Geometric formulation of the uncertainty principle
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Date
2014Author
Bosyk, Gustavo Martín
Osán, Tristán Martín
Lamberti, Pedro Walter
Portesi, Mariela
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A geometric approach to formulate the uncertainty principle between quantum observables acting on an N-dimensional Hilbert space is proposed. We consider the fidelity between a density operator associated with a quantum system and a projector associated with an observable, and interpret it as the probability of obtaining the outcome corresponding to that projector. We make use of fidelity-based metrics such as angle, Bures, and root infidelity to propose a measure of uncertainty. The triangle inequality allows us to derive a family of uncertainty relations. In the case of the angle metric, we recover the Landau-Pollak inequality for pure states and show, in a natural way, how to extend it to the case of mixed states in arbitrary dimension. In addition, we derive and compare alternative uncertainty relations when using other known fidelity-based metrics.
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Bosyk, G. M., Osán, T. M., Lamberti, P. W. y Portesi, M. (2014). Geometric formulation of the uncertainty principle. Physical Review A, 89 (3), 034101 https://doi.org/10.1103/PhysRevA.89.034101