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 and , the fidelity is generally defined as the quantity . In the special case where and represent pure quantum states, namely, and , the definition reduces to the squared overlap between the states: . If at least one of the two states is pure it reduces to: , where is the pure state. While not obvious from the general definition, the fidelity is symmetric: . Given two random variables with values (categorical random variables) and probabilities and , the fidelity of and 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 is the square of the inner product of and viewed as vectors in Euclidean space. Notice that if and only if . In general, . The measure is known as the Bhattacharyya coefficient. Given a classical measure of the distinguishability of two probability distributions, one can motivate a measure of distinguishability of two quantum states as follows. If an experimenter is attempting to determine whether a quantum state is either of two possibilities or , the most general possible measurement they can make on the state is a POVM, which is described by a set of Hermitian positive semidefinite operators . If the state given to the experimenter is , they will witness outcome with probability , and likewise with probability for . Their ability to distinguish between the quantum states and is then equivalent to their ability to distinguish between the classical probability distributions and .