# 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 |⟨*ϕ*, *ψ*⟩|2. For a pure state (vector *ψ*) and a mixed state (density matrix *ρ*) this generalizes to ⟨*ψ*, *ρ**ψ*⟩, 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 inJozsa94, where the term fidelity appears to have been used first.

However, one can also start from |⟨*ϕ*, *ψ*⟩|, leading to the alternative

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

used inNielsenChuang. 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 *ρ* = ∣*ψ*⟩⟨*ψ*∣ is pure, then *F*(*ρ*, *σ*) = ⟨*ψ*∣*σ*∣*ψ*⟩ and if both states are pure i.e. *ρ* = ∣*ψ*⟩⟨*ψ*∣ and *σ* = ∣*ϕ*⟩⟨*ϕ*∣, then *F*(*ρ*, *σ*) = ∣⟨*ψ*∣*ϕ*⟩∣2.

Other properties:

- 0 ≤
*F*(*ρ*,*σ*) ≤ 1 *F*(*ρ*,*σ*) =*F*(*σ*,*ρ*)*F*(*ρ*1 ⊗*ρ*2,*σ*1 ⊗*σ*2) =*F*(*ρ*1,*σ*1)*F*(*ρ*2,*σ*2)*F*(*U**ρ**U*† ,*U**σ**U*† ) =*F*(*ρ*,*σ*)*F*(*ρ*,*α**σ*1 + (1 −*α*)*σ*2) ≥*α**F*(*ρ*,*σ*1) + (1 −*α*)*F*(*ρ*,*σ*2),*α*∈ [0, 1]

### Bures distance

Fidelity can be used to define metric on the set of quantum states, so called *Bures distance*fuchs96phd *D**B*

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

and the *angle*NielsenChuang

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

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

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

### References

### See also

Category:Handbook of Quantum Information Category:Mathematical Structure Category:Linear Algebra