Bra-ket :

cos(0) = +1 |0> : sin(0) |1> = +0 |1> => +1 |0> +0 |0>

cos(90) = +0 |0> : sin(90) |1> = +1 |1> => +0 |1> +1 |0>

cos(180) = -1 |0> : sin(180) |1> = +0 |1> => -1 |0> +0 |0>

cos(270) = +0 |0> : sin(270) |1> = -1 |1> => +0 |1> -1 |0>

Probability :

cos(0)^2 = +1 |0> : sin(0)^2 |1> = +0 |1> => +1 |0> +0 |0>

cos(90)^2 = +0 |0> : sin(90)^2 |1> = +1 |1> => +0 |1> +1 |0>

cos(180)^2 = +1 |0> : sin(180)^2 |1> = +0 |1> => +1 |0> +0 |0>

cos(270)^2 = +0 |0> : sin(270)^2 |1> = -1 |1> => -0 |1> +1 |0>

As you can see the probability does not give indications on the sign of the coordinate. We take the consideration of Hadamard of +R|0>

+50% |0> + 50% |1> = 100% |0>

(+√0.5)^2 |0> + (+√0.5)^2 |1> = 100% |0>

And we take the consideration of Hadamard of +R|1>

+50% |0> + 50% |1> = 100% |1>

(+√0.5)^2 |0> + (-√0.5)^2 |1> = 100% |1>

The sum of probabilities can cause |1> to be canceled, and |0> to be added.

+50% |0> + 50% |0> = 100% |0>

+50% |1> – 50% |1> = 0% |1>

100% |0> 100% |1>

Since both are being used ..

+25% |0> + 25% |0> = +50% |0>

+25% |1> – 25% |1> = 0% |1>

+50% |0> +50% |1>

There is a discrepancy, this is because the second chance has become bra-ket! to obtain this result it is necessary to invert the process, not higher than the second one but under a square root!

+√25% |0> + √25% |0> = +100% |0>

+√25% |1> – √25% |1> = 0% |1>

+50% |0> + 50% |1>

With this I want to demonstrate that if the value of bra-ket, and also the probability, is not taken into account both, it is not possible to understand the result of the operations of the various gate.

sign(alpha) : alpha^2 |value>