Consider a $((3,5))$ pure state quantum secret sharing (QSS) scheme. For instance this paper: arXiv:quant-ph/9901025. If I divide any 5 shares to two sets then allways one of those two set are athorized meaning they have full information about the encoded state. Now let me discard one share and design a $((3,4))$ QSS scheme. In that case I have a mixed state QSS scheme. In this case a set with two shares is unauthorized. So if I divide the set of shares into two set each with two shares, non of these set of shares have any information about the encoded state. So my question is in this case where is the information? Whats espcial about pure states that somehow conservation of quantum information works for them?
This is because of the no-hiding theorem. If a quantum channel completely erases information, then its complementary channel should preserve all the information that the input state has. Therefore for any pure QSS scheme, the complementary set of any unauthorized set must be an authorized set. The vice-versa follows from the no-cloning theorem. The reason this doesn't apply to mixed QSS scheme is that it is actually possible to hide quantum information in multipartite quantum entanglement, as one can see from the possibility of $((n,2n-1))-$threshold pure QSS scheme. For example, in your example of $((3,4))-$threshold mixed QSS scheme, the complementary set for any unauthorized set is not the whole environment surrounding that unauthorized set. So the information flow is not wholely headed to the complementary set of the size 2, but to the purification system of the unauthorized set, which has the size of 3.