Born's rule

If a quantum state is:

$$|\psi\rangle = \sum_{i}\alpha_{i}|i\rangle$$

and you measure in the computational basis $\{|i\rangle\}$, then the probability of observing outcome $i$ is

$$P[i] = |\alpha_{i}|^{2}$$

In quantum mechanics:

• An obervable (e.g. position, energy, spin) is represented by a **Hermitian operator $\widehat{O}$ and it has eigenvectors $|v_{j}\rangle$ and real eigenvalues $\lambda_{j}$. if your staet $|\psi\rangle$ is decomposed as:

$$|\psi\rangle = \sum_{i}\beta_{j}|v_{j}\rangle$$

then the Born rule tells that

Measuring $\widehat{O}$ yields eigenvalue $\lambda_{j}$ with probability $|\beta_{j}|^{2}$

So let's connect betweeen eigenvalues and amplitudes. The eigenvalues are the possible outcomes you can observe and the amplitudes determine how likely each outcome is. Therefore, we can say that amplitudes encode probabilities and eigenvalues are the measured quantities (what you read on the device)

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Reference

1. Born rule - Wikipedia: https://en.wikipedia.org/wiki/Born_rule