I don't think we really know what "me" is yet, to any useful degree. We have no idea what causes our consciousness, or where it "lives" (if anywhere). We don't know if Star Trek-style transporters would actually kill you and create a new you somewhere else. We don't know if it's possible to transmit your consciousness elsewhere, leaving your body behind, a la Altered Carbon.
Sometimes we think of these as philosophical questions, but in many ways they are questions for science that we don't know how to tackle.
Define „I“. Are you mass (obviously) and information?
I'm talking about information that you „have“ in the sense of the one that is part, or maybe more importantly, makes up your mind. If we were able to entangle it with a (partial) information set somewhere else in space then we could achieve what I would call information traveling. And since at quantum level there seems to be something very similar, I hope this would allow us „traveling without mass“.
Entanglement doesn't mean information actually travels from point A to point B. Imagine it like tearing paper into 2 pieces without looking at it. Then both pieces are taken away. And then you look at the exact shape of the cut - and you know that the other piece of paper mirrors it exactly. There was no actual travel of information, you had it the whole time - you just looked at it when it was convenient. Quantum entanglement is nicer than that because it can also guarantee single usage (or the key doesn't work) but it's not a magical telecommunication channel, you still need radio/laser/...
Thanks for the paper analogy. First, I don't agree that if we bring the other half of the paper away that no information has traveled. What else is the shape of the cut than information? Second, do we know that no dual of entanglement exists, meaning that we start from one piece of the paper (far way) that we can somehow select and (re-)create the other half "here", thus being able to see its shape "here", all this not subject to the laws of mass physics? Sure this is all just thinking going on in my mind and I'm really not an expert in all these things; not at all. I just have questions that sometimes provoke ideas; which might be naive (due to lack of information).
Sometimes we think of these as philosophical questions, but in many ways they are questions for science that we don't know how to tackle.