where |b| = (b † b) 1/2 . It is important to note that the two density matrices being compared correspond to two different systems at the same time, not the same system at two different times. As discussed in ref. , this choice of construction ensures that the supports of the kernel matrix are orthogonal when the flux of information is negative, corresponding to an increase in entropy of the system and a loss of distinguishability. The factor of half is omitted in the Helstrom metric and is present here due to the simplifying assumption that each state is equally probable. The maximum and minimum values of the trace distance are observed when the supports are orthogonal or parallel, respectively which physically corresponds to completely distinguishable and completely indistinguishable quantum states.