Entropy production of doubly stochastic quantum channels (Q2795548)

In a stochastic channel, in general, the entropy \(S\) increases. For a signal that can take \(d\) possible values, the largest possible entropy \(\log(d)\) corresponds to the case when all these values are equally possible. As the signal passes through a channel, its entropy \(S_t\) at moment \(t\) increases with time \(t\). The value \(S_t-S\) of this increase depends on the original state \(\rho\). For some original quantum states, the entropy \(S\) was already close to \(\log(d)\), so the increase cannot be too large; for other states, the entropy \(S\) was much smaller than its largest possible value \(\log(d)\), so we may encounter a large increase. In view of this dependence, a reasonable way to gauge the quality of a quantum channel is by a \textit{relative} increase in entropy, i.e., by the ratio \((S_t-S)/(\log(d)-S)\).NEWLINENEWLINEThis ratio is less depending on the signal, but there is still some dependence. To get a number that characterizes the quality of the channel itself, it makes sense to take into account that in communications in general, to improve the efficiency, we often do not simply send a signal through a channel, we first perform an appropriate transformation (such as encoding) -- to decrease the negative effects of the channel. Specifically, we select signals for which the distortion is the smallest. From this viewpoint, as a reasonable measure of the quality of a channel, we can use the minimum of the ratios corresponding to all possible states.NEWLINENEWLINEThis minimum is rarely known. For most channels for which we have some information about their quality, we only know the lower bound \(C_t\) on this minimum, i.e., the value for which \(S_t-S\geq C_t\cdot (\log(d)-S)\). The current paper provides new improved lower bounds. For example, for a \textit{Pauli} channel, in which the state remains the same with the probability \(1-p_1-p_2-p_3\), and with probability \(p_i\), gets transformed into the corresponding Pauli matrix state \(\sigma_i\), the relative increase in entropy is bounded, from below, by the number \(C=2\min(p_1(p_ 2+p_3)\), \(p_2(p_1+p_3)\), \(p_3(p_1+p_2))\).NEWLINENEWLINETo provide these bounds, the authors consider \textit{logarithmic Sobolev constants} that describe how fast the entropy decreases in comparison to the relative (Kulbach) entropy between the current state and the maximum-entropy state.