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I am trying to get an intuitive idea of how the No-Cloning theorem affects Quantum computation. My understanding is that given a qubit $Q$ in superposition $Q_0 \left| 0 \right> + Q_1 \left| 1 \right>$, NCT states another Qubit $S$ cannot be designed such that $S$ is equivalent to the state of $Q$.

Now the catch is, what does Equivalent mean? It could mean either that:

  1. $S = S_0 \left| 0 \right> + S_1 \left| 1 \right>$ such that $S_0 = Q_0, S_1 = Q_1$.

  2. Or it could mean that $S = Q$, meaning that if $S$ is observed to be some value ( for example 0) then $Q$ MUST be that same value, and vice versa.

So it seems that point 2, occurs anyways in entangled systems (particularly cat-states), so I can eliminate that option and conclude that that No Cloning states, given a qubit $Q$, it's impossible to make another qubit $S$ such that:

$S = S_0 \left| 0 \right> + S_1 \left| 1 \right>$ such that $S_0 = Q_0, S_1 = Q_1$.

Is this correct?

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    $\begingroup$ it means if you have an unknown state then you cannot make a copy. Of course, if you know the state, you can manufacture duplicates. So your statement in the OP needs review. $\endgroup$ – Peter Diehr Apr 10 '16 at 1:27
  • $\begingroup$ You can know projections of the state and you can always copy those, if you like. Since the entire state is one of the possible superpositions of those projections, you can reconstruct all possible states that could have lead to those projections and duplicate those, it's just not going to be a unique solution. $\endgroup$ – CuriousOne Apr 10 '16 at 1:46
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    $\begingroup$ @PeterDiehr would you like to make that an answer? $\endgroup$ – frogeyedpeas Apr 10 '16 at 2:04
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    $\begingroup$ @CuriousOne, what do you mean by "projections"? $\endgroup$ – frogeyedpeas Apr 10 '16 at 2:04
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    $\begingroup$ @Phonon I've corrected and accepted answers I found most useful. Hopefully that clears the record :) $\endgroup$ – frogeyedpeas Apr 17 '16 at 8:39
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You need to use a more precise notion of the cloning process, in order to understand the general statement and its repercussions. I will give you some outline here (mainly following the explanations of B. Schumacher and M. Westmoreland given in the reference), with an emphasis on the most important aspects of it, but to fully appreciate the importance of the No-cloning Thm I highly recommend looking through the various ways you can prove it (I can show you some ways of proving it in this post, if you will see it necessary).

Main statement:

  • No-cloning theorem states that no unitary cloning machine exists that would clone arbitrary initial states.
  • A softer version would be: Quantum information cannot be copied exactly.

Repercussions if arbitrary cloning was possible: (not following any specific order)

  • If a hypothetical device exists that could duplicate the state of a quantum system, then an eavesdropper would be able to break the security of the $BB84$ key distribution protocol.
  • A cloning machine would allow to create multi-copy states $|x\rangle^{\otimes n}$ from a single state $|x\rangle.$ But take another single state $|y\rangle,$ create its corresponding multi-state $|y\rangle^{\otimes n},$ and you can overcome the basic distinguishability limitations of states in Quantum Mechanics, as multi-copy states can be better distinguished (correct term would be more reliably) than single states.

    Recall that the distinguishability of two states $x,y$ is given by their amount of overlap, i.e. $|\langle x | y\rangle|,$ the closer this is to vanishing, the better we can distinguish between the states. (If you're not familiar with the concept of multi-states being more reliably distinguishable, let me know).

  • The no-cloning theorem guarantees the no-communication theorem and thus prevents faster than-light communication using entangled states. (the no-communication theorem basically says that: if two parties have systems $A$ and $B$ respectively, and suppose their joint state is entangled $|\psi^{AB}\rangle,$ then: the two parties cannot transfer information to each other either by: choosing different measurements for their respective systems, or evolving their systems using different unitary time evolution operators.)

