The
no cloning theorem is a result of
quantum mechanicsQuantum mechanics is a set of principles describing the physical reality at the atomic level of matter and the subatomic . These descriptions include the simultaneous wave-like and particle-like behavior of both matter and radiation...
that forbids the creation of identical copies of an arbitrary unknown quantum state. It was stated by
WoottersWilliam Kent Wootters is an American physicist, and a leading contributor to the field of quantum information theory. He proved the no cloning theorem in a joint paper with Wojciech H. Zurek . He has also worked on the quantification of entanglement.He earned a B.S. from Stanford University in...
, Zurek, and Dieks in 1982, and has profound implications in
quantum computingA quantum computer is a device for computation that makes direct use of quantum mechanical phenomena, such as superposition and entanglement, to perform operations on data. The basic principle behind quantum computation is that quantum properties can be used to represent data and perform operations...
and related fields.
The state of one system can be
entangledQuantum entanglement, also called the quantum non-local connection, is a property of a quantum mechanical state of a system of two or more objects in which the quantum states of the constituting objects are linked together so that one object can no longer be adequately described without full...
with the state of another system. For instance, one can use the
Controlled NOT gateThe Controlled NOT gate is a quantum gate that is an essential component in the construction of a quantum computer. It can be used to disentangle EPR states...
and the Walsh-Hadamard gate to entangle two
qubitA quantum bit or qubit is a unit of quantum information. It is the quantum analogue of the classical bit. It is described by a state vector in a two-level quantum-mechanical system, which is formally equivalent to a two-dimensional vector space over the complex numbers.-Bit versus qubit:A bit is...
s.
The
no cloning theorem is a result of
quantum mechanicsQuantum mechanics is a set of principles describing the physical reality at the atomic level of matter and the subatomic . These descriptions include the simultaneous wave-like and particle-like behavior of both matter and radiation...
that forbids the creation of identical copies of an arbitrary unknown quantum state. It was stated by
WoottersWilliam Kent Wootters is an American physicist, and a leading contributor to the field of quantum information theory. He proved the no cloning theorem in a joint paper with Wojciech H. Zurek . He has also worked on the quantification of entanglement.He earned a B.S. from Stanford University in...
, Zurek, and Dieks in 1982, and has profound implications in
quantum computingA quantum computer is a device for computation that makes direct use of quantum mechanical phenomena, such as superposition and entanglement, to perform operations on data. The basic principle behind quantum computation is that quantum properties can be used to represent data and perform operations...
and related fields.
The state of one system can be
entangledQuantum entanglement, also called the quantum non-local connection, is a property of a quantum mechanical state of a system of two or more objects in which the quantum states of the constituting objects are linked together so that one object can no longer be adequately described without full...
with the state of another system. For instance, one can use the
Controlled NOT gateThe Controlled NOT gate is a quantum gate that is an essential component in the construction of a quantum computer. It can be used to disentangle EPR states...
and the Walsh-Hadamard gate to entangle two
qubitA quantum bit or qubit is a unit of quantum information. It is the quantum analogue of the classical bit. It is described by a state vector in a two-level quantum-mechanical system, which is formally equivalent to a two-dimensional vector space over the complex numbers.-Bit versus qubit:A bit is...
s. This is not cloning. No well-defined state can be attributed to a subsystem of an entangled state. Cloning is a process whose end result is a separable state with identical factors.
Informal proof
If it were possible to clone a quantum state, then one could violate the
uncertainty principleIn quantum mechanics, the Heisenberg uncertainty principle states that certain pairs of physical properties, like position and momentum, cannot both be known to arbitrary precision. That is, the more precisely one property is known, the less precisely the other can be known...
. For instance, one could clone particle A's state to particle B, then measure A's position and B's momentum with any desired precision. Since the uncertainty principle is believed to be a limitation on what there is to know about a particle (not just a limitation on specific techniques of measurement), it follows that quantum cloning is impossible.
Formal proof
Suppose the state of a quantum system A, which we wish to copy, is (see
bra-ket notationBra-ket notation is a standard notation for describing quantum states in the theory of quantum mechanics composed of angle brackets and vertical bars. It can also be used to denote abstract vectors and linear functionals in pure mathematics...
