Independent schools best for social mobility

You can't get it online but The Economist covers a study by Cambridge University which has found that independent schools are better at promoting social mobility than non-selective state schools. The study showed that each type of school sent the same proportion of students from working class backgrounds to Cambridge, but at independent schools just one in a hundred pupils come from working class backgrounds. They argue that this shows that these kind of schools are better at nurturing raw talent and therefore contribute more to social mobility than state schools. Why is that? More selection or a better environment? Would this be the case if the number of independent schools were allowed to increase? What it does help assert, I believe, is that the environment and general attitude in many state schools is dreadful. The challenge is to find a way to turn that round, and I can't help but think that an increase in the power of parents to choose schools would help, as it would focus the demands and attitude of both the parents & pupils and the school itself.