China, as China does not recognize the foreign nationalities of its nationals. The current practice is for Chinese consulates to issue "PRC Travel Documents" to dual-national Chinese children outside China. This is a travel document in booklet form just like a passport, and says on its Notes page "The bearer of this travel document is a citizen of the People's Republic of China." It is valid for 2 years, and can be used to enter and exit China during its period of validity. A new PRC Travel Document can be applied for at any PRC consulate outside China when the old one expires, as many times as needed (until the child is 18, according to reports). If the PRC Travel Document expires or is lost while in China and the person needs to exit China, they would apply for an Entry/Exit Permit to use to exit China, and then get a PRC Travel Document (if desired) at a PRC consulate abroad after they get abroad.
(It is unclear why PRC consulates do not just issue regular PRC passports to these dual-national children; they do issue regular PRC passports to single-national children. Both PRC Travel Documents and PRC passports identify the bearer as a PRC citizen, so there is no difference in that sense. I can see nothing in the PRC passport law that says issuance is affected by the possession of other nationalities. It seems to be just an unofficial unexplained policy. In practice though, there is a difference in usage between the two, as Chinese exit controls will let someone with a PRC Travel Document and a foreign passport leave China with no problems, but they will not let someone with PRC and foreign passports leave (and you would usually need to show the foreign passport as they want to see evidence you can enter the destination). Maybe this is a way of distinguishing "real" dual nationals, who would have a PRC Travel Document, from people who have already lost PRC citizenship by naturalizing abroad and are pretending to be PRC citizens, who would have a PRC passport.)
Complicating the situation a bit is that starting in 2015, there have been anecdotal reports that a few (but not all) Chinese consulates in the US have been refusing Chinese Travel Documents to dual-national children who have lived abroad for over a certain amount of time (like 18 months). There have been conflicting reports of this, sometimes describing the situation shifting back and forth by month at the same consulate. Because the situation is somewhat unclear, if your child is a dual national, you might want to fill out both a visa application and a Chinese Travel Document application just in case.
Me and wife are both Chinese with permanent residence in the UK but not UK citizen. Our daughter was born in Apr 2015 in the UK and she isn't allowed to be a Chinese national. So she is British and travels with Chinese visa on her UK passport.
However, this is a recent change. Before Apr 2015, children born in the UK to Chinese parents with permanent residence (but not naturalised) are allowed (by the Chinese government) to be Chinese (and if you so choose, they can't be British).
So the situation is not clear. It's best to call the embassy to find out.