No-cloning theorem

In physics, the no-cloning theorem states that it is impossible to create an identical copy of an arbitrary unknown quantum state. This no-go theorem of quantum mechanics was articulated by Wootters and Zurek[1] and Dieks[2] 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 result is a separable state with identical factors. According to Asher Peres[3] and David Kaiser,[4] the publication of the no-cloning theorem was prompted by a proposal of Nick Herbert[5] for a superluminal communication device using quantum entanglement.

The no-cloning theorem is normally stated and proven for pure states; the no-broadcast theorem generalizes this result to mixed states.

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.[6][7] 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 classical logic arises from Cartesian closed categories).

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, in quantum state B irrespective of the original state (see bra–ket notation). 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 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.

We can perform an observation, which irreversibly collapses the system into some eigenstate of an observable, corrupting the information contained in the qubit(s). This is obviously not what we want.

Alternatively, we could control the Hamiltonian of the combined system, and thus the time-evolution operator U(t) e.g for a time independent Hamiltonian, . Evolving up to some fixed time yields a unitary operator U on , the Hilbert space of the combined system . However, no such unitary operator U can clone all states:

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 unitary,

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.


Mixed states and nonunitary operations

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. 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. 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.

Arbitrary sets of states

Non-clonability can be seen as a property of arbitrary sets of quantum states. If we know that a system's state is one of the states in some set S, but we do not know which one, can we prepare another system in the same state? If the elements of S are pairwise orthogonal, the answer is always yes: for any such set there exists a measurement which will ascertain the exact state of the system without disturbing it, and once we know the state we can prepare another system in the same state.

On the other hand, if S contains a pair of elements that are not pairwise orthogonal, then an argument like that given above shows that the answer is no. So even if we can narrow down the state of a quantum system to just two possibilities, we still cannot clone it in general (unless the states happen to be orthogonal).

Another way of stating the no-cloning theorem is that amplification of a quantum signal can only happen with respect to some orthogonal basis. This is related to the emergence of the rules of classical probability via quantum decoherence.

No-cloning in a classical context

There is a classical analogue to the quantum no-cloning theorem, which might be stated as follows: given only the result of one flip of a (possibly biased) coin, we cannot simulate a second, independent toss of the same coin. The proof of this statement uses the linearity of classical probability, and has exactly the same structure as the proof of the quantum no-cloning theorem. Thus, in order to claim that no-cloning is a uniquely quantum result, some care is necessary in stating the theorem. One way of restricting the result to quantum mechanics is to restrict the states to pure states, where a pure state is defined to be one that is not a convex combination of other states. The classical pure states are pairwise orthogonal, but quantum pure states are not.

No-cloning in logic

In logic, the idea of no-cloning and no-deleting correspond to the notion of disallowing two rules of inference: the rule of weakening (monotonicity of entailment) and the rule of contraction (idempotency of entailment). The removal of these two rules of inference from classical logic results in linear logic, which is the form of logic that describes quantum systems (or, more generally, the behavior of tensor products on Hilbert spaces).


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.[9]

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

See also


  1. Wootters, William; Zurek, Wojciech (1982). "A Single Quantum Cannot be Cloned". Nature. 299: 802–803. Bibcode:1982Natur.299..802W. doi:10.1038/299802a0.
  2. Dieks, Dennis (1982). "Communication by EPR devices". Physics Letters A. 92 (6): 271–272. Bibcode:1982PhLA...92..271D. doi:10.1016/0375-9601(82)90084-6.
  3. Peres, Asher (2003). "How the No-Cloning Theorem Got its Name". Fortschritte der Physik. 51 (45): 458–461. arXiv:quant-ph/0205076Freely accessible. Bibcode:2003ForPh..51..458P. doi:10.1002/prop.200310062.
  4. Kaiser, David (2011). How the Hippies Saved Physics: Science, Counterculture, and the Quantum Revival. W. W. Norton. ISBN 978-0-393-07636-3.
  5. Herbert, Nick (1982). "FLASH—A superluminal communicator based upon a new kind of quantum measurement". Foundations of Physics. 12 (12): 1171–1179. Bibcode:1982FoPh...12.1171H. doi:10.1007/BF00729622.
  6. John Baez, Mike Stay, Physics, Topology, Logic and Computation: A Rosetta Stone (2009)
  7. Bob Coecke, Quantum Picturalism, (2009) ArXiv 0908.1787
  8. S. Abramsky, "No-Cloning in categorical quantum mechanics", (2008) Semantic Techniques for Quantum Computation, I. Mackie and S. Gay (eds), Cambridge University Press. arXiv:0910.2401
  9. Bužek V. and Hillery, M. Quantum Copying: Beyond the No-Cloning Theorem. Phys. Rev. A 54, 1844 (1996)

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