Semicontinuity bounds for the von Neumann entropy and partial majorization

We consider families of tight upper bounds on the difference $S(\rho )-S(\sigma )$ with the rank/energy constraint imposed on the state $\rho$ which are valid provided that the state $\rho$ partially majorizes the state $\sigma$ and is close to the state $\sigma$ w.r.t. the trace norm. The upper bounds within these families depend on the parameter $m$ of partial majorization. The upper bounds corresponding to $m=0$ coincide with the (unconditional) optimal semicontinuity bounds for the von Neumann entropy with the rank/energy constraint obtained in [Lett.Math.Phys.,113,121,35] and [arXiv:2410.02686]. State-dependent improvements of these semicontinuity bounds are proposed and analysed numerically. The notion of $\epsilon$-sufficient majorization dimension of the set of states with bounded energy is introduced and analysed. Classical versions of the above results formulated in terms of probability distributions and the Shannon entropy are also considered.