2
$\begingroup$

What is the output of a CNOT gate if both inputs are in superposition?

CNOT gate

For example, what happens if: $\left|x\right>=\alpha_x\left|0\right>+\beta_x\left|1\right>$ and $\left|y\right>=\alpha_y\left|0\right>+\beta_y\left|1\right>$. Note that the $\alpha$s and $\beta$s can have imaginary parts.

For another example, if:

$$\begin{gather} \alpha_x=0.6\times e^{i\theta_1} \\ \beta_x=0.8\times e^{i\theta_2} \\ \alpha_y = \frac{\sqrt3}{3}\times e^{i\theta_3} \\ \beta_y = \frac{\sqrt6}{3}\times e^{i\theta_4} \end{gather}$$

then what is $\left|x\oplus y\right>$?

$\endgroup$
1
  • 2
    $\begingroup$ You have to express $|x⟩|y⟩$ in the basis $|0⟩|0⟩, |0⟩|1⟩, |1⟩|0⟩, |1⟩|1⟩$, then apply the matrix C, and you got the output result in the same basis. You have to see then if it is a separable state or not (it seems not) $\endgroup$
    – Trimok
    Jun 11, 2013 at 19:26

1 Answer 1

1
$\begingroup$

Just to re-enforce the hint given by Trimok (Jun 11):

${\text{CNOT[ }} $
$\alpha_x \alpha_y \mid\!0, 0\rangle + \alpha_x \beta_y \mid\!0, 1\rangle + \beta_x \alpha_y \mid\!1, 0\rangle + \beta_x \beta_y \mid\!1, 1\rangle $
${\text{ ]}} := $
$\alpha_x \alpha_y \mid\!0, 0\rangle + \alpha_x \beta_y \mid\!0, 1\rangle + \beta_x \beta_y \mid\!1, 0\rangle + \beta_x \alpha_y \mid\!1, 1\rangle$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.