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In physics, the **no-cloning theorem** states that it is impossible to create an independent and identical copy of an arbitrary unknown quantum state, a statement which has profound implications in the field of quantum computing among others. The theorem is an evolution of the 1970 no-go theorem authored by James Park,^{ [1] } 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 ^{ [2] } as well as Dieks ^{ [3] } 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 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.

- History
- Theorem and proof
- Generalization
- Consequences
- Imperfect cloning
- See also
- References
- Other sources

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.^{ [4] }^{ [5] } 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 intuitionistic logic arises from Cartesian closed categories).

According to Asher Peres ^{ [6] } and David Kaiser,^{ [7] } the publication of the 1982 proof of the no-cloning theorem by Wootters and Zurek ^{ [2] } and by Dieks ^{ [3] } was prompted by a proposal of Nick Herbert ^{ [8] } for a superluminal communication device using quantum entanglement, and Giancarlo Ghirardi ^{ [9] } 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^{ [9] }). However, Ortigoso ^{ [10] } 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.^{ [1] }

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). 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 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.

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 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.

Take a qubit for example. It can be represented by two complex numbers, called probability amplitudes (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 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*.

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.^{[ clarification needed ]} 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.^{ [11] }^{ [12] } Similarly, an arbitrary quantum operation can be implemented via introducing an ancilla and performing a suitable unitary evolution.^{[ clarification needed ]} Thus the no-cloning theorem holds in full generality.

- The no-cloning theorem prevents the use of certain 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 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 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 or with equal probability. This would allow Alice and Bob to communicate classical bits between each other (possibly across space-like separations, violating causality). - Quantum states cannot be discriminated perfectly.
^{ [13] } - The no cloning theorem prevents an interpretation of the holographic principle for black holes 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.
^{ [14] }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 cobordisms.

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.^{ [15] }

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

In quantum mechanics, **bra–ket notation,** or **Dirac notation**, is ubiquitous. The notation uses the angle brackets, "" and "", and a vertical bar "", to construct "bras" and "kets".

**Quantum teleportation** is a technique for transferring quantum information from a sender at one location to a receiver some distance away. While teleportation is commonly portrayed in science fiction as a means to transfer physical objects from one location to the next, quantum teleportation only transfers quantum information. Moreover, the sender may not know the location of the recipient, and does not know which particular quantum state will be transferred.

**Quantum decoherence** is the loss of quantum coherence. In quantum mechanics, particles such as electrons are described by a wave function, a mathematical representation of the quantum state of a system; a probabilistic interpretation of the wave function is used to explain various quantum effects. As long as there exists a definite phase relation between different states, the system is said to be coherent. A definite phase relationship is necessary to perform quantum computing on quantum information encoded in quantum states. Coherence is preserved under the laws of quantum physics.

The **Fock space** is an algebraic construction used in quantum mechanics to construct the quantum states space of a variable or unknown number of identical particles from a single particle Hilbert space H. It is named after V. A. Fock who first introduced it in his 1932 paper "Konfigurationsraum und zweite Quantelung".

In quantum physics, a **measurement** is the testing or manipulation of a physical system to yield a numerical result. The predictions that quantum physics makes are in general probabilistic. The mathematical tools for making predictions about what measurement outcomes may occur were developed during the 20th century and make use of linear algebra and functional analysis.

In quantum computing and specifically the quantum circuit model of computation, a **quantum logic gate** is a basic quantum circuit operating on a small number of qubits. They are the building blocks of quantum circuits, like classical logic gates are for conventional digital circuits.

The **Bell states** or **EPR pairs** are specific quantum states of two qubits that represent the simplest examples of quantum entanglement; conceptually, they fall under the study of quantum information science. The Bell states are a form of entangled and normalized basis vectors. This normalization implies that the overall probability of the particle being in one of the mentioned states is 1: . Entanglement is a basis-independent result of superposition. Due to this superposition, measurement of the qubit will collapse it into one of its basis states with a given probability. Because of the entanglement, measurement of one qubit will assign one of two possible values to the other qubit instantly, where the value assigned depends on which Bell state the two qubits are in. Bell states can be generalized to represent specific quantum states of multi-qubit systems, such as the GHZ state for 3 or more subsystems.

In quantum information theory, a **quantum channel** is a communication channel which can transmit quantum information, as well as classical information. An example of quantum information is the state of a qubit. An example of classical information is a text document transmitted over the Internet.

**LOCC**, or **local operations and classical communication**, is a method in quantum information theory where a local (product) operation is performed on part of the system, and where the result of that operation is "communicated" classically to another part where usually another local operation is performed conditioned on the information received.

In functional analysis and quantum measurement theory, a **positive operator-valued measure** (**POVM**) is a measure whose values are positive semi-definite operators on a Hilbert space. POVMs are a generalisation of projection-valued measures (PVM) and, correspondingly, quantum measurements described by POVMs are a generalisation of quantum measurement described by PVMs.

