It expresses the probability that one state will pass a test to identify as the other.
As will be discussed in the following sections, this expression can be simplified in various cases of interest.
In any case it is advisable to clarify the adopted definition whenever the fidelity is employed.
is defined to be the quantity The fidelity deals with the marginal distribution of the random variables.
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
, 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
is then equivalent to their ability to distinguish between the classical probability distributions
A natural question is then to ask what is the POVM the makes the two distributions as distinguishable as possible, which in this context means to minimize the Bhattacharyya coefficient over the possible choices of POVM.
Formally, we are thus led to define the fidelity between quantum states as: It was shown by Fuchs and Caves[6] that the minimization in this expression can be computed explicitly, with solution the projective POVM corresponding to measuring in the eigenbasis of
, and results in the common explicit expression for the fidelity as
An equivalent expression for the fidelity between arbitrary states via the trace norm is: where the absolute value of an operator is here defined as
, which is positive semidefinite by construction and so the square roots of the eigenvalues are well defined.
Because the characteristic polynomial of a product of two matrices is independent of the order, the spectrum of a matrix product is invariant under cyclic permutation, and so these eigenvalues can instead be calculated from
[7] Reversing the trace property leads to If (at least) one of the two states is pure, for example
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.
More specifically, one can prove that the minimum is achieved by the projective POVM corresponding to measuring in the eigenbasis of the operator
[9] As was previously shown, the square root of the fidelity can be written as
The fidelity between two states can be shown to never decrease when a non-selective quantum operation
We can define the trace distance between two matrices A and B in terms of the trace norm by When A and B are both density operators, this is a quantum generalization of the statistical distance.
This is relevant because the trace distance provides upper and lower bounds on the fidelity as quantified by the Fuchs–van de Graaf inequalities,[11] Often the trace distance is easier to calculate or bound than the fidelity, so these relationships are quite useful.
We saw that for two pure states, their fidelity coincides with the overlap.
Uhlmann's theorem[12] generalizes this statement to mixed states, in terms of their purifications: Theorem Let ρ and σ be density matrices acting on Cn.
Let ρ1⁄2 be the unique positive square root of ρ and
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.
We will here provide an alternative, explicit way to prove Uhlmann's theorem.
Some immediate consequences of Uhlmann's theorem are So we can see that fidelity behaves almost like a metric.
This can be formalized and made useful by defining As the angle between the states