MatsumotoFidelity

MatsumotoFidelity
Computes the Matsumoto fidelity of two density matrices
Other toolboxes required	none
Related functions	Fidelity
Function category	Norms and distance measures
Usable within CVX?	yes (concave)

MatsumotoFidelity is a function that computes the Matsumoto fidelity between two quantum states $\rho$ and $\sigma$, defined by \[F(\rho,\sigma) := \mathrm{Tr}(\rho\#\sigma),\] where \[\rho\#\sigma := \rho^{1/2}\Big(\rho^{-1/2}\sigma\rho^{-1/2}\Big)\rho^{1/2}.\]

Syntax

• FID = MatsumotoFidelity(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 Matsumoto fidelity simply equals 1 if they are parallel and zero otherwise:

>> v = RandomStateVector(4);
>> w = RandomStateVector(4);
>> MatsumotoFidelity(v*v',w*w')
 
ans =
 
   1.7454e-05

This highlights one slight limitation of this function: it is only accurate to 4 or so decimal places when both inputs are non-invertible (it is much more accurate if at least one input is invertible).

Source code

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

1. %%  MATSUMOTOFIDELITY    Computes the Matsumoto fidelity of two density matrices

2. %   This function has two required input arguments:

3. %     RHO,SIGMA: density matrices

4. %

5. %   FID = MatsumotoFidelity(RHO,SIGMA) is the Matsumoto fidelity between

6. %   the two density matrices RHO and SIGMA, defined by tr(RHO#SIGMA), where

7. %   RHO#SIGMA is the matrix geometric mean. FID is a value between 0 and 1,

8. %   with 0 corresponding to matrices RHO and SIGMA with 0-intersection

9. %   ranges, and 1 corresponding to the case RHO = SIGMA.

10. %

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

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

14. %   authors: Nathaniel Johnston (nathaniel@njohnston.ca)

15. %   package: QETLAB

16. %   last updated: June 23, 2020

18. function fid = MatsumotoFidelity(rho,sigma)

20. sz_rho = size(rho);

22. % Do some error checking.

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

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

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

26.     error('MatsumotoFidelity:InvalidDims','RHO and SIGMA must be square.');

27. end

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

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

31. % or constraints of other CVX optimization problems.

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

33.     cvx_begin sdp quiet

34.         cvx_precision best

35.         variable X(sz_rho(1),sz_rho(1)) hermitian;

36.         maximize trace(X);

37.         subject to

38.             cons = [rho,X;X,sigma];

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

40.     cvx_end

42.     fid = cvx_optval;

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

45. % since this is much faster.

46. else

47.     if(abs(det(sigma)) > abs(det(rho))) % for numerical stability, invert the density matrix with larger determinant

48.         temp_rho = rho;

49.         rho = sigma;

50.         sigma = temp_rho;

51.     end

52.     rho = rho + 10^(-8)*eye(sz_rho(1)); % need rho to be invertilble -- this is a ham-fisted way of avoiding problems

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

54.     isq_rho = inv(sq_rho);

55.     isq_rho = (isq_rho+isq_rho')/2;

56.     [sq_fid,res] = sqrtm(isq_rho*sigma*isq_rho);

57.     fid = real(trace(sq_rho*sq_fid*sq_rho)); % finally, compute the fidelity

58. end