CritPt / Challenge_30_main / Let H_(b), H_(B), H_(f), H_(P) each be a finite-dimensional Hilbert space, with…

# Problem setup:
Let H_(b), H_(B), H_(f), H_(P) each be a finite-dimensional Hilbert space, with dimH_(b)>dimH_(B) and dim(H_(b)⊗H_(f))=dim(H_(B)⊗H_(P)).
Define the linear map V:H_(b)→H_(B) as follows,
\begin{equation}
V = \sqrt{\dim(H_P)} \langle 0|_P O |0\rangle_f,
\end{equation}
where |0⟩_(f) and |0⟩_(P) are fiducial states on H_(f) and H_(P), respectively, and O is an operator acting on H_(b)⊗H_(f) drawn at random from the orthogonal group O(d) with d=dim(H_(b)⊗H_(f)).

Let d_(P):=dimH_(P) and d_(B):=dimH_(B). And for any state |ϕ⟩,  denote its complex conjugate by $|\phi^*\rangle$.

# Main problem:

Given two states |ψ⟩_(b)∈H_(b) and |ϕ⟩_(b)∈H_(b), calculate the quantity
\begin{equation}
	\overline{\lvert \langle\phi|V^\dagger V|\psi\rangle \rvert^2},
\end{equation}
where the overline indicates the average over O.

# Problem setup:
Let $H_b$, $H_B$, $H_f$, $H_P$ each be a finite-dimensional Hilbert space, with $\dim H_b > \dim H_B$ and $\dim(H_b \otimes H_f) = \dim(H_B \otimes H_P)$.
Define the linear map $V: H_b \to H_B$ as follows,
\begin{equation}
V = \sqrt{\dim(H_P)} \langle 0|_P O |0\rangle_f,
\end{equation}
where $|0\rangle_f$ and $|0\rangle_P$ are fiducial states on $H_f$ and $H_P$, respectively, and $O$ is an operator acting on $H_b \otimes H_f$ drawn at random from the orthogonal group $O(d)$ with $d = \dim(H_b \otimes H_f)$.

Let $d_P := \dim H_P$ and $d_B := \dim H_B$. And for any state $|\phi\rangle$,  denote its complex conjugate by $|\phi^*\rangle$.

# Main problem:

Given two states $|\psi\rangle_b \in H_b$ and $|\phi\rangle_b \in H_b$, calculate the quantity
\begin{equation}
	\overline{\lvert \langle\phi|V^\dagger V|\psi\rangle \rvert^2},
\end{equation}
where the overline indicates the average over $O$.