Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2107.12431 · Physical Review A

Entropic uncertainty relations for mutually unbiased periodic coarse-grained observables resembling their discrete counterparts

Łukasz Rudnicki, Stephen Patrick Walborn

Published Oct 13, 2021• 3 citations
arXivHTMLPDF

Abstract

One of the most important and useful entropic uncertainty relations concerns a $d$-dimensional system and two mutually unbiased measurements. In such a setting, the sum of two information entropies is lower bounded by $lnd$. It has recently been shown that projective measurements subject to operational mutual unbiasedness can also be constructed in a continuous domain, with the help of periodic coarse graining. Here we consider the whole family of R\'enyi entropies applied to these discretized observables and prove that such a scheme does also admit the entropic uncertainty relations mentioned above.

DOI
10.1103/physreva.104.042210

Related papers

Majorization approach to entropic uncertainty relations for coarse-grained observables

Łukasz Rudnicki

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.

arXiv:1503.03682•Physical Review A•Mar 30, 2015•57 citations

Entropic uncertainty relations for incomplete sets of mutually unbiased observables

Adam Azarchs

Entropic uncertainty relations, based on sums of entropies of probability distributions arising from different measurements on a given pure state, can be seen as a generalization of the Heisenberg uncertainty relation that is in many cases a more useful way to quantify incompatibility between observables. Of particular interest are relationships between `mutually unbiased' observables, which are maximally incompatible. Lower bounds on the sum of entropies for sets of two such observables, and for complete sets of observables within a space of given dimension, have been found. This paper explores relations in the intermediate regime of large, but far from complete, sets of unbiased observables.

arXiv:quant-ph/0412083•Dec 10, 2004•10 citations

Mutual Unbiasedness in Coarse-Grained Continuous Variables

D. S. Tasca, P. Sánchez, Stephen Patrick Walborn +1 more

The notion of mutual unbiasedness for coarse-grained measurements of quantum continuous variable systems is considered. It is shown that while the procedure of "standard" coarse graining breaks the mutual unbiasedness between conjugate variables, this desired feature can be theoretically established and experimentally observed in periodic coarse graining. We illustrate our results in an optics experiment implementing Fraunhofer diffraction through a periodic diffraction grating, finding excellent agreement with the derived theory. Our results are an important step in developing a formal connection between discrete and continuous variable quantum mechanics.

arXiv:1709.00137•Physical Review Letters•Jan 24, 2018•22 citations

On Uncertainty Relations and Entanglement Detection with Mutually Unbiased Measurements

Alexey E. Rastegin

We formulate some properties of a set of several mutually unbiased measurements. These properties are used for deriving entropic uncertainty relations. Applications of mutually unbiased measurements in entanglement detection are also revisited. First, we estimate from above the sum of the indices of coincidence for several mutually unbiased measurements. Further, we derive entropic uncertainty relations in terms of the Rényi and Tsallis entropies. Both the state-dependent and state-independent formulations are obtained. Using the two sets of local mutually unbiased measurements, a method of entanglement detection in bipartite finite-dimensional systems may be realized. A certain trade-off between a sensitivity of the scheme and its experimental complexity is discussed.

arXiv:1407.7333•Open Systems & Information Dynamics•Mar 1, 2015•45 citations

Mutually unbiased coarse-grained measurements of two or more phase-space variables

E. C. Paul, Stephen Patrick Walborn, D. S. Tasca +1 more

Mutual unbiasedness of the eigenstates of phase-space operators---such as position and momentum, or their standard coarse-grained versions---exists only in the limiting case of infinite squeezing. In Phys. Rev. Lett. 120, 040403 (2018), it was shown that mutual unbiasedness can be recovered for periodic coarse graining of these two operators. Here we investigate mutual unbiasedness of coarse-grained measurements for more than two phase-space variables. We show that mutual unbiasedness can be recovered between periodic coarse graining of any two nonparallel phase-space operators. We illustrate these results through optics experiments, using the fractional Fourier transform to prepare and measure mutually unbiased phase-space variables. The differences between two and three mutually unbiased measurements is discussed. Our results contribute to bridging the gap between continuous and discrete quantum mechanics, and they could be useful in quantum-information protocols.

arXiv:1804.04480•Physical Review A•May 4, 2018•6 citations

Optimal uncertainty relations for extremely coarse-grained measurements

Łukasz Rudnicki, Stephen Patrick Walborn, Fabricio Toscano

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.

arXiv:1204.5170•Physical Review A•Apr 26, 2012•43 citations
One of the most important and useful entropic uncertainty relations concerns a ddd-dimensional system and two mutually unbiased measurements. In such a setting, the sum of two information entropies is lower bounded by lndlndlnd. It has recently been shown that projective measurements subject to operational mutual unbiasedness can also be constructed in a continuous domain, with the help of periodic coarse graining. Here we consider the whole family of R'enyi entropies applied to these discretized observables and prove that such a scheme does also admit the entropic uncertainty relations mentioned above.
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.
Entropic uncertainty relations, based on sums of entropies of probability distributions arising from different measurements on a given pure state, can be seen as a generalization of the Heisenberg uncertainty relation that is in many cases a more useful way to quantify incompatibility between observables. Of particular interest are relationships between `mutually unbiased' observables, which are maximally incompatible. Lower bounds on the sum of entropies for sets of two such observables, and for complete sets of observables within a space of given dimension, have been found. This paper explores relations in the intermediate regime of large, but far from complete, sets of unbiased observables.
The notion of mutual unbiasedness for coarse-grained measurements of quantum continuous variable systems is considered. It is shown that while the procedure of "standard" coarse graining breaks the mutual unbiasedness between conjugate variables, this desired feature can be theoretically established and experimentally observed in periodic coarse graining. We illustrate our results in an optics experiment implementing Fraunhofer diffraction through a periodic diffraction grating, finding excellent agreement with the derived theory. Our results are an important step in developing a formal connection between discrete and continuous variable quantum mechanics.
We formulate some properties of a set of several mutually unbiased measurements. These properties are used for deriving entropic uncertainty relations. Applications of mutually unbiased measurements in entanglement detection are also revisited. First, we estimate from above the sum of the indices of coincidence for several mutually unbiased measurements. Further, we derive entropic uncertainty relations in terms of the Rényi and Tsallis entropies. Both the state-dependent and state-independent formulations are obtained. Using the two sets of local mutually unbiased measurements, a method of entanglement detection in bipartite finite-dimensional systems may be realized. A certain trade-off between a sensitivity of the scheme and its experimental complexity is discussed.
Mutual unbiasedness of the eigenstates of phase-space operators---such as position and momentum, or their standard coarse-grained versions---exists only in the limiting case of infinite squeezing. In Phys. Rev. Lett. 120, 040403 (2018), it was shown that mutual unbiasedness can be recovered for periodic coarse graining of these two operators. Here we investigate mutual unbiasedness of coarse-grained measurements for more than two phase-space variables. We show that mutual unbiasedness can be recovered between periodic coarse graining of any two nonparallel phase-space operators. We illustrate these results through optics experiments, using the fractional Fourier transform to prepare and measure mutually unbiased phase-space variables. The differences between two and three mutually unbiased measurements is discussed. Our results contribute to bridging the gap between continuous and discrete quantum mechanics, and they could be useful in quantum-information protocols.
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.