No Cloning Theorem

Consider the following Qubit state

|ψ⟩  = a |0⟩ + b |1⟩     |a((|)^2) +  |b((|)^2) =1

You want to find out something about the state, so you make a measurement on that Qubit

Question: What are the possible results of your measurement?

Answer: You can only get the value 0 or 1, nothing else!   (e.g. 1/2)

Question:If  a=1/2  and you make a measurement with the result 0, what is the result of the next measurement immediately after that?

Answer: According to the collapse postulate a result of 0 will immediately force
the quantum state |ψ ⟩ into |0⟩. So the probability of finding 0 is 100%. (Quantum Zeno effect)

(* picture credit: Max Tegmark: Our Mathematical Universe  *)

Question: Suppose Alice has  a 1000 copies of |ψ⟩. She make a measurement on each,
how many of those measurement will lead to the result 0 ?  How many will result in the value 1 ?

Answer: She will find, approximately,  |1/2 ((|)^2) (1000) = 250 instances of 0.  There will be 750 instances where she finds the value 1.

Question: From the above set of measurements can Alice guess what the value of b (in the definition
of |ψ⟩ ) is ?

Answer: Sort of.  Since she knows, according to the Born rule, that |b((|)^2) = 3/4  she guesses the most simple form b=((Sqrt[3])/2). But would not  b = i ((Sqrt[3])/2), also work ? How many choices for b exist that will provide the identical ensemble of results obtained above?  Is there an experiment in which Alice might find the value of b?

Question: (i)  How many copies of |ψ⟩ does Alice need in order to find b?   (ii) Is there a copy machine that will allow Alice to make any number (e.g. 1000) of  copies from a single version of |ψ⟩ ?

Answer to (i) I don’t know. (ii) No!  The no-cloning theorem.

Suppose Alice has a Qubit state |ψ⟩ that she wants to make a copy of. If |ψ⟩ is encoded in a specific physical system (e.g. an electron with spin). Then by making a copy she needs to find another electron in some other state |ϕ ⟩ and copy the information contained in the first Qubit to this second
Qubit (electron). Diagrammatically Alice needs to do something like this

(1)     |ψ⟩|ϕ⟩  ⟶  |ψ⟩|ψ⟩
 
 If she achieves this then she now has two copies of |ψ⟩. The same information is encoded in both electrons. If she is successful in facilitating Eq. (1), then she should be able to make thousands  of copies of |ψ⟩ by taking a thousand electrons and repeating this among neighboring electrons so that at the end of the day she has taken the state

|ψ⟩|α⟩|β⟩ ...|ζ⟩  and turned it into |ψ⟩|ψ⟩|ψ⟩ ...|ψ⟩
 
 In this way Alice can now do the experiment discussed above. She first measures the state of electron 1, and writes down the result. She knows that electron1 ( or Qubit 1)  has collapsed  so that if she obtain the result 0 for that measurement the new state is
 
 |0⟩|ψ⟩|ψ⟩ ...|ψ⟩
 
 But not to worry.  Alice can now perform the measurement one the second electron and so on ...
 We will then have data that looks something like this
 
 01100101001 ....
 
 Each of these 0, and 1’s where the result of a measurement on the  identical system and Alice can then perform the statistics necessary to glean information on |ψ⟩.

Unfortunately, the no-cloning theorem says that Eq. (1) is impossible to implement.
Proof:  Eq. (1) is simply a gate operation but we know that gates are what kind of operators?

Answer: Unitary Operators

This means there should exist a gate U   ( U (SuperDagger[U])=(SuperDagger[U])U=1)  

(a)  U  |ψ⟩|ϕ⟩  =  |ψ⟩|ψ⟩

Now U should be universal, i.e. it should not depend on what |ψ⟩ is, so we can also
write Eq. (a) as

(b)  U  |Ω⟩|ϕ⟩  =  |Ω⟩|Ω⟩

Now lets take the conjugate of Eq. (b)  

(b)  ⟨ϕ|⟨Ω| (SuperDagger[U])  = ⟨Ω| ⟨Ω|

Taking the inner products of the respective right and left hand sides of these two equations

 ⟨ϕ|⟨Ω| (SuperDagger[U]) U  |ψ⟩|ϕ⟩ = ⟨Ω| ⟨Ω|  |ψ⟩|ψ⟩

Or since (SuperDagger[U]) U =1

 ⟨Ω |ψ⟩  = ((⟨Ω| ψ⟩)^2)

Since ⟨Ω|ψ⟩ is a complex number the only solution to the above equation is 

⟨Ω |ψ⟩ =0 
or
⟨Ω |ψ⟩=1 

however this condition is much too restrictive and so for any arbitrary state 
there exists no Universal copying machine U so that

U   |ψ⟩|ϕ⟩  =  |ψ⟩|ψ⟩

QED

Oh Yea?

How about I do this

1⊗ X  |1 ⟩|0⟩ ⟶ |1⟩|1⟩

Have I  copied the contents of  Qubit |1⟩ into the second Qubit on the r.h.s ?

Moral of the no-cloning theorem 

A quantum photocopy machine is not available. (Hey! isn’t that good news if you would like to protect
copyright material e.g. pictures, songs etc. )

Further implications. If Alice can’t make multiple copies of |ψ⟩ how can she ever hope to gain
information contained in |ψ⟩.

Quantum mechanics does not allow us to learn anything about an unknown quantum
state |ψ⟩

Question: However this does not mean that |ψ⟩  cannot be shared. If Alice has a Qubit |ψ⟩ can she send this, one and only, copy to Bob?

Answer: Yes

Quantum Teleportation

(Quantum Fedex, post etc.)