In
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
, the no-cloning theorem states that it is impossible to create an independent and identical copy of an arbitrary unknown
quantum state
In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution i ...
, a statement which has profound implications in the field of
quantum computing
Quantum computing is a type of computation whose operations can harness the phenomena of quantum mechanics, such as superposition, interference, and entanglement. Devices that perform quantum computations are known as quantum computers. Though ...
among others. The theorem is an evolution of the 1970
no-go theorem
In theoretical physics, a no-go theorem is a theorem that states that a particular situation is not physically possible. Specifically, the term describes results in quantum mechanics like Bell's theorem and the Kochen–Specker theorem that co ...
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
Wootters and
Zurek as well as
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
In computer science, the controlled NOT gate (also C-NOT or CNOT), controlled-''X'' gate'','' controlled-bit-flip gate, Feynman gate or controlled Pauli-X is a quantum logic gate that is an essential component in the construction of a gate-bas ...
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
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, ca ...
, 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
Linear logic is a substructural logic proposed by Jean-Yves Girard as a refinement of classical and intuitionistic logic, joining the dualities of the former with many of the constructive properties of the latter. Although the logic has also ...
as the logic of
quantum information theory (in the same sense that
intuitionistic logic
Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof. In particular, systems ...
arises from
Cartesian closed categories
In category theory, a category is Cartesian closed if, roughly speaking, any morphism defined on a product of two objects can be naturally identified with a morphism defined on one of the factors. These categories are particularly important in mat ...
).
History
According to
Asher Peres and
David Kaiser
David I. Kaiser is an American physicist and historian of science. He is Germeshausen Professor of the History of Science at the Massachusetts Institute of Technology (MIT), head of its Science, Technology, and Society program, and a full profes ...
, the publication of the 1982 proof of the no-cloning theorem by
Wootters and
Zurek and by
Dieks was prompted by a proposal of
Nick Herbert
Nicholas Le Quesne Herbert, Baron Herbert of South Downs, (born 7 April 1963) is a British Conservative Party politician and was the Member of Parliament (MP) for Arundel and South Downs from 2005 to 2019. He was Minister of State for Police ...
for a
superluminal communication
Superluminal communication is a hypothetical process in which information is sent at faster-than-light (FTL) speeds. The current scientific consensus is that faster-than-light communication is not possible, and to date it has not been achieved in ...
device using
quantum entanglement, and
Giancarlo Ghirardi
Giancarlo Ghirardi (28 October 1935 – 1 June 2018) was an Italian physicist and emeritus professor of theoretical physics at the University of Trieste.
He is well known for the Ghirardi–Rimini–Weber theory (GRW), which he proposed in 1985 t ...
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
). However,
Ortigoso 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.
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'', over the state
of quantum system ''B,'' for any original state
(see
bra–ket notation
In quantum mechanics, bra–ket notation, or Dirac notation, is used ubiquitously to denote quantum states. The notation uses angle brackets, and , and a vertical bar , to construct "bras" and "kets".
A ket is of the form , v \rangle. Mathem ...
). That is, beginning with the state
, we want to end up with the state
. 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 initial composite system is then described by the following
tensor product
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otime ...
:
:
(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
Observation is the active acquisition of information from a primary source. In living beings, observation employs the senses. In science, observation can also involve the perception and recording of data via the use of scientific instruments. The ...
, which irreversibly
collapses the system into some
eigenstate
In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution in ...
of an
observable
In physics, an observable is a physical quantity that can be measured. Examples include position and momentum. In systems governed by classical mechanics, it is a real-valued "function" on the set of all possible system states. In quantum phy ...
, 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
Time evolution is the change of state brought about by the passage of time, applicable to systems with internal state (also called ''stateful systems''). In this formulation, ''time'' is not required to be a continuous parameter, but may be di ...
''U''(''t''), e.g. for a time-independent Hamiltonian,
. Evolving up to some fixed time
yields a
unitary operator
In functional analysis, a unitary operator is a surjective bounded operator on a Hilbert space that preserves the inner product. Unitary operators are usually taken as operating ''on'' a Hilbert space, but the same notion serves to define the co ...
''U'' on
, the Hilbert space of the combined system. However, no such unitary operator ''U'' can clone all states.
The no-cloning theorem answers the following question in the negative: Is it possible to construct a unitary operator ''U'', acting on
, under which the state the system B is in always evolves into the state the system A is in, ''regardless'' of the state system A is in?
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 supposed to be unitary, we would have
:
Since the quantum state
is assumed to be normalized, we thus get
:
This implies that either
or
. Hence by the
Cauchy–Schwarz inequality
The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) is considered one of the most important and widely used inequalities in mathematics.
The inequality for sums was published by . The corresponding inequality f ...
either
or
is
orthogonal
In mathematics, orthogonality is the generalization of the geometric notion of '' perpendicularity''.
By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...
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.
Take a
qubit
In quantum computing, a qubit () or quantum bit is a basic unit of quantum information—the quantum version of the classic binary bit physically realized with a two-state device. A qubit is a two-state (or two-level) quantum-mechanical system, ...
for example. It can be represented by two
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s, called
probability amplitude
In quantum mechanics, a probability amplitude is a complex number used for describing the behaviour of systems. The modulus squared of this quantity represents a probability density.
