5.9 No-cloning, and other no-go theorems

Let us now look at something that the controlled-\texttt{NOT} seems to be doing but, in fact, isn’t. It is easy to see that the \texttt{c-NOT} can copy the bit value of the first qubit: |x\rangle|0\rangle \overset{\texttt{c-NOT}}{\longmapsto} |x\rangle|x\rangle \qquad\text{(for $x=0,1$)} so one might suppose that this gate could also be used to copy superpositions, such as |\psi\rangle = \alpha|0\rangle+\beta|1\rangle, so that |\psi\rangle|0\rangle \overset{\texttt{c-NOT}}{\longmapsto} |\psi\rangle|\psi\rangle for any |\psi\rangle. But this is not true!

The unitarity of the \texttt{c-NOT} means that it turns superpositions in the control qubit into entanglement of the control and the target: if the control qubit is in the a superposition state |\psi\rangle = \alpha|0\rangle+\beta|1\rangle (with \alpha,\beta\neq0), and the target is in |0\rangle, then the \texttt{c-NOT} gate generates the entangled state \big( \alpha|0\rangle+\beta|1\rangle \big) |0\rangle \overset{\texttt{c-NOT}}{\longmapsto} \alpha|00\rangle + \beta|11\rangle. In fact, it is impossible to clone an unknown quantum state, and we can prove this!

To prove this via contradiction, let us assume that we could build a universal quantum cloner, and then take any two normalised states |\psi\rangle and |\phi\rangle that are non-identical (i.e. |\langle\psi|\phi\rangle|\neq1) and non-orthogonal (i.e. \langle\psi|\phi\rangle\neq0). If we then run our hypothetical cloning machine we get \begin{aligned} |\psi\rangle|0\rangle|W\rangle &\longmapsto |\psi\rangle|\psi\rangle|W'\rangle \\|\phi\rangle|0\rangle|W\rangle &\longmapsto |\phi\rangle|\phi\rangle|W''\rangle \end{aligned} where the third system, initially in state |W\rangle, represents everything else (say, the internal state of the cloning machine). For this transformation to be unitary, it must preserve the inner product, and so we require that \langle\psi|\phi\rangle = \langle\psi|\phi\rangle^2 \langle W'|W''\rangle which can only be satisfied if |\langle\psi|\phi\rangle| is equal to 1 or 0, but this contradicts our assumptions!

Thus, states of qubits, unlike states of classical bits, cannot be faithfully cloned. Note that, in quantum teleportation, the original state must therefore be destroyed, since otherwise we would be producing a clone of an unknown quantum state. The no-cloning property of quantum states leads to interesting applications, of which quantum cryptography is one.

The no-cloning theorem. Universal quantum cloners are impossible.

The no-cloning theorem is one of many so-called “no-go” theorems in quantum information. We won’t look at all of them in depth, but it’s worth mentioning them here and giving a very rough idea of what each one says.

• No-teleportation. An arbitrary quantum state cannot be entirely expressed with classical information. In other words, the process of converting quantum information to classical information cannot be reversed: classical channels cannot transmit quantum information.

  This can be seen as a consequence of no-cloning: if we were able to turn a quantum state into classical information and then back again, we could simply clone the classical information and then get a cloned copy of our quantum state.

  The name is a bit confusing, because we have just seen that quantum teleportation is possible through the use of entanglement, but it refers to the idea of classical teleportation of quantum states.

  Note that the “converse” to this is possible though: if we start with some classical information then we can convert it to quantum information and then back again perfectly fine (for example, using the fact that orthogonal states can be perfectly distinguished).

• No-broadcasting. Given a single copy of a quantum state, it cannot be shared with two or more parties.

  This is an even more direct consequence of no-cloning: if we can’t copy a state, then we have no way of sharing it with multiple people. However, the real technical statement of this theorem involves non-pure states, which require the language of density operators to talk about — something that we will not see until Chapter 8.

  One particularly unexpected detail here is that the theorem is no longer true if we’re provided with more than one copy of the state to start with. For example, in a process known as superbroadcasting¹¹⁵, given four copies of an input state we can actually broadcast six copies!

• No-deleting. Given two copies of an arbitrary quantum state, it is impossible to delete one.

  You might hear people saying that the fact that we require our quantum operations to be unitary is to do with reversibility¹¹⁶, and so there is a general pattern in quantum theory where theorems will have time-dual versions, giving by taking the same theorem but imagining that time goes in the opposite direction. No-deleting is the time dual of no-cloning, and whereas the latter tells us that quantum states are pretty delicate, the former tells us that they are also in some sense rather robust.

  We might as well state this theorem a bit more precisely, because we have seen almost all of the necessary definitions already: given a qubit in an unknown state |\psi\rangle, there is no isometry V (Section 9.3) such that V\colon |\psi\rangle|\psi\rangle|W\rangle \longmapsto |\psi\rangle|0\rangle|W'\rangle with |W'\rangle being independent of |\psi\rangle. Just as for no-cloning, we can of course delete some qubits (for example those in orthogonal states, since these behave a lot like classical bits), but there is no V that works universally, for any arbitrary state |\psi\rangle.

• No-communication. An entangled state cannot be used to transmit information by measurement of a subsystem.

  We talk about this theorem in the context of a worked example in Exercise 5.14.3, and we delve into the details when we talk about Bell tests in Chapter 6, but it is basically the answer to Einstein’s worry about “spooky action at a distance” that we mentioned back in Section 5.1: the seemingly infinitely fast sending of information between entangled qubits cannot actually send any meaningful information, but only purely random bits.

  This theorem is actually stronger than no-cloning, in that we can prove no-cloning from no-communication.

  Yet again we see another example of how the quantum whole is much greater than the sum of its parts: no-teleportation says that classical channels alone cannot send quantum information; no-communication says that entanglement and measurement alone cannot send quantum information; the quantum teleportation protocol of Section 5.8 says that you can send quantum information if you combine both methods together.

• No-hiding. Quantum information cannot be lost, even through decoherence.

  This theorem is related to no-deletion, in that it shows the robustness of quantum states. In Chapter 13 we will study the notion of decoherence, which is sort of like “quantum noise”, and is one of the main problems faced when actually trying to design and build quantum computers in reality. The no-hiding theorem says that, when quantum information is “lost” through decoherence, it actually merely moves into the subspace corresponding to the environment — we might have lost it, but nature hasn’t.¹¹⁷

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115. This is shown in D’Ariano, Macchiavello, and Perinotti’s “Superbroadcasting of mixed states”, arXiv:quant-ph/0506251.↩︎

116. In fact, we’ll talk a bit about reversibility of computation in Section 10.1.↩︎

117. This theorem is of particular interest to physicists studying black holes, since it leads to the black hole information paradox.↩︎