My answer to your question would be yes. If you recreate a human brain at the atomic level you will create a second copy of the same consciousness that would be as much the same person as the original. That is provided you also create enough of the rest of the person for the brain to function. I would leave out the word "classical" because if you are talking about the atomic level then you are using quantum mechanics.
It is then interesting to ask how much detail you really need to make the same true. Surely getting every atom in the right place is over-kill? Would a simulated version of the brain be equally acceptable? Is it enough to just model the behavior of each neuron accurately? If so what happens if the copy becomes even more crude? Suppose it became possible to create general simulations of a human brain and train them to replicate your memories? Perhaps someone could just undergo a long interview with a computer in which they recall all their memories and discuss their opinions on everything they can think off, supplemented with photos of all the people and places they have known. If the computer could then use that information to create a version of them that would fool anyone, would it be a valid conscious copy? Suppose the copy is constructed from other people's memories of a person after they die instead of their own memories?
At some point on this scale of possibilities most people are going to say no, that is not a valid copy. I don't think there is a clear point at which that happens. I think that in the future all these things will become possible and it will start with the less perfect copies. Long before they figure out how to revive the people who had their head's frozen at death, it will be possible to create very convincing simulations of people just from an in-depth interview, or even produce a convincing virtual resurrection of someone who died. If you ask people now they are going to say that it will just be an imperfect simulation and not really a valid copy of consciousness. I think when it actually happens this is going to change the way people view consciousness and the self and there will be varying opinions about what it means philosophically. Some people will see a long interview as a much more reliable route to immortality than cryogenics.
I know people have asked questions like this but I don't think they have not gone far enough to anticipate the way it will actually pan out.