It didn't entirely fall apart. There is at least some evidence to support the idea that tenuous diplomatic relations between Byzantium and China were maintained. In a number of Chinese documents dating from the 6th - 10th Centuries, information pops up regarding the successor of Da Qin (the Chinese name for the Roman Empire, meaning "Great China"), which was renamed Fu-lin, meaning "wealthy successor". This information is quite detailed and may have been gathered by traders or envoys to Byzantium from China, or through intermediaries in the Muslim world. From these accounts, it is clear that the Chinese knew who they were dealing with, even if their understanding of the material was somewhat open to interpretation. Fordham University has posted up a lengthy set of translated passages regarding this relationship, and I will recreate one of the most telling passages here, dating from the Chiu-t'ang-shu of the 10th Century:

The country of Fu-lin [Byzantium], also called Ta-ts'in [Roman Syria], lies above the western sea [Indian Ocean]. In the southeast it borders on Po-si [Persia]. Its territory amounts to over 10,000 li. Of cities there are four hundred. Inhabited places are close together. The eaves, pillars, and window-bars of their palaces are frequently made with crystal and opaque glass. There are twelve honorable ministers who conjointly regulate government matters. They ordinarily let a man take a bag and follow the king's carriage. When the people have a complaint they throw a written statement into the bag. When the king comes back to the palace he decides between right and wrong. Their kings are not permanent rulers, but they select men of merit. If an extraordinary calamity visits the country, or if wind and rain come at the wrong time, he is deposed and another man is put in his stead. The king's cap is shaped like a bird raising its wings; its trimmings are beset with precious pearls; he wears silk-embroidered clothing, without a lapel in front. He sits on a throne with golden ornaments. He has a bird like a goose; its feathers are green, and it always sits on a cushion by the side of the king. Whenever anything poisonous has been put into the king's meals, the bird will crow. The walls of their capital are built of granite, and are of enormous height [the Theodosian triple walls]. The city[Constantinople] contains in all over 100,000 households [some 500,000 to 600,000 inhabitants]. In the south it faces the great sea. In the east of the city there is a large gate; its height is over twenty chang [over 235 feet]; it is beset with yellow gold [bronze] from top to bottom, and shines at a distance of several li. Coming from outside to the royal residence there are three large gates beset with all kinds of rare and precious stones. On the upper floor of the second gate they have suspended a large golden scale, twelve golden balls are suspended from the scale-stick by which the twelve hours of the day are shown. A human figure has been made all of gold of the size of a man standing upright, on whose side, whenever an hour has come, one of the golden balls will drop, the dingling sound of which makes known the divisions of the day without the slightest mistake [a clepsydra]. In the palaces, pillars are made of se-se [lapis lazuli], the floors of yellow gold [probably bronze], the leaves of folding doors of ivory, beams of fragrant wood. They have no tiles, but powdered plaster is rammed down into a floor above the house. This floor is perfectly firm and of glossy appearance like jade-stone. When, during the height of summer, the inhabitants are oppressed by heat, they lead water up and make it flow over the platform, spreading it all over the roof by a secret contrivance so that one sees and knows not how it is done, but simply hears the noise of a well on the roof; suddenly you see streams of water rushing down from the four eaves like a cataract; the draught caused thereby produces a cooling wind, which is due to this skilful contrivance [a common device in the Near East].

It is customary for men to have their hair cut and wear robes leaving the right arm bare. Women have no lapels on their dresses, they wear turbans of embroidered cloth. The possession of a great fortune confers superior rank on its owner. There are lambs which grow in the ground; the inhabitants wait till they are about to sprout, and then screen them off by building walls to prevent the beasts which are at large outside from eating them up. The navel of these lambs is connected with the ground; when it is forcibly cut the animal will die, but after the people have fixed the buds themselves' they frighten them by the steps of horses or the beating of drums, when the lambs will yield a sound of alarm, and the navel will be detached, and then the animal may be separated from the water-plant. The inhabitants are in the habit of cutting their hair and wearing embroidered clothing; they drive in small carriages with white canopies; when going in or out they beat drums and hoist flags, banners, and pennants. The country contains much gold, silver, and rare gems. There is the Yeh-kuang-pi [the jewel that shines at night]; the ming-yüeh-chu [the moon-shine pearl]; the hsieh-chi-hsi [the chicken-frightening rhinoceros stone]; large conches; the che-ch'u [mother-of-pearl], carnelian stones; the k'ung-ts'ui [Jadeite]; corals; amber; and all the valuable curiosities of the West are exported from this country.

