Due to the superficial symmetry set up, it can be hard to see exactly what's wrong with the argument, because to articulate the problem here requires some depth of political theory. Essentially, the argument slips in an implicit assumption that the State and the individual have identical moral restrictions on all categories of actions they may perform. But a fundamental idea of government in the first place is that the State may do certain things in a judicial capacity that are not generally permitted to individuals, such as deprivation of life , liberty, or property. This must of course be done under due process of law, by no means do I wish to imply any court action must be correct. But as a moral principle, judicial actions are simply accorded a different status, and even people trying for the above equivalence recognize this difference in other areas where one could attempt similarly facile comparisons.
Murder is an unlawful taking of a human life, execution a lawful one.
They are as similar and different as a theft and a fine, or kidnapping and prison terms. A great deal has been written on this subject. I want to step back a bit from the minutiae. I think it safe to point out that, given the complexity of studies that must be mustered by proponents in favor of the point, that if there is any such deterrent effect to be found, it is so small that extensive statistical analysis is necessary to detect it at all. Proponents sometimes talk as if the conjectured deterrence were a weighty factor, instead of a highly theoretical effect of miniscule influence if it even exists in the first place.