Flows of Rényi entropies

We demonstrate that the condensed matter quantum systems encompassing two reservoirs connected by a junction permit a natural definition of flows of conserved measures, i.e., Rényi entropies. Such flows are similar to the flows of physical conserved quantities such as charge and energy. We develop a perturbation technique that permits efficient computation of Rényi entropy flows and analyze second- and fourth-order contributions. Second-order approximation was shown to correspond directly to the transition events in the system and thereby to possess a set of intuitive features. The analysis of fourth-order corrections reveals a more complicated picture: The intuitive relations do not hold anymore, and the corrections exhibit divergencies in low-temperature limit, manifesting an intriguing nonanalytical dependence of the flows on coupling strength in the limit of weak couplings and vanishing temperatures.

Figure 1

Two leads (reservoirs) $A$ and $B$ in a typical quantum transport setup correspond to the bipartition $A\otimes B$. The junction connecting the two is associated with coupling Hamiltonian $H^{\left(AB\right)}$. In similarity with the flows of charge and energy, one can define the flows of Rényi entropies (Re-flows) in such setups.

Figure 2

Perturbation theory for a single density matrix on the Keldysh contour.

Figure 3

A diagram of perturbation theory for $S_M^{\left(A\right)}$ for $M=3$. It involves three parallel worlds. Reconnection of Keldysh contours for subspaces $A$ (black) and $B$ (white) accounts for partial trace over $B$ and matrix multiplication in $A$.

Figure 4

Second-order diagrams for time derivative of a Rényi entropy. The contributions come only from perturbations ${\widehat{H}}^{\left(AB\right)}$ in the same world; only this world is shown in each diagram. For all diagrams, the perturbations are taken at time moments $t$ and $t^{\prime }<t$. The letters at the contours label the states involved.

Figure 5

Master equation in a single world is obtained from the resummation of perturbation series whereby the timeline is separated into diagonal and nondiagonal (gray shaded) blocks. One can do this resummation in multiple worlds to compute the time evolution of Rényi entropies.

Figure 6

A fourth-order quantum diagram for Rényi entropy flows. The contributions come from perturbations ${\widehat{H}}^{\left(AB\right)}$ in two different worlds; only these two worlds are shown. The letters on the contours label the states involved.