Qubit

Learning Objectives

• Distinguish a classical bit from a qubit state.
• Use the computational basis states |0⟩ and |1⟩ correctly.
• Interpret amplitudes and measurement probabilities in the computational basis.

Scene

Alice enters a quiet observatory and finds a compass-like crystal. It does not behave like a normal needle that is already pointing north or south before she looks. The guide warns her: the crystal is not a metaphor for indecision; it follows quantum state rules.

Conceptual explanation

A classical bit is a variable that takes one of two values, 0 or 1. A qubit is represented by a state vector in a two-dimensional complex vector space. We usually choose basis vectors called the computational basis:

• |0⟩
• |1⟩

Any pure single-qubit state can be written as a linear combination of these basis states.

$$|\alpha|^2 + |\beta|^2 = 1$$

Measurement in the computational basis

If we measure |ψ⟩ in the computational basis, the Born rule gives:

$$P(0) = |\alpha|^2,\qquad P(1) = |\beta|^2.$$

The state after measurement is the observed basis state (|0⟩ or |1⟩) for that trial.

Common misconception: A qubit does not simply store two classical values at once.
The phrase “a qubit is both 0 and 1” is incomplete. A qubit is a state with amplitudes over basis vectors.

Advanced note: global phase. Multiplying a state by a global phase $e^{i\gamma}$ does not change measurement probabilities.

Worked intuition

Suppose $\alpha = \sqrt{0.8}$ and $\beta = i\sqrt{0.2}$. Then $P(0)=0.8$ and $P(1)=0.2$. The imaginary unit in $\beta$ matters later for interference, even though this single measurement probability only uses magnitudes.

Check Your Understanding

In computational-basis measurement of a qubit state, which statement is correct?

The amplitudes are directly the probabilities.Probabilities are the squared magnitudes of amplitudes and sum to 1.Measurement reveals both 0 and 1 at the same time.

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Summary

You now have the minimum formal model of a qubit: basis states, amplitude representation, normalization, and computational-basis measurement probabilities. Next, we study superposition more carefully and explain why phase matters.