Measurement uncertainty relations: characterising optimal error bounds for qubits

Abstract

In standard formulations of the uncertainty principle, two fundamental features are typically cast as impossibility statements: two noncommuting observables cannot in general both be sharply defined (for the same state), nor can they be measured jointly. The pioneers of quantum mechanics were acutely aware and puzzled by this fact, and it motivated Heisenberg to seek a mitigation, which he formulated in his seminal paper of 1927. He provided intuitive arguments to show that the values of, say, the position and momentum of a particle can at least be unsharply defined, and they can be measured together provided some approximation errors are allowed. Only now, nine decades later, a working theory of approximate joint measurements is taking shape, leading to rigorous and experimentally testable formulations of associated error tradeoff relations. Here we briefly review this new development, explaining the concepts and steps taken in the construction of optimal joint approximations of pairs of incompatible observables. As a case study, we deduce measurement uncertainty relations for qubit observables using two distinct error measures. We provide an operational interpretation of the error bounds and discuss some of the first experimental tests of such relations.

Final published version (J Phys A Topical Review), including new figure and discussion

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Cited by in corpus (8)

• Incompatible measurements in quantum information science
• What is nonclassical about uncertainty relations?
• Complementary observables in quantum mechanics
• On Quantum Uncertainty Relations and Uncertainty Regions
• Quantum incompatibility from the viewpoint of entanglement theory
• Uncertainty Relations Relative to Phase-Space Quantum Reference Frames
• Simplifying measurement uncertainty with quantum symmetries
• Entropic measurement uncertainty relations for all the infinite components of a spin vector