My interest in this is more the aspect of whether the ancient civilizations knew of the existence of each other and where would be the limits of the ancient "known" world. Although it is clear that the Silk Road linked China to Rome, I was curious why Ancient Greece is never mentioned, since it is so close to Rome.
You run into the problem of the ambiguity of your terms once again. What exactly to you mean by "known" to each other? Did anyone
in China know there were an extensive empire (the Roman Empire) far to the west? That seems likely. Did anyone in Rome know there was an extensive empire far to the east? That also seems likely. Did your average Pleb in the street know that? Far less likely. Did your average Chinese peasant know that? Far less likely.
As for Greece in the period of Alexander III of Macedon--it was probably fairly widely known that there were societies to the east that produced very valuable consumer goods. The Greeks, were, after all, successful merchants and traders. Whether or not they would have been able to distinguish, however, between India, which in addition to the gem stones produced there was also an entrpôt for spices from the islands of the east, and for the silks and bronzes of China--and China, the source of those silks, is far more doubtful. They probably only had a hazy idea of distant lands with valuable trade goods. I doubt if the Chinese, even those with arcane knowledge of the world, knew much about the Greeks, if they even knew of them at all. I also suspect they would not have cared.
The most educated and well-informed people in China, as a class, were the Mandarins. They were a class, and were for more than 3000 years, having established themselves as a power behind many thrones because of their literacy more than a thousand years before the first true Chinese empire was founded in about 200 BCE. However, quixotically, it was the policy of the Mandarins themselves to exclude foreigners and to prohibit trade with foreigners and travel to foreign lands. So the very people who were most likely to obtain information about foreign lands and peoples were responsible for the tradition of turning inward and ignoring the outside world. When, in the history of any particular dynasty, the members of the dynastic family gave up governance, and turned policy and governance over to Mandarins, foreign travel and foreign trade were banned--those who traded with foreigners in such times were considered criminal.
Mandarins may have known about Greece, but it is doubtful, because Greece was not even important in it's own world until a few centuries before Rome conquered them. A few Mandarins may have known about Rome, but it would have been specialist knowledge, and likely of little or no interest to other Mandarins. Other Chinese were, even including members of the ruling dynastic clan, very likely illiterate. It is possible that literacy was more common among Greeks (because, once again, so many were involved in commerce) than had ever been true in any civilization, but the majority of the population were bond servants or slaves, who likely weren't literate. Even for the literate, accounts of "the mysterious East" would have been few and hazy, and likely very unreliable. The same applies to the Roman Empire, with 80% or more of its population bondman or -woman or slave, and literacy no more common. The Romans had a similar, if less severe, attitude that nothing and no one outside their empire mattered as that which pertained among the Mandarins.
Probably some people in each society knew of the other (except perhaps that the Chinese were unaware of or indifferent to the Greeks), but is doubtful if anyone on either side of Eurasia had accurate information, or very much information at all.