The total variation distance between probability measures 
and 
on the same measurable space is 
, where the supremum
is over measurable sets. It equals half the total variation of the signed
measure 
on the whole space (Pollard 2005).
For probability mass functions 
and 
on a countable set, it is
 |
The distance lies between 0 and 1. It is zero precisely when the distributions agree and is 1 precisely when they are mutually singular. The trace distance of density matrices that are diagonal in a common orthonormal basis reduces to this classical distance.
