No cloning theorem
Encyclopedia
The no-cloning theorem is a result of quantum mechanics
that forbids the creation of identical copies of an arbitrary unknown quantum state. It was stated by Wootters
, Zurek, and Dieks
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 qubit
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.
(see bra-ket notation
). In order to make a copy, we take a system B with the same state space
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 product
, and its state is
There are only two ways to manipulate the composite system. We could perform an observation, which irreversibly collapses
the system into some eigenstate of the observable
, corrupting the information contained in the qubit. This is obviously not what we want. Alternatively, we could control the Hamiltonian
of the system, and thus the time evolution operator
U (for time independent Hamiltonian,
, and 
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
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. While orthogonal states in a specifically chosen basis 
, for example:
and
fit the requirement that
, this result does not hold for more general quantum states. Apparently U cannot clone a general quantum state. This proves the no-cloning 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.
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
.
of other states. The classical pure states are pairwise orthogonal, but quantum pure states are not.
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.
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
that forbids the creation of identical copies of an arbitrary unknown quantum state. It was stated by Wootters
William Wootters
William 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. It was also independently discovered by Dennis Dieks. He has also worked on the quantification of...
, Zurek, and Dieks
Dennis Dieks
Dennis Dieks is a Dutch physicist and philosopher of physics. In 1982 he proved the no-cloning theorem . In 1989 he proposed a new interpretation of quantum mechanics, later known as a version of the modal interpretation of quantum mechanics...
in 1982, and has profound implications in quantum computing
Quantum computer
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. Quantum computers are different from traditional computers based on transistors...
and related fields.
The state of one system can be entangled
Quantum entanglement
Quantum entanglement occurs when electrons, molecules even as large as "buckyballs", photons, etc., interact physically and then become separated; the type of interaction is such that each resulting member of a pair is properly described by the same quantum mechanical description , which is...
with the state of another system. For instance, one can use the Controlled NOT gate
Controlled NOT gate
The 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 qubit
Qubit
In quantum computing, a qubit or quantum bit is a unit of quantum information—the quantum analogue of the classical bit—with additional dimensions associated to the quantum properties of a physical atom....
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.
Proof
Suppose the state of a quantum system A, which we wish to copy, is (see bra-ket notationBra-ket notation
Bra-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 mathematics...
). In order to make a copy, we take a system B with the same state space
Hilbert space
The 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 productTensor product
In mathematics, the tensor product, denoted by ⊗, may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules, among many other structures or objects. In each case the significance of the symbol is the same: the most general...
, and its state is
There are only two ways to manipulate the composite system. We could perform an observation, which irreversibly collapses
Wavefunction collapse
In quantum mechanics, wave function collapse is the phenomenon in which a wave function—initially in a superposition of several different possible eigenstates—appears to reduce to a single one of those states after interaction with an observer...
the system into some eigenstate of the observable
Observable
In 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 Hamiltonian
Hamiltonian (quantum mechanics)
In quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, 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 operator
Mathematical formulation of quantum mechanics
The mathematical formulations of quantum mechanics are those mathematical formalisms that permit a rigorous description of quantum mechanics. Such are distinguished from mathematical formalisms for theories developed prior to the early 1900s by the use of abstract mathematical structures, such as...
U (for time independent Hamiltonian,
, and is called the generator of translations in time) up to some fixed time interval, which yields a unitary operatorUnitary operator
In functional analysis, a branch of mathematics, a unitary operator is a bounded linear operator U : H → H on a Hilbert space H satisfyingU^*U=UU^*=I...
. 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. While orthogonal states in a specifically chosen basis , for example:and
fit the requirement that
, this result does not hold for more general quantum states. Apparently 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 purifiedPurification of quantum state
In 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 operation
Quantum operation
In quantum mechanics, a quantum operation is a mathematical formalism used to describe a broad class of transformations that a quantum mechanical system can undergo. This was first discussed as a general stochastic transformation for a density matrix by George Sudarshan...
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. 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
Amplifier
Generally, an amplifier or simply amp, is a device for increasing the power of a signal.In popular use, the term usually describes an electronic amplifier, in which the input "signal" is usually a voltage or a current. In audio applications, amplifiers drive the loudspeakers used in PA systems to...
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
Quantum decoherence
In quantum mechanics, quantum decoherence is the loss of coherence or ordering of the phase angles between the components of a system in a quantum superposition. A consequence of this dephasing leads to classical or 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 combinationConvex combination
In convex geometry, a convex combination is a linear combination of points where all coefficients are non-negative and sum up to 1....
