Exercise 4.46

Consider a density matrix \(\rho\) for \(n-\)qubits system, the dimension of \(\rho\) should be \(2^n\times 2^n\), so there is totally \(2^n\times 2^n = 2^{2n} = 4^n\) complex elements in matrix \(\rho\). Since we need two real numbers to describe a complex number, so there should be \(2\times 4^n\). Note also that for density matrix, we have \({\rm Tr}\rho = 1\). With the constraint we could reduce the number of independent variable to \(2\times 4^n - 2\) since \({\rm Tr}\rho = 1\) means that (1) the sum of real components of diagonal elements is \(1\), and (2) the sum of imaginary components of diagonal elements is \(0\).