Fidelity

2020-09-14T05:46:55Z

Etanjk: The definition of fidelity needs to be squared after tracing according to Josza's paper.

{{Function
|name=Fidelity
|desc=Computes the (Uhlmann) fidelity of two density matrices
|rel=[[MatsumotoFidelity]]<br />[[MaximumOutputFidelity]]<br />[[TraceNorm]]
|cat=[[List of functions#Norms_and_distance_measures|Norms and distance measures]]
|upd=November 21, 2014
|cvx=yes (concave)}}
<tt>'''Fidelity'''</tt> is a [[List of functions|function]] that computes the Uhlmann fidelity between two quantum states $\rho$ and $\sigma$, defined as follows:
: <math>F(\rho,\sigma) := [\mathrm{Tr}\Big( \sqrt{ \sqrt{\rho}\sigma\sqrt{\rho}}\Big)]^2.</math>

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

==Syntax==
* <tt>FID = Fidelity(RHO,SIGMA)</tt>

==Argument descriptions==
* <tt>RHO</tt>: A density matrix.
* <tt>SIGMA</tt>: 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 $\big|\langle v|w \rangle\big|$:
<syntaxhighlight>
>> v = RandomStateVector(4);
>> w = RandomStateVector(4);
>> Fidelity(v*v',w*w')

ans =

0.6486

>> abs(v'*w)

ans =

0.6486
</syntaxhighlight>

===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 [[MaximumOutputFidelity|maximum output fidelity]] of two quantum channels:
<syntaxhighlight>
>> 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
</syntaxhighlight>

Of course, in this case it is more convenient to just use the <tt>[[MaximumOutputFidelity]]</tt> function directly:
<syntaxhighlight>
>> MaximumOutputFidelity(Phi,Psi)

ans =

0.9829
</syntaxhighlight>

{{SourceCode|name=Fidelity}}

QETLAB - User contributions [en]

Fidelity