# What does it mean that the elements of a generalized measurement are operators?

I have a very simple question on Generalized measurements.

The definition I was given is the one below:
A set of operator $$\left \{ M_{\alpha} \right \}$$ is called generalized measurements for an measurement operator $$M_{\alpha}$$ (not necessary projections operator) if $$\sum_{\alpha} M_{\alpha}^H M_{\alpha}$$

But does anyone have an other definition to help me understand?

• What does it means that $$M_{\alpha}$$ is an operator?
• It just has to be a linear operators ?
• Moreover all the different operator of the set $$\left \{ M_{\alpha} \right \}$$ has to measure the same quantity $$\alpha$$ but what does it mean? That means that each $$M_{\alpha}$$ is susceptible to give us a different possible value of the quantity $$\alpha$$ ? It is strange because when we only want operators to be Hermitian the same operator $$H$$ apply on a state can give all the possible values of the quantity (that are the eigenvalues of $$H$$) that the operator $$H$$ represent? (At least this is what I've learned).

More generally any more precise, even longer definition/explanation on what is Generalized measurements will be great.

$$\newcommand{\bra}[1]{\left<#1\right|}\newcommand{\ket}[1]{\left|#1\right>}\newcommand{\bk}[2]{\left<#1\middle|#2\right>}\newcommand{\bke}[3]{\left<#1\middle|#2\middle|#3\right>}$$ I will try to explain the generalized measurements, and I hope I address all of your questions in my explanation.

Quantum measurements are described by a set $$\{M_k\}$$ of measurement operators satisfying one constraint \begin{align} \sum_k M_k^\dagger M_k = I \,.\tag{1} \end{align}

Given a state $$\ket{\psi}$$, immediately after measurement, it becomes $$\ket{\psi_k}$$, $$$$\ket{\psi} \xrightarrow[]{\{M_k\}}\ket{\psi_k} \equiv \frac{M_k \ket{\psi}}{\sqrt{p_k}}\,,\tag{2}$$$$

with probability

\begin{align} p_k &= \bra{\psi}M^\dagger_kM_k\ket{\psi}\tag{3.1}\,,\\ &= \lVert M_k \ket{\psi} \rVert^2\tag{3.2}\,,\\ &\geq 0 \tag{3.3}\,. \end{align} Measurement outcome is the index $$k$$ of the state that resulted.

Assume if you perform measurement corresponding to $$\{ M_1, M_2, \cdots, M_k \}$$ on state $$\ket{\psi}$$, then there can be $$k$$ different outcomes which can occur. Any of these outcomes can occur at random with probabilistic odds given by Eq.$$(3.1)$$.

$$$$\ket{\psi}\xrightarrow[]{\{ M_1,M_2, \cdots, M_k \}}\begin{cases} \ket{\psi_1}, & \text{with probability p_1 ----- Getting outcome 1}\\ \ket{\psi_2}, & \text{with probability p_2 ----- Getting outcome 2}\\ \vdots\\ \ket{\psi_k}, & \text{with probability p_k ----- Getting outcome k} \end{cases}$$$$

You can think of these outcomes as some pointer device in your laboratory. If you have $$k$$ different possible outcomes for the measurement, in the lab, your apparatus will point to one of the $$k$$ possible points, and then you can say that $$k$$-outcome has happened, and then you can deduce that state of your system now is $$\ket{\psi_k}$$.

So, in general, these operators $$\{M_k\}$$ can be any linear operators, i.e. matrices, which satisfy the condition given in Eq. $$(1)$$.

However, there are special cases of this general mathematical framework of generalized measurements where, in each special case, apart from Eq.$$(1)$$, there are some additional conditions on measurement operators. For example, in Projective (Vonn Neumann) measurements, these measurement operators form a complete set of orthogonal projectors.

• So if I understood you well given a Hermitian operator is a special case. The set of $(M_{\alpha})$ of this operator will be the set of $(\sqrt{P_{\lambda_i}})$ with $P_{\lambda_i}$ a projector into the sub space corresponding to the e.v. $\lambda_i$? Commented Feb 10 at 11:28
• You are correct. We call that measuring an observable, but underlying mathematics is the same. Every hermitian operator is an observable. We can write the spectral decomposition of the $H$ as $$H = \lambda_k |k\rangle \langle k| \,,$$ where now, your possible outcomes are distinct eigenvalues $\{\lambda_k\}$ and measurement operators are the outer product of eigenvectors corresponding to that eigenvalue $\lambda_k$. $$M_k = P_k = |k\rangle \langle k|\,.$$ If there is degenercy in eigenvalues, then the corresponding projection measurement operator would be more than dimension 1. Commented Feb 10 at 19:22
Forgetting for a second the way you formalise generalised measurements (that is, POVM), ask yourself how a "measurement" should be formalised in quantum mechanics. A measurement is effectively some rule sending any quantum state into a probability distribution describing the probability with which you should see each of the possible outcomes. In other words, for each outcome, call it $$k$$, you need a rule (that is, a function) sending each input state $$\rho$$ into a corresponding probability. Let us denote this function with $$p_k$$, so that $$p_k(\rho)\in[0,1]$$ for all states $$\rho$$.
As it happens, this function is linear. So for each outcome $$k$$, we need a linear function $$p_k$$ sending density matrices $$\rho$$ (which are linear operators) into numbers $$p_k(\rho)$$. As per usual considerations about duality, any such linear functional can be represented as $$p_k(\rho)=\langle \mu_k,\rho\rangle\equiv \operatorname{tr}(\mu_k\rho)$$ for some Hermitian $$\mu_k$$. Due to the bijection between $$p_k$$ and $$\mu_k$$, we can directly say that a "measurement" is the operator $$\mu_k$$. Or better said, a "measurement" is the set of such operators $$\mu_k$$. The other conditions on $$\{ \mu_k\}$$ follow from the properties of $$p_k$$: we want $$\sum_k p_k(\rho)=1$$, which corresponds to $$\sum_k \mu_k=I$$, and we want $$p_k(\rho)\in[0,1]$$, which corresponds to $$0\le \mu_k \le I$$.
The operators $$\{\mu_k\}$$ form a so-called POVM, which is the most general way to describe a measurement in quantum mechanics. But note that this is meant in the sense of: the most general way to describe the relation between quantum states and measurement probabilities. In particular, POVMs do not deal with/describe the potential states you might still have after the measurement. If you want to describe post-measurement states, the better approach is imo to think of it in terms of a measurement channel. The "generalised measurements" discussed in N&C are a (IMO) somewhat cryptic way to do this. They relate with the POVM as described above via the simple relation $$\mu_k=M_k^\dagger M_k$$. However, the operators $$M_k$$ also describe a way to get post-measurement states via the usual rule also mentioned in the other answer. More formally, you can think of $$M_k$$ as the Kraus operators of the measurement channel describing the situation.