Nathaniel: Created page with "{{Function |name=MatsumotoFidelity |desc=Computes the Matsumoto fidelity of two density matrices |rel=Fidelity |cat=List of functions#Norms_and_distance_measures|Norms a..."

2020-06-23T14:42:34Z

Created page with "{{Function |name=MatsumotoFidelity |desc=Computes the Matsumoto fidelity of two density matrices |rel=Fidelity |cat=List of functions#Norms_and_distance_measures|Norms a..."

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{{Function
|name=MatsumotoFidelity
|desc=Computes the Matsumoto fidelity of two density matrices
|rel=[[Fidelity]]
|cat=[[List of functions#Norms_and_distance_measures|Norms and distance measures]]
|upd=June 23, 2020
|cvx=yes (concave)}}
<tt>'''MatsumotoFidelity'''</tt> is a [[List of functions|function]] that computes the Matsumoto fidelity<ref>K. Matsumoto. <em>Reverse test and quantum analogue of classical fidelity and generalized
fidelity</em>. E-print: [https://arxiv.org/abs/1006.0302 arXiv:1006.0302], 2010.</ref><ref>S. S. Cree and J. Sikora. <em>A fidelity measure for quantum states based on the matrix geometric mean</em>. E-print: [https://arxiv.org/abs/2006.06918 arXiv:2006.06918], 2020.</ref> between two quantum states $\rho$ and $\sigma$, defined by
: <math>F(\rho,\sigma) := \mathrm{Tr}(\rho\#\sigma),</math>
where
: <math>\rho\#\sigma := \rho^{1/2}\Big(\rho^{-1/2}\sigma\rho^{-1/2}\Big)\rho^{1/2}.</math>

==Syntax==
* <tt>FID = MatsumotoFidelity(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 Matsumoto fidelity simply equals 1 if they are parallel and zero otherwise:
<syntaxhighlight>
>> v = RandomStateVector(4);
>> w = RandomStateVector(4);
>> MatsumotoFidelity(v*v',w*w')

ans =

1.7454e-05
</syntaxhighlight>

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).

{{SourceCode|name=MatsumotoFidelity}}

==References==
<references />

MatsumotoFidelity - Revision history

Nathaniel: Created page with "{{Function |name=MatsumotoFidelity |desc=Computes the Matsumoto fidelity of two density matrices |rel=Fidelity |cat=List of functions#Norms_and_distance_measures|Norms a..."