  This is a Bachelor's thesis on quantum key distribution in the non-asymptotic
regime submitted to the Leibniz Universit\"at Hannover, Germany. It deals with
the question how to generate a key to encrypt messages between two honest
parties that an eavesdropper does not gain any information about. The purpose
of this thesis is to give an overview over the current state of the prepare and
measure quantum key distribution protocol and its security proof with a
specific focus on the non-asymptotic case. Therefore, we present the different
steps of the protocol and retrace them with the help of the BB84-protocol.
Afterwards, the security proof of this protocol is introduced and in the end,
we have a closer look at the quantum capacity and calculate it for the
amplitude damping channel.
