Projective Measurement

quantum-computing

Definition

Projective Measurement

A projective measurement is a measurement describes by a family of orthogonal projections

$$\{P_i{\}}_{i\in I}$$

that splits the Hilbert Space into mutually exclusive orthogonal branches.

Equivalently, a family of orthogonal projections

$$\{P_i{\}}_{i\in I}$$

is a projective measurement if:

1. $P_i^2=P_i$,
2. $P_i^{†}=P_i$,
3. $P_i{P}_j=0$ for $i\ne j$, and
4. ${\sum }_i{P}_i=I$.

Probability Rule

Probability Rule

Let $|\psi ⟩$ be a normalised quantum state and $\{P_i{\}}_{i\in I}$ a projective measurement. The Born rule assigns to outcome $i$ the probability

$$p(i)=⟨\psi |P_i|\psi ⟩.$$

Equivalently:

$$p(i)=\Vert P_i|\psi ⟩{\Vert }^2.$$

In words: the probability of branch $i$ equals the squared length of the component of $|\psi ⟩$ inside that branch.

The equivalence follows from the defining properties of a projective measurement: $P_i$ is idempotent ($P_i^2=P_i$) and self-adjoint ($P_i^{†}=P_i$).

$\Vert P_i|\psi ⟩{\Vert }^2=⟨\psi |P_i|\psi ⟩$

Expand the squared norm and reduce:

$$\begin{array}{rlrl}\Vert P_i|\psi ⟩{\Vert }^2& =⟨P_i\psi ⟩P_i\psi & & \\ & =⟨\psi ⟩P_i^{†}{P}_i\psi & & (⟨Ax⟩y=⟨x⟩A^{†}y)\\ & =⟨\psi ⟩P_i{P}_i\psi & & (P_i^{†}=P_i)\\ & =⟨\psi |P_i|\psi ⟩& & (P_i^2=P_i).\end{array}$$