The meaning of 'coupling'? In quantum mechanics if two quantities $A$ and $B$ are said to be coupled what does this actually mean?
I would guess that it means we have a term like $A\cdot B$ in the Hamiltonian but this is only a guess.
 A: Let me preface by saying that "coupling" is a favorite physicist word that is perhaps best described linguistically than rigorously; it's deployed in a few different situations.
In general, we say that a coupling exists in quantum mechanics if the evolution of one part of the system depends on another quantity, which could be either classical or quantum. I'll give one example for each.
Suppose the Hamiltonian of a two-level system is an internal Hamiltonian $H_\mathrm{int}$ and an additional part that depends on some external parameter, maybe $\theta$:
\begin{equation}
H = H_\mathrm{int} + \theta \sigma_z
\end{equation}
Here the $\sigma_z$ was arbitrary. The point is that this system's evolution depends directly on the parameter $\theta$--maybe it's an external magnetic field, or some other feature of the environment. In this case, we would generally say that the system is "coupled to $\theta$." (You'd often see this in a metrological context, where we might be interested in using a quantum system coupled to an external parameter to measure the parameter.) In this case, there is only one quantum object, evolving under $H$.
Another common system--maybe a little more general--would be the evolution of two different variables both treated quantum mechanically. The idea here is that there would be some operator $A$ characterizing one observable of interest, and another $B$ characterizing a second. Then the Hamiltonian might be:
\begin{equation}
H = H_A + H_B + H_{AB}
\end{equation}
Where $H_A$ doesn't contain any term depending on $B$, $H_B$ doesn't contain any term depending on $A$, and $H_{AB}$ might have terms like $A \cdot B$, $A^2 B$, etc. The reason why this couples the system is that if we now evaluate Heisenberg equations of motion $\dot{A} = \frac{i}{\hbar} \left[ H, A \right]$ we'll find that the $H_{AB}$ term will put terms depending on $B$ into $\dot{A}$ and vice versa. Therefore, solving the equations of motion will require describing both $A$ and $B$. On the other hand, if the equations "decouple" or we do something to decouple them ourselves, we can usually find a solution for $A(t)$ that doesn't depend on $B$ and vice versa.
This is all paralleled in classical mechanics, by the way, where we would call two variables coupled if they appeared in each others' equations of motion.
EDIT: Peter Shor points out that objects can be "indirectly coupled," which is correct but would usually require me to introduce another variable $C$. I think the most general statement of being coupled/uncoupled is asking whether the equations of motion can be solved independently of each other.
A: I think Entanglement may answer your question. Two systems are said to be entangled(coupled) if we cannot assign an independent and separate wavefunctions for each system, instead we define a composite system which is simply the tensor product of the original constitutes. To be precise, in the general case the wavefunction description of any quantum system is not satisfactory, so we use a more compact form to define the general state of any quantum system and this will be the density matrix which is simply a bijective map that has the power to define both quantum populations and coherences which is something that is lacking in the wavefunction treatment, also density matrices is most suitable when dealing with open quantum systems problems. 
So now you can assign a density matrix for each of your variables and another one for the composite of them both which is simply a tensor product and to say that they are coupled (entangled) is like saying that the matrix of the composite system cannot be written as the product of the original matrices.  
Any interaction between two systems within the context of open quantum systems will have some washed up correlation upon tracing out any of the degrees of freedom of any of the two systems, resulting in an entagled state. 
