By the end of this lesson, you should be able to:
- Distinguish a qubit from a classical bit.
- Interpret α and β as complex probability amplitudes.
- Relate θ and φ to a point on the Bloch sphere.
- Compute measurement probabilities from amplitudes.
From a bit to a qubit
A classical bit occupies one of two states, 0 or 1. A qubit is described by a normalized vector in a two-dimensional complex vector space. In the computational basis,
\[|\psi\rangle=\alpha|0\rangle+\beta|1\rangle,\qquad |\alpha|^2+|\beta|^2=1.\]
The complex numbers \(\alpha\) and \(\beta\) are called probability amplitudes. They are not probabilities themselves. Their squared magnitudes determine the probabilities of obtaining 0 or 1 when the qubit is measured in the computational basis.
The Bloch-sphere picture
Ignoring an overall global phase, every pure single-qubit state can be written as
\[|\psi\rangle=\cos\frac{\theta}{2}|0\rangle+e^{i\phi}\sin\frac{\theta}{2}|1\rangle.\]
The angles \(\theta\) and \(\phi\) locate the state on the Bloch sphere. Use the interactive laboratory above to move both angles and watch the state vector and amplitudes change together.
The Bloch sphere is a geometric representation of a single pure qubit state; it is not a picture of a tiny physical ball.
Measurement
If the state is measured in the computational basis, the probability of obtaining \(|0\rangle\) is \(|\alpha|^2\), while the probability of obtaining \(|1\rangle\) is \(|\beta|^2\). The act of measurement produces one classical outcome even though the pre-measurement state can contain both basis components.
What should stay with you
- A qubit is a normalized vector, not a bit that is simply “both 0 and 1.”
- Amplitudes can be complex; probabilities are real and nonnegative.
- The Bloch sphere represents pure single-qubit states up to global phase.
For |ψ⟩ = α|0⟩ + β|1⟩, which quantity is the probability of measuring 1?
Probability is the squared magnitude of the corresponding complex amplitude, so P(1)=|β|².