Notes on distinguishability of postselected computations

The framework of postselection is becoming more and more important in various recent directions in Quantum Computation research. Postselection renders simple computational models able to perform general quantum computation. This was first observed for the linear optics model [E. Knill, R. Laflamme, G. J. Milburn, Nature 409, 46 (2001)], and has since provided us with many near-term candidates for the quantum advantage, commuting computations [M. J. Bremner, R. Jozsa, D. J. Shepherd, Proc. R. Soc. A 467, 459 (2011)] being the first. To facilitate the discussion of errors in the presence of postselection, we define and characterize trace-induced distance and diamond distance of postselected computations. We show counterexamples to simple properties that one would expect of any distance measure; the properties of convexity (when considering only the pure-state inputs would suffice), contractivity, and subadditivity of errors. On the positive side, we prove that certain weaker versions of contractivity and subadditivity and a number of other properties are preserved in the postselected setting. We achieve this via a "conversion lemma" that translates any inequality from the standard to the postselected setting.