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Theorem in quantum information science
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, 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 William Wootters and Wojciech H. Zurek as well as Dennis Dieks 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. 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. 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).

According to Asher Peres and David Kaiser, the publication of the 1982 proof of the no cloning theorem by Wootters and Zurek for a superluminal communication device using quantum entanglement, and Giancarlo Ghirardi 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 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. Similarly, an arbitrary quantum operation can be implemented via introducing an ancilla and performing a suitable unitary evolution. Thus the no cloning theorem holds in full generality.

The no cloning theorem prevents the use of certain classical error correction techniques on quantum states. For example, backup copies of a state in the middle of a quantum computation cannot be created and used for correcting subsequent errors. Error correction is vital for practical quantum computing, and for some time it was unclear whether or not it was possible. In 1995, Shor and Steane showed that it is, by independently devising the first quantum error correcting codes, which circumvent the no cloning theorem. Similarly, cloning would violate the no teleportation theorem, which says that it is impossible to convert a quantum state into a sequence of classical bits (even an infinite sequence of bits), copy those bits to some new location, and recreate a copy of the original quantum state in the new location. This should not be confused with entanglement assisted teleportation, which does allow a quantum state to be destroyed in one location, and an exact copy to be recreated in another location. The no cloning theorem is implied by the no communication theorem, which states that quantum entanglement cannot be used to transmit classical information (whether superluminally, or slower). That is, cloning, together with entanglement, would allow such communication to occur. To see this, 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 or 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 produce the same result; otherwise, his measurements will have outcomes or with equal probability. This would allow Alice and Bob to communicate classical bits between each other (possibly across space like separations, violating causality). Quantum states cannot be discriminated perfectly. The no cloning theorem prevents an interpretation of the holographic principle for black holes as meaning that there are two copies of information, one lying at the event horizon and the other in the black hole interior. This leads to more radical interpretations, such as black hole complementarity. The no cloning theorem applies to all dagger compact categories: there is no universal cloning morphism for any non trivial category of this kind. Although the theorem is inherent in the definition of this category, it is not trivial to see that this is so; the insight is important, as this category includes things that are not finite dimensional Hilbert spaces, including the category of sets and relations and the category of cobordisms.

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

V. Buzek and M. Hillery, Quantum cloning, Physics World 14 (11) (2001), pp. 25–29.