View source for No-cloning theorem

{{Short description|Theorem in quantum information science}}
{{quantum}}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 computer|quantum computing]] among others. The theorem is an evolution of the 1970 [[no-go theorem]] authored by James L. Park,<ref name="park">{{cite journal
 | last1 = Park | first1 = James L.
 | year = 1970
 | title = The concept of transition in quantum mechanics
 | journal = [[Foundations of Physics]]
 | volume = 1 | issue = 1
 | pages = 23–33
 |doi = 10.1007/BF00708652 | bibcode = 1970FoPh....1...23P| citeseerx = 10.1.1.623.5267
 | s2cid = 55890485
 }}</ref><ref>{{Cite web  |title=James L. Park Main Scientific Publications |url=https://quantum-thermodynamics.unibs.it/james_l_park_publications.htm  }}</ref> 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 [[William Wootters]] and  [[Wojciech H. Zurek]]<ref name="wootterszurek">{{cite journal
 | last1 = Wootters | first1 = William 
 | last2 = Zurek| first2 = Wojciech 
 | year = 1982
 | title = A Single Quantum Cannot be Cloned
 | journal = [[Nature (journal)|Nature]]
 | volume = 299 | issue = 5886 
 | pages = 802–803
 |bibcode = 1982Natur.299..802W |doi = 10.1038/299802a0 | s2cid = 4339227 
 }}</ref> as well as [[Dennis Dieks]]<ref name="dieks">{{cite journal
 | last = Dieks| first = Dennis
 | year = 1982
 | title = Communication by EPR devices
 | journal = [[Physics Letters A]]
 | volume = 92 | issue = 6 | pages = 271–272
 |bibcode = 1982PhLA...92..271D |doi = 10.1016/0375-9601(82)90084-6 | citeseerx = 10.1.1.654.7183
 | hdl = 1874/16932
 }}</ref> the same year). The aforementioned theorems do not preclude the state of one system becoming [[quantum entanglement|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 [[Hadamard transform#Quantum computing applications|Walsh–Hadamard gate]] to entangle two [[qubit]]s 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 state]]s whereas the generalized statement regarding [[mixed state (physics)|mixed states]] is known as the [[no-broadcast theorem]]. The no-cloning theorem has a time-reversed [[dual (category theory)|dual]], the [[no-deleting theorem]].

== History ==

According to [[Asher Peres]]<ref>{{cite journal|last=Peres|first=Asher|year=2003|title=How the No-Cloning Theorem Got its Name|journal=[[Fortschritte der Physik]]|volume=51|issue=45|pages=458–461|bibcode=2003ForPh..51..458P|doi=10.1002/prop.200310062|arxiv=quant-ph/0205076|s2cid=16588882}}</ref> and [[David Kaiser (physicist)|David Kaiser]],<ref>{{cite book
 | last = Kaiser | first = David | author-link = David Kaiser (physicist)
 | year = 2011
 | title = How the Hippies Saved Physics: Science, Counterculture, and the Quantum Revival
 | publisher = [[W. W. Norton]] | isbn = 978-0-393-07636-3}}</ref> the publication of the 1982 proof of the no-cloning theorem by [[William Wootters|Wootters]] and [[Wojciech H. Zurek|Zurek]]<ref name = "wootterszurek"/> and by [[Dennis Dieks|Dieks]]<ref name = "dieks" /> was prompted by a proposal of [[Nick Herbert (physicist)|Nick Herbert]]<ref>{{cite journal | last = Herbert|first=Nick|year=1982|title = FLASH—A superluminal communicator based upon a new kind of quantum measurement|journal = [[Foundations of Physics]]|volume =12|issue=12|pages=1171–1179|bibcode=1982FoPh...12.1171H|doi=10.1007/BF00729622|s2cid = 123118337}}</ref> for a [[superluminal communication]] device using quantum entanglement, and [[Giancarlo Ghirardi]]<ref name="Ghirardi2013">{{Citation|last = Ghirardi
| first = GianCarlo
| chapter = Entanglement, Nonlocality, Superluminal Signaling and Cloning
| date = 2013
| editor-last  = Bracken
| editor-first = Paul
| title  = Advances in Quantum Mechanics
| publisher = IntechOpen
| publication-date = April 3, 2013
| arxiv = 1305.2305
| doi = 10.5772/56429
| isbn = 978-953-51-1089-7
| s2cid = 118778014
}}</ref> 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<ref name="Ghirardi2013" />). However, Juan Ortigoso<ref>{{cite journal
 | last1=Ortigoso|first1=Juan|year = 2018|title = Twelve years before the quantum no-cloning theorem
 | journal = [[American Journal of Physics]]
 | volume = 86 | issue = 3 
 | pages = 201–205
 |doi = 10.1119/1.5021356 | arxiv = 1707.06910| bibcode = 2018AmJPh..86..201O| s2cid = 119192142 
 }}</ref> 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.<ref name = "park" />

