###### tags: `Quantum Calculation`

# Fidelity(忠実度)

Fidelity (忠実度 or フィデリティ in Japanaese) is a quanitiy representing the similarity of two states.

## Definition

### Fidelities of 2 pure states
\begin{eqnarray}
F(\left|\psi'\right>,\left|\psi\right>) =: \left| \left< \psi' \middle| \psi \right>\right|.
\end{eqnarray}

This can be interpreted as inner product of two vectors on Hilbert space.

### Generalization to mixed states
Let the two mixed state be $\rho,\rho'$, and pure state via the purification be $\psi,\psi'$.
One wants to define fidility by $\left| \left< \psi' \middle| \psi \right>\right|$. However **the purification of mixed state is not unique**, the filelity of mixed states are defined by the maximum of fidelity of the pure states, namely,
\begin{eqnarray}
F(\rho',\rho) =: \max_{\left|\psi'\right>,\left|\psi\right>} \left|\left< \psi' \middle| \psi \right>\right|.
\end{eqnarray}

The "max" on the RHS seems to be complicated, thus we try to simplify.

##
\begin{eqnarray}
\rho_{A} = \sum_{k=1}^{d} \lambda_{k} \left|k\right>_ A \left<k\right|_ A
\end{eqnarray}

\begin{eqnarray}
\left|\psi\right>_ A = \sum_{k=1}^{d} \sqrt{\lambda_{k}} \ket{k}_ {A} \ket{k}_{R}
\end{eqnarray}

## referenses
[量子情報理論の基本：フィデリティ](https://qiita.com/SamN/items/634b11b6faf8d713cff1)