Fidelity of quantum states

In quantum information theory, fidelity is a measure of the "closeness" of two quantum states. It 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 random variables X, Y with values (1...n) and probabilities p = (p₁...p_n) and q = (q₁...q_n). The fidelity of X and Y is defined to be the quantity

The fidelity deals with the marginal distribution of the random variables. It says nothing about the joint distribution of those variables. In other words, the fidelity F(X,Y) is the inner product of ${\displaystyle ({\sqrt {p_{1}}},\cdots ,{\sqrt {p_{n}}})}$ and ${\displaystyle ({\sqrt {q_{1}}},\cdots ,{\sqrt {q_{n}}})}$ viewed as vectors in Euclidean space. Notice that F(X,Y) = 1 if and only if p = q. In general, ${\displaystyle 0\leq F(X,Y)\leq 1}$. This measure is known as the Bhattacharyya coefficient.

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