Within the framework of resource theory, the research on quantum coherence has not only furnished a rigorous and systematic methodology for investigating quantum coherence and developing coherence quantification methods, but also tackled key issues regarding the conversion and practical utilization of coherence. Notably, in practical applications, the order of the diagonal elements in an incoherent state is directly linked to the core requirements of thermodynamics under noisy conditions, and constitutes a necessary prerequisite for translating the coherence theory into practical use. Addressing this, we establish a resource theory framework with $\pi$-ordered incoherent states as free states. Concretely, we explore quantum resource theory in which incoherent states featuring specific diagonal element orderings qualify as free states, propose suitable measures for quantifying the resource states, and construct valid witnesses for their identification. Comparative analysis with concrete examples reveals that the variation trends of $\pi$-ordered coherence in certain quantum states are asynchronous with their coherence characteristics. This further underscores the independent significance of the $\pi$-ordered coherence resource theory, separate from the general coherence resource theory.

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• Quantum coherence & coherence measures
• Quantum correlations, foundations & formalism