Von Neumann entropy in terms of the tangle

[barnflakes]

The Von Neumann entropy is $$\mathcal{S}(|\psi\rangle) = -Tr[\rho_a ln \rho a]$$. The linear entropy $$S_L = \frac{l}{l-1}(1 - Tr[\rho_a^2])$$ For l =2 the linear entropy is written $$4Det(\rho_A)$$ which is also called the tangle $$\tau$$. I understand this just fine, I can show that. Now it says the Von Neumann can be written:

$$\mathcal{S}(|\psi\rangle) = -xln_{2}x - (1-x)ln_{2}(1-x)$$ where $$x = (1+\sqrt{1-\tau})/2$$

I don't know how to show this last step? Anyone offer any insight? This is for a 2-dimensional case if that isn't clear from the above.

 

[barnflakes]

Nobody know?