  In this work, we reveal a new type of impossibility discovered in our recent
research which forbids comparing the closeness of multiple unknown quantum
states with any non-trivial threshold in a perfect or an unambiguous way. This
impossibility is distinct from the existing impossibilities in that it is a
"collective" impossibility on multiple quantum states while most other "no-go"
theorems concern with only one single state each time, i.e., it is an
impossibility on a non-local quantum operation. This novel impossibility may
provide a new insight into the nature of quantum mechanics and it implies more
limitations on quantum information tasks than the existing "no-go" theorems.
