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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[1][2] between two quantum states $\rho$ and $\sigma$, defined by \[F(\rho,\sigma) := \mathrm{Tr}(\rho\#\sigma),\] where \[\rho\#\sigma := \rho^{1/2}\Big(\rho^{-1/2}\sigma\rho^{-1/2}\Big)\rho^{1/2}.\]


  • FID = MatsumotoFidelity(RHO,SIGMA)

Argument descriptions

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


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 =


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 here to view this function's source code on github.


  1. K. Matsumoto. Reverse test and quantum analogue of classical fidelity and generalized fidelity. E-print: arXiv:1006.0302, 2010.
  2. S. S. Cree and J. Sikora. A fidelity measure for quantum states based on the matrix geometric mean. E-print: arXiv:2006.06918, 2020.