Measurement Effect

quantum-computing quantum-mechanics

Definition

Measurement Effect

Let be a complex Hilbert space. A measurement effect is a positive semidefinite operator satisfying

Equivalently,

The number is the Born-rule probability assigned to the measurement outcome represented by .

Effect as One Outcome

A positive operator-valued measure is a family of effects

whose total effect is the identity operator:

Each represents one possible measurement outcome. For a pure state ,

For a mixed state represented by a density operator ,

Why

The condition

ensures

The condition

means

which ensures

for normalised .

Thus an effect is exactly an operator whose expectation value is always a valid probability.

Projectors as Sharp Effects

Every orthogonal projection is a measurement effect:

Projection effects are sharp: for states inside the projected subspace, the outcome has probability ; for states orthogonal to it, the outcome has probability .

General effects need not be projections. They can represent noisy or unsharp outcomes.

Qubit Example

For , define

The eigenvalues of are and , so

For

the effect assigns probability

If , then is a sharp projection. If , then is a noisy effect rather than a projection.