# Fidelity

## Quantum Fidelity

Fidelity is a popular measure of distance between density operators. It is not a metric, but has some useful properties and it can be used to defined a metric on this space of density matrices, known as Bures metric.

Fidelity as a distance measure between pure states used to be called "transition probability". For two states given by unit vectors φ,ψ it is $\vert\langle\phi,\psi\rangle\vert^2$. For a pure state (vector ψ) and a mixed state (density matrix ρ) this generalizes to $\langle\psi,\rho\psi\rangle$, and for two density matrices ρ,σ it is generalized as the largest fidelity between any two purifications of the given states. According to a theorem by Uhlmann, this leads to the expression

$F(\rho,\sigma)=\left(\textrm{tr}\sqrt{\sqrt{\rho}\sigma\sqrt{\rho}}\right)^2$

This is precisely the expression used by Richard Jozsa in<bibref>Jozsa94</bibref>, where the term fidelity appears to have been used first.

However, one can also start from $\vert\langle\phi,\psi\rangle\vert$, leading to the alternative

$F'(\rho,\sigma)=\textrm{tr}\sqrt{\sqrt{\rho}\sigma\sqrt{\rho}}$

used in<bibref>NielsenChuang</bibref>. This second quantity is sometimes denoted as $\sqrt{F}$ and called square root fidelity. It has no interpretation as a probability, but appears in some estimates in a simpler way.

## Basic properties

If $\rho=|\psi\rangle\langle\psi|$ is pure, then $F(\rho,\sigma)=\langle\psi|\sigma|\psi\rangle$ and if both states are pure i.e. $\rho=|\psi\rangle\langle\psi|$ and $\sigma=|\phi\rangle\langle\phi|$, then $F(\rho,\sigma)=|\langle\psi|\phi\rangle|^2$.

Other properties:

1. $0\leq F(\rho,\sigma)\leq 1$
2. $F(\rho,\sigma)=F(\sigma,\rho)\,$
3. $F(\rho_1\otimes\rho_2,\sigma_1\otimes\sigma_2)=F(\rho_1,\sigma_1)F(\rho_2,\sigma_2)$
4. $F(U \rho U^\dagger,U\sigma U^\dagger)=F(\rho,\sigma)$
5. $F(\rho,\alpha\sigma_1+(1-\alpha)\sigma_2)\geq \alpha F(\rho,\sigma_1)+(1-\alpha)F(\rho,\sigma_2),\ \alpha\in[0,1]$

## Bures distance

Fidelity can be used to define metric on the set of quantum states, so called Bures distance<bibref>fuchs96phd</bibref> DB

$D_B(\rho,\sigma) = \sqrt{2-2\sqrt{F(\rho,\sigma)}}$

and the angle<bibref>NielsenChuang</bibref>

$D_A(\rho,\sigma) = \arccos\sqrt{F(\rho,\sigma)}.$

The quantity DB(ρ,σ) is the minimal distance between purifications of ρ and σ using a common environment.

## Classical fidelity

Fidelity is also defined for classical probability distributions. Let {pi} and {qi} where i = 1,2,...,n be probability distributions. The fidelity between p and q is defined as

$F'(\{p_i\},\{q_i\})=\sum_{i=1}^n\sqrt{p_i,q_i}.$

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