Coherifying quantum channels

Abstract

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 $Φ^{\mathcal{C}}$ 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 $J_Φ^{\mathcal{C}}$ of the optimally coherified channel is mixed, and the dynamics must necessarily be irreversible. To assess the extent to which an optimal process $Φ^{\mathcal{C}}$ is indeterministic we find explicit bounds on the entropy and purity of $J_Φ^{\mathcal{C}}$, and relate the latter to the unitarity of $Φ^{\mathcal{C}}$. 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 $d$.