How do you experimentally realise the operator of the form $\hat{A}\hat{B}+\hat{B}\hat{A}$? I was reading a paper in which the authors use the operator of the form $\hat{A}\hat{B}+\hat{B}\hat{A}$  and it is implied to be experimentally realisable. 
(i.e either creating an apparatus to measure this operator or/and using some method to calculate the expectation value) 
My question is about how in general this operator be experimentally realised in terms of some combination of measurements of $\hat{A}$ and $\hat{B}$ where $\hat{A}$ and $\hat{B}$ are hermitian operators that are measurable?
Edit: 
If $\hat{A}$ and $\hat{B}$ commute we can calculate the $<\hat{A}\hat{B}>$ by just multiplying the measurements of $\hat{A}$ and $\hat{B}$ and then taking the average value. 
Therefore my question reduces to non- commuting observables only. 
 A: 
My question is about how in general this operator be experimentally realisable in terms of some combination of measurements of $\hat{A}$ and $\hat{B}$ where $\hat{A}$ and $\hat{B}$ are hermitian operators that are experimentally realisable?

There is no reason to believe that a measurement of $\hat{A}\hat{B}+\hat{B}\hat{A}$ can be realized as a combination of the measurements of $\hat{A}$ and $\hat{B}$ unless, of course, $[\hat{A},\hat{B}]=0$. 
If the two operators commute, you can simply measure one and then the other. The measurement of $\hat{B}$ won't take the state out of the eigensubspace of $\hat{A}$ into which it was projected as a result of the measurement of $\hat{A}$ and thus, you can simultaneously measure both, and thus, any combination thereof.
However, if the two operators don't commute, you cannot measure something like $\hat{A}\hat{B}+\hat{A}\hat{B}$ using only the measurement of $\hat{A}$ and $\hat{B}$ for they simply cannot be measured simultaneously (as they don't share an eigenbasis). 
Notice that the formalism of quantum mechanics as such allows for the measurement of any Hermitian operator but it might be incredibly complex to actually measure some arbitrarily cooked up Hermitian operator. Physically meaningful operators are almost always the ones associated with symmetries.

Of course, if you have a large number of copies of the state, you can simply deduce the wavefunction of the state using observations of a tomographically complete set of observables and then you can calculate the expectation value of the operator $\hat{A}\hat{B}+\hat{B}\hat{A}$ over the given state without needing to actually measure it. However, if you're interested in actually measuring the state and collapsing it to one of the eigenstates of $\hat{A}\hat{B}+\hat{B}\hat{A}$, you have to build a clever enough experiment to do so if $[\hat{A},\hat{B}]\neq 0$. 

Important Point Raised by @NorbertSchuch 
Even in the case when $\hat{A}$ and $\hat{B}$ commute, while it is true that (twice) the product of the eigenvalues obtained by the measurement of $\hat{A}$ and $\hat{B}$ would give the same eigenvalue as the measurement of $\hat{A}\hat{B}+\hat{B}\hat{A}$ would give on the same state, it's not necessary that the post-measurement state would be the same.
To steal the nice example that @NorbertSchuch provided in the comment, consider the state $\left\vert\uparrow\uparrow\right\rangle+\left|\downarrow\downarrow\right\rangle$. A measurement of $\hat{\sigma}_z\otimes\hat{\sigma}_z$ would give the eigenvalue $1$ and the post-measurement state would simply remain $|\uparrow\uparrow\rangle+|\downarrow\downarrow\rangle$ as it is an eigenstate of $\hat{\sigma}_z\otimes\hat{\sigma}_z$. However, if you make the simultaneous measurements of $\hat{\sigma}_z\otimes\hat{\mathbb{I}}$ and  $\hat{\mathbb{I}}\otimes\hat{\sigma}_z$ then the resulting state would be either $|\uparrow\uparrow\rangle$ or $|\downarrow\downarrow\rangle$. Notice that in either case, we would still get the product of the eigenvalues to be $1$ in agreement with the direct measurement of $\hat{\sigma}_z\otimes\hat{\sigma}_z$. 
A: The smallest explicit example I could come up with where this can be done for sure, a bit silly maybe: let
$$A = \begin{pmatrix}1&0\\0&-1\end{pmatrix},\ \ \ \ \ B = \begin{pmatrix}1&-i\\i&1\end{pmatrix}.$$
Then $A$ and $B$ don't commute, and
$$AB + BA = \begin{pmatrix}2&0\\0&-2\end{pmatrix}.$$
This means that you can just measure $A$ and multiply the result by 2 to get a measurement of $AB + BA$.
If you want to realize this physically, you could for example interpret these as measuring the spin of a spin 1/2 particle in the usual basis. Then $A$ measures (a fixed multiple $2/\hbar$ of) the spin in $z$-direction, and $B$ measures a different function of spin in $y$-direction, namely 0 for up and 2 for down. 
Their anticommutator measures $4/\hbar$ times the spin in $z$-direction, so you could just change the configuration file of the device measuring spin in $z$-direction.
A: Such quantities are measured more frequently than it may seem: indeed, any real (i.e. measurable) correlation function of two non-commuting quantities is given by
$$K(t, t_1) = \langle \frac{1}{2}\left[\hat{A}(t), \hat{B}(t_1)\right]_+\rangle =\frac{1}{2}\left[
\langle\hat{A}(t)\hat{B}(t_1)\rangle + \langle\hat{B}(t_1)\hat{A}(t)\rangle\right], 
$$
which is routinely measured in experiments, e.g., in spin resonance.
Even when this quantity is not measured directly, it is easily obtained via the response function (aka susceptibility)
$$\varphi(t, t_1) = \langle \frac{i}{\hbar}\left[\hat{A}(t), \hat{B}(t_1)\right]_-\rangle,$$
to which it is related via the fluctuation-dissipation theorem
$$K(\omega)=\hbar\coth\left(\frac{\hbar\omega}{k_BT}\right)Im\varphi(\omega).$$
A: Some physical quantities such as energy, momentum, position, are associated with corresponding operators. It does not follow that arbitrary operators have a meaningful physical quantity associated with them.
If A, B do not commute, it is likely that measurement of A prevents simultaneous measurement of B. For example, measurement of position prevents simultaneous measurement of momentum. Operator $\hat{x}\hat{p}_x$ is not associated with any common useful physical quantity and there is no common way to measure it.
