Fine-grained uncertainty relations for several quantum measurements

Abstract

We study fine-grained uncertainty relations for several quantum measurements in a finite-dimensional Hilbert space. The proposed approach is based on 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.

13 pages, no figures. Minor improvements. The bibliography is extended. To appear in Quantum Inf. Processing

References in corpus (7)

• The uncertainty principle determines the non-locality of quantum mechanics
• Higher-dimensional orbital angular momentum based quantum key distribution with mutually unbiased bases
• Strong Majorization Entropic Uncertainty Relations
• Mutually unbiased measurements in finite dimensions
• Entropic uncertainty relations and locking: tight bounds for mutually unbiased bases
• Quantum and classical entropic uncertainty relations
• A transform of complementary aspects with applications to entropic uncertainty relations

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