Fidelity

Fidelity
Computes the fidelity of two density matrices
Other toolboxes required	none
Related functions	TraceNorm
Function category	Norms and distance measures

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

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

Source code

Click on "expand" to the right to view the MATLAB source code for this function.

1. %%  FIDELITY    Computes the (Uhlmann) fidelity of two density matrices

2. %   This function has two required input arguments:

3. %     RHO,SIGMA: density matrices

4. %

5. %   FID = Fidelity(RHO,SIGMA) is the fidelity between the two density

6. %   matrices RHO and SIGMA, defined by ||sqrt(RHO)*sqrt(SIGMA)||_1, where

7. %   ||.||_1 denotes the trace norm. FID is a value between 0 and 1, with 0

8. %   corresponding to matrices RHO and SIGMA with orthogonal support, and 1

9. %   corresponding to the case RHO = SIGMA.

10. %

11. %   URL: http://www.qetlab.com/Fidelity

13. %   requires: nothing (only uses CVX if called as an input to CVX)

14. %   authors: Vincent Russo (vrusso@uwaterloo.ca)

15. %            Nathaniel Johnston (nathaniel@njohnston.ca)

16. %   package: QETLAB

17. %   last updated: November 21, 2014

19. function fid = Fidelity(rho,sigma)

21. sz_rho = size(rho);

23. % Do some error checking.

24. if(~all(sz_rho == size(sigma)))

25.     error('Fidelity:InvalidDims','RHO and SIGMA must be matrices of the same size.');

26. elseif(sz_rho(1) ~= sz_rho(2))

27.     error('Fidelity:InvalidDims','RHO and SIGMA must be square.');

28. end

30. % If rho or sigma is a CVX variable then compute fidelity via semidefinite

31. % programming, so that this function can be used in the objective function

32. % or constraints of other CVX optimization problems.

33. if(isa(rho,'cvx') || isa(sigma,'cvx'))

34.     cvx_begin sdp quiet

35.         cvx_precision best

36.         variable X(sz_rho(1),sz_rho(1)) complex;

37.         maximize trace(X) + trace(X');

38.         subject to

39.             cons = [rho,X;X',sigma];

40.             cons + cons' >= 0; % avoid some numerical problems: CVX often thinks things aren't symmetric without this

41.     cvx_end

43.     fid = cvx_optval/2;

45. % If rho and sigma are *not* CVX variables, compute fidelity normally,

46. % since this is much faster.

47. else

48.     [sq_rho,res] = sqrtm(rho); % need "res" parameter to suppress MATLAB singularity warning

49.     sq_prod = sq_rho*sigma*sq_rho;

50.     [sq_fid,res] = sqrtm((sq_prod+sq_prod')/2); % needed to avoid numerical problems

51.     fid = real(trace(sq_fid)); % finally, compute the fidelity

52. end