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Register a new userExperimental test of the majorization uncertainty relation with mixed states
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The uncertainty relation lies at the heart of quantum theory and behaves as a non-classical constraint on the indeterminacies of incompatible observables in a system. In the literature, many experiments have been devoted to the test of the uncertainty relations which mainly focus on the pure states. Here we present an experimental investigation on the optimal majorization uncertainty for mixed states by means of the coherent light. The polarization states with adjustable mixedness are prepared by the combination of two coherent beams, and we test the majorization uncertainty relation for three incompatible observables using the prepared mixed states. The experimental results show that the direct sum majorization uncertainty relations are tight and optimal for general mixed systems.
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Uncertainty relation is one of the fundamental principle in quantum mechanics and plays an important role in quantum information science. We experimentally test the error-disturbance uncertainty relation (EDR) with continuous variables for Gaussian states. Two conjugate continuous-variable observables, amplitude and phase quadratures of an optical mode, are measured simultaneously by using a heterodyne measurement system. The EDR with continuous variables for a coherent state, a squeezed state and a thermal state are verified experimentally. Our experimental results demonstrate that Heisenbergs EDR with continuous variables is violated, yet Ozawas and Branciards EDR with continuous variables are validated.
Incompatible observables can be approximated by compatible observables in joint measurement or measured sequentially, with constrained accuracy as implied by Heisenbergs original formulation of the uncertainty principle. Recently, Busch, Lahti, and Werner proposed inaccuracy trade-off relations based on statistical distances between probability distributions of measurement outcomes [Phys. Rev. Lett. 111, 160405 (2013); Phys. Rev. A 89, 012129 (2014)]. Here we reform their theoretical framework, derive an improved relation for qubit measurement, and perform an experimental test on a spin system. The relation reveals that the worst-case inaccuracy is tightly bounded from below by the incompatibility of target observables, and is verified by the experiment employing joint measurement in which two compatible but typically non-commutative observables on one qubit are measured simultaneously.
Uncertainty relation is not only of fundamental importance to quantum mechanics, but also crucial to the quantum information technology. Recently, majorization formulation of uncertainty relations (MURs) have been widely studied, ranging from two measurements to multiple measurements. Here, for the first time, we experimentally investigate MURs for two measurements and multiple measurements in the high-dimensional systems, and study the intrinsic distinction between direct-product MURs and direct-sum MURs. The experimental results reveal that by taking different nonnegative Schur-concave functions as uncertainty measure, the two types of MURs have their own particular advantages, and also verify that there exists certain case where three-measurement majorization uncertainty relation is much stronger than the one obtained by summing pairwise two-measurement uncertainty relations. Our work not only fills the gap of experimental studies of majorization uncertainty relations, but also represents an advance in quantitatively understanding and experimental verification of majorization uncertainty relations which are universal and capture the essence of uncertainty in quantum theory.
Heisenbergs original uncertainty relation is related to measurement effect, which is different from the preparation uncertainty relation. However, it has been shown that Heisenbergs error-disturbance uncertainty relation can be violated in some cases. We experimentally test the error-tradeoff uncertainty relation by using a continuous-variable Einstein-Podolsky-Rosen (EPR) entangled state. Based on the quantum correlation between the two entangled optical beams, the errors on amplitude and phase quadratures of one EPR optical beam coming from joint measurement are estimated respectively, which are used to verify the error-tradeoff relation. Especially, the error-tradeoff relation for error-free measurement of one observable is verified in our experiment. We also verify the error-tradeoff relations for nonzero errors and mixed state by introducing loss on one EPR beam. Our experimental results demonstrate that Heisenbergs error-tradeoff uncertainty relation is violated in some cases for a continuous-variable system, while the Ozawas and Brainciards relations are valid.
We derive an entropic uncertainty relation for generalized positive-operator-valued measure (POVM) measurements via a direct-sum majorization relation using Schur concavity of entropic quantities in a finite-dimensional Hilbert space. Our approach provides a significant improvement of the uncertainty bound compared with previous majorization-based approaches [S. Friendland, V. Gheorghiu and G. Gour, Phys. Rev. Lett. 111, 230401 (2013); A. E. Rastegin and K. .Zyczkowski, J. Phys. A, 49, 355301 (2016)], particularly by extending the direct-sum majorization relation first introduced in [L. Rudnicki, Z. Pucha{l}a and K. .{Z}yczkowski, Phys. Rev. A 89, 052115 (2014)]. We illustrate the usefulness of our uncertainty relations by considering a pair of qubit observables in a two-dimensional system and randomly chosen unsharp observables in a three-dimensional system. We also demonstrate that our bound tends to be stronger than the generalized Maassen--Uffink bound with an increase in the unsharpness effect. Furthermore, we extend our approach to the case of multiple POVM measurements, thus making it possible to establish entropic uncertainty relations involving more than two observables.
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