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lecturenotes4 - CSE 599d Quantum Computing The No-Cloning...

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CSE 599d - Quantum Computing The No-Cloning Theorem, Classical Teleportation and Quantum Teleportation, Superdense Coding Dave Bacon Department of Computer Science & Engineering, University of Washington I. THE NO-CLONING THEOREM Cloning has been in the news a lot lately. But today we are not going to talk about that type of cloning, but instead of a process of duplicating quantum information. That we are not talking about cloning DNA (for our results will not be positive) is rather fortunate for the Raelians! The no-cloning theorem is one of the earlier results in the study of quantum information. It has an interesting history, some of which is written down in a paper by Asher Peres, “How the No-Cloning Theorem Got Its Name” which is available online at http://arxiv.org/abs/quant-ph/0205076 and is generally attributed to Wootters, Zurek, and Dieks in 1982. (When I was an undergraduate at Caltech, they had an automated system for requesting copies of articles when you were searching their publication database. I discovered, probably around my junior year, the papers of William Wootters, promptly ordered the automated system to print out every paper Wootters had ever written at that time and my life hasn’t been the same ever since! Wootters thesis is, in my opinion, one of the most interesting resutls I’ve ever encountered.) So what is the no-cloning theorem? Suppose that we have in our lab a qubit in an unknown quantum state | ψ = α | 0 + β | 1 . Actually it is perhaps better to say that some external referee knows a description of this quantum state, but we in the lab don’t know this quantum state. Now we can construct all sort of machines (unitaries and measurements) which act on this qubit. The question posed in the no-cloning theorem is whether it is possible to design a machine which, for all possible actual states | ψ , is able to take this state | ψ and create two copies of this state | ψ ⊗ | ψ , hence “cloning” the quantum state. Let’s prove a simple version of the no-cloning theorem. We want to show in this version that there is no unitary operation which can enact the evolution | ψ ⊗ | 0 → | ψ ⊗ | ψ for all possible states | ψ . To see this, suppose that there exists such a unitary. Then it must be able to clone | 0 and | 1 : U | 0 ⊗ | 0 = | 0 ⊗ | 0 and U | 1 ⊗ | 0 = | 1 ⊗ | 1 Then if this is true, by the linearity of quantum theory, U 1 2 ( | 0 + | 1 ) ⊗ | 0 = 1 2 ( | 0 ⊗ | 0 + | 1 ⊗ | 1 ) which is not equal to 1 2 ( | 0 + | 1 ) 1 2 ( | 0 + | 1 ) which is what we would require if this were a cloning unitary. Thus we have a contradiction: no such unitary can exist. We’ve shown the no-cloning theorem for qubits and for just unitary transformations. In fact there is a more general version of this theorem, called the no-broadcasting theorem which deals with an even more general situation. Suffice it to say, in all of these formulations a quantum machine which can perfectly clone all quantum states is shown to not be constructible. An interesting question which we won’t talk about too much, but which has led to a lot of very nice results is what about
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