Intuition on positive-operator valued measures (POVM)

Question

I'm having a little trouble understanding what positive-operator valued measure (POVM) are- in particular why/how they are non-negative. For instance, if they just represent measurements, what about something like measuring spin- it can take on a negative value, but I imagine it is also a POVM. What am I missing?

Answer 1

Let me expand a bit on the intuition part and write down an example. This is all essentially already covered by yuggib's answer.

Your confusion about positive operator valued measures, as also pointed out, is that they are not to be confused with measurement outcomes. The problem with measurement outcomes is that they are rather arbitrary. Often, they rely on a scale (e.g. position relies on a frame of reference), hence you can fix that scale differently and change the measurement outcomes. They are, in a way, something to be determined within an experiment.

Thus, as a theoretician, you are not really interested in the measurement outcomes most of the time, but you are just interested in knowing when you obtain different outcomes and most importantly in knowing their probabilities. And this is, where POVMs come in.

Let's suppose we are on a Hilbert space. A positive operator-valued measure is a map that takes some Borel set (mostly the reals as outcomes of the experiment) and maps it to positive operators. They must be positive (semi)definite, because given a state, there is a measure (i.e. a map from the Borel set to $[0,1]$) associated to the operator via the scalar product.

For instance, let's take your spin experiment: We want to measure the spin of an electron. The outcomes are $-1/2,1/2$. We can define a positive operator valued measure in the following way:

Let $\mathcal{H}$ be our Hilbert space with our state being $\rho\in\mathcal{B}(\mathcal{H})$ (the corresponding density matrix in the bounded operators - here, we could just take $\mathbb{C}^2$ for the spin, but maybe we want to have the whole density matrix with a lot of other information in it). Let $\mathcal{B}$ be the Borel sigma algebra of the set $\{-1/2,1/2\}$, i.e. the set of subsets of this set. Then the positive operator valued measure is a map

$$ \mathcal{P}:\mathcal{B}\to \mathcal{B}(\mathcal{H}) $$

and it is defined via:

$$ \mathcal{P}(\{-1/2\})=P,\quad \mathcal{P}(\{1/2\})=1-P\quad \mathcal{P}(\emptyset)=0 \quad \mathcal{P}(\{-1/2,1/2\})=1$$

where $P$ is some positive operator associated to the spin measurement (maybe the Pauli $Z$ when we consider $\mathbb{C}^2$) The measure associated to this for the given state $\rho$ is

$$ \mu_{\rho}(U):=\operatorname{tr}(\mathcal{P}(U)\rho) \qquad \forall U\in \mathcal{B}$$

This is supposed to be a probability measure. If you for example take the subset $\{-1/2\}$ then $\mu_{\rho}(\{-1/2\})$ gives you the probability that if you measure $\rho$, you'll get the outcome $-1/2$. This explains our definitions above: The empty set should be mapped to zero, the whole space should be mapped to 1 and everything should be nonnegative and between zero and one. Therefore, the operators must be positive and sum up to one!

Answer 2

To expand the comment, spectral measures, or projection valued measures are introduced to characterize self-adjoint operators.

They are families of orthogonal projections on the Hilbert space that, when acting on vectors suitably, define a measure. If you denote by $\{P_\lambda\}_{\lambda\in\mathbb{R}}$ this family, a self adjoint operator $A$ corresponding to it can be written as $$A=\int_{\mathbb{R}} \lambda dP_\lambda$$

The projections are positive operators, since a measure is usually positive when measuring (Borel) subsets of reals (positive volume). Nevertheless, if we integrate a function, you may obtain negative values. So integrating $\lambda$ w.r.t. the spectral measure, you may get negative values (in accordance with the measurement of negative-valued observables).

For a general operator-valued measure with the same interpretation, you also need positivity of the operators that generate the measure. Hence the denomination "positive" operator valued measures.

(warning: This is how I understand it, but I'm not an expert on the argument, so I may be wrong.)

Answer 3

The measurement procedure described by the measurement is as follows: You have a quantum state and do some measurement on it. Doing a measurement means you get one of a set of results. For example, if you measure the component of the spin of an electron in a certain direction, you have two possible results: Spin up, or spin down. The state of the electron determines which of the results occurs with which probability.

Note that in the above description, I did not assign any values to those events. Of course we know that "spin up" corresponds to a spin component of $\hbar/2$, and "spin down" to a spin component of $-\hbar/2$, but for the description of the measurement itself that's irrelevant; all that counts is that detector 1 clicks if we got a spin up particle, and detector 2 clicks if we got a spin down particle.

Also when considering POVMs, we are not interested in what the state if after the measurement (indeed, if we measure photons, the state after a measurement typically is "the photon no longer exists"). All we are interested in is which event occurs with which probability.

Also note that the measurement procedure might be more involved. For example, a measurement could be "let the object in question interact with another object in a certain way, and then measure that other object". In that case, a measurement result cannot in general be associated with a specific pure state of the measured object. For example, you can easily have a POVM with three outcomes for the spin of a spin-1/2-particle. None of the three outcomes then can be associated with a specific unique spin value.

Indeed, those considerations alone are sufficient to derive the properties of a POVM:

First, the POVM is described by a set of functions $f_k$ mapping the input state $\rho$ to the probability $p_k=f_k(\rho)$ to get measurement result $k$ when measuring this input state.

First, consider the case that you are provided the state $\rho_A$ with probability $p_A$ and the state $\rho_B$ with probability $p_B$. Then the total state you are provided with is $$\rho = p_A\rho_A + p_B\rho_B. \tag{1}$$

Now, according to the rules of probability, if , then the probability of getting measurement resukt $k$ is $$f_k(\rho) = p_A\cdot f_k(\rho_A) + p_B\cdot f_k(\rho_B)\tag{2}$$

Now inserting (1) in (2), you get linearity of the functions $f_k$. But every linear function from a bounded operator to a number can be written as trace $$f_k(\rho) = \operatorname{tr}(E_k\rho)$$ for some operator $E_k$.

The properties of $E_k$ can then be easily derived from the properties of probabilities:

Since probabilities are real, and density matrices are Hermitian, it follows that the $E_k$ also have to be Hermitian.

Since probabilities are always nonnegative, we have for any $\rho$ $$0 \le p_k = \operatorname{tr}(E_k\rho)$$ therefore each $E_k$ has to be positive.

Also, since we always get one of the measurement results, for any state $\rho$ the probabilities of the different results must add up to $1$, that is $$1 = \sum_k f_k(\rho) = \sum_k \operatorname{tr}(E_k\rho) = \operatorname{tr}\left(\left(\sum_kE_k\right)\rho\right)$$

which implies that $$\sum_kE_k = I$$ where $I$ is the identity operator.