Fidelity of quantum states

In quantum mechanics, notably in quantum information theory, fidelity is a measure of the "closeness" of two quantum states. It expresses the probability that one state will pass a test to identify as the other. The fidelity is not a metric on the space of density matrices, but it can be used to define the Bures metric on this space.

Given two density operators ${\displaystyle \rho }$ and ${\displaystyle \sigma }$, the fidelity is generally defined as the quantity ${\displaystyle F(\rho ,\sigma )=\left(\operatorname {tr} {\sqrt {{\sqrt {\rho }}\sigma {\sqrt {\rho }}}}\right)^{2}}$. In the special case where ${\displaystyle \rho }$ and ${\displaystyle \sigma }$ represent pure quantum states, namely, ${\displaystyle \rho =|\psi _{\rho }\rangle \!\langle \psi _{\rho }|}$ and ${\displaystyle \sigma =|\psi _{\sigma }\rangle \!\langle \psi _{\sigma }|}$, the definition reduces to the squared overlap between the states: ${\displaystyle F(\rho ,\sigma )=|\langle \psi _{\rho }|\psi _{\sigma }\rangle |^{2}}$. While not obvious from the general definition, the fidelity is symmetric: ${\displaystyle F(\rho ,\sigma )=F(\sigma ,\rho )}$.