Incompatibility of Observables as State-Independent Bound of Uncertainty Relations

Abstract

For a pair of observables, they are called "incompatible", if and only if the commutator between them does not vanish, which represents one of the key features in quantum mechanics. The question is, how can we characterize the incompatibility among three or more observables? Here we explore one possible route towards this goal through Heisenberg's uncertainty relations, which impose fundamental constraints on the measurement precisions for incompatible observables. Specifically, we quantify the incompatibility by the optimal state-independent bounds of additive variance-based uncertainty relations. In this way, the degree of incompatibility becomes an intrinsic property among the operators, but not on the quantum state. To justify our case, we focus on the incompatibility of spin systems. For an arbitrary setting of two or three linearly-independent Pauli-spin operators, the incompatibility is analytically solved, the spins are maximally incompatible if and only if they are orthogonal to each other. On the other hand, the measure of incompatibility represents a versatile tool for applications such as testing entanglement of bipartite states, and EPR-steering criteria.

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Cited by in corpus (8)

• Quasi-Fine-Grained Uncertainty Relations
• Uncertainty regions of observables and state-independent uncertainty relations
• Near-Optimal Variance-Based Uncertainty Relations
• Improved unitary uncertainty relations
• Strong Majorization Uncertainty Relations: Theory and Experiment
• Probability density functions of quantum mechanical observable uncertainties
• Uncertainty relation and the constrained quadratic programming
• Frustration graph formalism for qudit observables