Positive Operator-Valued Measure

quantum-computing quantum-mechanics

Definition

Positive Operator-Valued Measure

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

$$\{E_j:\mathcal{H}\to \mathcal{H}{\}}_{j\in J}$$

such that

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

Each $E_j$ is called a measurement effect. For a normalised quantum state $|\psi ⟩$, the Born rule gives

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

Positivity

The condition $E_j\ge 0$ means that $E_j$ is a positive semidefinite operator:

$$⟨\psi |E_j|\psi ⟩\ge 0\forall |\psi ⟩\in \mathcal{H}.$$

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

Normalisation

For a normalised quantum state $|\psi ⟩$,

$$\begin{array}{rl}\sum _{j\in J}\mathrm{Pr}(j\mid \psi )& =\sum _{j\in J}⟨\psi |E_j|\psi ⟩\\ & =⟨\psi |\left(\sum _{j\in J}{E}_j\right)|\psi ⟩\\ & =⟨\psi |I|\psi ⟩\\ & =⟨\psi |\psi ⟩\\ & =1.\end{array}$$

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 $\{P_j{\}}_{j\in J}$ satisfying

$$P_j=P_j^{†},P_j^2=P_j,P_j{P}_k=0\text{ for }j\ne k,\sum _{j\in J}{P}_j=I.$$

Every projective measurement is a POVM by setting

$$E_j:=P_j.$$

The converse is false. In a general POVM, the effects $E_j$ 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 $\{M_j{\}}_{j\in J}$ such that

$$E_j=M_j^{†}{M}_j.$$

Then

$$\mathrm{Pr}(j\mid \psi )=⟨\psi |M_j^{†}{M}_j|\psi ⟩=\Vert M_j|\psi ⟩{\Vert }^2.$$

After observing outcome $j$, the corresponding post-measurement state is

$$|\psi ⟩\mapsto \frac{M_j|\psi ⟩}{\sqrt{\mathrm{Pr}(j\mid \psi )}}$$

provided $\mathrm{Pr}(j\mid \psi )>0$.

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

Noisy Computational-Basis Measurement

Let $0\le \epsilon \le 1$ and define on one qubit

$$\begin{array}{rl}{E}_0& =(1-\epsilon )|0⟩⟨0|+\epsilon |1⟩⟨1|,\\ E_1& =\epsilon |0⟩⟨0|+(1-\epsilon )|1⟩⟨1|.\end{array}$$

Then

$$E_0+E_1=|0⟩⟨0|+|1⟩⟨1|=I.$$

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

$$\mathrm{spec}(E_0)=\mathrm{spec}(E_1)=\{\epsilon ,1-\epsilon \}.$$

For

$$|\psi ⟩=\alpha |0⟩+\beta |1⟩,$$

the outcome probabilities are

$$\begin{array}{rl}\mathrm{Pr}(0\mid \psi )& =(1-\epsilon )|\alpha {|}^2+\epsilon |\beta {|}^2,\\ \mathrm{Pr}(1\mid \psi )& =\epsilon |\alpha {|}^2+(1-\epsilon )|\beta {|}^2.\end{array}$$

For $0<\epsilon <1$, this is not a projective measurement because $E_0^2\ne E_0$ and $E_1^2\ne E_1$.