In physics, in the area of quantum information theory, a **Greenberger–Horne–Zeilinger state** is a certain type of entangled quantum state that involves at least three subsystems. It was first studied by Daniel Greenberger, Michael Horne and Anton Zeilinger in 1989. Extremely non-classical properties of the state have been observed.

In physics, the **no-deleting theorem** of quantum information theory is a no-go theorem which states that, in general, given two copies of some arbitrary quantum state, it is impossible to delete one of the copies. It is a time-reversed dual to the no-cloning theorem, which states that arbitrary states cannot be copied. This theorem seems remarkable, because, in many senses, quantum states are fragile; the theorem asserts that, in a particular case, they are also robust. Physicist Arun K. Pati along with Samuel L. Braunstein proved this theorem.

A **decoherence-free subspace** (**DFS**) is a subspace of a quantum system's Hilbert space that is invariant to non-unitary dynamics. Alternatively stated, they are a small section of the system Hilbert space where the system is decoupled from the environment and thus its evolution is completely unitary. DFSs can also be characterized as a special class of quantum error correcting codes. In this representation they are *passive* error-preventing codes since these subspaces are encoded with information that (possibly) won't require any *active* stabilization methods. These subspaces prevent destructive environmental interactions by isolating quantum information. As such, they are an important subject in quantum computing, where (coherent) control of quantum systems is the desired goal. Decoherence creates problems in this regard by causing loss of coherence between the quantum states of a system and therefore the decay of their interference terms, thus leading to loss of information from the (open) quantum system to the surrounding environment. Since quantum computers cannot be isolated from their environment and information can be lost, the study of DFSs is important for the implementation of quantum computers into the real world.

A **symmetric, informationally complete, positive operator-valued measure** (SIC-POVM) is a special case of a generalized measurement on a Hilbert space, used in the field of quantum mechanics. A measurement of the prescribed form satisfies certain defining qualities that makes it an interesting candidate for a "standard quantum measurement", utilized in the study of foundational quantum mechanics, most notably in QBism. Furthermore, it has been shown that applications exist in quantum state tomography and quantum cryptography, and a possible connection has been discovered with Hilbert's twelfth problem.

A **quantum t-design** is a probability distribution over either pure quantum states or unitary operators which can duplicate properties of the probability distribution over the Haar measure for polynomials of degree t or less. Specifically, the average of any polynomial function of degree t over the design is exactly the same as the average over Haar measure. Here the Haar measure is a uniform probability distribution over all quantum states or over all unitary operators. Quantum t-designs are so called because they are analogous to t-designs in classical statistics, which arose historically in connection with the problem of design of experiments. Two particularly important types of t-designs in quantum mechanics are projective and unitary t-designs.

**Entanglement distillation** is the transformation of *N* copies of an arbitrary entangled state into some number of approximately pure Bell pairs, using only *local operations and classical communication* (LOCC).

In quantum mechanics, **weak measurements** are a type of quantum measurement that results in an observer obtaining very little information about the system on average, but also disturbs the state very little. From Busch's theorem the system is necessarily disturbed by the measurement. In the literature weak measurements are also known as unsharp, fuzzy, dull, noisy, approximate, and gentle measurements. Additionally weak measurements are often confused with the distinct but related concept of the weak value.

The **no-hiding theorem** states that if information is lost from a system via decoherence, then it moves to the subspace of the environment and it cannot remain in the correlation between the system and the environment. This is a fundamental consequence of the linearity and unitarity of quantum mechanics. Thus, information is never lost. This has implications in black hole information paradox and in fact any process that tends to lose information completely. The no-hiding theorem is robust to imperfection in the physical process that seemingly destroys the original information.

In quantum information theory and quantum optics, the **Schrödinger–HJW theorem** is a result about the realization of a mixed state of a quantum system as an ensemble of pure quantum states and the relation between the corresponding purifications of the density operators. The theorem is named after physicists and mathematicians Erwin Schrödinger, Lane P. Hughston, Richard Jozsa and William Wootters. The result was also found independently by Nicolas Hadjisavvas building upon work by Ed Jaynes, while a significant part of it was likewise independently discovered by N. David Mermin. Thanks to its complicated history, it is also known by various other names such as the **GHJW theorem**, the **HJW theorem**, and the **purification theorem**.

In quantum information science, **monogamy** is a fundamental property of quantum entanglement that describes the fact that entanglement cannot be freely shared between arbitrarily many parties.

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*Advances in Quantum Mechanics*, IntechOpen (published April 3, 2013), arXiv: 1305.2305 , doi:10.5772/56429 - ↑ Ortigoso, Juan (2018). "Twelve years before the quantum no-cloning theorem".
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*Semantic Techniques for Quantum Computation*, I. Mackie and S. Gay (eds), Cambridge University Press. arXiv : 0910.2401 - ↑ Bužek, V.; Hillery, M. (1996). "Quantum Copying: Beyond the No-Cloning Theorem".
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- V. Buzek and M. Hillery,
*Quantum cloning*, Physics World 14 (11) (2001), pp. 25–29.

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