Products of weak values: Uncertainty relations, complementarity, and incompatibility

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Hall, Michael
Pati, Arun Kumar
Wu, Junde
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2016
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Abstract

The products of weak values of quantum observables are shown to be of value in deriving quantum uncertainty and complementarity relations, for both weak and strong measurement statistics. First, a “product representation formula” allows the standard Heisenberg uncertainty relation to be derived from a classical uncertainty relation for complex random variables. We show this formula also leads to strong uncertainty relations for unitary operators and underlies an interpretation of weak values as optimal (complex) estimates of quantum observables. Furthermore, we show that two incompatible observables that are weakly and strongly measured in a weak measurement context obey a complementarity relation under the interchange of these observables, in the form of an upper bound on the product of the corresponding weak values. Moreover, general tradeoff relations between weak purity, quantum purity, and quantum incompatibility, and also between weak and strong joint probability distributions, are obtained based on products of real and imaginary components of weak values, where these relations quantify the degree to which weak probabilities can take anomalous values in a given context.

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Physical Review A
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93
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© 2016 American Physical Society. This is the author-manuscript version of this paper. Reproduced in accordance with the copyright policy of the publisher. Please refer to the journal's website for access to the definitive, published version.
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Quantum optics and quantum optomechanics
Quantum physics not elsewhere classified
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