Trade-off relations in open quantum dynamics via Robertson and Maccone-Pati uncertainty relations

The Heisenberg uncertainty relation, together with its generalization by Robertson, serves as a fundamental concept in quantum mechanics, encapsulating that non-commutative pairs of observable cannot be measured precisely. In this Letter, we explore the Robertson uncertainty relation to demonstrate its effectiveness in establishing a series of thermodynamic uncertainty relations and quantum speed limits in open quantum dynamics. The derivation utilizes the scaled continuous matrix product state representation that maps the time evolution of quantum continuous measurement to the time evolution of the system and field. Specifically, we consider the Maccone-Pati uncertainty relation, a refinement of the Robertson uncertainty relation, to derive thermodynamic uncertainty relations and quantum speed limits within open quantum dynamics scenarios. These newly derived relations, which use a state orthogonal to the initial state, yield tighter bounds than the previously known bounds. Our findings not only reinforce the significance of the Robertson uncertainty relation, but also expand its applicability to identify uncertainty relations in open quantum dynamics.