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{{quantum}}
'''Twierdzenie o zakazie klonowania''', wynikające z [[Mechanika kwantowa|mechaniki kwantowej]], jest zasadą, zabraniającą stworzenia identycznych kopii dowolnego nieznanego [[Stan kwantowy|stanu kantowego]]. Zostało udowodnione przez Williama Wootersa i [[Wojciech Żurek|Wojciecha Żurka]] oraz niezależnie Dennisa Dieksa w roku 1982 i znalazło zastosowanie w komputerach kwantowych i innych dziedzinach.

Stany układów można że sobą [[Stany splątane|splątać]]. Na przykład, można użyć [[Bramka CNOT|bramki CNOT]] i [[Bramka Hadamarda|bramki Hadamarda]] do splątania dwóch [[Kubit|kubitów]], lecz nie jest to klonowaniem. Żaden stan określony nie może zostać przypisany do podsystemu stanu splątanego. Klonowanie jest procesem, którego wynikiem są stany niesplątane z identycznymi wynikami pomiarów.

Według izraelskiego fizyka Ashera Peresa, publikacja twierdzenia o zakazie klonowania opierała się na pomyśle Nicka Herberta na [[Prędkość nadświetlna|nadświetlne]] urządzenie komunikacyjne wykorzystujące splątanie kwantowe.

== Proof ==
Suppose the state of a quantum system A, which we wish to copy, is <math>|\psi\rangle_A</math> (see [[bra-ket notation]]). In order to make a copy, we take a system B with the same [[Hilbert space|state space]] and initial state <math>|e\rangle_B</math>. The initial, or blank, state must be independent of <math>|\psi\rangle_A</math>, of which we have no prior knowledge. The composite system is then described by the [[tensor product]], and its state is

:<math>|\psi\rangle_A |e\rangle_B.\,</math>

There are only two ways to manipulate the composite system. We could perform an observation, which irreversibly [[Wavefunction collapse|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 (quantum mechanics)|Hamiltonian]] of the system, and thus the [[mathematical formulation of quantum mechanics|time evolution operator]] ''U'' (for a time independent Hamiltonian, <math>U(t)=e^{-iHt/\hbar}</math>, where <math>-H/\hbar</math> 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

:<math>U |\phi\rangle_A |e\rangle_B = |\phi\rangle_A |\phi\rangle_B \, </math>

for all possible states <math>| \phi \rangle</math> in the state space (including <math>|\psi\rangle</math>). Since ''U'' is unitary, it preserves the inner product:

:<math>
\langle e|_B  \langle \phi|_A  |\psi\rangle_A |e\rangle_B 
= \langle e|_B  \langle \phi|_A U^{\dagger} U |\psi\rangle_A |e\rangle_B 
= \langle \phi|_B \langle \phi|_A |\psi\rangle_A |\psi\rangle_B,
</math>

and since quantum mechanical states are assumed to be normalized, it follows that

:<math>\langle \phi | \psi \rangle =  \langle \phi | \psi \rangle ^2. \, </math>

This implies that either <math>\phi=\psi</math> (in which case <math>\langle \phi | \psi \rangle = 1</math>) or <math>\phi</math> is [[orthogonal]] to <math>\psi</math> (in which case <math>\langle \phi | \psi \rangle = 0</math>). However, this is not the case for two arbitrary states. While orthogonal states in a specifically chosen basis <math> \{ |0\rangle, |1\rangle \} </math>, for example:

:<math> | \phi \rangle = {1 \over \sqrt{2}} \bigg( | 0 \rangle + | 1 \rangle \bigg) </math>

and

:<math> | \psi \rangle = {1 \over \sqrt{2}} \bigg( | 0 \rangle - | 1 \rangle \bigg) </math>

fit the requirement that <math>\langle \phi | \psi \rangle =  \langle \phi | \psi \rangle ^2 </math>, this result does not hold for more general quantum states. Apparently ''U'' cannot clone a ''general'' quantum state. This proves the no-cloning theorem.

== Generalizations ==

=== 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 [[purification of quantum state|purified]]. 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.

=== 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 [[amplifier|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]].

=== 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 combination]] of other states. The classical pure states are pairwise orthogonal, but quantum pure states are not.

== Wnioski ==
Zasada nie klonowania zabrania używania klasycznych kodów korekcyjnych do stanów kwantowych. Przykładowo, nie można zrobić kopii zapasowej stanu w komputerze kwantowym aby zapobiec kolejnym błędom. Korekcja błędów jest kluczowa dla obliczeń kwantowych, i przez pewien czas uważano to za ograniczenie na komputer kwantowy. W 1995 [[Peter Shor]] oraz [[Andrew Steane]]  odświeżyli koncepcję obliczeń kwantowych w oparciu o kwantowe kody korekcyjne, które obchodzą ograniczenia zasady nie klonowania. 

Podobnie, klonowanie może naruszyć zakaz klasycznej teleportacji. 

*Similarly, cloning would violate the [[no teleportation theorem]], which says classical teleportation (not to be confused with [[quantum teleportation|entanglement-assisted teleportation]]) is impossible. In other words, quantum states cannot be measured reliably.

*The no-cloning theorem does not prevent [[superluminal communication]] via [[quantum entanglement]], as cloning is a sufficient condition for such communication, but not a necessary one. Nevertheless, 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 will produce the same result; otherwise, his measurements will have outcomes +1/2 and &minus;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 principle]] for [[black hole]]s as meaning we have two copies of information lying at the [[event horizon]] and the black hole interior simultaneously. This leads us to more radical interpretations like [[black hole complementarity]].

== Niedoskonałe klonowanie ==

Mimo że jest niemożliwe zrobienie idealnych kopii nieznanego stanu kwantowego, można wytworzyć niedoskonałe kopie. Proces ten może być przeprowadzony poprzez użycie większego systemu pomocniczego który będzie klonowany i zastosować [[Przekształcenie unitarne|przekształcenie unitarne]] do całego systemu. Jeżeli przekształcenie jest wybrane poprawne niektóre elementy układu mogą ewoluować w przybliżone kopie wyjściowego systemu. [[Klonowanie kwantowe]] może zostać użyte do ataków podsłuchujących na protokoły kryptograficzne.

==See also==
* [[Fundamental Fysiks Group]]
* [[No-broadcast theorem]]
* [[Quantum entanglement]]
* [[Quantum cloning]]
* [[Quantum information]]
* [[Quantum no-deleting theorem]]
* [[Quantum teleportation]]
* [[Uncertainty principle]]

==References==
<references/>

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

[[:Category:Quantum information science]]
[[:Category:Physics theorems]]
[[:Category:Articles containing proofs]]

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