Projective Measurement

quantum-computing

Definition

Projective Measurement

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

that splits the Hilbert Space into mutually exclusive orthogonal branches.

Equivalently, a family of orthogonal projections

is a projective measurement if:

1. ,
2. ,
3. for , and
4. .

Probability Rule

Probability Rule

Let be a normalised quantum state and a projective measurement. The Born rule assigns to outcome the probability

Equivalently:

In words: the probability of branch equals the squared length of the component of inside that branch.

The equivalence follows from the defining properties of a projective measurement: is idempotent () and self-adjoint ().

Expand the squared norm and reduce: