Fidelity

== Quantum Fidelity == Fidelity is a popular [[Distance measures between states|measure of distance]] between [[density operator]]s. 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 $\backslash phi,\backslash psi$ it is $\backslash vert\backslash langle\backslash phi,\backslash psi\backslash rangle\backslash vert^2$. For a [[pure state]] (vector $\backslash psi$) and a [[mixed state]] (density matrix $\backslash rho$) this generalizes to $\backslash langle\backslash psi,\backslash rho\backslash psi\backslash rangle$, and for two density matrices $\backslash rho,\backslash sigma$ 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 :$F(\backslash rho,\backslash sigma)=\backslash left(\backslash textrm\{tr\}\backslash sqrt\{\backslash sqrt\{\backslash rho\}\backslash sigma\backslash sqrt\{\backslash rho\}\}\backslash right)^2$ 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 $\backslash vert\backslash langle\backslash phi,\backslash psi\backslash rangle\backslash vert$, leading to the alternative :$F\text{'}(\backslash rho,\backslash sigma)=\backslash textrm\{tr\}\backslash sqrt\{\backslash sqrt\{\backslash rho\}\backslash sigma\backslash sqrt\{\backslash rho\}\}$ used inNielsenChuang. This second quantity is sometimes denoted as $\backslash sqrt\{F\}$ and called ''square root fidelity''. It has no interpretation as a probability, but appears in some estimates in a simpler way. == Basic properties == If $\backslash rho=|\backslash psi\backslash rangle\backslash langle\backslash psi|$ is pure, then $F(\backslash rho,\backslash sigma)=\backslash langle\backslash psi|\backslash sigma|\backslash psi\backslash rangle$ and if both states are pure i.e. $\backslash rho=|\backslash psi\backslash rangle\backslash langle\backslash psi|$ and $\backslash sigma=|\backslash phi\backslash rangle\backslash langle\backslash phi|$, then $F(\backslash rho,\backslash sigma)=|\backslash langle\backslash psi|\backslash phi\backslash rangle|^2$. Other properties: # $0\backslash leq\; F(\backslash rho,\backslash sigma)\backslash leq\; 1$ # $F(\backslash rho,\backslash sigma)=F(\backslash sigma,\backslash rho)\backslash ,$ # $F(\backslash rho\_1\backslash otimes\backslash rho\_2,\backslash sigma\_1\backslash otimes\backslash sigma\_2)=F(\backslash rho\_1,\backslash sigma\_1)F(\backslash rho\_2,\backslash sigma\_2)$ # $F(U\; \backslash rho\; U^\backslash dagger,U\backslash sigma\; U^\backslash dagger)=F(\backslash rho,\backslash sigma)$ # $F(\backslash rho,\backslash alpha\backslash sigma\_1+(1-\backslash alpha)\backslash sigma\_2)\backslash geq\; \backslash alpha\; F(\backslash rho,\backslash sigma\_1)+(1-\backslash alpha)F(\backslash rho,\backslash sigma\_2),\backslash \; \backslash alpha\backslash in[0,1]$ == Bures distance == Fidelity can be used to define metric on the set of quantum states, so called ''Bures distance''fuchs96phd $D\_B$ :$D\_B(\backslash rho,\backslash sigma)\; =\; \backslash sqrt\{2-2\backslash sqrt\{F(\backslash rho,\backslash sigma)\}\}$ and the ''angle''NielsenChuang :$D\_A(\backslash rho,\backslash sigma)\; =\; \backslash arccos\backslash sqrt\{F(\backslash rho,\backslash sigma)\}.$ The quantity $D\_B(\backslash rho,\backslash sigma)$ is the minimal distance between purifications of $\backslash rho$ and $\backslash sigma$ using a common environment. == Classical fidelity == Fidelity is also defined for classical probability distributions. Let $\backslash \{p\_i\backslash \}$ and $\backslash \{q\_i\backslash \}$ where $i=1,2,...,n$ be probability distributions. The fidelity between p and q is defined as $F\text{'}(\backslash \{p\_i\backslash \},\backslash \{q\_i\backslash \})=\backslash sum\_\{i=1\}^n\backslash sqrt\{p\_i,q\_i\}.$ == References == == See also == * [[Trace distance]] * [[Trace norm]] * [[Superfidelity]] [[Category:Handbook of Quantum Information]] [[Category:Mathematical Structure]] [[Category:Linear Algebra]]