Born Rule

The fundamental rule that connects quantum amplitudes to measurement probabilities: probability equals amplitude squared.

The Born rule is the bridge between the abstract mathematics of quantum mechanics and experimental observations. It tells us how to calculate the probability of getting a particular measurement outcome.

The Rule

For a quantum state $|\psi ⟩$, the probability of measuring outcome $|k⟩$ is:

$$P(k)=|⟨k|\psi ⟩{|}^2$$

In other words: probability = |amplitude|²

Example

Consider a qubit in state: $$|\psi ⟩=\frac{1}{\sqrt{3}}|0⟩+\sqrt{\frac{2}{3}}|1⟩$$

The measurement probabilities are:

• $P(0)=|1/\sqrt{3}{|}^2=1/3\approx 33\%$
• $P(1)=|\sqrt{2/3}{|}^2=2/3\approx 67\%$

Note that probabilities sum to 1 (which is why we need the normalization condition).

Why Squared?

The Born rule is a postulate of quantum mechanics. It’s not derived from deeper principles. Some interpretations (like many-worlds) attempt to derive it, but we accept it because it matches experimental results with extraordinary precision.

The squaring is key:

• Amplitudes can be negative or complex
• Probabilities must be non-negative
• The absolute value squared ensures non-negative probabilities

For Density Matrices

For mixed states described by density matrix $\rho$:

$$P(k)=⟨k|\rho |k⟩=\text{Tr}(|k⟩⟨k|\rho )$$

Why It Matters

The Born rule is:

• The only way to connect quantum theory to experiment
• Why quantum amplitudes interfering can affect measurement outcomes
• Fundamental to understanding why quantum algorithms work

See also: Measurement, Quantum State, Superposition