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Fidelity

1 Quantum Fidelity
2 Basic properties
3 Bures distance
4 Classical fidelity
5 Related papers
6 See also

Quantum Fidelity

Fidelity is a popular measure of distance between density operators. It is not a metric, but has some useful properties.

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 [1], 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 [2]. 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

Other properties:

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

Bures distance

Fidelity can be used to define metric on the set of quantum states, so called Bures distance $D B$

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

and the angle [2]

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

The quantity $D B (ρ,σ)$ is the minimal distance between purifications of $ρ$ and $σ$ using a common enviroment.

Classical fidelity

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