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:1605.04174 · Physical Review Letters

Simultaneous Measurement of Complementary Observables with Compressive Sensing

Gregory A. Howland, James Schneeloch, Daniel J. Lum, John C. Howell

Published Jun 26, 2014• 33 citations
arXivHTMLPDF

Abstract

The more information a measurement provides about a quantum system's position statistics, the less information a subsequent measurement can provide about the system's momentum statistics. This information trade-off is embodied in the entropic formulation of the uncertainty principle. Traditionally, uncertainly relations correspond to resolution limits; increasing a detector's position sensitivity decreases its momentum sensitivity and vice versa. However, this is not required in general; for example, position information can instead be extracted at the cost of noise in momentum. Using random, partial projections in position followed by strong measurements in momentum, we efficiently determine the transverse-position and transverse-momentum distributions of an unknown optical field with a single set of measurements. The momentum distribution is directly imaged, while the position distribution is recovered using compressive sensing. At no point do we violate uncertainty relations; rather, we economize the use of information we obtain.

DOI
10.1103/physrevlett.112.253602

Related papers

Complementary imaging with compressive sensing

Gregory A. Howland, James Schneeloch, Daniel J. Lum +2 more

Measurements on quantum systems are always constrained by uncertainty relations. For traditional, projective measurements, uncertainty relations correspond to resolution limitations; a detector's position resolution is increased at the cost of its momentum resolution and vice-versa. However, many experiments in quantum measurement are now exploring non- or partially-projective measurements. For these techniques, measurement disturbance need not manifest as a blurring in the complementary domain. Here, we describe a technique for complementary imaging | obtaining sharp position and momentum distributions of a transverse optical field with a single set of measurements. Our technique consists of random, partially-projective filtering in position followed by projective measurements in momentum. The partial-projections extract information about position at the cost of injecting a small amount of noise into the momentum distribution, which can still be directly imaged. The position distribution is recovered via compressive sensing.

Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•May 21, 2015•0 citations

Information Exclusion Principle for Complementary Observables

Michael J. W. Hall

The sum of the information gains corresponding to measurements of position and momentum is bounded by $\mathrm{log}2\ensuremath{\Delta}X\ensuremath{\Delta}P/\ensuremath{\hbar}$ for a quantum ensemble with position and momentum uncertainties $\ensuremath{\Delta}X$ and $\ensuremath{\Delta}P$. The bound implies the Heisenberg uncertainty principle and that the gain of position information can be maximized only at the expense of momentum information, and vice versa. This exclusion principle for the information contents of complementary observables is extended to finite Hilbert spaces, and to the quadrature, number, and phase observables of bosonic fields degraded by Gaussian noise.

Physical Review Letters•Apr 24, 1995•88 citations

Position-momentum uncertainty relations in the presence of quantum memory

Fabian Furrer, Mario Berta, Marco Tomamichel +2 more

A prominent formulation of the uncertainty principle identifies the fundamental quantum feature that no particle may be prepared with certain outcomes for both position and momentum measurements. Often the statistical uncertainties are thereby measured in terms of entropies providing a clear operational interpretation in information theory and cryptography. Recently, entropic uncertainty relations have been used to show that the uncertainty can be reduced in the presence of entanglement and to prove security of quantum cryptographic tasks. However, much of this recent progress has been focused on observables with only a finite number of outcomes not including Heisenberg’s original setting of position and momentum observables. Here, we show entropic uncertainty relations for general observables with discrete but infinite or continuous spectrum that take into account the power of an entangled observer. As an illustration, we evaluate the uncertainty relations for position and momentum measurements, which is operationally significant in that it implies security of a quantum key distribution scheme based on homodyne detection of squeezed Gaussian states.

arXiv:1308.4527•Journal of Mathematical Physics•Dec 1, 2014•67 citations

Compressively Characterizing High-Dimensional Entangled States with Complementary, Random Filtering

Gregory A. Howland, Samuel H. Knarr, James Schneeloch +2 more

In quantum mechanics, Heisenberg's uncertainty principle prevents the determination of precise, simultaneous measurements of two quantities. A new approach extracts position and momentum information from the same group of entangled photons by dramatically undersampling the system.

arXiv:1605.04792•Physical Review X•May 12, 2016•23 citations

Measurement Uncertainty Relations for Position and Momentum: Relative Entropy Formulation

Alberto Barchielli, Matteo Gregoratti, Alessandro Toigo

Heisenberg’s uncertainty principle has recently led to general measurement uncertainty relations for quantum systems: incompatible observables can be measured jointly or in sequence only with some unavoidable approximation, which can be quantified in various ways. The relative entropy is the natural theoretical quantifier of the information loss when a `true’ probability distribution is replaced by an approximating one. In this paper, we provide a lower bound for the amount of information that is lost by replacing the distributions of the sharp position and momentum observables, as they could be obtained with two separate experiments, by the marginals of any smeared joint measurement. The bound is obtained by introducing an entropic error function, and optimizing it over a suitable class of covariant approximate joint measurements. We fully exploit two cases of target observables: (1) n-dimensional position and momentum vectors; (2) two components of position and momentum along different directions. In (1), we connect the quantum bound to the dimension n; in (2), going from parallel to orthogonal directions, we show the transition from highly incompatible observables to compatible ones. For simplicity, we develop the theory only for Gaussian states and measurements.

arXiv:1705.09949•Entropy•Jun 24, 2017•16 citations

Determining complementary properties using weak-measurement: uncertainty, predictability, and disturbance