). In order to make a copy, we take a system B with the same
state spaceThe mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
and initial state . The initial, or blank, state must be independent of , of which we have no prior knowledge. The composite system is then described by the
tensor productIn mathematics, the tensor product, denoted by ⊗, may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules. In each case the significance of the symbol is the same: the most general bilinear operation. In some contexts, this...
, and its state is
There are only two ways to manipulate the composite system. We could perform an observation, which irreversibly
collapsesIn quantum mechanics, wave function collapse is the process by which a wave function, initially in a superposition of different eigenstates, appears to reduce to a single one of the states after interaction with an observer...
the system into some eigenstate of the
observableIn physics, particularly in quantum physics, a system observable is a property of the system state that can be determined by some sequence of physical operations. For example, these operations might involve submitting the system to various electromagnetic fields and eventually reading a value off...
, corrupting the information contained in the qubit. This is obviously not what we want. Alternatively, we could control the
HamiltonianIn quantum mechanics, the Hamiltonian H is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...
of the system, and thus the
time evolution operatorThe mathematical formulation of quantum mechanics is the body of mathematical formalisms which permits a rigorous description of quantum mechanics. It is distinguished from mathematical formalisms for theories developed prior to the early 1900s by the use of abstract mathematical structures, such...
U up to some fixed time interval, which yields a
unitary operatorIn functional analysis, a branch of mathematics, a unitary operator is a bounded linear operator U : H → H on a Hilbert space H satisfying...
. Then
U acts as a copier provided that
and
for all and . By definition of unitary operator,
U preserves the inner product:
i.e.
This implies that either or is orthogonal to which is not true in general. Therefore
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
purifiedIn quantum mechanics, especially quantum information, purification refers to the fact that every mixed state acting on finite dimensional Hilbert spaces can be viewed as the reduced state of some pure state....
. Similarly, an arbitrary
quantum operationIn quantum mechanics, a quantum operation is a mathematical formalism used to describe a broad class of transformations that a quantum mechanical system can undergo...
can be implemented via introducing an ancilla and perform 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. 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
amplificationGenerally, an amplifier or simply amp, is any device that changes, usually increases, the amplitude of a signal. The relationship of the input to the output of an amplifier—usually expressed as a function of the input frequency—is called the transfer function of the amplifier, and the magnitude of...
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 decoherenceIn quantum mechanics, quantum decoherence is the mechanism by which quantum systems interact with their environments to exhibit probabilistically additive behavior...
.
No-cloning in a classical context
There is a classical analogue to the quantum no-cloning theorem, which we might state 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 if we wish 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 combinationA convex combination is a linear combination of points where all coefficients are non-negative and sum up to 1. All possible convex combinations will be within the convex hull of the given points...
of other states. The classical pure states are pairwise orthogonal, but quantum pure states are not.
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
A quantum computer is a device for computation that makes direct use of quantum mechanical phenomena, such as superposition and entanglement, to perform operations on data. The basic principle behind quantum computation is that quantum properties can be used to represent data and perform operations...
, 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, ShorPeter Williston Shor is an American professor of applied mathematics at MIT, most famous for his work on quantum computation, in particular for devising Shor's algorithm, a quantum algorithm for factoring exponentially faster than the best currently-known algorithm running on a classical...
and SteaneAndrew Martin Steane is lecturer of Physics at the University of Oxford. He is also a fellow of Exeter College, Oxford.He was a student at St Edmund Hall, Oxford where he obtained his MA and DPhil....
revived the prospects of quantum computing by independently devising the first quantum error correctingQuantum error correction is used in quantum computing to protect quantum information from errors due to decoherence and other quantum noise. Quantum error correction is essential if one is to achieve fault-tolerant quantum computation that can deal not only with noise on stored quantum...
codes, which circumvent the no cloning theorem.
- In contrast, the no cloning theorem is a vital ingredient in quantum cryptography
Quantum cryptography, or quantum key distribution , uses quantum mechanics to guarantee secure communication. It enables two parties to produce a shared random bit string known only to them, which can be used as a key to encrypt and decrypt messages....
