What is the relation between POVMs and observables (as Hermitian operators)?

One way of looking at the relationship between POVMs and observables arises from identifying their counterparts in the theory of probability of which quantum mechanics can be thought of as an extension. It is easier to identify the counterparts if we temporarily restrict our attention to a special type of POVMs known as projection-valued measures or PVMs.

Below, we summarize four concepts: probability measure, random variable, PVM and observable in a way that highlights the high degree of similarity between probability measures and PVMs on one hand and random variables and observables on the other.

Probability measures and random variables

In elementary probability we construct random variables as tables that assign probabilities to possible outcomes, e.g. a random variable taking values $lambda_k$ for $k=1,dots,n$ with respective probabilities $p_k$ is described by

$$ begin{array}{c|cccccc} X & lambda_1 & lambda_2 & dots & lambda_n hline p(X) & p_1 & p_2 & dots & p_n end{array} $$

In the more abstract approach based on measure theory the probabilistic structure is separated out by introducing the probability space $(Omega, mathcal{F}, P)$ where the sample space $Omega$ is the set of all possible outcomes of a random experiment, $mathcal{F}$ is a $sigma$-algebra of events on $Omega$ and $P: mathcal{F} to mathbb{R}_{ge0}$ is a probability measure.

A random variable $X$ is defined as a measurable function from $Omega$ to a measurable space, e.g. $mathbb{R}$. This splits the above table by inserting the sample space

$$ begin{array}{c|cccccc} X & lambda_1 & lambda_2 & dots & lambda_n hline Omega & omega_1 & omega_2 & dots & omega_n hline P & p_1 & p_2 & dots & p_n end{array} $$

(Technically, $P$ is defined on $mathcal{F}$ but when $Omega$ is finite and $mathcal{F}=mathcal{P}(Omega)$ additivity implies that $P$ is uniquely defined by its values on the singleton subsets of $Omega$.)

In this view a random variable can be thought of as a random experiment which is represented by the probability measure $P$ and which yields an abstract outcome $omega_k$ followed by post-processing which is described by the measurable function $X$ and which finds the experiment result $lambda_k := X(omega_k)$. Moreover, if the range of $X$ is finite, the random variable can be written as

$$ X = sum_klambda_kmathbb{1}_{A_k}tag1 $$

where $mathbb{1}_E$ is the indicator function of a set $Esubset Omega$ and the sets $A_k$ form a partitioning of $Omega$, i.e. they are disjoint subsets of $Omega$ whose union is $Omega$. Note that in terms of the indicator functions, the last condition can be stated as $mathbb{1}_{A_i}mathbb{1}_{A_j} = delta_{ij}mathbb{1}_{A_i}$ and $sum_kmathbb{1}_{A_k} = 1$.

Projection-valued measures and observables

In elementary quantum mechanics we construct observables by assigning projectors to possible outcomes and combining them to form a Hermitian operator, e.g. an observable $X$ taking values $lambda_k$ for $k=1,dots,n$ with respective projectors $|kranglelangle k|$ is $X=sum_klambda_k|kranglelangle k|$ or in table format

$$ begin{array}{c|cccccc} X & lambda_1 & lambda_2 & dots & lambda_n hline pi & |1ranglelangle 1| & |2ranglelangle 2| & dots & |nranglelangle n|end{array} $$

As before, in the more abstract approach using measure theory we introduce a sample space $Omega$ and a $sigma$-algebra $mathcal{F}$. In place of a probability measure $P: mathcal{F} to mathbb{R}_{ge0}$, we define a projection-valued measure $pi: mathcal{F} to L_H(mathcal{H})$ where $L_H(mathcal{H})$ is the space of Hermitian operators on a Hilbert space $mathcal{H}$ and $pi(E)$ is a projector for every event $Einmathcal{F}$. Moreover, for any partitioning $A_k$ of $Omega$ we require that $pi(A_i)pi(A_j) = delta_{ij}pi(A_i)$ and $sum_kpi(A_k)=I$. For each state $|psirangle$, this gives us a probability measure $P_psi: mathcal{F} to mathbb{R}_{ge0}$ defined as $P_psi(E) = langlepsi|pi(E)|psirangle$.

Defining $Omega = {omega_1, dots, omega_n}$, $mathcal{F}=mathcal{P}(Omega)$ and $pi(A) = sum_{omega_kin A}|kranglelangle k|$ we can then write $X=sum_klambda_kpi({omega_k})$ or in table format

$$ begin{array}{c|cccccc} X & lambda_1 & lambda_2 & dots & lambda_n hline Omega & omega_1 & omega_2 & dots & omega_n hline pi & |1ranglelangle 1| & |2ranglelangle 2| & dots & |nranglelangle n| end{array} $$

(As before, technically, $pi$ is defined on $mathcal{F}$ but when $Omega$ is finite and $mathcal{F}=mathcal{P}(Omega)$ additivity implies that $pi$ is uniquely defined by its values on the singleton subsets of $Omega$.)

In this view an observable can be thought of as a projective measurement which is represented by the PVM $pi$ and which given a state $|psirangle$ yields an abstract outcome $omega_k$ followed by post-processing which is described by the eigendecomposition of $X$ and which finds the measurement result $lambda_k$. Moreover, if the spectrum of $X$ is finite, the observable can be written as

$$ X = sum_klambda_kPi_ktag2 $$

where $Pi_k=pi({omega_k})$ is the projector on the eigenspace of $X$ associated with eigenvalue $lambda_k$. Note that the projectors are orthogonal, i.e. $Pi_iPi_j = delta_{ij}Pi_i$ and $sum_kPi_k = I$.

Observables vs POVMs

The correspondence above shows that there is indeed merit in thinking of an observable as equivalent to a special kind of POVM - namely a PVM - together with post-processing of measurement results. Moreover, the correspondence explains why POVMs are of independent interest outside of their role in specifying observables. Specifically, POVMs are more general than PVMs since they are not limited to describing projective measurements. This situation also finds its mirror image in measure theory: just as probability measures are not the only interesting type of scalar measures so the PVMs are not the only interesting type of POVMs. This situation may be represented graphically as

$$ begin{array}{|c|c|} hline text{random variables} & hline text{probability measures} & text{other scalar measures} hline end{array} begin{array}{|c|c|} hline text{observables} & hline text{PVMs} & text{other POVMs} hline end{array} $$

where positioning of one cell above another is to be interpreted as "builds upon" and where empty cells highlight the fact that the type of construction used to form random variables on top of probability measures and observables on top of PVMs does not easily generalize to other scalar measures and POVMs.

This relationship explains why both observables and POVMs are useful and encountered regularly. On one hand, observables like random variables provide a higher level language than measures which includes convenient shortcuts for computing quantities such as mean and standard deviation (useful to express results such as the Heisenberg uncertainty principle). On the other, they are less general since they only model projective measurements.

Note that general measurement may be emulated by projective measurement together with auxiliary subsystem and unitary evolution. Therefore, in a physical sense observables and POVMs describe the same fundamental physical reality. The utility of POVMs lies in convenient description of information theoretic aspects of processes more complex than projective measurement.

Remark on shared limitations

Finally, note that general quantum measurements (as described e.g. in section 2.2.3 on page 84 in Nielsen & Chuang) capture a wider class of processes than both observables and POVMs. Specifically, the letter two model measurement statistics, but fail to provide the most general way to compute post-measurement state.