  Continuous variable quantum key distribution with discrete modulation has the
potential to provide information-theoretic security using widely available
optical elements and existing telecom infrastructure. While their
implementation is significantly simpler than that for protocols based on
Gaussian modulation, proving their finite-size security against coherent
attacks poses a challenge. In this work we prove finite-size security against
coherent attacks for a discrete-modulated quantum key distribution protocol
involving four coherent states and heterodyne detection. To do so, and contrary
to most of the existing schemes, we first discretize all the continuous
variables generated during the protocol. This allows us to use the entropy
accumulation theorem, a tool that has previously been used in the setting of
discrete variables, to construct the finite-size security proof. We then
compute the corresponding finite-key rates through semi-definite programming
and under a photon-number cutoff. Our analysis provides asymptotic rates in the
range of $0.1-10^{-4}$ bits per round for distances up to hundred kilometres,
while in the finite case and for realistic parameters, we get of the order of
$10$ Gbits of secret key after $n\sim10^{11}$ rounds and distances of few tens
of kilometres.
