What exactly does No cloning mean, in the context of Quantum Computing?

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?

Answer

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.

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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.)