  We introduce a quantum convolution and a conceptual framework to study states
in discrete-variable (DV) quantum systems. All our results suggest that
stabilizer states play a role in DV quantum systems similar to the role
Gaussian states play in continuous-variable systems; hence we suggest the name
''discrete quantum Gaussians'' for stabilizer states. For example, we prove
that the convolution of two stabilizer states is another stabilizer state, and
that stabilizer states extremize both quantum entropy and Fisher information.
We establish a ''maximal entropy principle,'' a ''second law of thermodynamics
for quantum convolution,'' and a quantum central limit theorem (QCLT). The
latter is based on iterating the convolution of a zero-mean quantum state,
which we prove converges to a stabilizer state. We bound the exponential rate
of convergence of the QCLT by the ''magic gap,'' defined by the support of the
characteristic function of the state. We elaborate our general results with a
discussion of some examples, as well as extending many of them to quantum
channels.
