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

Physical Review A

Quantum uncertainties and Heisenberg-like uncertainty relations for a weak measurement scheme involving two arbitrary noncommuting observables

Milan Šindelka, Nimrod Moiseyev

Published Jan 23, 2018• 4 citations
Publisher

Abstract

We study theoretically a weak measurement of an observable $A$ and its impact on a subsequent strong measurement of another observable $B$. We assume that both measurements are described by the Gaussian point detectors. It is shown that a decrease of the system-detector interaction during the measurement of $A$ results in an increase of the fluctuations of $A$, while the mean value of $A$ is kept unaltered. The ratio ${(\mathrm{\ensuremath{\Delta}}A)}_{\mathrm{weak}\phantom{\rule{0.28em}{0ex}}\phantom{\rule{0.16em}{0ex}}\mathrm{measurement}}/{(\mathrm{\ensuremath{\Delta}}A)}_{\mathrm{strong}\phantom{\rule{0.28em}{0ex}}\phantom{\rule{0.16em}{0ex}}\mathrm{measurement}}$ turns out to be universal, i.e., independent of the physical system, its quantum state, and the quantity measured. Most of our attention is focused on analyzing outcomes of the subsequent strong measurement of the quantity $B$, namely, on the mean value $\overline{B}$ and the variance ${(\mathrm{\ensuremath{\Delta}}B)}^{2}$. We derive explicit analytic corrections for $\overline{B}$ and ${(\mathrm{\ensuremath{\Delta}}B)}^{2}$ arising due to the prior weak measurement of $A$. Importantly, the leading-order impact of the weak measurement of $A$ on $\overline{B}$ and ${(\mathrm{\ensuremath{\Delta}}B)}^{2}$ is encoded in the terms ${\phantom{\rule{0.16em}{0ex}}}_{s}\ensuremath{\langle}\ensuremath{\psi}|[\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{A},[\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{A},{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{B}}^{k}]]{|\ensuremath{\psi}\ensuremath{\rangle}}_{s}$, where $k=1,2$ and ${|\ensuremath{\psi}\ensuremath{\rangle}}_{s}$ is the quantum state vector of our system before the measurements. Our additional contribution consists of deriving Heisenberg-like uncertainty relations for the product ${(\mathrm{\ensuremath{\Delta}}A)}_{\mathrm{weak}\phantom{\rule{0.28em}{0ex}}\phantom{\rule{0.16em}{0ex}}\mathrm{measurement}}\phantom{\rule{0.28em}{0ex}}{(\mathrm{\ensuremath{\Delta}}B)}_{\mathrm{strong}\phantom{\rule{0.28em}{0ex}}\phantom{\rule{0.16em}{0ex}}\mathrm{measurement}}$ and analyzing the case of minimum uncertainty. Finally, our theoretical developments are illustrated numerically via studying the $s$ states of the hydrogen atom, assuming $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{A}=\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{r}$ (atomic radius) and $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{B}=\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}$ (energy).

DOI
10.1103/physreva.97.012122

Related papers

Uncertainty relations for time-averaged weak values

Eli Pollak, Salvador Miret-Artés

Time averaging of weak values using the quantum transition path time probability distribution enables us to establish a general uncertainty relation for the weak values of two not necessarily Hermitian operators. This new relation is a weak value analog of the Schr\"odinger strong value uncertainty relation. It leads to the conclusion that it is possible to determine with high accuracy the simultaneous mean weak values of noncommuting operators by judicious choice of the pre- and postselected states even when the postselected state is not an eigenfunction of one of the respective operators. When the time fluctuations of the two weak values are proportional to each other there is no uncertainty limitation on their variances and, in principle, their means can be determined with arbitrary precision even though their corresponding operators do not commute. To exemplify these properties we consider specific weak value uncertainty relations for the time-energy, coordinate-momentum, and coordinate--kinetic-energy pairs. In addition we analyze spin operators and the Stern-Gerlach experiment in weak and strong inhomogeneous magnetic fields. This classic case leads to anomalous spin values when the field is weak. The weak value uncertainty relation implies that anomalous spin values are associated with large variances so that their measurement demands increased signal averaging. These examples establish the importance of considering the time dependence of weak values in scattering experiments.

arXiv:1808.06384•Physical Review A•Jan 8, 2019•6 citations

Modelling of weak quantum measurements consistent with conservation laws

Stanisław Sołtan, Mateusz Fra̧czak, Wolfgang Belzig +1 more

Abstract We discuss quantum mechanical detection models in the weak limit in the context of conservation laws of physical quantities. In particular, we analyze what kind of system–detector interaction can preserve the global conservation or the related symmetry, and how the final measurement on the detector affects the measured observable of the systems and its presumed conservation. It turns out that the order of noncommuting measurements results in observable differences on the level of third-order correlations functions.