, as it forbids eavesdroppers from creating copies of a transmitted quantum cryptographic key.
- The no-cloning theorem protects the uncertainty principle
In quantum mechanics, the Heisenberg uncertainty principle states that certain pairs of physical properties, like position and momentum, cannot both be known to arbitrary precision. That is, the more precisely one property is known, the less precisely the other can be known...
in quantum mechanicsQuantum mechanics is a set of principles describing the physical reality at the atomic level of matter and the subatomic . These descriptions include the simultaneous wave-like and particle-like behavior of both matter and radiation...
. 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.
Consider how the possibility of cloning leads to a violation of the uncertainty principle on an example of uncertainty of all spin-components. It's known, that if one spin component has been measured, then values of others are uncertain. But if we are able to produce as much copies of given spin state as we want, we can perform one measurement on each copy to find average x, y, z projectors. This will lead to determination of spin state with any accuracy. In general various spin states on Bloch sphere are non-orthogonal, thus the proof above is hold here for prohibition of cloning spin states.
- Similarly, cloning would violate the no teleportation theorem
In quantum information theory, the no teleportation theorem states that quantum information cannot be measured with complete accuracy.-Formulation:The term quantum information refers to information stored in the state of a quantum system...
, which says classical teleportation (not to be confused with entanglement-assisted teleportationQuantum teleportation, or entanglement-assisted teleportation, is a technique used to transfer information on a quantum level, usually from one particle to another particle in another location via quantum entanglement...
) is impossible. In other words, quantum states cannot be measured reliably.
- The no cloning theorem does not prevent superluminal communication
Superluminal communication is the term used to describe the hypothetical process by which one might send information at faster-than-light speeds...
via quantum entanglementQuantum entanglement, also called the quantum non-local connection, is a property of a quantum mechanical state of a system of two or more objects in which the quantum states of the constituting objects are linked together so that one object can no longer be adequately described without full...
, as cloning is a sufficient condition for such communication, but not a necessary one. Nevertheless, consider the EPR thought experimentIn quantum mechanics, the EPR paradox is a thought experiment which challenged long-held ideas about the relation between the observed values of physical quantities and the values that can be accounted for by a physical theory...
, and suppose quantum states could be cloned. Assume parts of a maximally entangled Bell stateThe Bell states are a concept in quantum information science and represent the simplest possible examples of entanglement. They are named after John S. Bell, as they are the subject of his famous Bell inequality. A Bell pair is a pair of qubits which jointly are in a Bell state, that is, entangled...
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 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.
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 transformationInformally, a unitary transformation is a transformation that respects the dot product: the dot product of two vectors before the transformation is equal to their dot product after the 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.
See also
- No-broadcast theorem
The no-broadcast theorem is a result in quantum information theory. In the case of pure quantum states, it is a corollary of the no cloning theorem: since quantum states cannot be copied in general, they cannot be broadcast. For mixed states, it generalizes no-cloning.The no-cloning theorem says...
- Quantum teleportation
Quantum teleportation, or entanglement-assisted teleportation, is a technique used to transfer information on a quantum level, usually from one particle to another particle in another location via quantum entanglement...
- Quantum entanglement
Quantum entanglement, also called the quantum non-local connection, is a property of a quantum mechanical state of a system of two or more objects in which the quantum states of the constituting objects are linked together so that one object can no longer be adequately described without full...
- Quantum information
In quantum mechanics, quantum information is physical information that is held in the "state" of a quantum system. The most popular unit of quantum information is the qubit, a two-level quantum system...
- Uncertainty principle
In quantum mechanics, the Heisenberg uncertainty principle states that certain pairs of physical properties, like position and momentum, cannot both be known to arbitrary precision. That is, the more precisely one property is known, the less precisely the other can be known...
Other sources
- W.K. Wootters and W.H. Zurek, A Single Quantum Cannot be Cloned, Nature 299 (1982), pp. 802–803.
- D. Dieks, Communication by EPR devices, Physics Letters A, vol. 92(6) (1982), pp. 271–272.
- V. Buzek and M. Hillery, Quantum cloning, Physics World 14 (11) (2001), pp. 25–29.