Sequential measurements of non-commuting observables with quantum controlled interactions

Abstract

The origin of non-classical correlations is difficult to identify since the uncertainty principle requires that information obtained about one observable invariably results in the disturbance of any other non-commuting observable. Here, this problem is addressed by investigating the uncertainty trade-off between measurement errors and disturbance for measurement interactions controlled by the state of a single qubit, where the measurement is described by a quantum coherent superposition of a fully projective measurement and the identity operation. It is shown that the measurement statistics obtained from a quantum controlled measurement of A followed by a projective measurement of B can be explained in terms of a simple combination of resolution and back-action errors acting on an intrinsic joint probability of the non-commuting observables defined by the input state of the system. These intrinsic joint probabilities are consistent with the complex-valued joint probabilities recently observed in weak measurements of quantum systems and provide direct evidence of non-commutativity in the form of imaginary correlations between the non-commuting operators. In quantum controlled measurements, these imaginary correlations can be converted into well-defined contributions to the real measurement statistics, allowing a direct experimental observation of the less intuitive aspects of quantum theory.

12 pages, complete re-write of the original 5 page letter, including better explanations of the error statistics and the origin and meaning of imaginary probability in quantum statistics

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• Phase transitions in sequential weak measurements
• Non-Abelian transport distinguishes three usually equivalent notions of entropy production
• Interferometry of quantum correlation functions to access quasiprobability distribution of work
• A framework for measuring weak values without weak interactions and its diagrammatic representation
• On the fundamental role of dynamics in quantum physics
• Observation of non-classical correlations in sequential measurements of photon polarization
• Contextuality, Coherences, and Quantum Cheshire Cats
• Operational formulation of weak values without probe systems
• Local measurement uncertainties impose a limit on non-local quantum correlations
• Determining complementary properties using weak-measurement: uncertainty, predictability, and disturbance
• A Scheme for Performing Strong and Weak Sequential Measurements of Non-commuting Observables
• Characterization of measurement uncertainties using the correlations between local outcomes obtained from maximally entangled pairs
• Memory effects in a sequence of measurements of non-commuting observables
• Quasiprobability distributions with weak measurements
• Systematic errors in direct state measurements with quantum controlled measurements
• Statistical signatures of quantum contextuality
• Orthogonalization speed-up from quantum coherence after a sudden quench