More precise definition of the cloning problem:

There are three elements involved, the initial state (input) to be copied $(1)$, a blank state onto which we want to create the copy $(2)$ and a machine that plays the role of the cloning device $(3)$. The composite system is then $(123).$

Suppose the state of $(2)$ is $|0\rangle,$ state of $(1)$ being $|\Phi\rangle$ and the starting state of $(3)$ is $|D_i\rangle.$ Let us denote the action of the cloning device by the unitary operator $U.$ It is important to point out that the starting state of the composite system $(23)$ and the action $U$ is independent of the state to be copied, i.e. system $(1).$ Our starting composite state is then:

$$ |123\rangle_i = |\Phi\rangle \otimes |0\rangle \otimes |D_i\rangle $$ By applying $U$, thus after cloning, the state of $(1)$ is unchanged, but upon success of the cloning, the state of $(2)$ must be exactly that of $(1).$ So

$$ U|123\rangle_i = |\Phi\rangle \otimes |\Phi\rangle \otimes |D_f\rangle $$

Given this description, the no-cloning theorem says that such $U$ does not exist for arbitrary states of $(1).$

Hints on the proofs:

  • One way would be to use the principle of superposition to show that such cloning is not possible, by showing that if the device is to work for two orthogonal states, it would create entangled outputs for their superposition. (thus the subsystems are no longer even in pure states)
  • Another way would be going back to the concept of distinguishability between non-orthogonal states, and using the fact that unitary time evolution preserve inner products, thus showing that the cloning device is impossible as it would allow considerable improvement on the distinguishability.

Reference: A highly recommended reference, also for further reading on all this matter, would be the book of Quantum Processes, systems & Information by Benjamin Schumacher and Michael Westmoreland. (Of relevance here, are chapters 4 and 7.)

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  • $\begingroup$ I think that there is a subtlety in your answer. The no cloning theorem as I understand it (which might be wrong) states that a universal cloning machine $U$ cannot exist. In which universal means that given $U$ and two arbitrary states $\Psi \rangle$ and $\Phi \rangle$ are cloned via the same $U$. But it doesn't say anything about finding a certain $U$ that copies only a particular state. $\endgroup$ – Jasimud Aug 1 '16 at 5:53
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The No-Cloning Theorem means that if you have an unknown state then it is not possible to make an identical copy.

The original reference is to Wooters, A single quantum cannot be cloned. Also see Wooters and Zurek on the no-cloning theorem, 2009.

Of course, if you know the state, you can manufacture duplicates; or if you have many identical copies of the unknown state, provided by some quantum machine, you could take a complete set of measurements, such as projections onto each of the eigenspaces, and use this information to attempt a recreation. But this isn't possible to do with a single unknown state.

Thus in classical computing one can exchange two variables with statements like this:

C=A; A=B; B=C;

But the very first statement is not possible in quantum computing due to the No-Cloning Theorem. It is possible to copy certain prepared states. For example, pairs of orthogonal states can be copied by means of CNOT gates. But these are known states; arbitrary states cannot be cloned.

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  • $\begingroup$ I think it's very misleading to suggest that you can't swap qubits in a classical way. You can swap two qubits with A ^= B; B ^= A; A ^= B, where ^= means CNOT. The problem with C=A is not that you're copying A, but that you're erasing the old value of C, which is irreversible (unless it had a known value). $\endgroup$ – benrg Jul 31 '16 at 3:04
  • $\begingroup$ @bengr: your proposed CNOT gate implementation of cloning works for orthogonal states, but cannot copy general states. I've updated the answer slightly. For example, see physics.stackexchange.com/questions/43141/… $\endgroup$ – Peter Diehr Aug 1 '16 at 0:54
  • $\begingroup$ What I was trying to say is that you are conflating copying and cloning. C=A is a copy, not a clone. Or at least, it can be a copy. This is a classical algorithm and classically it is a copy. The classical algorithm fails because assignment is irreversible, not because it's nonlinear. Cloning is nonlinear. They're forbidden but for different reasons. $\endgroup$ – benrg Aug 1 '16 at 2:53
  • $\begingroup$ (I think the XOR-swap example just confused the issue. But my point was that if you take a classical swapping algorithm that is similar to assignment-swap but reversible, its quantum version works on arbitrary states with no preferred basis—even though, like the assignment-swap algorithm, it temporarily copies bits. This was supposed to demonstrate that the assignment swap fails only because of irreversibility, not because of any copying-related problem.) $\endgroup$ – benrg Aug 1 '16 at 2:55
  • $\begingroup$ Cloning and copying are the same thing. You start with one, then you have two. But it cannot be done with unitary operations for the general case. $\endgroup$ – Peter Diehr Aug 1 '16 at 3:10

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