One fundamental feature of quantum entanglement is the monogamy which shows that the entanglement of two systems limits the ability of either system to entangle with a third one. Such an understanding is only well described in the qubit systems, but it remains an open question for high-dimensional quantum systems. Here the relative entropy of entanglement and the negativity are respectively used to quantify the entanglement. Based on the resource theory of coherence, we found that the monogamy inequalities in arbitrarily finite-dimensional systems can be successfully established for the entanglement induced by quantum coherence. Moreover, the similar inequalities are also constructed for the distribution of quantum coherence.

The standard deviation (SD) quantifies the spread of the observed values on a measurement of an observable. In this paper, we study the distribution of SD among the different components of a superposition state. It is found that the SD of an observable on a superposition state can be well bounded by the SDs of the superposed states. We also show that the bounds also serve as good bounds on coherence of a superposition state. As a further generalization, we give an alternative definition of incompatibility of two observables subject to a given state and show how the incompatibility subject to a superposition state is distributed.

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