 This value \(2\sqrt{2}\) appearing in equations (19) and (21) is the maximum violation of the CHSH inequality for any quantum state, as shown by Tsirelson (Cirel’son 1980). This bound on quantum violations of the CHSH inequality is called the Tsirelson bound. As Gisin (1991) and Popescu and Rohrlich (1992) independently demonstrated, for any pure entangled quantum state of a pair of systems observables can be found yielding a violation of the CHSH inequality. Popescu and Rohrlich (1992) also show that the maximum amount of violation is achieved with a quantum state of maximum degree of entanglement. Incidentally, this is not true for mixed states; there are entangled mixed states not violating any Bell inequality (Werner 1989).