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arXiv:1204.5170 · Physical Review A

Optimal uncertainty relations for extremely coarse-grained measurements

Łukasz Rudnicki, Stephen Patrick Walborn, Fabricio Toscano

Published Apr 26, 2012• 43 citations
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Abstract

We derive two quantum uncertainty relations for position and momentum coarse-grained measurements. Building on previous results, we first improve the lower bound for uncertainty relations using the R\'enyi entropy, particularly in the case of coarse-grained measurements. We then sharpen a Heisenberg-like uncertainty relation derived previously in [Europhys. Lett. 97, 38003 (2012)] that uses variances and reduces to the usual one in the case of infinite precision measurements. Our sharpened uncertainty relation is meaningful for any amount of coarse graining. That is, there is always a nontrivial uncertainty relation for coarse-grained measurement of the noncommuting observables, even in the limit of extremely large coarse graining.

DOI
10.1103/physreva.85.042115

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We derive two quantum uncertainty relations for position and momentum coarse-grained measurements. Building on previous results, we first improve the lower bound for uncertainty relations using the R'enyi entropy, particularly in the case of coarse-grained measurements. We then sharpen a Heisenberg-like uncertainty relation derived previously in [Europhys. Lett. 97, 38003 (2012)] that uses variances and reduces to the usual one in the case of infinite precision measurements. Our sharpened uncertainty relation is meaningful for any amount of coarse graining. That is, there is always a nontrivial uncertainty relation for coarse-grained measurement of the noncommuting observables, even in the limit of extremely large coarse graining.
We improve the entropic uncertainty relations for position and momentum coarse-grained measurements. We derive the continuous, coarse-grained counterparts of the discrete uncertainty relations based on the concept of majorization. The entropic inequalities obtained involve two R\'enyi entropies of the same order, and thus go beyond the standard scenario with conjugated parameters. In a special case describing the sum of two Shannon entropies, the majorization-based bounds significantly outperform the currently known results in the regime of larger coarse graining, and might thus be useful for entanglement detection in continuous variables.
We ask which is the best strategy to reveal uncertainty relations between complementary observables of a continuous variable system for coarse-grained measurements. This leads to the derivation of new uncertainty relations for coarse-grained measurements that are always valid, even for detectors with low precision. These relations should be particularly relevant in experimental demonstrations of squeezing in quantum optics, quantum state reconstruction, and the development of trustworthy entanglement criteria.
Uncertainty relations involving incompatible observables are one of the cornerstones of quantum mechanics. Aside from their fundamental significance, they play an important role in practical applications, such as detection of quantum correlations and security requirements in quantum cryptography. In continuous variable systems, the spectra of the relevant observables form a continuum and this necessitates the coarse graining of measurements. However, these coarse-grained observables do not necessarily obey the same uncertainty relations as the original ones, a fact that can lead to false results when considering applications. That is, one cannot naively replace the original observables in the uncertainty relation for the coarse-grained observables and expect consistent results. As such, several uncertainty relations that are specifically designed for coarse-grained observables have been developed. In recognition of the 90th anniversary of the seminal Heisenberg uncertainty relation, celebrated last year, and all the subsequent work since then, here we give a review of the state of the art of coarse-grained uncertainty relations in continuous variable quantum systems, as well as their applications to fundamental quantum physics and quantum information tasks. Our review is meant to be balanced in its content, since both theoretical considerations and experimental perspectives are put on an equal footing.
The limitation on obtaining precise outcomes of measurements performed on two noncommuting observables of a particle as set by the uncertainty principle in its entropic form can be reduced in the presence of quantum memory. We derive a new entropic uncertainty relation based on fine graining, which leads to an ultimate limit on the precision achievable in measurements performed on two incompatible observables in the presence of quantum memory. We show that our derived uncertainty relation tightens the lower bound set by entropic uncertainty for members of the class of two-qubit states with maximally mixed marginals, while accounting for the recent experimental results using maximally entangled pure states and mixed Bell-diagonal states. An implication of our uncertainty relation on the security of quantum key generation protocols is pointed out.
While quantum metrology enables measurement precision beyond classical limits, its performance is often susceptible to experimental imperfections. Most prior studies have focused on imperfections in quantum states and operations. Here, we investigate the effect of coarse graining in quantum measurement through both theoretical analysis and experimental demonstration. Using an interferometer with a squeezed vacuum and a laser input, we analyze how coarse graining in homodyne detection affects the precision of phase estimation. We evaluate the Fisher information under various coarse-graining conditions and determine, in each case, an optimal estimation strategy that saturates the Cramér-Rao bound. Remarkably, even extremely coarse-grained measurement -- with only two bins -- enables phase estimation beyond the standard quantum limit and even achieves a precision that follows the Heisenberg scaling. We experimentally demonstrate quantum-enhanced phase estimation under coarse-grained homodyne detection. To determine an optimal estimation strategy, we employ the method of moments and present calibration procedures that enable its application to general experimental settings. Using only two bins, we observe a quantum enhancement of 1.2 dB compared to the classical method using the ideal measurement, improving towards 3.8 dB as the bin number increases. These results highlight a practical pathway to achieving quantum enhancement under the presence of severe experimental imperfections.
Uncertainty relations involving incompatible observables are one of the cornerstones of quantum mechanics. Aside from their fundamental significance, they play an important role in practical applications, such as detection of quantum correlations and security requirements in quantum cryptography. In continuous variable systems, the spectra of the relevant observables form a continuum and this necessitates the coarse graining of measurements. However, these coarse-grained observables do not necessarily obey the same uncertainty relations as the original ones, a fact that can lead to false results when considering applications. That is, one cannot naively replace the original observables in the uncertainty relation for the coarse-grained observables and expect consistent results. As such, several uncertainty relations that are specifically designed for coarse-grained observables have been developed. In recognition of the 90th anniversary of the seminal Heisenberg uncertainty relation, celebrated last year, and all the subsequent work since then, here we give a review of the state of the art of coarse-grained uncertainty relations in continuous variable quantum systems, as well as their applications to fundamental quantum physics and quantum information tasks. Our review is meant to be balanced in its content, since both theoretical considerations and experimental perspectives are put on an equal footing.