The European Physical Journal Special Topics•Jun 1, 2021•0 citations

Uncertainty and Complementarity Relations in Weak Measurement

Arun Kumar Pati, Junde Wu

We prove uncertainty relations that quantitatively express the impossibility of jointly sharp preparation of pre- and post-selected quantum states for measuring incompatible observables during the weak measurement. By defining a suitable operator whose average in the pre-selected quantum state gives the weak value, we show that one can have new uncertainty relations for variances of two such operators corresponding to two non-commuting observables. These generalize the recent stronger uncertainty relations that give non-trivial lower bounds for the sum of variances of two observables which fully capture the concept of incompatible observables. Furthermore, we show that weak values for two non-commuting projection operators obey a complementarity relation. Specifically, we show that for a pre-selected state if we measure a projector corresponding to an observable $A$ weakly followed by the strong measurement of another observable $B$ (for the post-selection) and, for the same pre-selected state we measure a projector corresponding to an observable $B$ weakly followed by the strong measurement of the observable $A$ (for the post-selection), then the product of these two weak values is always less than one. This shows that even though individually they are complex and can be large, their product is always bounded.

arXiv:1411.7218•Nov 26, 2014•3 citations

Uncertainties in successive measurements

Jacques Distler B, Sònia Paban

When you measure an observable, $A$, in quantum mechanics, the state of the system changes. This, in turn, affects the quantum-mechanical uncertainty in some noncommuting observable, $B$. The standard uncertainty relation puts a lower bound on the uncertainty of $B$ in the initial state. What is relevant for a subsequent measurement of $B$, however, is the uncertainty of $B$ in the postmeasurement state. We re-examine this problem, both in the case where $A$ has a pure point spectrum and in the case where $A$ has a continuous spectrum. In the latter case, the need to include a finite detector resolution, as part of what it means to measure such an observable, has dramatic implications for the result of successive measurements. Ozawa, [Phys. Rev. A 67, 042105 (2003)] proposed an inequality satisfied in the case of successive measurements. Among our results, we show that his inequality is ineffective (can never come close to being saturated). For the cases of interest, we compute a sharper lower bound.

arXiv:1211.4169•Physical Review A•Jun 21, 2013•29 citations

Quantum uncertainty relation using coherence

Xiao Yuan, Ge Bai, Tianyi Peng +1 more

Measurement outcomes of a quantum state can be genuinely random (unpredictable) according to the basic laws of quantum mechanics. The Heisenberg-Robertson uncertainty relation puts constraints on the accuracy of two noncommuting observables. The existing uncertainty relations adopt variance or entropic measures, which are functions of observed outcome distributions, to quantify the uncertainty. According to recent studies of quantum coherence, such uncertainty measures contain both classical (predictable) and quantum (unpredictable) components. In order to extract out the quantum effects, we define quantum uncertainty to be the coherence of the state on the measurement basis. We discover a quantum uncertainty relation using coherence between two measurement noncommuting bases. Furthermore, we analytically derive the quantum uncertainty relation for the qubit case with three widely adopted coherence measures, the relative entropy of coherence, the coherence of formation, and the ${l}_{1}$ norm of coherence.

arXiv:1612.02573•Physical Review A•Sep 11, 2017•79 citations

Influence of simultaneous weak measurements in Heisenberg uncertainty relation

Tao Wei, You-Yang Xu

Abstract In quantum mechanics the simultaneous measurement of non-commuter variables has attracted widespread attention. Here we propose a scheme through simultaneously weak measurement to achieve the approximated weak values of two non-commuting variables. It can be observed that the measurement error is better than the Heisenberg uncertainty relation. Furthermore, the effects of free Hamiltonian and higher-order expansion of the time evolution operator are considered.

