Much research is required on how you "prove" that quantum encryption schemes are secure. Cryptographers have developed many ways of proving that new schemes are secure. If you attend a cryptography course it won't be long before you are introduced to the concept of semantic security and the ubiquitous "game" where you an attack attempts to use plain text and cipher text to break the scheme.
Before proceeding it worth a very brief detour to clear up a common misunderstanding: the threat from quantum computers to public key encryption is not the same as quantum encryption. For an introduction to early quantum encryption (quantum key distribution) you can start here. Also "post quantum encryption" is simply those schemes being developed that are resistant to the threat posed by quantum computers.
The concept of semantic security first emerged in 1982. It is a bit cumbersome which is why it was shown only two years later (by the same researchers) that semantic security was essentially the same as another concept called "ciphertext indistinguishability". It is a simple but powerful concept where an attacker cannot distinguish between two separate ciphertexts to determine which contains each of two messages. This is a much more intuitive means by which the adversary game can be run and it is considered a fundamental requirement if an encryption scheme is to be considered provably secure.
