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## 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 . For a pure state (vector ψ) and a mixed state (density matrix ρ) this generalizes to , 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

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 , leading to the alternative

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

## Basic properties

If is pure, then and if both states are pure *i.e.* and , then .

Other properties:

## Bures distance

Fidelity can be used to define metric on the set of quantum states, so called *Bures distance*<bibref>fuchs96phd</bibref> *D*_{B}

and the *angle*<bibref>NielsenChuang</bibref>

The quantity *D*_{B}(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 {*p*_{i}} and {*q*_{i}}
where *i* = 1,2,...,*n* be probability distributions. The fidelity between *p* and *q*
is defined as

## References

<bibreferences/>