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: .

Motivation 

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 .  Naturally, the experimenter will choose the best POVM they can find, so this motivates defining the quantum fidelity as the squared Bhattacharyya coefficient when extremized over all possible POVMs :

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

where, for a positive semidefinite matrix ,   denotes its unique positive square root, as given by the spectral theorem. The Euclidean inner product from the classical definition is replaced by the Hilbert–Schmidt inner product.

Some of the important properties of the quantum state fidelity are:

 Symmetry. .
 Bounded values. For any  and , , and .
 Consistency with fidelity between probability distributions. If  and  commute, the definition simplifies to where  are the eigenvalues of , respectively. To see this, remember that if  then they can be diagonalized in the same basis: so that 
 Simplified expressions for pure states. If  is pure, , then . This follows from If both  and  are pure,  and , then . This follows immediately from the above expression for  pure.

 Equivalent expression.
An equivalent expression for the fidelity may be written, using the trace norm

where the absolute value of an operator is here defined as .

 Explicit expression for qubits.
If  and  are both qubit states, the fidelity can be computed as

Qubit state means that  and  are represented by two-dimensional matrices. This result follows noticing that  is a positive semidefinite operator, hence , where  and  are the (nonnegative) eigenvalues of . If  (or ) is pure, this result is simplified further to  since  for pure states.

Alternative definition 
Some authors use an alternative definition  and call this quantity fidelity. The definition of  however is more common. To avoid confusion,  could be called "square root fidelity". In any case it is advisable to clarify the adopted definition whenever the fidelity is employed.

Other properties

Unitary invariance 

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

for any unitary operator .

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 ρ be the unique positive square root of ρ and

be a purification of ρ (therefore  is an orthonormal basis), then the following equality holds:

where  is a purification of σ. Therefore, in general, the fidelity is the maximum overlap between purifications.

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

and σ 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)). Furthermore, equality is achieved if U* is the unitary operator in the polar decomposition of A. From this follows directly Uhlmann's theorem.

Proof with explicit decompositions 
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

Consequences 
Some immediate consequences of Uhlmann's theorem are
 Fidelity is symmetric in its arguments, i.e. F (ρ,σ) = F (σ,ρ). Note that this is not obvious from the original definition.
 F (ρ,σ) lies in [0,1], by the Cauchy–Schwarz inequality.
 F (ρ,σ) = 1 if and only if ρ = σ, since Ψρ = Ψσ implies ρ = σ.

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  and . It follows from the above properties that  is non-negative, symmetric in its inputs, and is equal to zero if and only if . Furthermore, it can be proved that it obeys the triangle inequality, so this angle is a metric on the state space: the Fubini–Study metric.

Relationship with the fidelity between the corresponding probability distributions 
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 .

Proof of inequality 
As was previously shown, the square root of the fidelity can be written as which is equivalent to the existence of a unitary operator  such that

Remembering that  holds true for any POVM, we can then writewhere in the last step we used Cauchy-Schwarz inequality as in .

Behavior under quantum operations 
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 .

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.

References 

 Quantiki: Fidelity

Quantum information science