Uncertainty and certainty relations for successive projective measurements of a qubit in terms of Tsallis' entropies

TL;DR

This paper investigates uncertainty and certainty relations for two successive projective measurements on a qubit, using Tsallis' entropies, and explores how these relations extend to higher dimensions and relate to mutual unbiasedness.

Contribution

It introduces new bounds for entropic uncertainties in successive measurements of qubits and extends some results to finite-dimensional systems, highlighting the role of mutual unbiasedness.

Findings

Derived minimal and maximal entropic uncertainty bounds for qubit measurements.

Extended some uncertainty relations to higher-dimensional systems.

Connected uncertainty relations with the concept of mutual unbiasedness.

Abstract

We study uncertainty and certainty relations for two successive measurements of two-dimensional observables. Uncertainties in successive measurement are considered within the following two scenarios. In the first scenario, the second measurement is performed on the quantum state generated after the first measurement with completely erased information. In the second scenario, the second measurement is performed on the post-first-measurement state conditioned on the actual measurement outcome. Induced quantum uncertainties are characterized by means of the Tsallis entropies. For two successive projective measurement of a qubit, we obtain minimal and maximal values of related entropic measures of induced uncertainties. Some conclusions found in the second scenario are extended to arbitrary finite dimensionality. In particular, a connection with mutual unbiasedness is emphasized.