Determine the effect of system coherence in the unsaturated population-change regime

Determine whether introducing coherence into the system enhances or reduces ergotropic injection when the population change satisfies $(\Delta'-\Delta)\Delta<0$ in the central-block regime with an incoherent but athermal environment.

Background

The paper analyzes a qubit system interacting with a qubit environment through energy-conserving dynamics in the central-block regime. When the environment is incoherent but athermal, the authors show that system coherence cannot enhance ergotropic injection in the parameter regime where the incoherent-state result saturates the relevant triangle-inequality bound, namely when (Δ′−Δ)Δ≥0(\Delta'-\Delta)\Delta\geq 0.

For the complementary regime (Δ′−Δ)Δ<0(\Delta'-\Delta)\Delta<0, that bound is not saturated by the incoherent state. Consequently, the argument used in Proposition 2 cannot determine whether adding system coherence is beneficial or detrimental. The unresolved issue is therefore to characterize the role of system coherence in this remaining parameter regime.

References

When $(\Delta'-\Delta)\Delta <0$, however, the triangle inequality is not saturated for the incoherent state, and the above argument does not suffice to establish whether coherence is advantageous or disadvantageous. Thus, no general conclusion regarding the role of coherence can be drawn in that regime.

— Towards minimal conditions for ergotropy injection in open quantum systems  (2609.04457 - Singh et al., 3 Sep 2026) in Remark following Proposition 2, Section 5.2 (Initial States with System Coherence Alone)