We study the quantum-mechanical uncertainty relation originating from the successive measurement of two observables $\hat{A}$ and $\hat{B}$ , with eigenvalues $a_n$ and $b_m$ , respectively, performed on the same system. We use an extension of the von Neumann model of measurement, in which two probes interact with the same system proper at two successive times, so we can exhibit how the disturbing effect of the first interaction affects the second measurement. Detecting the statistical properties of the second {\em probe} variable $Q_2$ conditioned on the first {\em probe} measurement yielding $Q_1$ we obtain information on the statistical distribution of the {\em system} variable $b_m$ conditioned on having found the system variable $a_n$ in the interval $δa$ around $a^{(n)}$ . The width of this statistical distribution as function of $δa$ constitutes an {\em uncertainty relation}. We find a general connection of this uncertainty relation with the commutator of the two observables that have been measured successively. We illustrate this relation for the successive measurement of position and momentum in the discrete and in the continuous cases and, within a model, for the successive measurement of a more general class of observables.We study the quantum-mechanical uncertainty relation originating from the successive measurement of two observables $\hat{A}$ and $\hat{B}$, with eigenvalues $a_n$ and $b_m$, respectively, performed on the same system. We use an extension of the von Neumann model of measurement, in which two probes interact with the same system proper at two successive times, so we can exhibit how the disturbing effect of the first interaction affects the second measurement. Detecting the statistical properties of the second {\em probe} variable $Q_2$ conditioned on the first {\em probe} measurement yielding $Q_1$ we obtain information on the statistical distribution of the {\em system} variable $b_m$ conditioned on having found the system variable $a_n$ in the interval $δa$ around $a^{(n)}$. The width of this statistical distribution as function of $δa$ constitutes an {\em uncertainty relation}. We find a general connection of this uncertainty relation with the commutator of the two observables that have been measured successively. We illustrate this relation for the successive measurement of position and momentum in the discrete and in the continuous cases and, within a model, for the successive measurement of a more general class of observables.

arXiv:2006.00347May 30, 20201 citation

Uncertainty relation and the measurement error – perturbation relation

Yurii I. Vorontsov

Uspekhi Fizicheskih NaukJan 1, 20052 citations

Entropic uncertainty relation for successive measurements

Kyunghyun Baek, Wonmin Son

In quantum mechanics, the uncertainty is fundamental concept, so its underlying meaning has been argued intensively due to its weird properties. On this account, there has been various fomulations expressing the uncertainty principle in dierent ways. In this paper, the new entropic uncertainty relations will be suggested showing dierent aspects of uncertainty as we consider successive measurements. Already, Deutsch and Unk suggested entropic uncertainty relations, so our work will be to modify it as focusing on another aspect (error-disurbance) of uncertainry. The uncertainty relations Heisenberg is the rst man who suggested the uncertainty principle. In 1927 [1], he proposed the uncertainty relation,In quantum mechanics, the uncertainty is fundamental concept, so its underlying meaning has been argued intensively due to its weird properties. On this account, there has been various fomulations expressing the uncertainty principle in dierent ways. In this paper, the new entropic uncertainty relations will be suggested showing dierent aspects of uncertainty as we consider successive measurements. Already, Deutsch and Unk suggested entropic uncertainty relations, so our work will be to modify it as focusing on another aspect (error-disurbance) of uncertainry. The uncertainty relations Heisenberg is the rst man who suggested the uncertainty principle. In 1927 [1], he proposed the uncertainty relation,

Jan 1, 20120 citations

Uncertainty relations in simultaneous measurements for arbitrary observables

Shiro Ishikawa

Reports on Mathematical PhysicsJun 1, 199165 citations

Entropic uncertainty relations for successive measurements of canonically conjugate observables

Alexey E. Rastegin

Uncertainties in successive measurements of general canonically conjugate variables are examined. Such operators are approached within a limiting procedure of the Pegg–Barnett type. Dealing with unbounded observables, we should take into account a finiteness of detector resolution. An appropriate reformulation of two scenarios of successive measurements is proposed and motivated. Uncertainties are characterized by means of generalized entropies of both the Rényi and Tsallis types. The Rényi and Tsallis formulations of uncertainty relations are obtained for both the scenarios of successive measurements of canonically conjugate operators. Entropic uncertainty relations for the cases of position and momentum are separately discussed. imageUncertainties in successive measurements of general canonically conjugate variables are examined. Such operators are approached within a limiting procedure of the Pegg–Barnett type. Dealing with unbounded observables, we should take into account a finiteness of detector resolution. An appropriate reformulation of two scenarios of successive measurements is proposed and motivated. Uncertainties are characterized by means of generalized entropies of both the Rényi and Tsallis types. The Rényi and Tsallis formulations of uncertainty relations are obtained for both the scenarios of successive measurements of canonically conjugate operators. Entropic uncertainty relations for the cases of position and momentum are separately discussed. image

arXiv:1604.08682Annalen der PhysikOct 17, 201626 citations

We study the quantum-mechanical uncertainty relation originating from the successive measurement of two observables

\hat{A}

and

\hat{B}

, with eigenvalues

a_n

and

b_m

, respectively, performed on the same system. We use an extension of the von Neumann model of measurement, in which two probes interact with the same system proper at two successive times, so we can exhibit how the disturbing effect of the first interaction affects the second measurement. Detecting the statistical properties of the second {\em probe} variable

Q_2

conditioned on the first {\em probe} measurement yielding

Q_1

we obtain information on the statistical distribution of the {\em system} variable

b_m

conditioned on having found the system variable

a_n

in the interval

δa

around

a^{(n)}

. The width of this statistical distribution as function of

δa

constitutes an {\em uncertainty relation}. We find a general connection of this uncertainty relation with the commutator of the two observables that have been measured successively. We illustrate this relation for the successive measurement of position and momentum in the discrete and in the continuous cases and, within a model, for the successive measurement of a more general class of observables.