Born Rule

quantum-computing quantum-mechanics

Definition

Born Rule

Let $\mathcal{H}$ be a complex Hilbert space, let $|\psi ⟩\in \mathcal{H}$ be a normalised quantum state, and let $\{E_j{\}}_{j\in J}$ be a positive operator-valued measure, meaning

$$E_j\ge 0\text{and}\sum _{j\in J}{E}_j=I.$$

The Born rule assigns to measurement outcome $j$ the probability

$$\mathrm{Pr}(j\mid \psi )=⟨\psi |E_j|\psi ⟩.$$

For a projective measurement in an orthonormal basis $B=\{|b_i⟩{\}}_{i\in I}$, this becomes

$$\mathrm{Pr}(b_i\mid \psi )={\left|⟨b_i|\psi ⟩\right|}^2.$$

What the Rule Does Not Say

The Born rule gives only the probabilities of measurement outcomes. It does not by itself specify the post-measurement state.

For projective measurement, the usual state-update rule is the projection postulate:

$$|\psi ⟩\mapsto \frac{P_j|\psi ⟩}{\sqrt{⟨\psi |P_j|\psi ⟩}}$$

after observing outcome $j$, provided $⟨\psi |P_j|\psi ⟩>0$.