Quantum Amplitudes and Probabilities

The qubit state vector contains amplitudes rather than measurement probabilities.

The amplitudes belong to waves, because in quantum mechanics, the behavior of quantum particles is described by wave functions.

Waves have three characteristics:

• The wavelength is the distance over which the wave’s shape repeats.
• The phase of a wave is the position on the waveform cycle at a certain point.
• The amplitude of a wave is the distance between its center and its crest.

The following figure depicts these three characteristics.

As we can see in the figure, amplitudes can be positive or negative. Whether the amplitude is positive or negative depends on the imaginary point $x$. If you chose a different point $x^*$, the same wave would have a negative amplitude.

If we take two identical yet shifted waves, one might have a positive amplitude, whereas the other has a negative at point $x$. These two waves differ in their phase. However, when we measure either one of these two waves, they are alike. We won’t see any difference. Their effects on the measurement probabilities are the same.

Mathematically, the probability of measuring the qubit as 0 or 1 is the square of the corresponding amplitude. It doesn’t matter whether the amplitude is positive or negative.

$|\psi\rangle = \alpha|0\rangle + \beta|1\rangle = \begin{bmatrix}\alpha\\\beta\end{bmatrix}$ ...