Research

We establish a tight upper bound for the difference in von Neumann entropies between two quantum states, $\rho_1$ and $\rho_2$. This bound is expressed in terms of the von Neumann entropies of the mutually orthogonal states derived from the Jordan-Hahn decomposition of the difference operator $(\rho_1 - \rho_2)$. This yields a novel entropic inequality that implies the well-known Audenaert-Fannes inequality amongst other inequalities for the relative entropy.

The exponential convergence to invariant subspaces of quantum Markov semigroups plays a crucial role in quantum information theory. In this paper, we investigate perturbations of quantum dynamical semigroups operating on continuous variable (CV) systems that admit an invariant subspace. To establish this analysis on mathematically sound foundations, we first prove a generation theorem for quantum Markov semigroups on CV systems under physically motivated assumptions about the generator structure.

In this work, we prove uniform continuity bounds for entropic quantities related to the sandwiched Rényi divergences such as the sandwiched Rényi conditional entropy. We follow three different approaches: The first one exploits the sub-/superadditivity and joint concavity/convexity of the exponential of the divergence while the second uses norm-like properties of a power of the exponential inspired by norms on interpolation spaces. The third approach mixes the former two to achiev favourable scaling in all parameter regimes.

Building on the continuity proofs for conditional entropy established by Alicki, Fannes, and Winter, we introduce the almost locally affine (ALAFF) method. This approach enables us to derive continuity bounds for a broad class of entropic quantities underlying divergences that exhibit simultaneous almost joint convexity and concavity. We showcase the use on the Umegaki relative entropy rederiving known bounds and derived the first bounds for the Belavkin-Staszewski relative entropy.

We study the problem of sampling from and preparing quantum Gibbs states of local commuting Hamiltonians on hypercubic lattices of arbitrary dimension. We prove that any such Gibbs state which satisfies a clustering condition that we coin decay of matrix-valued quantum conditional mutual information (MCMI) can be quasi-optimally prepared on a quantum computer in a normalised Wasserstein distance of order one. Under an additional polynomial uniform bound on the local gap we further show optimal preparation in trace distance.

We show that translation invariant, local spin chains in thermal equilibrium have a correlation structure in which individual regions are strongly correlated at most with their near vicinity. We quantify this with alternative notions of the conditional mutual information, defined through the so-called Belavkin-Staszewski relative entropy. With this result we can proof efficiency of an MPO reconstruction of the Gibbs states of such systems.