One possible stance referred to by Drux is that with perfect copying none of the resulting instances are more real than the others (Parfit). This is rational and OK as far as it goes. But it's only an observation about the immediate reality, and does not go into consequences.
Two other answers referred to by Alexander S. King are that mind is supernatural (Descartes, nonsense), and that there is no "I" or "you" (Buddha, Hume, Russel, again nonsense, as, for example, I can attest that I exist). Alexander also referred to Hofstadter giving essentially the realistic view of Parfit. Out of the five philosophers mentioned in the answers before this, then, two have apparently had reasonable rational views on the matter.
Going one step further then Perfit/Hostadter, or at least beyond what's been referred from them so far, we can imagine the copy on Mars doing some criminal act immediately after the copying. It could for example be fraud, but to make it interesting let's say that it's murder. Then there are two questions that at least springs readily to my mind:
Is the instance on Earth legally responsible for the Mars actions?
There's almost no difference between the two minds at the time of the murder. So what one of them would do, the other would also do, placed in the same situation. Therefore the one on Earth would have committed the murder if he (or she) just were placed in the appropriate situation. If the purpose of punishment is to some degree correction of a mind that's otherwise prone to criminal behavior, then it stands to reason to apply that to the instance on Earth. But this involves punishing someone who's done nothing wrong.
Is the instance on Earth responsible if the Mars one murders and suicides?
Someone is bound to hit on the idea of creating a body+mind clone via teleportation, where that clone murders someone and then commits suicide. Having planned and executed this, can the original be punished? After all, one probably can't prove that it was planned, that he/she was an accessory or used the clone merely as an instrument to commit the murder.
Since much of the purpose of law is to protect society, I guess that if this hypothetical situation were ever to develop then the law would view the person on Earth as responsible, at least for some time after copying. But that does not necessarily imply a view where the one on Earth is viewed as the same person (or to a large degree the same) as the one on Mars. Rather, it might just mean that the law would hold anyone responsible for making a criminal act much more likely by doing the teleportation without being very sure that one would never do anything really criminal within such and such a short timespan.
I think these considerations might appear to extremely far-fetched, due to the OP's choice of mechanism for mind cloning. But it's worth noting that any reasonable calculation places an upper limit of about 40 years on the appearance of computational power sufficient to do a real time human brain simulation, which indicates that machine intelligences (although possibly not very mature) will appear well before that. And with a machine intelligence cloning the mind might be a practical proposition, rather then science-fiction-ish.
Then there will be a need for suitable legislation. E.g., one would need legal mechanisms to deal with ownership and responsibility.