  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.
