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Probability Amplitude

A probability amplitude is the complex number attached to each basis state in a quantum superposition; its squared magnitude gives the probability of measuring that outcome, and its phase enables interference.

What it means

A quantum state |ψ⟩ = Σᵢ cᵢ|i⟩ assigns a complex number cᵢ — the probability amplitude — to each basis state |i⟩.The Born rule says the probability of measuring outcome i is P(i) = |cᵢ|², and normalization requires Σᵢ|cᵢ|² = 1.Amplitudes carry strictly more information than probabilities: being complex, they have both a magnitude and a phase.The magnitude sets how likely an outcome is; the relative phase between amplitudes determines how they interfere when combined — in phase they reinforce, out of phase they cancel.This is why quantum computation is often described as computing with amplitudes rather than probabilities: classical probabilities are non-negative and can only add, whereas amplitudes can be negative or complex and can therefore cancel each other.A quantum algorithm's job is to steer amplitude — moving it away from wrong answers and piling it onto right ones — before measurement converts amplitudes into probabilities.

Everyday analogy

Think of each possible answer as having a secret arrow. The longer the arrow, the more likely that answer is when you look. But arrows also point in directions — and two arrows pointing opposite ways cancel out to nothing, while two arrows pointing the same way make one big arrow. Quantum computers win by turning the wrong answers' arrows against each other and lining up the right answer's arrows.
An amplitude is like the height and timing of a wave at the beach: the height (magnitude) tells you how big the splash can be, but whether two waves make a giant splash or flat water depends on their timing (phase) relative to each other.

Common misconceptions

  • An amplitude is NOT a probability — it is a complex number that can be negative or imaginary. Only its squared magnitude |c|² is a probability. Probabilities can never cancel each other; amplitudes can, and that cancellation (interference) is the whole point.
  • Amplitudes are not directly observable in a single measurement — a measurement yields one outcome. Estimating amplitudes requires statistics over many identically prepared copies (e.g., state tomography).
  • A global phase (multiplying the whole state by e^{iθ}) has no physical effect — only relative phases between amplitudes are physically meaningful.

Key takeaways

  • Each basis state in a superposition carries a complex amplitude; the Born rule P(i) = |cᵢ|² converts amplitudes into measurement probabilities.
  • Amplitudes must satisfy the normalization condition Σᵢ|cᵢ|² = 1.
  • The phase of an amplitude does not change that outcome's probability by itself, but relative phases control interference when amplitudes combine.
  • Quantum algorithms manipulate amplitudes — including their signs and phases — which is something classical probability distributions cannot do.

Check your understanding

A qubit is in the state |ψ⟩ = (1/√2)|0⟩ − (1/√2)|1⟩. What is the probability of measuring |1⟩?

  1. A.−1/2
  2. B.1/2
  3. C.1/√2
  4. D.0
Show the answer

Answer: B. 1/2

Why: The amplitude of |1⟩ is −1/√2. Probability is the squared magnitude: |−1/√2|² = 1/2. Note that the minus sign does not make the probability negative — probabilities come from |c|², while the sign (phase) matters for interference.

Builds on

Superposition

Primary source: Nielsen & Chuang, Quantum Computation and Quantum Information (2010), doi:10.1017/CBO9780511976667

Graded 2026-07-10 (human sign-off): established — probability amplitudes and the Born rule per Nielsen & Chuang (2010) §1.2/§2.2 and Preskill Ph219. Pending human grading.

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