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.
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Motivation
Given two random variables X, Y with values (1...n) and probabilities p = (p1...pn) and q = (q1...qn). 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
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
It was shown by Fuchs and Caves that this manifestly symmetric definition is equivalent to the simple asymmetric formula given in the next section.
Definition
Given two density matrices ρ and σ, the fidelity is defined by
By M½ of a positive semidefinite matrix M, we mean its unique positive square root given by the spectral theorem. The Euclidean inner product from the classical definition is replaced by the Hilbert–Schmidt inner product. When the states are classical, i.e. when ρ and σ commute, the definition coincides with that for probability distributions.
An equivalent definition is given by
where the norm is the trace norm (sum of the singular values). This definition has the advantage that it clearly shows that the fidelity is symmetric in its two arguments.
For two qubit states
Notice by definition F is non-negative, and F(ρ,ρ) = 1. In the following section it will be shown that it can be no larger than 1.
In the original 1994 paper of Jozsa the name 'fidelity' was used for the quantity
Pure states
Suppose that one of the states is pure:
If the other state is also pure,
This is sometimes called the overlap between two states. If, say,
Commuting states
Let ρ and σ be two density matrices that commute. Therefore they can be simultaneously diagonalized by unitary matrices, and we can write
for some orthonormal basis
This shows that, heuristically, fidelity of quantum states is a genuine extension of the notion from probability theory.
Unitary invariance
Direct calculation shows that the fidelity is preserved by unitary evolution, i.e.
for any unitary operator U.
Uhlmann's theorem
We saw that for two pure states, their fidelity coincides with the overlap. Uhlmann's theorem generalizes this statement to mixed states, in terms of their purifications:
Theorem Let ρ and σ be density matrices acting on Cn. Let ρ½ be the unique positive square root of ρ and
be a purification of ρ (therefore
where
Proof: A simple proof can be sketched as follows. Let
and σ½ 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)½). Furthermore, equality is achieved if U* is the unitary operator in the polar decomposition of A. From this follows directly Uhlmann's theorem.
Consequences
Some immediate consequences of Uhlmann's theorem are
So we can see that fidelity behaves almost like a metric. This can be formalised and made useful by defining
As the angle between the states
Relationship to trace distance
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,
Often the trace distance is easier to calculate or bound than the fidelity, so these relationships are quite useful. In the case that at least one of the states is a pure state Ψ, the lower bound can be tightened.