Dephasing superchannels

We characterize a class of environmental noises that decrease the coherent properties of quantum channels by introducing and analyzing the properties of dephasing superchannels. These are defined as superchannels that affect only nonclassical properties of a quantum channel $\mathcal{E}$, i.e., they leave invariant the transition probabilities induced by $\mathcal{E}$ in the distinguished basis. We prove that such superchannels ${\mathrm{\Xi }}_C$ form a particular subclass of Schur-product supermaps that act on the Jamiołkowski state $J\left(\mathcal{E}\right)$ of a channel $\mathcal{E}$ via a Schur product, $J^{\prime }=J\circ C$. We also find physical realizations of general ${\mathrm{\Xi }}_C$ through pre- and postprocessing employing dephasing channels with memory, and we show that memory plays a nontrivial role for quantum systems of dimension $d>2$. Moreover, we prove that the coherence-generating power of a general quantum channel is a monotone under dephasing superchannels. Finally, we analyze the effect that dephasing noise can have on a quantum channel $\mathcal{E}$ by investigating the number of distinguishable channels that $\mathcal{E}$ can be mapped to by a family of dephasing superchannels. More precisely, we upper-bound this number in terms of hypothesis-testing channel divergence between $\mathcal{E}$ and its fully dephased version, and we also relate it to the robustness of coherence of $\mathcal{E}$.

Figure 1

Dephasing channels and superchannels. The set of quantum states ${\mathrm{\Omega }}_q$ (density matrices) is projected onto the set of classical states ${\mathrm{\Omega }}_c$ (probability distributions) via the completely dephasing channel that removes all coherences in the distinguished basis, but keeps the occupations unchanged. General dephasing channels are those maps between quantum states that affect coherences but do not change occupations, i.e., they keep the classical (completely dephased) version of the state invariant. Analogously, the set of quantum channels ${\mathcal{T}}_q$ (completely positive trace-preserving maps) is projected onto the set of classical channels ${\mathcal{T}}_c$ (stochastic matrices) via a completely dephasing superchannel that removes all coherent properties of the channel but keeps the transition probabilities in the distinguished basis unchanged. General dephasing superchannels form the subset of quantum superchannels ${\mathcal{S}}_q$ that affect the coherent properties of the channel but do not change transition probabilities, i.e., they keep the classical (completely superdephased) version of the channel invariant.