Europhysics Letters (EPL)•Mar 1, 2023•0 citations
We study theoretically a weak measurement of an observable AAA and its impact on a subsequent strong measurement of another observable BBB. We assume that both measurements are described by the Gaussian point detectors. It is shown that a decrease of the system-detector interaction during the measurement of AAA results in an increase of the fluctuations of AAA, while the mean value of AAA is kept unaltered. The ratio (\ensuremathΔA)weakmeasurement/(\ensuremathΔA)strongmeasurement{(\mathrm{\ensuremath{\Delta}}A)}_{\mathrm{weak}\phantom{\rule{0.28em}{0ex}}\phantom{\rule{0.16em}{0ex}}\mathrm{measurement}}/{(\mathrm{\ensuremath{\Delta}}A)}_{\mathrm{strong}\phantom{\rule{0.28em}{0ex}}\phantom{\rule{0.16em}{0ex}}\mathrm{measurement}}(\ensuremathΔA)weakmeasurement​/(\ensuremathΔA)strongmeasurement​ turns out to be universal, i.e., independent of the physical system, its quantum state, and the quantity measured. Most of our attention is focused on analyzing outcomes of the subsequent strong measurement of the quantity BBB, namely, on the mean value B‾\overline{B}B and the variance (\ensuremathΔB)2{(\mathrm{\ensuremath{\Delta}}B)}^{2}(\ensuremathΔB)2. We derive explicit analytic corrections for B‾\overline{B}B and (\ensuremathΔB)2{(\mathrm{\ensuremath{\Delta}}B)}^{2}(\ensuremathΔB)2 arising due to the prior weak measurement of AAA. Importantly, the leading-order impact of the weak measurement of AAA on B‾\overline{B}B and (\ensuremathΔB)2{(\mathrm{\ensuremath{\Delta}}B)}^{2}(\ensuremathΔB)2 is encoded in the terms s\ensuremath⟨\ensuremathψ∣[A\ifmmode^\elseˆ\fi,[A\ifmmode^\elseˆ\fi,B\ifmmode^\elseˆ\fik]]∣\ensuremathψ\ensuremath⟩s{\phantom{\rule{0.16em}{0ex}}}_{s}\ensuremath{\langle}\ensuremath{\psi}|[\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{A},[\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{A},{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{B}}^{k}]]{|\ensuremath{\psi}\ensuremath{\rangle}}_{s}s​\ensuremath⟨\ensuremathψ∣[A\ifmmode^\elseˆ\fi​,[A\ifmmode^\elseˆ\fi​,B\ifmmode^\elseˆ\fi​k]]∣\ensuremathψ\ensuremath⟩s​, where k=1,2k=1,2k=1,2 and ∣\ensuremathψ\ensuremath⟩s{|\ensuremath{\psi}\ensuremath{\rangle}}_{s}∣\ensuremathψ\ensuremath⟩s​ is the quantum state vector of our system before the measurements. Our additional contribution consists of deriving Heisenberg-like uncertainty relations for the product (\ensuremathΔA)weakmeasurement(\ensuremathΔB)strongmeasurement{(\mathrm{\ensuremath{\Delta}}A)}_{\mathrm{weak}\phantom{\rule{0.28em}{0ex}}\phantom{\rule{0.16em}{0ex}}\mathrm{measurement}}\phantom{\rule{0.28em}{0ex}}{(\mathrm{\ensuremath{\Delta}}B)}_{\mathrm{strong}\phantom{\rule{0.28em}{0ex}}\phantom{\rule{0.16em}{0ex}}\mathrm{measurement}}(\ensuremathΔA)weakmeasurement​(\ensuremathΔB)strongmeasurement​ and analyzing the case of minimum uncertainty. Finally, our theoretical developments are illustrated numerically via studying the sss states of the hydrogen atom, assuming A\ifmmode^\elseˆ\fi=r\ifmmode^\elseˆ\fi\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{A}=\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{r}A\ifmmode^\elseˆ\fi​=r\ifmmode^\elseˆ\fi (atomic radius) and B\ifmmode^\elseˆ\fi=H\ifmmode^\elseˆ\fi\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{B}=\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}B\ifmmode^\elseˆ\fi​=H\ifmmode^\elseˆ\fi​ (energy).
Time averaging of weak values using the quantum transition path time probability distribution enables us to establish a general uncertainty relation for the weak values of two not necessarily Hermitian operators. This new relation is a weak value analog of the Schr\"odinger strong value uncertainty relation. It leads to the conclusion that it is possible to determine with high accuracy the simultaneous mean weak values of noncommuting operators by judicious choice of the pre- and postselected states even when the postselected state is not an eigenfunction of one of the respective operators. When the time fluctuations of the two weak values are proportional to each other there is no uncertainty limitation on their variances and, in principle, their means can be determined with arbitrary precision even though their corresponding operators do not commute. To exemplify these properties we consider specific weak value uncertainty relations for the time-energy, coordinate-momentum, and coordinate--kinetic-energy pairs. In addition we analyze spin operators and the Stern-Gerlach experiment in weak and strong inhomogeneous magnetic fields. This classic case leads to anomalous spin values when the field is weak. The weak value uncertainty relation implies that anomalous spin values are associated with large variances so that their measurement demands increased signal averaging. These examples establish the importance of considering the time dependence of weak values in scattering experiments.
