# Fidelity

 Other toolboxes required Fidelity Computes the (Uhlmann) fidelity of two density matrices none MatsumotoFidelityMaximumOutputFidelityTraceNorm Norms and distance measures yes (concave)

Fidelity is a function that computes the Uhlmann fidelity between two quantum states $\rho$ and $\sigma$, defined as follows: $F(\rho,\sigma) := [\mathrm{Tr}\Big( \sqrt{ \sqrt{\rho}\sigma\sqrt{\rho}}\Big)]^2.$

Note that, in some sources, "fidelity" refers to the square of this quantity.

## Syntax

• FID = Fidelity(RHO,SIGMA)

## Argument descriptions

• RHO: A density matrix.
• SIGMA: A density matrix.

## Examples

### Pure states

If $\rho = |v\rangle\langle v|$ and $\sigma = |w\rangle\langle w|$ are both pure states then their fidelity simply equals $\big|\langle v|w \rangle\big|$:

>> v = RandomStateVector(4);
>> w = RandomStateVector(4);
>> Fidelity(v*v',w*w')

ans =

0.6486

>> abs(v'*w)

ans =

0.6486

### Can be used with CVX

The fidelity function is a jointly concave function, and it can be used in the objective function or constraints of a CVX optimization problem. For example, the following code computes the maximum output fidelity of two quantum channels:

>> Phi = RandomSuperoperator(3); % generate a random channel
>> Psi = RandomSuperoperator(3); % generate another one
>> cvx_begin sdp quiet
variable rho(3,3) hermitian;
variable sigma(3,3) hermitian;

maximize Fidelity(ApplyMap(rho,Phi),ApplyMap(sigma,Psi))

subject to % the constraints here just force rho and sigma to be density matrices
trace(rho) == 1;
trace(sigma) == 1;
rho >= 0;
sigma >= 0;
cvx_end
>> cvx_optval

cvx_optval =

0.9829

Of course, in this case it is more convenient to just use the MaximumOutputFidelity function directly:

>> MaximumOutputFidelity(Phi,Psi)

ans =

0.9829