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We mainly study the block-coherence measures based on resource theory ofblock-coherence and the coherence measures based on positive-operator-valuedmeasures (POVM). Several block-coherence measures including a block-coherencemeasure based on maximum relative entropy, the one-shot block coherence costunder the maximally block-incoherent operations, and a coherence measure basedon coherent rank have been introduced and the relationships between theseblock-coherence measures have been obtained. We also give the definition of themaximally block-coherent state and describe the deterministic coherencedilution process by constructing block-incoherent operations. Based on the POVMcoherence resource theory, we propose a POVM-based coherence measure by usingthe known scheme of building POVM-based coherence measures from block-coherencemeasures, and the one-shot block coherence cost under the maximallyPOVM-incoherent operations. The relationship between the POVM-based coherencemeasure and the one-shot block coherence cost under the maximallyPOVM-incoherent operations is analysed.

The block-coherence measures and the coherence measures based on positive-operator-valued measures