Interestingly, the Byzantine side of the story is much less documented. We know that several Roman envoys were sent to China during Late Antiquity, including delegates from the courts of the Emperors Carus or Diocletian. Contact with Byzantine traders continued to occur throughout the Medieval period, although reporting on the Byzantine side is essentially non-existent. The reasons for the lack of evidence may simply be due to untranslated or lost material, or perhaps these events were not seen as noteworthy enough of mention to Byzantine writers. Regardless, at some point there were delegations sent (although they are strangely not reported as far as I am aware in Byzantine sources), but as is reported in the following passage, the Chinese were apparently visited by someone in the 11th Century from the Court of Michael VII:

The country of Fu-lin [Byzantium]. South-east of it you go to Mei-lu-ku [Kilikia ("Cilicia")]; north you go to the sea [Black Sea]; both forty days' journey; west you go to the sea [Mediterranean], thirty days' journey; in the east, starting from western Ta-shih, you come to Yu-tien [Khoten], Hui-ho and Ch'ing-t'ang, and finally reach Zhongguo [China]. They have during former dynasties not sent tribute to our court. During the tenth month of the fourth year of the period Yuan-feng [November, 1081 C.E.], their king, Mieh-li-i-ling-kai-sa [Michael VII Parapinaces Caesar], first sent the ta-shou-ling [a high official] Ni-si-tu-ling-si-meng-p'an to offer as tribute saddled horses, sword-blades, and real pearls. He said: the climate of this country is very cold; houses there have no tiles; the products are gold, silver, pearls, western silk cloth, cows, sheep, horses' camels with single humps, pears, almonds, dates, pa-lan [a kind of date], millet, and wheat. They make wine from grapes; their musical instruments are the lute, the hu-ch'in [the "tea-pot-shaped lute"]; the hsiao-pi-li[a kind of flageolet]; and the p'ien-ku ["side drum"]. The king dresses in red and yellow robes, and wears a turban of silken cloth interwoven with gold thread. In the third month every year he goes to the Temple of Fou-shih [ "Temple of Buddha", here meaning either Muhammed or Christ; in other places the Qu'ran is described as Fou-ching".], to sit on a red couch [palanquin?] which he gets the people to lift. His honored servants [ministers, courtiers, priests?] are dressed like the king, but wear blue, green, purple, white mottled, red, yellow, or brown stuff, wear turbans and ride on horseback. The towns and the country districts are each under the jurisdiction of a shou-ling [chief, sheik?]. Twice a year, during the summer and autumn, they must offer money and cloth [chin-ku-po]. In their criminal decisions they distinguish between great and small offences. Light offences are punished by several tens of blows with the bamboo; heavy offences with up to 200 blows; capital punishment is administered by putting the culprit into a feather bag which is thrown into the sea. They are not bent on making war to neighboring countries, and in the case of small difficulties try to settle matters by correspondence; but when important interests are at stake they will also send out an army. They cast gold and silver coins' without holes, however; on the pile they cut the words Mi-le-fou, which is a king's name. The people are forbidden to counterfeit the coin. During the sixth year of Yuan-yu [1091 C.E.] they sent two embassies, and their king was presented, by imperial order, with 200 pieces of cloth, pairs of white gold vases, and clothing with gold bound in a girdle.

Why the Byzantine Emperor would have decided to contact the Chinese court in this period is unknown to me, but it seems that there must have been at least some precedent during the height of Macedonian power that allowed it to occur. According to this Chinese source, Alexios I should have sent and received something in return, but this is not reported in any Byzantine sources as far as I am aware. Why this would have occurred seems to not line up with the events occurring in Byzantium around this time, but I would not completely rule it out. Anyways, the body of evidence is tenuous and the topic needs a lot more investigation on the whole, but it is certainly interesting and worthy of further study.