# MatsumotoFidelity

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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

^{[1]}

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

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

## References

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