Information theoretical lower limit on the size of public and private keys
The private key can be reduced to $k$-bit in any asymmetric cryptosystem (encryption or signature) with a resistance to brute force of $O(2^k)$ steps (on a conventional, non-quantum computer). Proof sketch: replace the RNG used for key generation by a PRNG with at least $k$-bit security and seeded with a $k$-bit key. And that bound can't be much improved since there must be at least $O(2^k)$ private keys to avoid a brute force search. Note: that does not apply to already generated private keys.

On a quantum computer, that goes reducing private key to $2k$-bit for $O(2^k)$-something quantum resistance. Argument: a symmetric cryptosystem secure to $O(4^k)$ classical steps is widely believed to require at least $O(2^k)$-something (time or/and qubits) on a quantum computer.



The public key must be at least about $k$-bit in any asymmetric cryptosystem (encryption or signature) with a resistance $O(2^k)$ to brute force (on a conventional, non-quantum computer). Argument: we could use the above compressed form of the private key, and there can't be much less public keys than there are private keys, since that would allow finding by enumeration with effort $O(2^k)$ a private key matching a given public key.

But there is doubt that a $k$-bit public key is achievable. All known asymmetric cryptosystems actually have their public key $2k$-bit or more, for security as above. $2k$-bit is achieved by schemes based on the Discrete Logarithm in an Elliptic Curve group, and that's a plausible lower bound for the public key size. I vaguely remember having seen it conjectured at sci.crypt(.fr?), but without a convincing argument; and I fail to dig it up.