Show that the purity P (ρA) is equal to the following expression:P (ρA) = Tr {(ρA ⊗ ρA′ ) FAA′ } . (4.106)where system A′ has a Hilbert space structure isomorphic to that of system A and FAA′ isthe swap operator that has the following action on kets in A and A′:∀x, y FAA′ |x〉A|y〉A′ = |y〉A|x〉A′ . (4.107)(One can in fact show more generally that Tr {f (ρA)} = Tr {(f (ρA) ⊗ IA′ ) FAA′ } for anyfunction f on the operators in system A.)

To show that the purity P (ρA) is equal to the expression P (ρA) = Tr {(ρA ⊗ ρA′ ) FAA′ }, we need to understand the definitions and properties of the terms involved.

1. Purity of a quantum state: The purity of a quantum state ρA is defined as Tr(ρA^2). It is a measure of the "mixedness" of a state, with a purity of 1 indicating a pure state and a purity less than 1 indicating a mixed state.

2. Hilbert space structure: A Hilbert space is a complex inner product space that is complete, meaning that it contains all its limit points. In this case, system A′ has a Hilbert space structure isomorphic to that of system A, meaning that there is a one-to-one correspondence between the points in the two spaces that preserves the structure of the spaces.

3. Swap operator: The swap operator FAA′ is defined by its action on kets in A and A′: ∀x, y FAA′ |x〉A|y〉A′ = |y〉A|x〉A′. This operator swaps the states of systems A and A′.

Now, let's prove the given expression:

We start with the definition of the purity of ρA:

P (ρA) = Tr(ρA^2).

We can rewrite this as:

P (ρA) = Tr{(ρA ⊗ ρA′)(ρA ⊗ ρA′)}.

Now, we apply the swap operator FAA′:

P (ρA) = Tr{(ρA ⊗ ρA′) FAA′ (ρA ⊗ ρA′) FAA′}.

Since the swap operator is its own inverse (FAA′^2 = I), we can simplify this to:

P (ρA) = Tr{(ρA ⊗ ρA′) FAA′}.

So, we have shown that the purity P (ρA) is equal to the expression P (ρA) = Tr {(ρA ⊗ ρA′ ) FAA′ }.