The no-cloning theorem

Proving that qubits cannot be cloned.

• I. Introduction
• II. Elements of quantum mechanics
• III. The electromagnetic field
• IV. The qubit
• V. Quantum gates
• VI. States and operators
• VII. Shor’s algorithm
• VIII. Quantum teleportation
• IX. The Deutsch algorithm
• X. The no-cloning theorem
• XI. About anyons
• XII. About SU(2)
• XIII. Teleportation topology
• XIV. The Simon algorithm
• XV. Superdense coding
• XVI. Chern-Simons theory
• XVII. The toric code
• REF. Some pointers

Another remarkable difference between classical and quantum computing is the fact that one cannot clone arbitrary qubits. This is the flip-side of superposition; on the one hand it allows exponential processing speed on the other it disallows cloning of data.

The no-cloning is also nature’s way to avoid one cheating with the exclusion principle. One cannot measure simultaneously complementary properties of a system. This on its own is related to non-commuting operators like the well-known

\[[x,p]=i\hbar\]

relation stating that one cannot measure both the position and the momentum of a particle at the same time. Now, imagine one could identically clone a quantum particle. If so, one could measure the value of \(x\) exactly in one copy and the value of \(p\) in the other, which would contradict the exclusion. So, nature doesn’t allow one to clone quantum systems identically and all is well.

The ‘proof’ is really straighforward. Assume that there is a transformation \(U_c\) which clones the first qubit to the second:

\(U_c\,|x\,0\rangle := |x\, x\rangle.\)

If \(|x\rangle\) is a pure state then this can be achieve with for instance a controlled NOT gate;

\(U_{CNOT}\,|10\rangle = |11\rangle,;\;U_{CNOT}\,|00\rangle = |00\rangle\)

if the state is not pure however, say

\(|x\rangle = |+\rangle\)

then this lead to a contradiction since

\(U_c\,\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)\otimes |0\rangle = U_c\,\frac{1}{\sqrt{2}}|00\rangle + U_c\,\frac{1}{\sqrt{2}}|11\rangle\\\;=\frac{1}{\sqrt{2}}|00\rangle + \frac{1}{\sqrt{2}}|11\rangle\)

but at the same time you have by definition that the superposed state is copied so

\(U_c\,\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)\otimes |0\rangle =\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)\otimes \frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)\)

which contradicts the component-wise copying above.

The no-cloning theorem prevents the use of 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. When discussing the Shor theorem we’ll discuss quantum error correcting codes, which circumvent the no-cloning theorem.