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Fine-grained uncertainty relations for several quantum measurements

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Abstract

We study fine-grained uncertainty relations for several quantum measurements in a finite-dimensional Hilbert space. The proposed approach is based on the exact calculation or estimation of the spectral norms of corresponding positive matrices. Fine-grained uncertainty relations of the state-independent form are derived for an arbitrary set of mutually unbiased bases. Such relations are extended with a recent notion of mutually unbiased measurements. The case of so-called mutually biased bases is considered in a similar manner. We also discuss a formulation of fine-grained uncertainty relations in the case of generalized measurements. The general approach is then applied to two measurements related to state discrimination. The case of three rank-one projective measurements is further examined in details. In particular, we consider fine-grained uncertainty relations for mutually unbiased bases in three-dimensional Hilbert space.

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  1. Department of Theoretical Physics, Irkutsk State University, Gagarin Bv. 20, 664003, Irkutsk, Russia

    Alexey E. Rastegin

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  1. Alexey E. Rastegin
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Rastegin, A.E. Fine-grained uncertainty relations for several quantum measurements. Quantum Inf Process 14, 783–800 (2015). https://doi.org/10.1007/s11128-014-0869-9

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