Positive Operator-Valued Measure

quantum-computing quantum-mechanics

Definition

Positive Operator-Valued Measure

Let be a complex Hilbert space and let be a finite or countable set of measurement outcomes. A positive operator-valued measure (POVM) on is a family of linear operators

such that

Each is called a measurement effect. For a normalised quantum state , the Born rule gives

Positivity

The condition means that is a positive semidefinite operator:

This ensures that every assigned outcome probability is non-negative.

Normalisation

For a normalised quantum state ,

Thus a POVM produces a valid probability distribution over its outcomes.

Relation to Projective Measurement

A projective measurement is a special case of a POVM. It consists of orthogonal projections satisfying

Every projective measurement is a POVM by setting

The converse is false. In a general POVM, the effects need not be projections and need not be mutually orthogonal.

Measurement Operators

A POVM determines outcome probabilities, but not uniquely the post-measurement state. To describe the state update, one chooses measurement operators such that

Then

After observing outcome , the corresponding post-measurement state is

provided .

Different choices of can induce the same POVM effects , so a POVM is primarily a probability rule rather than a complete physical measurement dynamics.

Noisy Computational-Basis Measurement

Let and define on one qubit

Then

Both operators are positive because their eigenvalues are non-negative:

For

the outcome probabilities are

For , this is not a projective measurement because and .