Abstract We discuss quantum mechanical detection models in the weak limit in the context of conservation laws of physical quantities. In particular, we analyze what kind of system–detector interaction can preserve the global conservation or the related symmetry, and how the final measurement on the detector affects the measured observable of the systems and its presumed conservation. It turns out that the order of noncommuting measurements results in observable differences on the level of third-order correlations functions.
We prove uncertainty relations that quantitatively express the impossibility of jointly sharp preparation of pre- and post-selected quantum states for measuring incompatible observables during the weak measurement. By defining a suitable operator whose average in the pre-selected quantum state gives the weak value, we show that one can have new uncertainty relations for variances of two such operators corresponding to two non-commuting observables. These generalize the recent stronger uncertainty relations that give non-trivial lower bounds for the sum of variances of two observables which fully capture the concept of incompatible observables. Furthermore, we show that weak values for two non-commuting projection operators obey a complementarity relation. Specifically, we show that for a pre-selected state if we measure a projector corresponding to an observable AAA weakly followed by the strong measurement of another observable BBB (for the post-selection) and, for the same pre-selected state we measure a projector corresponding to an observable BBB weakly followed by the strong measurement of the observable AAA (for the post-selection), then the product of these two weak values is always less than one. This shows that even though individually they are complex and can be large, their product is always bounded.
When you measure an observable, AAA, in quantum mechanics, the state of the system changes. This, in turn, affects the quantum-mechanical uncertainty in some noncommuting observable, BBB. The standard uncertainty relation puts a lower bound on the uncertainty of BBB in the initial state. What is relevant for a subsequent measurement of BBB, however, is the uncertainty of BBB in the postmeasurement state. We re-examine this problem, both in the case where AAA has a pure point spectrum and in the case where AAA has a continuous spectrum. In the latter case, the need to include a finite detector resolution, as part of what it means to measure such an observable, has dramatic implications for the result of successive measurements. Ozawa, [Phys. Rev. A 67, 042105 (2003)] proposed an inequality satisfied in the case of successive measurements. Among our results, we show that his inequality is ineffective (can never come close to being saturated). For the cases of interest, we compute a sharper lower bound.
Measurement outcomes of a quantum state can be genuinely random (unpredictable) according to the basic laws of quantum mechanics. The Heisenberg-Robertson uncertainty relation puts constraints on the accuracy of two noncommuting observables. The existing uncertainty relations adopt variance or entropic measures, which are functions of observed outcome distributions, to quantify the uncertainty. According to recent studies of quantum coherence, such uncertainty measures contain both classical (predictable) and quantum (unpredictable) components. In order to extract out the quantum effects, we define quantum uncertainty to be the coherence of the state on the measurement basis. We discover a quantum uncertainty relation using coherence between two measurement noncommuting bases. Furthermore, we analytically derive the quantum uncertainty relation for the qubit case with three widely adopted coherence measures, the relative entropy of coherence, the coherence of formation, and the l1{l}_{1}l1​ norm of coherence.