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

Keywords

• coherence
• quantum channels
• decoherence
• coherence of a channel
• stochastic maps
• irreversibility