== Theorem and proof ==
Suppose we have two quantum systems ''A'' and ''B'' with a common Hilbert space <math>H = H_A = H_B</math>. Suppose we want to have a procedure to copy the state <math>|\phi\rangle_A</math> of quantum system ''A'', over the state <math>|e\rangle_B</math> of quantum system ''B,'' for any original state <math>|\phi\rangle_A</math> (see [[bra–ket notation]]). That is, beginning with the state <math>|\phi\rangle_A \otimes |e\rangle_B </math>, we want to end up with the state <math>|\phi\rangle_A \otimes |\phi\rangle_B </math>. To make a "copy" of the state ''A'', we combine it with system ''B'' in some unknown initial, or blank, state <math>|e\rangle_B</math> independent of <math>|\phi\rangle_A</math>, of which we have no prior knowledge.

The state of the initial composite system is then described by the following [[tensor product]]:
<math display="block">|\phi\rangle_A \otimes |e\rangle_B.</math>
(in the following we will omit the <math>\otimes</math> symbol and keep it implicit).

There are only two permissible [[quantum operation]]s with which we may manipulate the composite system:

* We can perform an [[Measurement in quantum mechanics|observation]], which irreversibly [[wave function collapse|collapses]] the system into some [[eigenstate]] of an [[observable]], corrupting the information contained in the quantum-mechanical system. This is obviously not what we want.
* Alternatively, we could control the [[Hamiltonian (quantum mechanics)|Hamiltonian]] of the ''combined'' system, and thus the [[mathematical formulation of quantum mechanics|time-evolution operator]] ''U''(''t''), e.g. for a time-independent Hamiltonian, {{nowrap|<math>U(t) = e^{-iHt/\hbar}</math>.}} Evolving up to some fixed time <math>t_0</math> yields a [[unitary operator]] ''U'' on {{nowrap|<math>H \otimes H</math>,}} 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 <math>H_A \otimes H_B = H \otimes H</math>, 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?

{{math theorem | math_statement = There is no unitary operator ''U'' on <math>H \otimes H</math> such that for all normalised states <math>|\phi \rangle_A</math> and <math>|e\rangle_B</math> in <math>H</math>
<math display="block">U(|\phi\rangle_A |e\rangle_B) = e^{i \alpha(\phi,e)} |\phi\rangle_A |\phi\rangle_B</math>
for some real number <math>\alpha</math> depending on <math>\phi</math> and <math>e</math>.}}

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 [[projective space|projectivised Hilbert space]].

To prove the theorem, we select an arbitrary pair of states <math>|\phi\rangle_A</math> and <math>|\psi\rangle_A</math> in the Hilbert space <math>H</math>. Because ''U'' is supposed to be unitary, we would have
<math display="block">
 \langle \phi| \psi\rangle \langle e | e \rangle \equiv
 \langle \phi|_A \langle e|_B |\psi\rangle_A |e\rangle_B =
 \langle \phi|_A \langle e|_B U^\dagger U |\psi\rangle_A |e\rangle_B =
 e^{-i(\alpha(\phi, e) - \alpha(\psi, e))} \langle \phi|_A \langle \phi|_B  |\psi\rangle_A |\psi\rangle_B \equiv
 e^{-i(\alpha(\phi, e) - \alpha(\psi, e))} \langle \phi |\psi\rangle^2.
</math>
Since the  quantum state <math>|e\rangle</math> is assumed to be normalized, we thus get
<math display="block"> |\langle \phi | \psi \rangle|^2 = |\langle \phi | \psi \rangle|.</math>

