The trace distance between density matrices 
and 
is
It lies between 0 and 1 and is a metric on the set of density matrices. Since 
is Hermitian, it is half the sum of the absolute values
of its eigenvalues. For diagonal density
matrices it reduces to total variation
distance between the corresponding probability
distributions. Quantum channels cannot increase
trace distance.
See also
Density Matrix,
Eigenvalue,
Quantum Channel,
Total
Variation Distance
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References
Watrous, J. The Theory of Quantum Information. Cambridge,
England: Cambridge University Press, 2018. https://doi.org/10.1017/9781316848142.
Cite this as:
Weisstein, Eric W. "Trace Distance." From
MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/TraceDistance.html
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