Uncertainty under Quantum Measures and Quantum Memory

Abstract

The uncertainty principle restricts potential information one gains about physical properties of the measured particle. However, if the particle is prepared in entanglement with a quantum memory, the corresponding entropic uncertainty relation will vary. Based on the knowledge of correlations between the measured particle and quantum memory, we have investigated the entropic uncertainty relations for two and multiple measurements, and generalized the lower bounds on the sum of Shannon entropies without quantum side information to those that allow quantum memory. In particular, we have obtained generalization of Kaniewski-Tomamichel-Wehner's bound for effective measures and majorization bounds for noneffective measures to allow quantum side information. Furthermore, we have derived several strong bounds for the entropic uncertainty relations in the presence of quantum memory for two and multiple measurements. Finally, potential applications of our results to entanglement witnesses are discussed via the entropic uncertainty relation in the absence of quantum memory.

12 pages, 4 figures

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• Improved quantum entropic uncertainty relations
• Strong unitary uncertainty relations
• Incompatibility of Observables as State-Independent Bound of Uncertainty Relations
• Conditional entropic uncertainty relations for Tsallis entropies
• Quasi-Fine-Grained Uncertainty Relations
• Near-Optimal Variance-Based Uncertainty Relations
• Strong Majorization Uncertainty Relations: Theory and Experiment
• Experimental investigation of majorization uncertainty relations in the high-dimensional systems