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,
Diamond Norm,
Eigenvalue,
Quantum
Channel,
Total Variation Distance,
Trace Norm
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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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