Consider an irreducible continuous-time Markov chain $X_t$ on finite state space with $Q$-matrix $Q=(q_{ij}, i,j=1,2,\cdots, N)$, in which $q_{ij}=q_{ji}$. Fix initial state $j$, let $P_i(t)$ be the probability that the chain is in $i$ at time $t$, i.e., $P_i(t)=\mathbb{P}(X_t=i)$. Define $$E(t)=-\sum_{i=1}^{N}P_i(t)\log P_i(t).$$ The aim is to prove $E(t)$ is non-decreasing in $t$.
I have thought this problem for several days and have no idea.
The only thing I am able to do is to take the derivative of $E(t)$ with respect to $t$, but it seems no use because the expression is more complicated such that it is difficult to judge the sign.
I wold appreciate it if someone could give me some useful advice on solving this problem. Thanks very much!