Friday, 4 July 2014

An arbitrary quantum cannot be cloned

One of the early important results in the study of quantum information is the no-cloning theorem, which tells us that there is no quantum operation that allows us to create multiple copies of an arbitrary quantum state.

This property is very different from what we expect from classical information, which you may reproduce as many times as you wish. For example, you can send a PDF file by email to many recipients while keeping a copy to yourself. The important point is that whatever the contents may be, you can make a duplicate of it.

Now consider a cloning machine M for qubits that can produce identical copies of the states |u) and |d):

M |u) |0) = M |u) |u),
M |d) |0) = M |d) |d) ,

where |0) denotes any fixed initial state for M. This is necessary in pretty much the same way you would need a blank piece of paper before you can photocopy a printed document.

Suppose we want to produce a copy of the state

|q) = a |u) + b|d).

If we feed |q) as an input to machine M, we would get

M |q) |0) = M [ a |u) |0) + b |d) |0) ] 
                  =  a [M |u) |0)] + b [M|d) |0)]
                  = a |u) |u) + b |d) |d).

If M could make a clone of |q), we expect the output |q) |q),

|q) |q) = [a |u) + b|d)] [a |u) + b|d)] 
             = a^2 |u) |u) + ab |u) |d) + ba |d) |u) +b^2 |d) |d),

which is not the same as 

M |q) |0) =  a |u) |u) + b |d) |d).

unless a = 0 or b = 0, which just means the input state to M must be either |u) or |d).

What this tells us is that a quantum cloning machine works only for states that are fully distinguishable by measurement, since we know that |u) and |d) are orthogonal quantum states, and we can discriminate one from the other by a measurement in the basis {|u), |d)}

Attempt to construct a quantum cloning machine M for qubits. Assuming the machine produces exact copies of the states |u) and |d), if we give M the arbitrary input state |q), it outputs a state that is generally entangled and is not the same as having two copies of |q), except when |q) = |u) or  |q) = |d).

You may wonder if we can even have a cloning machine like M, since we actually assume at the start that we have such a machine. There is no problem in doing this because if all our states are guaranteed to be either |u) and |d), then we are effectively dealing with classical information. You can think of |u) as something like bit "0" and |d) as something like bit "1", and we know we can tell |u) and |d) apart by measurement, so it will be just like differentiating between the bits 0 and 1.

This means, for example, that it is possible to construct a cloning machine M' such that 

M' |+) |0) = M' |+) |+),
M' |-) |0) = M' |-) |-) ,


|+) =  [ |0) + |1) ] /sqrt(2), 
|-) =  [ |0) - |1) ] /sqrt(2),

since this is another basis of orthogonal states for qubits.

It is worth noting that the generalization actually holds true: quantum information that is encoded exclusively in orthogonal (hence, fully distinguishable) states behave like classical information.

What has been shown is that duplicating an arbitrary quantum state with a fixed machine is not possible. However, it is possible to construct a quantum cloning machine if we are willing to make some concessions. One way is to have a machine that sometimes produces exact copies but fails on other times. Another way is to have a machine that does not quite replicate the input but always produces a close approximation of the input state. 

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