Tight state-independent uncertainty relations for qubits

TL;DR

This paper develops a method to derive tight, state-independent uncertainty relations for qubits, applicable to any number of observables, and connects these to entropic measures, improving understanding of quantum measurement limitations.

Contribution

It introduces a general approach for tight state-independent uncertainty relations for qubits, including explicit relations for pairs and triples of measurements, and links them to entropic measures.

Findings

Derived tight state-independent uncertainty relations for qubits.

Explicit relations for pairs and triples of qubit measurements.

Connected uncertainty relations to entropic measures.

Abstract

The well-known Robertson-Schr\"odinger uncertainty relations have state-dependent lower bounds which are trivial for certain states. We present a general approach to deriving tight state-independent uncertainty relations for qubit measurements that completely characterise the obtainable uncertainty values. This approach can give such relations for any number of observables, and we do so explicitly for arbitrary pairs and triples of qubit measurements. We show how these relations can be transformed into equivalent tight entropic uncertainty relations. More generally, they can be expressed in terms of any measure of uncertainty that can be written as a function of the expectation value of the observable for a given state.