Probability amplitudes provide a relationship between the qu ...
s (
normalised to 1), that is three real numbers (two polar angles and one radius). Copying three numbers on a classical computer using any
copy and paste operation is trivial (up to a finite precision) but the problem manifests if the qubit is unitarily transformed (e.g. by the
Hadamard quantum gate) to be polarised (which
unitary transformation
In mathematics, a unitary transformation is a transformation that preserves the inner product: the inner product of two vectors before the transformation is equal to their inner product after the transformation.
Formal definition
More precisely, ...
is a
surjective isometry). In such a case the qubit can be represented by just two real numbers (one polar angle and one radius equal to 1), while the value of the third can be arbitrary in such a representation. Yet a
realisation of a qubit (polarisation-encoded photon, for example) is capable of storing the whole qubit information support within its "structure". Thus no single universal unitary evolution ''U'' can clone an arbitrary quantum state according to the no-cloning theorem. It would have to depend on the transformed qubit (initial) state and thus would not have been ''universal''.
Generalization
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," i.e. treated as a pure state of a larger system. 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.
Consequences
*The no-cloning theorem prevents the use of certain classical
error correction
In information theory and coding theory with applications in computer science and telecommunication, error detection and correction (EDAC) or error control are techniques that enable reliable delivery of digital data over unreliable communi ...
techniques on quantum states. For example, backup copies of a state in the middle of a
quantum computation
Quantum computing is a type of computation whose operations can harness the phenomena of quantum mechanics, such as superposition, interference, and entanglement. Devices that perform quantum computations are known as quantum computers. Though ...
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
The Bell states or EPR pairs are specific quantum states of two qubits that represent the simplest (and maximal) examples of quantum entanglement; conceptually, they fall under the study of quantum information science. The Bell states are a form ...
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
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why dif ...
separations, violating
causality
Causality (also referred to as causation, or cause and effect) is influence by which one event, process, state, or object (''a'' ''cause'') contributes to the production of another event, process, state, or object (an ''effect'') where the cau ...
).
* Quantum states cannot be discriminated perfectly.
* The no cloning theorem prevents an interpretation of the
holographic principle
The holographic principle is an axiom in string theories and a supposed property of quantum gravity that states that the description of a volume of space can be thought of as encoded on a lower-dimensional boundary to the region — such as a ...
for
black hole
A black hole is a region of spacetime where gravity is so strong that nothing, including light or other electromagnetic waves, has enough energy to escape it. The theory of general relativity predicts that a sufficiently compact mass can def ...
s 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
cobordism
In mathematics, cobordism is a fundamental equivalence relation on the class of compact manifolds of the same dimension, set up using the concept of the boundary (French '' bord'', giving ''cobordism'') of a manifold. Two manifolds of the same di ...
s.
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
In mathematics, a unitary transformation is a transformation that preserves the inner product: the inner product of two vectors before the transformation is equal to their inner product after the transformation.
Formal definition
More precisely, ...
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
Quantum cloning is a process that takes an arbitrary, unknown quantum state and makes an exact copy without altering the original state in any way. Quantum cloning is forbidden by the laws of quantum mechanics as shown by the no cloning theorem, w ...
can be used as an
eavesdropping attack
Eavesdropping is the act of secretly or stealthily listening to the private conversation or communications of others without their consent in order to gather information.
Etymology
The verb ''eavesdrop'' is a back-formation from the noun ''eaves ...
on
quantum cryptography
Quantum cryptography is the science of exploiting quantum mechanical properties to perform cryptographic tasks. The best known example of quantum cryptography is quantum key distribution which offers an information-theoretically secure solution ...
protocols, among other uses in quantum information science.
See also
*
Fundamental Fysiks Group
The 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 ...
*
Monogamy of entanglement
*
No-broadcast theorem
*
No-communication theorem
*
No-deleting theorem
*
No-hiding theorem
*
Quantum entanglement
*
Quantum cloning
Quantum cloning is a process that takes an arbitrary, unknown quantum state and makes an exact copy without altering the original state in any way. Quantum cloning is forbidden by the laws of quantum mechanics as shown by the no cloning theorem, w ...
*
Quantum information
Quantum information is the information of the state of a quantum system. It is the basic entity of study in quantum information theory, and can be manipulated using quantum information processing techniques. Quantum information refers to both ...
*
Quantum teleportation
*
Stronger uncertainty relations
Heisenberg's uncertainty relation is one of the fundamental results in quantum mechanics. Later Robertson proved the uncertainty relation for two general non-commuting observables, which was strengthened by Schrödinger. However, the conventional ...
*
Uncertainty principle
In quantum mechanics, the uncertainty principle (also known as Heisenberg's uncertainty principle) is any of a variety of mathematical inequalities asserting a fundamental limit to the accuracy with which the values for certain pairs of physic ...
References
Other sources
* V. Buzek and M. Hillery, ''Quantum cloning'', Physics World 14 (11) (2001), pp. 25–29.
{{Quantum computing
Quantum information science
Theorems in quantum mechanics
Articles containing proofs
No-go theorems