Can a No-Knowledge Proof exist?

A Knowledge Proof is a mechanism by which a ‘prover’ entity can show to a ‘verifier’ entity that it possesses some certain information without revealing what that specific information is (within some arbitrarily small tolerance of certainty).

Is it possible for a ‘prover’ to show to a ‘verifier’ that it does not possess some certain information (without the ‘prover’ being computationally limited, etc.)?