Born Rule (Measurement Probability)
Probability of measuring outcome |k⟩ from a quantum state |ψ⟩.
Formula
Variables
Example
|ψ⟩ = (√0.3)|0⟩ + (√0.7)|1⟩ → P(0) = 30 %, P(1) = 70 %
How it works
In quantum mechanics, the quantity \(\langle k|\psi\rangle\) is the amplitude for finding the system in state \(|k\rangle\) when it is prepared in \(|\psi\rangle\). The Born rule says the actual measurement probability is the squared magnitude of that amplitude, because complex amplitudes can interfere and only their absolute square produces a real, nonnegative probability. This is the bridge from the abstract state vector to experimentally observed frequencies.
When to use it
Use this formula whenever you want the probability of a measurement outcome in a chosen basis, such as reading a qubit in the computational basis or projecting onto any specific state. It is also the starting point for calculating outcome distributions after applying quantum gates.
Worked example, step by step
Suppose a qubit is in the state \(|\psi\rangle = \frac{1}{2}|0\rangle + \frac{\sqrt{3}}{2}|1\rangle\). To find the probability of measuring \(|0\rangle\), compute the amplitude: \(\langle 0|\psi\rangle = \frac{1}{2}\). Then square its magnitude: \(P(|0\rangle) = \left|\frac{1}{2}\right|^2 = \frac{1}{4} = 0.25\). To find the probability of measuring \(|1\rangle\), compute the amplitude: \(\langle 1|\psi\rangle = \frac{\sqrt{3}}{2}\). Square its magnitude: \(P(|1\rangle) = \left|\frac{\sqrt{3}}{2}\right|^2 = \frac{3}{4} = 0.75\). Check: \(0.25 + 0.75 = 1\), so the probabilities are consistent.
Common mistakes to avoid
- Forgetting to square the magnitude and using the amplitude directly as the probability
- Ignoring complex numbers and dropping the absolute value bars
- Using the wrong basis state \(|k\rangle\) for the measurement being performed
Did You Know?
Max Born won the 1954 Nobel Prize for this single insight — that quantum amplitudes give probabilities via their squared modulus.
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Bell State (EPR Pair)
IntermediateMaximally entangled two-qubit state — measuring one qubit instantly determines the other.