No-cloning theorem

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In physics, the no-cloning theorem is a no-go theorem of quantum mechanics that forbids the creation of identical copies of an arbitrary unknown quantum state. It was stated 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[edit]

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

|\psi\rangle_A \otimes |e\rangle_B \, .

(in the following we will omit the  \otimes symbol and keep it implicit) There are only two ways to manipulate the composite system. We could 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 system, and thus the time evolution operator U (for a time independent Hamiltonian, U(t)=e^{-iHt/\hbar}, where -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

U |\phi\rangle_A |e\rangle_B = |\phi\rangle_A |\phi\rangle_B \,

for all possible states | \phi \rangle in the state space (including |\psi\rangle). Since U is unitary, it preserves the inner product:


\langle e|_B  \langle \phi|_A  |\psi\rangle_A |e\rangle_B 
= \langle e|_B  \langle \phi|_A U^{\dagger} U |\psi\rangle_A |e\rangle_B 
= \langle \phi|_B \langle \phi|_A |\psi\rangle_A |\psi\rangle_B,

and since quantum mechanical states are assumed to be normalized, it follows that

\langle \phi | \psi \rangle =  \langle \phi | \psi \rangle ^2. \,

This implies that either \phi=\psi (in which case \langle \phi | \psi \rangle = 1) or \phi is orthogonal to \psi (in which case \langle \phi | \psi \rangle = 0). However, this cannot be the case for two arbitrary states, although it can hold for certain specific cases. So, for example, choosing orthogonal states in a basis, specifically:  \{ |0\rangle, |1\rangle \} ,

 | \phi \rangle = {1 \over \sqrt{2}} \bigg( | 0 \rangle + | 1 \rangle \bigg)

and

 | \psi \rangle = {1 \over \sqrt{2}} \bigg( | 0 \rangle - | 1 \rangle \bigg)

fit the requirement that \langle \phi | \psi \rangle =  \langle \phi | \psi \rangle ^2 , this result does not hold for more general quantum states. Apparently U cannot clone a general quantum state. This proves the no-cloning theorem.

Generalizations[edit]

Mixed states and nonunitary operations[edit]

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[edit]

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. If S contains two elements that are not pairwise orthogonal (in particular, the set of all quantum states includes such pairs) then an argument like that given above shows that the answer is no.

The cardinality of an unclonable set of states may be as small as two, 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 classical probability rules in quantum decoherence.

No-cloning in a classical context[edit]

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.

Consequences[edit]

  • The no-cloning theorem prevents the use of 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 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.
  • The no-cloning theorem is implied by the no-communication theorem, which states that quantum information channels cannot be used to transmit classical information (whether superluminally, or slower). That is, cloning is a sufficient condition for 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 |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 |z+\rangle_B or |z-\rangle_B with equal probability. This would allow Alice and Bob to communicate classical bits between each other (possibly across space-like separations, violating causality).

Imperfect cloning[edit]

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[edit]

References[edit]

  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/0205076. 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
  9. ^ Bužek V. and Hillery, M. Quantum Copying: Beyond the No-Cloning Theorem. Phys. Rev. A 54, 1844 (1996)

Other sources[edit]

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