This implies that either <math>|\langle \phi | \psi \rangle| = 1</math> or <math>|\langle \phi | \psi \rangle| = 0</math>. Hence by the [[Cauchy–Schwarz inequality]] either <math>|\phi\rangle = e^{i\beta}|\psi\rangle</math> or <math>|\phi\rangle</math> is [[orthogonality|orthogonal]] to <math>|\psi\rangle</math>. 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 number]]s, called [[probability amplitude]]s ([[Qubit#Qubit states|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 [[Quantum logic gate#Hadamard gate|Hadamard quantum gate]]) to be polarised (which [[Unitary operator|unitary transformation]] is a [[Isometry#Introduction|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 [[Qubit#Physical implementations|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 (physics)|mixed state]], it can be [[purification of quantum state|"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.<ref>{{Cite journal|last1=Barnum|first1=Howard|last2=Caves|first2=Carlton M.|last3=Fuchs|first3=Christopher A.|last4=Jozsa|first4=Richard|last5=Schumacher|first5=Benjamin|date=1996-04-08|title=Noncommuting Mixed States Cannot Be Broadcast|journal=Physical Review Letters|volume=76|issue=15|pages=2818–2821|doi=10.1103/PhysRevLett.76.2818|pmid=10060796|arxiv=quant-ph/9511010|bibcode=1996PhRvL..76.2818B|s2cid=11724387}}</ref><ref>{{Cite journal|last1=Kalev|first1=Amir|last2=Hen|first2=Itay|date=2008-05-29|title=No-Broadcasting Theorem and Its Classical Counterpart|journal=Physical Review Letters|volume=100|issue=21|article-number=210502|doi=10.1103/PhysRevLett.100.210502|pmid=18518590|arxiv=0704.1754|bibcode=2008PhRvL.100u0502K|s2cid=40349990}}</ref>  Similarly, an arbitrary [[quantum operation]] can be implemented via introducing an [[Ancilla bit|ancilla]] and performing a suitable unitary evolution.{{clarify|reason=What other quantum operations are there? The phrasing of the proof in the previous section contains the implicit claim that we had them all when we had unitary ones and Hermitian ones. Now there are more? These new operations should be spelled out. The linked article is a grab bag.|date=May 2015}}  Thus the no-cloning theorem holds in full generality.

== Consequences ==
*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 computer|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, [[Peter Shor|Shor]] and [[Andrew Steane|Steane]] showed that it is, by independently devising the first [[quantum error correction|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 [[quantum teleportation|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 paradox|EPR thought experiment]], and suppose quantum states could be cloned. Assume parts of a [[Maximally entangled state|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 <math>|z+\rangle_B</math> or <math>|z-\rangle_B</math>.  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 <math>|z+\rangle_B</math> or <math>|z-\rangle_B</math> with equal probability. This would allow Alice and Bob to communicate classical bits between each other (possibly across [[space-like]] separations, violating [[causality]]).
* The no cloning theorem prevents an interpretation of the [[holographic principle]] for [[black hole]]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]].

== 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]] 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.<ref>{{cite journal |last1=Bužek |first1=V. |last2=Hillery |first2=M. |title=Quantum Copying: Beyond the No-Cloning Theorem |journal=Phys. Rev. A |volume=54 |issue=3 |pages=1844–1852 |year=1996 |doi=10.1103/PhysRevA.54.1844 |pmid=9913670 |arxiv=quant-ph/9607018 |bibcode=1996PhRvA..54.1844B |s2cid=1446565 }}</ref>

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

==See also==
{{cols|colwidth=26em}}
* [[Fundamental Fysiks Group]]
* [[Monogamy of entanglement]]
* [[No-broadcast theorem]]
* [[No-communication theorem]]
* [[No-deleting theorem]]
* [[No-hiding theorem]]
* [[Quantum entanglement]]
* [[Quantum cloning]]
* [[Quantum information]]
* [[Quantum teleportation]]
* [[Stronger uncertainty relations]]
* [[Uncertainty principle]]
{{colend}}

==References==
<references/>

== Other sources ==
* V. Buzek and M. Hillery, ''Quantum cloning'', Physics World 14 (11) (2001), pp.&nbsp;25&ndash;29.

{{Quantum computing}}

[[Category:Quantum information science]]
[[Category:Theorems in quantum mechanics]]
[[Category:Articles containing proofs]]
[[Category:No-go theorems]]