I was just not understanding the need to create this system all the way to China, in the Roman Empire this system makes sense however communication with the Chinese Emperor doesn't give you any profit and wastes a lot of money.
If you can successfully create this system (which is not given), have it being reliable and it is not too costly then yes it would make sense, however I doubt that those three requirements are fulfilled.
I did not say it needed to be created as such. I was merely projecting a potential long term evolution, within a few centuries of the first lines being built.
Closer to home, Rome would have great use of fast communications of a Chappe telegraph network to improve border defense, which was notoriously difficult with thousands of kilometers of border to watch and defend. Being able to communicate all along its length within a day from report going up to orders flowing down would have allowed a much more efficient use of manpower, be it at theater or empire level. Structurally, it also allows a greater degree of control of the central government in Rome over the provinces, which is something any empire spanning thousands of kilometers is much needing.
The same applies to the Persian and Chinese empires.
As the technology spreads, and the knowledge of even the basics of such a communication network, it's pretty much guaranteed the Persians first would copy the Romans, and you'd get lines of relays reaching the far corners of the Persian empire in Khwarezm, the Caucasus and upper Mesopotamia down to the capitals of Ctesiphon and Susa.
As for China, besides the obvious need to administer a wide country, their control of the Tarim basin, fluctuating according to the fortunes of such or such dynasty, would be greatly made easier with such a line. The Chinese would thus get this technology from the Persians, with whom they had relations; technology, knowledge exchange was a thing back in the ancient times, along the Silk Road (
silkworms smuggled out of China by Byzantine spies in the 6th century, or paper through the Arabs in the 8th century), so this technology making its way from Persia to Luoyang would be plausible. The Han dynasty, or the Tang, would probably build a network of semaphores, and obviously one line to the Tarim basin.
All you are left between these three, Roman, Persian and Chinese networks, are short trunks without relays: over the Euphrates frontier between the Romans and the Persians, and over the Tian Shan mountains between Persian Khwarezm and Chinese Tarim. So far, I'm only speaking about organic growth of distinct semaphore networks, and the short gap there would be in between, gaps that would not be all to implausible to see closed.
Due to diplomatic and geographic proximities between Roman and Persian empires, it's pretty much a given a connection between Rome and Ctesiphon would happen, a "
red telephone" between Imperator and Shahanshah. Central Asia would take more time since it was remote and hard to control for both Chinese and Persians, but the existence of the semaphore line across Persia to Ctesiphon, facilitating Persian control in the region, or Chinese control in the Tarim, would help bring about such occasions.