Simultaneous measurement of two non-commuting quantum variables: Solution of a dynamical model

Abstract

The possibility of performing simultaneous measurements in quantum mechanics is investigated in the context of the Curie-Weiss model for a projective measurement. Concretely, we consider a spin-$\frac{1}{2}$ system simultaneously interacting with two magnets, which act as measuring apparatuses of two different spin components. We work out the dynamics of this process and determine the final state of the measuring apparatuses, from which we can find the probabilities of the four possible outcomes of the measurements. The measurement is found to be non-ideal, as (i) the joint statistics do not coincide with the one obtained by separately measuring each spin component, and (ii) the density matrix of the spin does not collapse in either of the measured observables. However, we give an operational interpretation of the process as a generalised quantum measurement, and show that it is fully informative: The expected value of the measured spin components can be found with arbitrary precision for sufficiently many runs of the experiment.

24 pages, 9 figures; close to published version

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• Understanding quantum mechanics: a review and synthesis in precise language
• Minimizing back-action through entangled measurements
• Simultaneous weak measurement of non-commuting observables
• Certifying Non-Classical Behavior for Negative Keldysh Quasi-Probabilities
• Past observable dynamics of a continuously monitored qubit
• Uniqueness of the Observable Leaving Redundant Imprints in the Environment in the Context of Quantum Darwinism
• The Random Fractional Matching Problem
• Dynamics of quantum measurements employing two Curie-Weiss apparatuses
• Dynamics of ideal quantum measurement of a spin 1 with a Curie-Weiss magnet