No cloning theorem

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The no cloning theorem is a result of quantum mechanics which 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.

Note that 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 does not constitute cloning since no well-defined state can be attributed to a subsystem of an entangled state. The term cloning refers to a process whose end result is a separable state whose factors are identical .

Contents

1 Proof

1.1 Note

2 Consequences
3 Imperfect cloning
4 See also
5 References

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

[|\psi\rangle_A |e\rangle_B.]

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 up to some fixed time interval, which is an unitary operator. Then U acts as a copier provided

[U |\psi\rangle_A |e\rangle_B = |\psi\rangle_A |\psi\rangle_B]

and

[U |\phi\rangle_A |e\rangle_B = |\phi\rangle_A |\phi\rangle_B]

for all [| \phi \rangle] and [| \phi \rangle]. By definition of unitary operator, U preserves the inner product:

, i.e.

[\langle \phi | \psi \rangle = \langle \phi | \psi \rangle ^2.]

This is clearly not true in general. Therefore no such U exists. This proves the no cloning theorem.

Note

In the statement of the theorem, two assumptions were made: the state to be copied is a pure state and the proposed copier acts via unitary time evolution. These assumptions cause no loss of generality. If the state to be copied is a mixed state, it can be purified. Similarly, an arbitrary quantum operation can be implemented via introducing an ancilla and perform a suitable unitary evolution. Thus the no cloning theorem holds in full generality.

Consequences

The no cloning theorem prevents us from using classical error correction techniques on quantum states. For example, we cannot create backup copies of a state in the middle of a quantum computation, and use them to correct subsequent errors. Error correction is vital for practical quantum computing, and for some time this was thought to be a fatal limitation. In 1995, Shor and Steane revived the prospects of quantum computing by independently devising the first quantum error correcting codes, which circumvent the no cloning theorem.

In contrast, the no cloning theorem is a vital ingredient in quantum cryptography, as it forbids eavesdroppers from creating copies of a transmitted quantum cryptographic key.

The no-cloning theorem protects the uncertainty principle in quantum mechanics. If one could clone an unknown state, then one could make as many copies of it as one wished, and measure each dynamical variable with arbitrary precision, thereby bypassing the uncertainty principle. This is prevented by the no cloning theorem.

Similarly, cloning would violate the no teleportation theorem, which says classical teleportation (not to be confused with entanglement-assisted teleportation) is impossible. In other words, quantum states cannot be measured reliably.

The no cloning theorem prevents superluminal communication via quantum entanglement, as cloning is a sufficient condition for such communication. Consider the EPR thought experiment, and suppose quantum states could be cloned. Assume parts of a maximally entangled Bell state are distributed to Alice and Bob. Alice could send bits to Bob in the following way: If Alice wishes to transmit a "0", she measures the spin of her electron in the z direction, collapsing Bob's state to either [|z+\rangle_B] or [|z-\rangle_B]. To transmit "1", Alice does nothing to her qubit. Bob creates many copies of his electron's state, and measures the spin of each copy in the z direction. Bob will know that Alice has transmitted a "0" if all his measurements will produce the same result; otherwise, his measurements will have outcomes +1/2 and -1/2 with equal probability. This would allow Alice and Bob to communicate across space-like separations, potentially violating causality.

Imperfect cloning

Even though it is impossible to make perfect copies of an unknown quantum state, it is possible to produce imperfect copies. This can be done by coupling a larger auxiliary system to the system that is to be cloned, and applying a unitary transformation to the combined system. If the unitary transformation is chosen correctly, several components of the combined system will evolve into approximate copies of the original system. Imperfect cloning can be used as an eavesdropping attack on quantum cryptography protocols, among other uses in quantum information science.

References

Wootters, W.K. and Zurek, W.H.: A Single Quantum Cannot be Cloned. Nature 299 (1982), pp. 802-803
Dieks, D.: Communication by EPR devices. Physics Letters A, vol. 92(6) (1982), pp. 271-272
Buzek, V. and Hillery, M.: "Quantum cloning". Physics World 14 (11) (2001), pp. 25-29