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 computationQuantum computerA 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. Quantum computers are different from traditional computers based on transistors...
, 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 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 SteaneAndrew Martin Steane is Professor 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 correctionQuantum 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.
- Similarly, cloning would violate the no teleportation theoremNo teleportation theoremIn 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 teleportationQuantum teleportation, or entanglement-assisted teleportation, is a process by which a qubit can be transmitted exactly from one location to another, without the qubit being transmitted through the intervening space...
) is impossible. In other words, quantum states cannot be measured reliably.
- The no-cloning theorem does not prevent superluminal communication via quantum entanglementQuantum entanglementQuantum entanglement occurs when electrons, molecules even as large as "buckyballs", photons, etc., interact physically and then become separated; the type of interaction is such that each resulting member of a pair is properly described by the same quantum mechanical description , which is...
, as cloning is a sufficient condition for such communication, but not a necessary one. Nevertheless, consider the EPR thought experimentEPR paradoxThe EPR paradox is a topic in quantum physics and the philosophy of science concerning the measurement and description of microscopic systems by the methods of quantum physics...
, and suppose quantum states could be cloned. Assume parts of a maximally entangled Bell stateBell 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. An EPR 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.
- The no cloning theorem prevents us from viewing the holographic principleHolographic principleThe holographic principle is a property of quantum gravity and string theories which states that the description of a volume of space can be thought of as encoded on a boundary to the region—preferably a light-like boundary like a gravitational horizon...
for black holeBlack holeA black hole is a region of spacetime from which nothing, not even light, can escape. The theory of general relativity predicts that a sufficiently compact mass will deform spacetime to form a black hole. Around a black hole there is a mathematically defined surface called an event horizon that...
s as meaning we have two copies of information lying at the event horizonEvent horizonIn general relativity, an event horizon is a boundary in spacetime beyond which events cannot affect an outside observer. In layman's terms it is defined as "the point of no return" i.e. the point at which the gravitational pull becomes so great as to make escape impossible. The most common case...
and the black hole interior simultaneously. This leads us to more radical interpretations like black hole complementarityBlack hole complementarityBlack hole complementarity is a conjectured solution to the black hole information paradox, proposed by Leonard Susskind and Gerard 't Hooft.Ever since Stephen Hawking suggested information is lost in evaporating black hole once it passes through the event horizon and is inevitably destroyed at the...
.
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 transformationUnitary transformation
In mathematics, a unitary transformation may be informally defined as a transformation that respects the inner product: the inner product of two vectors before the transformation is equal to their inner 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
Quantum cloning
Quantum cloning is the process that takes an arbitrary, unknown quantum state and makes an exact copy without altering the original state in any way...
can be used as an eavesdropping attack on quantum cryptography protocols, among other uses in quantum information science.
See also
- Fundamental Fysiks GroupFundamental Fysiks GroupThe Fundamental Fysiks Group was founded in San Francisco in May 1975 by two physicists, Elizabeth Rauscher and George Weissmann, at the time both graduate students at the University of California, Berkeley. The group held informal discussions on Friday afternoons to explore the philosophical...
- No-broadcast theoremNo-broadcast theoremThe 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...
- Quantum entanglementQuantum entanglementQuantum entanglement occurs when electrons, molecules even as large as "buckyballs", photons, etc., interact physically and then become separated; the type of interaction is such that each resulting member of a pair is properly described by the same quantum mechanical description , which is...
- Quantum cloningQuantum cloningQuantum cloning is the process that takes an arbitrary, unknown quantum state and makes an exact copy without altering the original state in any way...
- Quantum informationQuantum informationIn 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...
- Quantum no-deleting theoremQuantum no-deleting theoremQuantum states are fragile in one sense and also robust in another sense. Quantum theory tells us that given a single quantum state it is impossible to determine it exactly. One needs an infinite number of identically prepared quantum states to know a state exactly. This has remarkable...
- Quantum teleportationQuantum teleportationQuantum teleportation, or entanglement-assisted teleportation, is a process by which a qubit can be transmitted exactly from one location to another, without the qubit being transmitted through the intervening space...
- Uncertainty principleUncertainty principleIn quantum mechanics, the Heisenberg uncertainty principle states a fundamental limit on the accuracy with which certain pairs of physical properties of a particle, such as position and momentum, can be simultaneously 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.