  Elucidating the fundamental limitations inherent in physical systems is a
central subject in physics. For important thermodynamic operations such as
information erasure, cooling, and copying, resources like time and energetic
cost must be expended to achieve the desired outcome within a predetermined
error margin. In this study, we introduce the concept of {\it separated
states}, which consist of fully unoccupied and occupied states. This concept
generalizes many critical states involved in relevant thermodynamic operations.
We uncover a three-way trade-off relation between {\it time}, {\it cost}, and
{\it error} for a general class of thermodynamic operations aimed at creating
separated states, simply expressed as $\tau{\cal C}\varepsilon_{\tau}\ge
1-\eta$. This fundamental relation is applicable to diverse thermodynamic
operations, including information erasure, cooling, and copying. It provides a
profound quantification of the unattainability principle in the third law of
thermodynamics in a general form. Building upon this relation, we explore the
quantitative limitations governing cooling operations, the preparation of
separated states, and a no-go theorem for exact classical copying. Furthermore,
we extend these findings to the quantum regime, encompassing both Markovian and
non-Markovian dynamics. Specifically, within Lindblad dynamics, we derive a
similar three-way trade-off relation that quantifies the cost of achieving a
pure state with a given error. The generalization to general quantum dynamics
involving a system coupled to a finite bath implies that heat dissipation
becomes infinite as the quantum system is exactly cooled down to the ground
state or perfectly reset to a pure state, thereby resolving an open question
regarding the thermodynamic cost of information erasure.
