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Bulletin of the American Physical Society

Simultaneous measurement of non-commuting observables in circuit QED: Theory

Leigh S. Martin, Shay Hacohen-Gourgy, Emmanuel Flurin, K. Birgitta Whaley, Irfan A. Siddiqi

Published Mar 15, 2016• 0 citations
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The fact that not all quantum observables are jointly measurable is one of the major differences between quantum and classical theory. In the former, non-commuting observables can only be simultaneously measured with limited precision. We report on an experimental implementation of such a simultaneous measurement of two non-commuting observables based on the framework of weak von Neumann measurements. We use a photonic setup where the polarisation degree of freedom acts as the system and the two components of the transversal position correspond to the pointers of the measurement apparatuses. In addition, the theory shows that these weak measurements demonstrate a counter-intuitive non-monotonicity: weaker measurements can potentially reveal more information than stronger ones.
Simultaneous measurement of non-commuting observables deepens the understanding of Heisenberg’s uncertainty principle and has significant applications to quantum feedback control theory. Here we use techniques from machine learning to define a measure for the degree of non-commuting behavior of quantum measurements coming from the initial and final states of the measurements. Both the measurement strength effect and the angle effect between two observables have been constructed by using the K-means clustering method to reflect the defined non-commutativity measure. The rationality of the measure is demonstrated and its application in obtaining the measurement limitation that physically realizable is presented. Our work helps to advance the understanding of quantum non-commuting measurement and quantum control theory.