Coherifying quantum channels
Kamil Korzekwa, Stanisław Czachórski, Zbigniew Puchała, Karol Życzkowski · New Journal of Physics · 2018
Is it always possible to explain random stochastic transitions between states of a finite-dimensional system as arising from the deterministic quantum evolution of the system? If not, then what is the minimal amount of randomness required by quantum theory to explain a given stochastic process? Here, we address this problem by studying possible coherifications of a quantum channel Φ, i.e., we look for channels Φ C that induce the same classical transitions T , but are ‘more coherent’. To quantify the coherence of a channel Φ we measure the coherence of the corresponding Jamiołkowski state J Φ . We show that the classical transition matrix T can be coherified to reversible unitary dynamics if and only if T is unistochastic. Otherwise the Jamiołkowski state J Φ C of the optimally coherified channel is mixed, and the dynamics must necessarily be irreversible. To assess the extent to which an optimal process Φ C is indeterministic we find explicit bounds on the entropy and purity of J Φ C , and relate the latter to the unitarity of Φ C . We also find optimal coherifications for several classes of channels, including all one-qubit channels. Finally, we provide a non-optimal coherification procedure that works for an arbitrary channel Φ and reduces its rank (the minimal number of required Kraus operators) from d 2 to d .