With our human replicator machine, I guess we can (theoretically) suppose that the original person might be scanned, and the information saved, down to the atomic level. But we can't record the quantum states of the individual atoms or subatomic particles without changing them in the process, so it'd be difficult to make a duplicate in whom all the constituent particles have the same quantum state as the original.
Just as we can't send information by checking the polarisation of entangled particles, I can't think of a way of using the divergent quantum states in the original person and the copy to identify which is which.
But, philosophically speaking, might the continuity of (albeit ever-changing) quantum states in the original (which we will leave unchanged, as there is no point in attempting to record and duplicate them) mean that the original person has the greater claim to continuity of identity?
I don't know if building our duplicate with quantum states necessarily different to those of the original will have any biological, and possibly cognitive, effects. While we're familiar with the idea of individual quantum states affecting macroscopic outcomes- Schrödinger's cat- maybe in large objects without too many unstable nuclei, the numerous quantum states are just background noise, rarely affecting activity at the molecular level, or at the level of neuronal firing etc.
Mathematician Roger Penrose has claimed that consciousness is based (directly) on a quantum-level substrate, but this is a minority view and his proposed mechanism- quantum effects in microtubules- has arguably been refuted by Tegmark and others,
https://en.wikipedia.org/wiki/Roger_Penrose.
A different issue, maybe getting us back toward interstellar exploration:
Suppose we send our duplicator, or a machine capable of building a duplicator from local resources, via some conventional nuts-n-bolts method* to a habitable planet** of some nearby star.
A suitably-qualified and trained volunteer is scanned, and the information describing their physical being is transmitted at
c to the replicator. -I'll skate over the fact that's a lot of information that would need to be received perfectly; might take a while.
Our copy awakes, and sharing the motivation and training of the original, starts their duties.
But after some time, our clonenaut*** starts getting a bit despondent. Whatever information they send to Earth will take years to get there, and any grateful response years to come back. Entertainment is limited. (I guess we might suppose that the descriptions of several people are sent, and saved by the replicator, so it can construct a few people in short order).
Though habitable in a broad sense, the alien environment presents a host of challenges and hazards, some unforeseen, that require near-constant consideration and efforts to mitigate their impact.
There is no external help, and any request for additional equipment or personnel will take many years to be fulfilled.
Time to relax is rare, and every so often a new threat or problem arises, taking a toll on the equipment, health and mood of the clonenauts, who have memories of their lives on Earth and realise there's no going back.
They start thinking, "Why did I volunteer for
this?"- and realise,
they didn't.
Unlike, say, a "conventional" astronaut, a serviceman or a scientist in Antarctica, the original volunteers are not facing the hardships and risks of their decision to take part. They remain safe at home, their "others", who were never in a position to give informed consent, face a lifetime of struggle without any prospect of returning to the home of which they have fond memories.
The clone of a specialist plumber who arrived sixteen years into the program, after the initial arrivals found that their plastic piping was prematurely ageing due to some local factor, is furious that life in the colony has been misrepresented to her (subjectively a few days before but actually some years earlier), and starts having issues about her memory of happily receiving a huge payment for "her" participation.
We should take action now, no exploitation of clonenauts!
And support for the future inhabitants of generation starships if they decide to turn round and come back- they didn't consent to a life of duty and privation for themselves and their children.
*Which- as many above posts argue- would be at best immensely difficult, and take a very long time as far as we can tell.
**Assuming one exists.
***(c) John J.