Fidelity of quantum states

Term in quantum mechanics
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.

Direct calculation shows that the fidelity is preserved by unitary evolution, i. e.

for any unitary operator .

Let be an arbitrary positive operator valued measure (POVM); that is, a set of positive semidefinite operators satisfying Then, for any pair of states and , we have

where in the last step we denoted with and the probability distributions obtained by measuring with the POVM
This shows that the square root of the fidelity between two quantum states is upper bounded by the Bhattacharyya coefficient between the corresponding probability distributions in any possible POVM. Indeed, it is more generally true that where , and the minimum is taken over all possible POVMs. More specifically, one can prove that the minimum is achieved by the projective POVM corresponding to measuring in the eigenbasis of the operator .

The fidelity between two states can be shown to never decrease when a non selective quantum operation is applied to the states: for any trace preserving completely positive map .

A simple proof can be sketched as follows. Let denote the vector

and σ1/2 be the unique positive square root of σ. We see that, due to the unitary freedom in square root factorizations and choosing orthonormal bases, an arbitrary purification of σ is of the form

where Vi's are unitary operators. Now we directly calculate

But in general, for any square matrix A and unitary U, it is true that |tr(AU)| ≤ tr((A*A)1/2). Furthermore, equality is achieved if U* is the unitary operator in the polar decomposition of A. From this follows directly Uhlmann's theorem.

We will here provide an alternative, explicit way to prove Uhlmann's theorem. Let and be purifications of and , respectively. To start, let us show that
The general form of the purifications of the states is:were are the eigenvectors of , and are arbitrary orthonormal bases. The overlap between the purifications iswhere the unitary matrix is defined asThe conclusion is now reached via using the inequality : Note that this inequality is the triangle inequality applied to the singular values of the matrix. Indeed, for a generic matrix and unitary , we havewhere are the (always real and non negative) singular values of , as in the singular value decomposition. The inequality is saturated and becomes an equality when , that is, when and thus The above shows that when the purifications and are such that Because this choice is possible regardless of the states, we can finally conclude that