The Case Against Capital Punishment

Despite being sacked by her predecessor, Priti Patel has returned to Boris Johnson’s cabinet as home secretary which led to a chorus of concerns regarding her past in human rights and a 2011 clip of her calling for capital punishment has resurfaced.
In 2011, while during a discussion around Capital Punishment on BBC's Question Time, Private Eye editor Ian Hislop said that the inaccuracy of sentencing in the UK would mean innocent people would be killed by the state and referenced several cases where people were incorrectly charged with murder only to be cleared years later.
In response, Patel argued capital punishment can act as a 'deterrent' to serious crime and you would need strong proof to convict, and then execute people which Hislop responded that the people he mentioned would be dead as they were thought guilty originally.
'The point is' Patel continued 'it’s about having deterrents' to which Hislop interrupted with 'It’s not a deterrent killing the wrong people'. 
According to Amnesty International, 'scientists agree, by an overwhelming majority, that the death penalty has no deterrent effect' although the greatest argument against that has to be that in the 53 countries that still have the death penalty, murders still happen.
I can't see how another death, in the case of Capital Punishment legalised murder by the state, make anything better because the tragic thing is that cutting short another life doesn´t bring back anybody killed or heal any of the pain, what it does do is fulfil the lust for revenge.
The logic that we must take their life because they took a life never made much sense to me, two wrongs don't make a right, i would much rather leave them sitting in their cell, slowly rotting and regretting everyday of their miserable incarcerated life as their naturally allotted time on this planet slowly ebbs away.