No-cloning theorem as an instance of quantum incompatibility

In my first blog post, I will continue where Daniel ended his. Namely, I will give a more detailed explanation of what quantum incompatibility can mean. But this will still be just an example.

Suppose a quantum channel AB takes two systems as an input and gives two out. We keep the second input state fixed at all times, and call this a blank state. The situation looks something like this:


If we ignore one of the output states, we obtain a channel that takes in one input state and gives one out. We can either ignore the first or the second output, hence there are the two possibilities:


If the starting channel AB is specified, then we can calculate channels A and B. But we can also consider the inverse problem :

Suppose two quantum channels C1 and C2 are given. Is there a bipartite quantum channel C (and a blank state) such that C1 = ‘part A of C’ and C2 = ‘part B of C’?

The answer is: not always. Two channels can be incompatible! An exemplary instance is the one where both C1 and C2 are the identity channels. This case would look as follows:


The fact that two identity channels are not compatible in the previously specified way is known as the No-cloning theorem, now a standard piece in any quantum theory course.

The intention of this post was simply to point out that the No-cloning theorem can be seen as an instance of quantum incompatibility. Looking at it from this perspective puts it into a bigger frame.



About heinosaari
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