Generalized Observables as POVMs

A positive-operator-valued measure (POVM) assigns an effect to every measurable outcome event. Its countable additivity makes the trace rule a genuine probability measure, including for continuous outcomes. The canonical POVMs article supplies the operational introduction; this page states the analytic definition and explains why a formal operator-valued probability density is not always available.

Helpful background. Effects supplies the interval $0\leq E\leq I$; normal states supplies continuity of trace pairings.

A normalized positive operator measure

Let $(\Omega,\Sigma)$ be a measurable space. A POVM is a map $E:\Sigma\to\mathcal B(\mathcal H)$ satisfying $E(\Delta)\geq0$, $E(\Omega)=I$, and weak countable additivity:

$$\left\langle\phi,E\!\left(\bigcup_n\Delta_n\right)\psi\right\rangle =\sum_n\langle\phi,E(\Delta_n)\psi\rangle$$

for disjoint measurable sets $\Delta_n$ and arbitrary vectors $\phi,\psi$. Positivity implies $0\leq E(\Delta)\leq I$, because $E(\Omega\setminus\Delta)=I-E(\Delta)$.

For a disjoint union, the finite partial sums form an increasing sequence of positive operators bounded above by $I$. They converge strongly to their supremum; weak additivity identifies this supremum with the union effect. Thus, for normalized positive measures, one may equivalently require strong convergence of the event sums. Operator-norm convergence is not required.

For a density operator $\rho$,

$$p_\rho(\Delta)=\operatorname{Tr}(\rho E(\Delta))$$

is a countably additive probability measure. Expand $\rho$ spectrally and exchange nonnegative sums, or invoke normality, to prove countable additivity. Every prediction is therefore a scalar probability statement even when the measure takes operator values.

What differs from a sharp observable

For a PVM, effects are projections and obey $P(\Delta\cap\Gamma)=P(\Delta)P(\Gamma)$. For a general POVM there is no such multiplication rule. Effects assigned to disjoint events need not have orthogonal ranges, and effects assigned to different events need not commute.

For instance, a qubit trine POVM has three effects

$$E_k=\tfrac13(I+n_k\cdot\sigma), \qquad k=1,2,3,$$

where the three unit Bloch vectors lie in one plane at angles $120^\circ$ apart. Each effect has eigenvalues $0,2/3$, and $\sum_kn_k=0$ ensures $\sum_kE_k=I$. The effects are not projections and do not commute pairwise. Three outcomes are possible on a two-dimensional space without requiring three mutually orthogonal state vectors.

For an input with Bloch vector $r$, the probabilities are $p_k=(1+n_k\cdot r)/3$. Because the $n_k$ span only a plane, this measurement does not determine the perpendicular component of $r$. Generalized measurement and informational completeness are distinct properties.

Integrating outcome functions

For a bounded measurable real function $f$, define $\int f\,dE$ first for simple functions, then by bounded approximation and weak operator limits. The map is positive and unital, and

$$-\|f\|_\infty I\leq\int f\,dE\leq\|f\|_\infty I.$$

Its defining prediction is

$$\operatorname{Tr}\!\left(\rho\int f\,dE\right) =\int f\,dp_\rho.$$

Unlike a PVM’s functional calculus, this map is generally not multiplicative. In particular, $\int f^2\,dE$ need not equal $(\int f\,dE)^2$. Their difference is positive for real bounded $f$, as follows from the dilation argument in Unsharp Measurements.

For unbounded $f$, scalar moments and operator integrals require domain conditions. The POVM’s normalization does not guarantee integrability of arbitrary outcome labels.

A first moment does not determine the measurement

On outcomes $\{-1,0,1\}$, consider

$$E_t(\{-1\})=E_t(\{1\})=\tfrac t2I, \qquad E_t(\{0\})=(1-t)I, \quad0\leq t\leq1.$$

Every member has first moment zero, yet its second moment is $tI$ and its outcome probabilities depend on $t$. Thus replacing a general POVM by its first-moment operator loses measurement information. The self-adjoint-operator/PVM correspondence cannot be transferred unchanged to POVMs.

Likewise, notation such as $E(dx)=M(x)\,dx$ is an extra assumption. In infinite dimensions, a scalar dominating measure need not produce an everywhere bounded operator-valued density. The event measure is the primary object. If a valid density is established, predictions may be integrated through it; otherwise the measure notation avoids inventing nonexistent pointwise effects.

Exercises

1. Smear a PVM $P$ with a Markov kernel $k(x,\Delta)$, a probability measure in $\Delta$ for each $x$ and a measurable function of $x$ for each event. Show that $E(\Delta)=\int k(x,\Delta)P(dx)$ is a POVM.

Solution

The integrand lies in $[0,1]$, giving positivity and the effect bound. Kernel normalization gives $E(\Omega)=I$. Countable additivity follows by scalar monotone convergence inside each vector spectral measure, then polarization. This models classical detector noise added to projective measurements.

2. For the trine POVM, find two distinct input states with identical probabilities.

Solution

Choose Bloch vectors $r=\pm\hat z$ perpendicular to the trine plane. Every dot product $n_k\cdot r$ vanishes, so both states give the uniform distribution $1/3$. The measurement misses one Bloch coordinate.

References

• Busch, Paul, Pekka Lahti, Juha-Pekka Pellonpää, and Kari Ylinen, Quantum Measurement, Springer, 2016 — operator measures, integration, and generalized observables. Publisher.
• Holevo, Alexander S., Quantum Systems, Channels, Information: A Mathematical Introduction, 2nd revised and expanded ed., De Gruyter, 2019 — POVMs and outcome statistics in Hilbert space. Publisher.