Perfectly secret cipher can leak about the key?
mikeazo's answer clearly covers what the question asks. However, I want to go further and answer this:

How much information does a perfectly secure cipher leak about the key?

Exactly all the excessive information of the key that does not help in finding anything about the message.

In simple words: if the key length is $k$ bits and the message is $m$ bits long then in case of a perfectly secure cipher by knowing the ciphertext you know exactly $(k-m)$ bits of the key.

Formally: By knowing a ciphertext which is ciphered using a perfectly secure cipher the uncertainty about the key is exactly equal to the uncertainty about the message.

Or using entropy function:

$$H(K|C)=H(M)$$

Where $H(K)$ is the entropy of the key, $H(C)$ is the entropy of the ciphertext, and $H(M)$ is the entropy of the message.

The equation above can be deduced from the following equations:

$H(M)=H(M|C)$, $H(M|C,K)=0$