Magnitudes and phases of coefficients of a qubit

1

Quantum mechanics is based on the idea of waves, and waves have both a magnitude and a phase?

$$|\psi\rangle = i\alpha|0\rangle + \beta|1\rangle.$$

Does $\alpha$ and $\beta$ represent magnitude and $i$ represent phase?

Or how do we represent phase? Or is it something else?

guest

Posted 2020-02-14T02:37:03.720

Reputation: 215

Question was closed 2020-07-11T21:20:53.200

You can find everything you need to understand this in this Wikipedia article: https://en.wikipedia.org/wiki/Qubit#Qubit_States

And if you need some background on complex numbers, see here: https://en.wikipedia.org/wiki/Complex_number

– dlyongemallo – 2020-02-14T04:07:22.523

Answers

3

It is more common to write a qubit as

$$ |\psi\rangle = \alpha|0\rangle + \beta|1\rangle, $$

where $\alpha, \beta \in \mathbb{C}$, i.e. to omit $i$ and get it in complex number $\alpha$.

Parameters $\alpha$ and $\beta$ are called complex amplitudes, $|\alpha|^2$ is a probability of measuring state $|0\rangle$ and $|\beta|^2$ is a probability of measuring state $|1\rangle$. So, $\alpha$ and $\beta$ (or rather square of their absolute values) can be called "magnitude".

Any qubit can be rewritten as

$$ |\psi\rangle = \cos(\theta/2)|0\rangle + \mathrm{e}^{i\phi}\sin(\theta/2)|1\rangle, $$

where $\theta$ and $\phi$ are coordinates on so-called Bloch sphere.

In this notation, $\cos^2(\theta/2)$ and $\sin^2(\theta/2)$ are probabilities ("magnitudes") of measuring $|0\rangle$ and $|1\rangle$, respecitively. Parameter $\phi$ is a phase (or to be precise, relative phase).

Martin Vesely

Posted 2020-02-14T02:37:03.720

Reputation: 7 763