Current technology is very far from being able to clone a person or recreating their mind in fashion similar to the ship of Theseus (quantum teleportation and genetic cloning are more hype that near-future possibility).
The most likely possibility given the current level of technology is that computer simulation and AI technology are advanced enough that we are able to upload a copy of a person's personality to a computer that is faithful enough to the original that it can be considered a virtual ship of Theseus recreation of that person.
Ray Kurzweil is already trying to recreate his dead father from memories, and talks about digital minds in his book "How to Create a Mind". He says that using digital mind creating technology, we will achieve immortality.
Derek Parfit gives the philosophical version of the Ship of Theseus paradox applied to the question of the mind in his book "Reasons and Persons", and he calls it the Teleporter Paradox, even if the technology for this is very farfetched.
Update per the OP comments:
The OP mentions the example of interconnected brains of 4 rats as an example of the bundle theory of mind. The interconnected mind example case is not a direct application of the ship of theseus paradox applied to mental states and personal identity. However both cases (i.e. mental ship of theseus and interconnected brains) are related to the question of personal identity and the bundle theory.
The mental version of the Ship of Theseus would proceed as follows: if we were to gradually replace each individual component of a person's mind or brain (keep in mind that mind and brain are not necessarily the same thing), will the end result of that replacement process still be the same person? or will it be a different person?
Alternatively if we gradually replaced each individual component of that person's mind of brain, and we use the old components to build a second mind or brain, will the result be 2 copies of the same person? or will it be 2 separate people?
On the other hand the 4 rats with interconnected brains example that the OP mentions leads to the following question: does this experiment lead to 4 interconnected minds or does it lead to on large mind instantiated on 4 separate brains?
As I mentioned above the 2 questions are related in the sense that they both challenge our preconceptions about personal identity (that there is a central and unique locus of personal identity located in us).
If Hume (and other's) bundle theory is correct, then the answers are:
In the mental Ship of Theseus case, the result will be 2 separate people. This is the conclusion that Derek Parfit arrived at in his teleportation example.
In the 3 rat brains example, the end results would be one single mind combining the memories and sensations of all 3 physical rat brains. This is supported by experimental evidence from the opposite case, i.e. split brain patients. In cases where certain connections within a single patients brain are severed, the patient starts to exhibit symptoms of having two separate personalities. Some have gone so far as to interpret this as there being "two people" in one body. See here, here and here. Presumably if splitting one brain into 2 leads to 2 different people, then adding 3 brains together leads to 1 new person.
There is another way of looking at the 4 rats experiment however, which could lead to a different conclusion: If one subscribes to the computational theory of mind, that the brain is a computer and the mind is just the software to this computer, then when we connect the 4 rat brains together, are we dealing with 4 interconnected computers like the internet ? or with one distributed computing cluster running on 4 processors?
Going back to the split brain case: Splitting one brain into to 2 leads to 2 personalities, but splitting the brain into 1000 doesn't lead to a 1000 personalities. It only leads to a broken brain. There a point beyond which you cannot break down the brain into separate computationally autonomous components, a minimum architecture is necessary for the brain to support a single personality.
The point I am trying to make here is that there are limits to how far you can take the bundle theory: obviously one thought or one sensation doesn't constitute a single mind, nor can any arbitrary combination of minds lead linearly to one single mind. If the computational theory of mind holds, then there are certain architectural limitations to what constitutes and mind and what is just a bunch of random memories and sensations.
The authors of the article on the rat brain seem to be hinting that they are looking at it from that point of view, as opposed to the first case where the bundle theory of mind implies that minds can be arbitrarily divided and added up like heaps of sand.
From the Newsweek article:
The study concluded that the networked brains, through sharing the burden of problem solving and attacking the challenge from different perspectives and approaches, were equally if not more efficient than a single brain in executing the same task. This processes is analogous to the increasing the number of web servers to spread out the workload and increase the efficiency of a computer system through collective input.
- Ship of theseus and the 4 rat brains experiment are different but related concepts that both challenge the concept of personal identity, especially in the context of the bundle theory.
- The 4 rat brains experiment might be seen as supporting the bundle theory of the mind.
- However, if one takes into account the idea that the brain is a computer and the mind is the software, there are limits to how one can divide and add mind and still always end up with new minds/personalities in the process.