blogged about the Sampul Tapestry, which looks Greek, was probably made circa 200 BC, but was excavated in western China buried with one of the Tarim Basin mummies. It probably made its way there via Bactria, or through trade along the Silk Road.
There is plenty of evidence for trade between East and West in Antiquity, although interestingly those studying Asia concentrate on the overland routes, whilst those studying Rome tend to prefer to look at the sea routes since we have an ancient manual for reaching India by boat, the Periplus Maris Erythraei, that survives.
Plenty of evidence of trade with Rome has been uncovered in India, including pottery and Roman coins (alas a huge hoard was found a few years ago, but the Indian gentleman melted them down for their silver content).
JSTOR). We know that Pompeii was destroyed in AD 79, so this provides us with a terminus ante quem for the figurine to have been imported from the Indian subcontinent. Similar first century ivories were found at Begram in Afghanistan (here). Although we know of active sea trade with India under Nero, this figure is likely to have come in overland via the Central Asian route.
What makes the figure particularly interesting is that she shows how Greeks of Alexander the Great first took images of Aphrodite to Bactria, which influenced local Classicising images of Buddhist and Hindu figures - and then centuries later one such figure returned to the Roman West where it was excavated.
The Begram ivories were excavated a long time ago, so their stratification is disputed and they have been dated by scholars from the first to the third centuries AD, because of the scarcity of images in India. Because of the strong parallels to the Pompeii ivory, they can now be firmly assigned to the first century AD - unfortunately many of them were destroyed under the Taliban.
For an Indian who went to Eleusis under Augustus, and was buried there see: An Indian at Eleusis
Homer Dubs' theory of Romans settling in China, and their descendants living on, was disproven by DNA. This does not mean that the Romans and Chinese did not have contact - some Romans arrived in China in AD 161, claiming to be an embassy from the Emperor. Various groups of Chinese set off for Western Europe - one only made it as far as Antioch, convinced that such a great city must have been the Roman capital, another may have made it to Augustus' Rome. Most Roman-Chinese contact was indirect, and the Roman glass found in China and South-East Asia probably arrived there through traders and intermediaries.
Byzantine gold coins were found in tombs throughout China, as this map shows:
Lin Ying wrote a fascinating article about these, and the British Museum has more examples, such as the 'coin' below brought back by Aurel Stein from Astana, where many similar examples were found (source).
The only surviving map of the Roman world, the Tabula Peutingeriana copied by a Medieval Monk, of course ends in the East with a depiction of Indian and China.