How many real numbers are required to describe density matrix for $n$ qubits?

(All of these coming from the topic of simulation of quantum systems) A density matrix $$\rho$$ Which describe state of $$n$$ qubits will have $$2^{n} \times 2^{n}$$ size. We have couple of conditions like

1. $$\mathrm{tr}(\rho) = 1$$
2. $$\rho$$ is positive.

Then in this case we only need to specify $$\dfrac{2^{n}(2^{n}-1)}{2}$$ for off-diagonal elements and for the diagonal ones we need $$2^{n}-1$$ Terms. So total terms required $$\dfrac{2^{n}(2^{n}-1)}{2} + 2^{n}-1$$. Now each of these terms can be complex numbers. And for defining a complex number we need one real number (assuming $$e^{i\theta}$$ form and defining theta to a good enough approximation).

Total number of independent real numbers - $$\dfrac{2^{n}(2^{n}-1)}{2} + 2^{n}-1 = 4^{n}+\dfrac{2^{n}}{2}-1$$

But in Nielsen Chuang it was asked to proof

Exercise 4.46: (Exponential complexity growth of quantum systems) Let $$\rho$$ be a density matrix describing the state of $$n$$ qubits. Show that describing $$ρ$$ requires $$4^{n} − 1$$ independent real numbers.

Who is correct here?

• Isn't it the case that the information content in $\rho$ (assuming pure state) is equivalent to the information content of $2^n-1$ complex amplitudes, meaning you can "convert" from one representation to the other? Each complex number can be described by 2 real numbers, hence Nielsen & Chuang's formula. Feb 21, 2021 at 11:55
• – glS
Feb 21, 2021 at 11:59
• Thanks @glS. That's helpful. Feb 21, 2021 at 12:11
• @AttilaKun Thanks for your help Feb 21, 2021 at 12:11

2. You need two real numbers to specify a complex number. In polar form it is $$r e^{i \theta}$$, $$r> 0$$ and $$\theta \in (-\pi,\pi]$$. Or in Cartesian form $$x + i y$$ for $$x,y\in \mathbb{R}$$.
After taking into account these two points you should get $$4^n - 1$$.