No solution to puzzle 8
I confess that I do not know the solution to this puzzle. Dr. Who got himself in a bloody mess.
On one hand, Dr. Who has nothing to worry about. The duplication machine would not "duplicate" his old self even if it recreates his body particle for particle. Indeed, one should also duplicate quantum states of all these particles. Such cloning is strongly forbidden by the no cloning theorem that holds for any closed system. Of course, Dr. Who is not a closed system, but this makes it only worse: Dr. Who is entagled with the rest of the Universe and his "duplication" requires recreation of these entanglements. Interestingly, even in classical physics the exact cloning is generally impossible in a dynamical system: an arbitrary probability distribution associated with the system that is being cloned cannot be copied onto the clone while leaving the original distribution unperturbed. A cloning machine that preserves the statistical independence of the two Dr. Whos is ruled out for any ensemble whose dynamics is Liouvillian, except for the special case when the distributions are non-overlapping. See http://dare.ubvu.vu.nl/bitstream/1871/10137/1/L59.pdf These no-cloning theorems suggest that perfect cloning cannot be achieved for the majority of classical and all of quantum systems. Furthermore, there are inequalities suggesting that the more clones one attempts to make, the poorer is their quality. For optimal cloning of a large number of qubits, the fidelity (the overlap between the source and the target states) is tending to 1/2 (single copy); for a single qubit it is 5/6.
In short, Dr. Who is doomed to be unique -- and he is substantially different from his clone. However, this does not bring us closer to solving the puzzle.
Indeed, knowing that his clone is different from his old self does not help Dr. Who to find out whether he is the source or the target. He needs a way of detecting this difference against the record of his old self. That is an insurmountable problem because any such record is yet another copy, which is different from both his old self and the clone. The paradox is that eventhough cloning is imperfect there still appears to be no way of telling with certainty which Dr. Who was the source and which one was the target - for a closed system.
For an open system one can try to entagle Dr. Who with another system and test for this entanglement after cloning. The problem with this approach is that we do not know anything about the duplication machine: it might interfere with the entangled state and break the entanglement.
I do not know the solution to this paradox and I wonder if such a method exists. I've suggested this puzzle to a few people studying quantum information, but nobody was able to suggest the strategy for Dr. Who.
Can you?
On one hand, Dr. Who has nothing to worry about. The duplication machine would not "duplicate" his old self even if it recreates his body particle for particle. Indeed, one should also duplicate quantum states of all these particles. Such cloning is strongly forbidden by the no cloning theorem that holds for any closed system. Of course, Dr. Who is not a closed system, but this makes it only worse: Dr. Who is entagled with the rest of the Universe and his "duplication" requires recreation of these entanglements. Interestingly, even in classical physics the exact cloning is generally impossible in a dynamical system: an arbitrary probability distribution associated with the system that is being cloned cannot be copied onto the clone while leaving the original distribution unperturbed. A cloning machine that preserves the statistical independence of the two Dr. Whos is ruled out for any ensemble whose dynamics is Liouvillian, except for the special case when the distributions are non-overlapping. See http://dare.ubvu.vu.nl/bitstream/1871/10137/1/L59.pdf These no-cloning theorems suggest that perfect cloning cannot be achieved for the majority of classical and all of quantum systems. Furthermore, there are inequalities suggesting that the more clones one attempts to make, the poorer is their quality. For optimal cloning of a large number of qubits, the fidelity (the overlap between the source and the target states) is tending to 1/2 (single copy); for a single qubit it is 5/6.
In short, Dr. Who is doomed to be unique -- and he is substantially different from his clone. However, this does not bring us closer to solving the puzzle.
Indeed, knowing that his clone is different from his old self does not help Dr. Who to find out whether he is the source or the target. He needs a way of detecting this difference against the record of his old self. That is an insurmountable problem because any such record is yet another copy, which is different from both his old self and the clone. The paradox is that eventhough cloning is imperfect there still appears to be no way of telling with certainty which Dr. Who was the source and which one was the target - for a closed system.
For an open system one can try to entagle Dr. Who with another system and test for this entanglement after cloning. The problem with this approach is that we do not know anything about the duplication machine: it might interfere with the entangled state and break the entanglement.
I do not know the solution to this paradox and I wonder if such a method exists. I've suggested this puzzle to a few people studying quantum information, but nobody was able to suggest the strategy for Dr. Who.
Can you?