Guillaume Thekkadath, Felix Hufnagel, Jeff S. Lundeen

It is often said that measuring a system's position must disturb the complementary property, momentum, by some minimum amount due to the Heisenberg uncertainty principle. Using a 'weak-measurement', this disturbance can be reduced. One might expect this comes at the cost of also reducing the measurement's precision. However, it was recently demonstrated that a sequence consisting of a weak position measurement followed by a regular momentum measurement can probe a quantum system at a single point, with zero width, in position-momentum space. Here, we study this 'joint weak-measurement' and reconcile its compatibility with the uncertainty principle. While a single trial probes the system with a resolution that can saturate Heisenberg's limit, we show that averaging over many trials can be used to surpass this limit. The weak-measurement does not trade away precision, but rather another type of uncertainty called 'predictability' which quantifies the certainty of retrodicting the measurement's outcome.

arXiv:1809.05984•New Journal of Physics•Oct 31, 2018•11 citations
The more information a measurement provides about a quantum system's position statistics, the less information a subsequent measurement can provide about the system's momentum statistics. This information trade-off is embodied in the entropic formulation of the uncertainty principle. Traditionally, uncertainly relations correspond to resolution limits; increasing a detector's position sensitivity decreases its momentum sensitivity and vice versa. However, this is not required in general; for example, position information can instead be extracted at the cost of noise in momentum. Using random, partial projections in position followed by strong measurements in momentum, we efficiently determine the transverse-position and transverse-momentum distributions of an unknown optical field with a single set of measurements. The momentum distribution is directly imaged, while the position distribution is recovered using compressive sensing. At no point do we violate uncertainty relations; rather, we economize the use of information we obtain.
Measurements on quantum systems are always constrained by uncertainty relations. For traditional, projective measurements, uncertainty relations correspond to resolution limitations; a detector's position resolution is increased at the cost of its momentum resolution and vice-versa. However, many experiments in quantum measurement are now exploring non- or partially-projective measurements. For these techniques, measurement disturbance need not manifest as a blurring in the complementary domain. Here, we describe a technique for complementary imaging | obtaining sharp position and momentum distributions of a transverse optical field with a single set of measurements. Our technique consists of random, partially-projective filtering in position followed by projective measurements in momentum. The partial-projections extract information about position at the cost of injecting a small amount of noise into the momentum distribution, which can still be directly imaged. The position distribution is recovered via compressive sensing.
The sum of the information gains corresponding to measurements of position and momentum is bounded by $\mathrm{log}2\ensuremath{\Delta}X\ensuremath{\Delta}P/\ensuremath{\hbar}$ for a quantum ensemble with position and momentum uncertainties $\ensuremath{\Delta}X$ and $\ensuremath{\Delta}P$. The bound implies the Heisenberg uncertainty principle and that the gain of position information can be maximized only at the expense of momentum information, and vice versa. This exclusion principle for the information contents of complementary observables is extended to finite Hilbert spaces, and to the quadrature, number, and phase observables of bosonic fields degraded by Gaussian noise.
A prominent formulation of the uncertainty principle identifies the fundamental quantum feature that no particle may be prepared with certain outcomes for both position and momentum measurements. Often the statistical uncertainties are thereby measured in terms of entropies providing a clear operational interpretation in information theory and cryptography. Recently, entropic uncertainty relations have been used to show that the uncertainty can be reduced in the presence of entanglement and to prove security of quantum cryptographic tasks. However, much of this recent progress has been focused on observables with only a finite number of outcomes not including Heisenberg’s original setting of position and momentum observables. Here, we show entropic uncertainty relations for general observables with discrete but infinite or continuous spectrum that take into account the power of an entangled observer. As an illustration, we evaluate the uncertainty relations for position and momentum measurements, which is operationally significant in that it implies security of a quantum key distribution scheme based on homodyne detection of squeezed Gaussian states.
Heisenberg’s uncertainty principle has recently led to general measurement uncertainty relations for quantum systems: incompatible observables can be measured jointly or in sequence only with some unavoidable approximation, which can be quantified in various ways. The relative entropy is the natural theoretical quantifier of the information loss when a `true’ probability distribution is replaced by an approximating one. In this paper, we provide a lower bound for the amount of information that is lost by replacing the distributions of the sharp position and momentum observables, as they could be obtained with two separate experiments, by the marginals of any smeared joint measurement. The bound is obtained by introducing an entropic error function, and optimizing it over a suitable class of covariant approximate joint measurements. We fully exploit two cases of target observables: (1) n-dimensional position and momentum vectors; (2) two components of position and momentum along different directions. In (1), we connect the quantum bound to the dimension n; in (2), going from parallel to orthogonal directions, we show the transition from highly incompatible observables to compatible ones. For simplicity, we develop the theory only for Gaussian states and measurements.
It is often said that measuring a system's position must disturb the complementary property, momentum, by some minimum amount due to the Heisenberg uncertainty principle. Using a 'weak-measurement', this disturbance can be reduced. One might expect this comes at the cost of also reducing the measurement's precision. However, it was recently demonstrated that a sequence consisting of a weak position measurement followed by a regular momentum measurement can probe a quantum system at a single point, with zero width, in position-momentum space. Here, we study this 'joint weak-measurement' and reconcile its compatibility with the uncertainty principle. While a single trial probes the system with a resolution that can saturate Heisenberg's limit, we show that averaging over many trials can be used to surpass this limit. The weak-measurement does not trade away precision, but rather another type of uncertainty called 'predictability' which quantifies the certainty of retrodicting the measurement's outcome.