Heisenberg Uncertainty Principle (Nielsen and Chuang Box 2.4)

Question

I'm trying to follow Nielsen and Chuang Book on Quantum Computation and Quantum Information. There is Box 2.4 on the Heisenberg Uncertainty Principle. I got stuck pretty fast. In that box they define: $$ \left<\psi | AB|\psi \right> = x+iy $$ where $x$ an $y$ are real. They note that the commutator $\left<\psi \left | [A,B]\right |\psi \right> = 2iy$ and the anticommutator is $\left<\psi |\{ A,B \}|\psi \right> = 2x$. Thus, the book says that this implies the following. $$ \left |\left<\psi \left | [A,B]\right |\psi \right>\right|^2 + \left| \left<\psi | \left\{ A,B\right\}|\psi \right>\right|^2=4\left| \left<\psi | AB|\psi \right>\right|^2 $$ I'm trying to proof that statement, but I cannot figure it out. I tried two ways.

1. I expand the lhs of the equation to obtain the rhs.

$$ \begin{matrix} \left |\left<\psi \left | [A,B]\right |\psi \right>\right|^2 + \left| \left<\psi | \left\{ A,B\right\}|\psi \right>\right|^2=\\ \left |\left<\psi \left | AB-BA\right |\psi \right>\right|^2 + \left| \left<\psi | AB+BA|\psi \right>\right|^2= \\ \left |\left<\psi \left | AB\right |\psi \right>-\left<\psi \left | BA\right |\psi \right>\right|^2 + \left |\left<\psi \left | AB\right |\psi \right>+\left<\psi \left | BA\right |\psi \right>\right|^2 =\\ \left<\psi \left | AB\right |\psi \right>^2-2\left<\psi \left | AB\right |\psi \right>\left<\psi \left | BA\right |\psi \right> + \left<\psi \left | BA\right |\psi \right>^2+\left<\psi \left | AB\right |\psi \right>^2+2\left<\psi \left | AB\right |\psi \right>\left<\psi \left | BA\right |\psi \right> + \left<\psi \left | BA\right |\psi \right>^2 =\\ 2\left<\psi \left | AB\right |\psi \right>^2+2\left<\psi \left | BA\right |\psi \right>^2 \end{matrix} $$

Which doesn't seem equal to $4\left| \left<\psi | AB|\psi \right>\right|^2$ (unless it commutes but I guess it is not the case, is it?).

2. I expand from $x$ and $y$'s definition.

Expanding lhs: $$ \begin{matrix} \left |\left<\psi \left | [A,B]\right |\psi \right>\right|^2 + \left| \left<\psi | \left\{ A,B\right\}|\psi \right>\right|^2=\\ |2iy|^2+|2x|^2\\ -4y^2+4x^2 \end{matrix} $$ Expanding rhs: $$ \begin{matrix} 4\left| \left<\psi | AB|\psi \right>\right|^2=\\ |x+iy|^2=\\ x^2+2ixy-y^2 \end{matrix} $$

Maybe I'm missing something, but it the lhs, the expansion seems real and the rhs the expansion seems complex.

I feel like missing something obvious, but I'm failing to find an answer.

Answer 1

What you seem to be missing is that $|\cdot |$ refers to the modulus of a complex number. For a complex number $z=x+iy$ we have $|z|=\sqrt{x^2+y^2}=\sqrt{z\cdot\overline{z}}$ where $\overline{z}=x-iy$ is the conjugate of $z$. So (in your second attempt for example): $$|x+iy|^2=(x+iy)(x-iy)=x^2-ixy+iyx+y^2=x^2+y^2 \color{red}{\neq} x^2+2ixy-y^2$$ The same thing went wrong in your first calculation since you treated the modulus the same way you would treat the absolute value for real numbers.

EDIT: To see why $\langle\psi|BA\psi\rangle=x-iy$, assuming $A$ and $B$ are Hermitian operators, write: $$\langle\psi|BA|\psi\rangle=\langle A^\dagger B^\dagger \psi|\psi\rangle = \langle AB\psi|\psi\rangle = \overline{\langle\psi|AB|\psi\rangle}=x-iy$$