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We study fine-grained uncertainty relations for several quantum measurementsin a finite-dimensional Hilbert space. The proposed approach is based on exactcalculation or estimation of the spectral norms of corresponding positivematrices. Fine-grained uncertainty relations of the state-independent form arederived for an arbitrary set of mutually unbiased bases. Such relations areextended with a recent notion of mutually unbiased measurements. The case ofso-called mutually biased bases is considered in a similar manner. We alsodiscuss a formulation of fine-grained uncertainty relations in the case ofgeneralized measurements. The general approach is then applied to twomeasurements related to state discrimination. The case of three rank-oneprojective measurements is further examined in details. In particular, weconsider fine-grained uncertainty relations for mutually unbiased bases inthree-dimensional Hilbert space.

Fine-grained uncertainty relations for several quantum measurements