Time evolution postulate YouTube

Simply stated, the time evolution postulate states that the time evolution of a quantum state is linear. Although stated in the form of a postulate, this is an experimental fact.

Suppose some operator, $T$ say, transforms, over an as yet unspecified period of time, states into other states, then the postulate says that $T$ must be linear. Thus, $$T\left(\alpha \ket{a} + \beta \ket{b}\right) = \alpha T\ket{a} + \beta T\ket{b}$$

More simply, for an unspecified transformation, $$\left.\begin{matrix}\ket{a} \rightarrow \ket{a'}\\\ket{b} \rightarrow \ket{b'}\end{matrix}\right\} \Rightarrow \alpha \ket{a} + \beta \ket{b} \rightarrow \alpha \ket{a'} + \beta \ket{b'}$$

We shall be using this postulate in the two-slit experiment in the next lecture, but we can consider a simple application now.

No-cloning theorem

Hypothetically, if we were able to build a cloning machine, then, given an arbitrary system, we would want it to pump out pairs of indentical copies of that system, each having the same state vector as the original system and these new pairs would have to be product states.

That is, for a given state, $\ket{a}$, say, $$\ket{a} \rightarrow \ket{a} \otimes \ket{a}$$

We show that this leads to a contradiction, meaning that no such cloning machine can exist, which is the statement of the theorem.

Consider cloning the state $$\ket{a} = \frac{\ket{u} + \ket{d}}{\sqrt{2}}$$

The new pair of states would then be $$\begin{align*}\ket{a} \otimes \ket{a} &= \left(\frac{\ket{u} + \ket{d}}{\sqrt{2}}\right) \otimes \left(\frac{\ket{u} + \ket{d}}{\sqrt{2}}\right)\\ &= \frac{\ket{uu} + \ket{ud} + \ket{du} + \ket{dd}}{2}\end{align*}$$

But, if we consider how $\ket{a}$ must transform according to the time evolution postulate, as a linear combination of $\ket{u}, \ket{d}$ then, $$\left.\begin{matrix}\ket{u} \rightarrow \ket{uu}\\\ket{d} \rightarrow \ket{dd}\end{matrix}\right\} \Rightarrow \ket{a} \rightarrow \frac{\ket{uu} + \ket{dd}}{\sqrt{2}} \neq \ket{a} \otimes \ket{a}$$

The conclusion is that a machine that could clone arbitrary states is impossible.

Nb. It is theoretically possible to build a machine that can clone a particular state or states, but not a machine that can clone arbitrary states.