No-cloning theorem

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In physics, the no-cloning theorem states that it is impossible to create an independent and identical copy of an arbitrary unknown quantum state, a statement which has profound implications in the field of quantum computing among others. The theorem is an evolution of the 1970 no-go theorem authored by James Park, [1] in which he demonstrates that a non-disturbing measurement scheme which is both simple and perfect cannot exist (the same result would be independently derived in 1982 by Wootters and Zurek [2] as well as Dieks [3] the same year). The aforementioned theorems do not preclude the state of one system becoming entangled with the state of another as cloning specifically refers to the creation of a separable state with identical factors. For example, one might use the controlled NOT gate and the Walsh–Hadamard gate to entangle two qubits without violating the no-cloning theorem as no well-defined state may be defined in terms of a subsystem of an entangled state. The no-cloning theorem (as generally understood) concerns only pure states whereas the generalized statement regarding mixed states is known as the no-broadcast theorem.

Contents

The no-cloning theorem has a time-reversed dual, the no-deleting theorem. Together, these underpin the interpretation of quantum mechanics in terms of category theory, and, in particular, as a dagger compact category. [4] [5] This formulation, known as categorical quantum mechanics, allows, in turn, a connection to be made from quantum mechanics to linear logic as the logic of quantum information theory (in the same sense that intuitionistic logic arises from Cartesian closed categories).

History

According to Asher Peres [6] and David Kaiser, [7] the publication of the 1982 proof of the no-cloning theorem by Wootters and Zurek [2] and by Dieks [3] was prompted by a proposal of Nick Herbert [8] for a superluminal communication device using quantum entanglement, and Giancarlo Ghirardi [9] had proven the theorem 18 months prior to the published proof by Wootters and Zurek in his referee report to said proposal (as evidenced by a letter from the editor [9] ). However, Ortigoso [10] pointed out in 2018 that a complete proof along with an interpretation in terms of the lack of simple nondisturbing measurements in quantum mechanics was already delivered by Park in 1970. [1]

Theorem and proof

Suppose we have two quantum systems A and B with a common Hilbert space . Suppose we want to have a procedure to copy the state of quantum system A, over the state of quantum system B, for any original state (see bra–ket notation). That is, beginning with the state , we want to end up with the state . To make a "copy" of the state A, we combine it with system B in some unknown initial, or blank, state independent of , of which we have no prior knowledge.

The state of the initial composite system is then described by the following tensor product:

(in the following we will omit the symbol and keep it implicit).

There are only two permissible quantum operations with which we may manipulate the composite system:

The no-cloning theorem answers the following question in the negative: Is it possible to construct a unitary operator U, acting on , under which the state the system B is in always evolves into the state the system A is in, regardless of the state system A is in?

Theorem: There is no unitary operator U on such that for all normalised states and in

for some real number depending on and .

The extra phase factor expresses the fact that a quantum-mechanical state defines a normalised vector in Hilbert space only up to a phase factor i.e. as an element of projectivised Hilbert space.

To prove the theorem, we select an arbitrary pair of states and in the Hilbert space . Because U is supposed to be unitary, we would have

Since the quantum state is assumed to be normalized, we thus get

This implies that either or . Hence by the Cauchy–Schwarz inequality either or is orthogonal to . However, this cannot be the case for two arbitrary states. Therefore, a single universal U cannot clone a general quantum state. This proves the no-cloning theorem.

Take a qubit for example. It can be represented by two complex numbers, called probability amplitudes (normalised to 1), that is three real numbers (two polar angles and one radius). Copying three numbers on a classical computer using any copy and paste operation is trivial (up to a finite precision) but the problem manifests if the qubit is unitarily transformed (e.g. by the Hadamard quantum gate) to be polarised (which unitary transformation is a surjective isometry). In such a case the qubit can be represented by just two real numbers (one polar angle and one radius equal to 1), while the value of the third can be arbitrary in such a representation. Yet a realisation of a qubit (polarisation-encoded photon, for example) is capable of storing the whole qubit information support within its "structure". Thus no single universal unitary evolution U can clone an arbitrary quantum state according to the no-cloning theorem. It would have to depend on the transformed qubit (initial) state and thus would not have been universal.

Generalization

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.[ clarification needed ] Alternately, a different proof can be given that works directly with mixed states; in this case, the theorem is often known as the no-broadcast theorem. [11] [12] Similarly, an arbitrary quantum operation can be implemented via introducing an ancilla and performing a suitable unitary evolution.[ clarification needed ] Thus the no-cloning theorem holds in full generality.

Consequences

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. In 1996, V. Buzek and M. Hillery showed that a universal cloning machine can make a clone of an unknown state with the surprisingly high fidelity of 5/6. [15]

Imperfect quantum cloning can be used as an eavesdropping attack on quantum cryptography protocols, among other uses in quantum information science.

See also

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References

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