No cloning theorem

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The no-cloning theorem is a result of quantum mechanics that forbids the creation of identical copies of an arbitrary unknown quantum state. It was stated by Wootters, Zurek, and Dieks in 1982, and has profound implications in quantum computing and related fields.

The state of one system can be entangled with the state of another system. For instance, one can use the Controlled NOT gate and the Walsh-Hadamard gate to entangle two qubits. This is not cloning. No well-defined state can be attributed to a subsystem of an entangled state. Cloning is a process whose end result is a separable state with identical factors.

Contents

Proof

Suppose the state of a quantum system A, which we wish to copy, is |\psi\rangle_A (see bra-ket notation). In order to make a copy, we take a system B with the same state space and initial state |e\rangle_B. The initial, or blank, state must be independent of |\psi\rangle_A, of which we have no prior knowledge. The composite system is then described by the tensor product, and its state is

There are only two ways to manipulate the composite system. We could perform an observation, which irreversibly collapses the system into some eigenstate of the observable, corrupting the information contained in the qubit. This is obviously not what we want. Alternatively, we could control the Hamiltonian of the system, and thus the time evolution operator U (for time independent Hamiltonian, U(t)=e^{-iHt/\hbar}, and -H/\hbar is called the generator of translations in time) up to some fixed time interval, which yields a unitary operator. Then U acts as a copier provided that

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