No-cloning theorem

The no-cloning theorem is a significant result of quantum physics . Accordingly, it is not possible to build a system that perfectly copies any qubit onto another qubit without changing the original one. On the one hand, the theorem can be seen as a consequence of the unitarity of quantum mechanical time evolution operators or the linearity of operators .

The no-cloning theorem has far-reaching consequences for quantum computing . On the one hand, classic error correction codes that are based on copying the information to be transmitted cannot be used. On the other hand, nobody can eavesdrop on a corresponding transmission of information unnoticed because they would have to make a copy of the transmitted qubits. This property forms the basis of quantum cryptography .

The trigger for the discovery of the no-cloning theorem was a work by Nick Herbert , in which he showed how faster-than-light information transfer would be possible by copying qubits. William Wootters and Wojciech Zurek published the no-cloning theorem in 1982 and thus showed that this way, no faster-than-light information transfer can take place.

proof

To prove the no-cloning theorem, it is assumed that a quantum mechanical method exists that can perfectly copy any qubits. This assumption will then lead to an objection.

Let and be any two states that are to be copied to an independent state . Since scalar products (and probabilities) are to be obtained, the necessary procedure can only be described by a unitary mapping . This must have the following properties for copying: ${\displaystyle |\phi \rangle }$ ${\displaystyle |\psi \rangle }$ ${\displaystyle |k\rangle }$ ${\displaystyle U}$

${\displaystyle U(|\phi \rangle \otimes |k\rangle )=|\phi \rangle \otimes |\phi \rangle }$

${\displaystyle U(|\psi \rangle \otimes |k\rangle )=|\psi \rangle \otimes |\psi \rangle }$

The following two equations can be given for the scalar product : ${\displaystyle \langle U(\phi \otimes k)|U(\psi \otimes k)\rangle }$

${\displaystyle \langle U(\phi \otimes k)|U(\psi \otimes k)\rangle =\langle \phi \otimes \phi |\psi \otimes \psi \rangle }$

${\displaystyle \langle U(\phi \otimes k)|U(\psi \otimes k)\rangle =\langle \phi \otimes k|\psi \otimes k\rangle }$

The first equation follows by inserting the above equations, while the second equation results because unitary maps do not change the scalar product. Thus one obtains

${\displaystyle \langle \phi \otimes \phi |\psi \otimes \psi \rangle =\langle \phi \otimes k|\psi \otimes k\rangle ,}$

as well as due to the compatibility of the scalar product and the tensor product

${\displaystyle \langle \phi |\psi \rangle \langle \phi |\psi \rangle =\langle \phi |\psi \rangle \langle k|k\rangle \,.}$

So there follows ${\displaystyle \langle k|k\rangle =1}$

${\displaystyle \langle \phi |\psi \rangle ^{2}=\langle \phi |\psi \rangle .}$

This equation only has the solutions and . This means that either is (if ) or and are orthogonal (if ). This means that a quantum mechanical process that is able to copy a state can at best copy all orthogonal states. However, it is not possible to copy any states. ${\displaystyle \langle \phi |\psi \rangle =0}$ ${\displaystyle \langle \phi |\psi \rangle =1}$ ${\displaystyle \phi =\psi }$ ${\displaystyle \langle \phi |\psi \rangle =1}$ ${\displaystyle \phi }$ ${\displaystyle \psi }$ ${\displaystyle \langle \phi |\psi \rangle =0}$ ${\displaystyle \phi }$ ${\displaystyle \phi }$

An alternative proof that makes use of the linearity of can be formulated as follows: ${\displaystyle U}$

Be the to state which should be copied to. We develop into any base  : ${\displaystyle |\phi \rangle }$ ${\displaystyle |k\rangle }$ ${\displaystyle |\phi \rangle }$ ${\displaystyle |\phi _{j}\rangle }$

${\displaystyle |\phi \rangle =\sum _{j}a_{j}|\phi _{j}\rangle }$

with any expansion coefficient . With this development follows in the application of ${\displaystyle a_{j}}$ ${\displaystyle U}$

${\displaystyle U(|\phi \rangle \otimes |k\rangle )=|\phi \rangle \otimes |\phi \rangle =\sum _{j}a_{j}|\phi _{j}\rangle \otimes \sum _{j}a_{j}|\phi _{j}\rangle }$

Since an arbitrary state is to be copied, the following must also apply for the individual basis vectors : ${\displaystyle U}$ ${\displaystyle \phi _{j}}$

${\displaystyle U(|\phi _{j}\rangle \otimes |k\rangle )=|\phi _{j}\rangle \otimes |\phi _{j}\rangle }$

However, this implies for the copying process of ${\displaystyle |\phi \rangle }$

${\displaystyle U(|\phi \rangle \otimes |k\rangle )=U\left(\sum _{j}a_{j}|\phi _{j}\rangle \otimes |k\rangle \right){\stackrel {\text{Lin.}}{=}}\sum _{j}a_{j}U(|\phi _{j}\rangle \otimes |k\rangle )=\sum _{j}a_{j}|\phi _{j}\rangle \otimes |\phi _{j}\rangle }$

where we used the linearity of . It does, however ${\displaystyle U}$

${\displaystyle \sum _{j}a_{j}|\phi _{j}\rangle \otimes \sum _{j}a_{j}|\phi _{j}\rangle \neq \sum _{j}a_{j}|\phi _{j}\rangle \otimes |\phi _{j}\rangle }$

which refutes the existence of such a thing . ${\displaystyle U}$

swell

1. ↑ Dagmar Bruß: Quantum Information. Fischer Taschenbuch Verlag, Frankfurt am Main 2003, ISBN 3-596-15563-0 , pp. 35-40
2. ↑ Matthias Homeister: Understanding Quantum Computing. Vieweg, Wiesbaden 2005, ISBN 3-528-05921-4 , pp. 81-84
3. ↑ Moses Fayngold, Vadim Fayngold: Quantum Mechanics and Quantum Information. Wiley-VCH, ISBN 978-3-527-40647-0 , pp. 609-610.