Theoretical question with regard to weak password based KDF & high entropy input

I know this question is theoretical however I would like some thoughts from a security perceptive.

Take this thought:

1) I generate a high entropy, cryptographically secure string of bits (256-bits of entropy) 2) I use this entropy as a password fed into a weak PBKFD, like PBKDF1 3) I use the output as a key, should I expect this key provides 256-bit entropy?

Extra question, should I instead use a strong KDF, like PBKDF2, would I then be able to expect no drop in entropy?

PBKDF2 by definition should stretch entropy right? not decrease it.