Entanglement is in the eye of the beholder (4×4 operator, or subsystem division). Yes?
Yes, but that is a pretty useless observation.
The formal definition of an entangled state of a bipartite quantum system with state space $\mathcal H = \mathcal H_A\otimes \mathcal H_B$ is as follows:
* a separable state is one whose density matrix can be separated as a sum of tensor products of individual density matrices, i.e. if $\rho\in \mathcal B(\mathcal H)$ is the density matrix of the system, $\rho$ is separable if and only if there exist density matrices $\rho_{A,i}\in \mathcal B(\mathcal H_A)$ and $\rho_{B,i}\in \mathcal B(\mathcal H_B)$ and weights $p_i\geq 0$ such that $$ \rho = \sum_i p_i \rho_{A,i}\otimes \rho_{B,i}. $$
* an entangled state is any state that is not separable.
For clarity, entanglement is an intrinsic property of the state, together with the partition of the state space into tensor factors.
If you're willing to re-factorize your total state space into some other tensor-product factorization, then a state that's entangled in the $A$, $B$ bipartite scheme is indeed liable to be seen as separable in some alternative $A'$, $B'$ factorization.
However, if you're able to re-factorize your total state space in such a way, then that tells you that your initial split into parties wasn't very meaningful to begin with. In real-world scenarios, we use entanglement as a relevant concept for bipartite systems where the tensor-product factorization of the state space (i.e. 976
the splitting of the system into the two "parties" alluded to in "bipartite") is fixed from the context and cannot be changed easily. If you see it used in a context where that's not the case (ahem) then any conclusions drawn from the entanglement are correspondingly weakened.
One useful way to see this is by noting that the theory of entanglement is, very often, best thought of as a resource theory. Resource theories are great ways to analyze situations where you have one class of operations which is easy to implement but which might be insufficient to achieve some pre-specified goal. Other good examples are thermodynamics (where the operations are energy-conserving processes, and the resource is entropy) and gaussianity (where the operations are linear optical operations); in entanglement, the class of free operations is that of Local Operations and Classical Communication, generally abbreviated as LOCC, and it is obviously tied strictly to a splitting of the system into parties which can operate 'locally' and which can communicate classically.
Resource theories, of course, are only useful when the resource they describe is actually valuable, and when their restricted operations are in fact hard to implement: just as the study of thermodynamics is pretty useless if you have a magical black box that can inject and remove energy from any part of your system at your command, the study of entanglement is pretty meaningless if you have free access to non-LOCC unitary operations that cut across the A-to-B split.
That doesn't mean that you can't talk about entanglement in such a situation, like e.g. the spins of two electrons which are in bound states in the same atom or molecule, but if the re-factorization is physically possible in anything like a reasonable sense, then the conclusions that stem from the presence of the entanglement will be correspondingly trivialized.
But more importantly, if you look at real-world usage, it is always of the form
this system is entangled with that system.
Under your re-factorization, the first part of that sentence to lose its meaning is not "entangled", it's "system".
(The answer below addresses a specific interpretation of v6 of the question, which was, frankly, much more interesting than the current version. I'm keeping it around because of that.)
What <PERSON> provides is known as an entanglement witness, and here you do get some amount of "eye of the beholder" behaviour. 28