The uncertainty relation for continuous variables due to Bya\l{}inicki-Birula and Mycielski [I. Bia\l{}ynicki-Birula and J. Mycielski, Commun. Math. Phys. 44, 129 (1975)] expresses the complementarity between two $n$ -tuples of canonically conjugate variables $({x}_{1},{x}_{2},...,{x}_{n})$ and $({p}_{1},{p}_{2},...,{p}_{n})$ in terms of Shannon differential entropy. Here we consider the generalization to variables that are not canonically conjugate and derive an entropic uncertainty relation expressing the balance between any two $n$ -variable Gaussian projective measurements. The bound on entropies is expressed in terms of the determinant of a matrix of commutators between the measured variables. This uncertainty relation also captures the complementarity between any two incompatible linear canonical transforms, the bound being written in terms of the corresponding symplectic matrices in phase space. Finally, we extend this uncertainty relation to R\'enyi entropies and also prove a covariance-based uncertainty relation which generalizes the Robertson relation.The uncertainty relation for continuous variables due to Bya\l{}inicki-Birula and Mycielski [I. Bia\l{}ynicki-Birula and J. Mycielski, Commun. Math. Phys. 44, 129 (1975)] expresses the complementarity between two $n$-tuples of canonically conjugate variables $({x}_{1},{x}_{2},...,{x}_{n})$ and $({p}_{1},{p}_{2},...,{p}_{n})$ in terms of Shannon differential entropy. Here we consider the generalization to variables that are not canonically conjugate and derive an entropic uncertainty relation expressing the balance between any two $n$-variable Gaussian projective measurements. The bound on entropies is expressed in terms of the determinant of a matrix of commutators between the measured variables. This uncertainty relation also captures the complementarity between any two incompatible linear canonical transforms, the bound being written in terms of the corresponding symplectic matrices in phase space. Finally, we extend this uncertainty relation to R\'enyi entropies and also prove a covariance-based uncertainty relation which generalizes the Robertson relation.

arXiv:1711.04566Physical Review AJan 12, 20183 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

Entropic uncertainty and certainty relations for complementary observables

Jorge Sánchez

Physics Letters AFeb 1, 1993143 citations

Entropic Uncertainty Relations in Quantum Physics

Iwo Białynicki‐Birula, Łukasz Rudnicki

arXiv:1001.4668Springer eBooksJan 1, 2011125 citations

Number-phase uncertainty relations in terms of generalized entropies

Alexey E. Rastegin

Number-phase uncertainty relations are formulated in terms of unified entropies which form a family of two-parametric extensions of the Shannon entropy. For two generalized measurements, unified-entropy uncertainty relations are given in both the state-dependent and state-independent forms. A few examples are discussed as well. Using the Pegg--Barnett formalism and the Riesz theorem, we obtain a nontrivial inequality between norm-like functionals of generated probability distributions in finite dimensions. The principal point is that we take the infinite-dimensional limit right for this inequality. Hence number-phase uncertainty relations with finite phase resolutions are expressed in terms of the unified entropies. Especially important case of multiphoton coherent states is separately considered. We also give some entropic bounds in which the corresponding integrals of probability density functions are involved.Number-phase uncertainty relations are formulated in terms of unified entropies which form a family of two-parametric extensions of the Shannon entropy. For two generalized measurements, unified-entropy uncertainty relations are given in both the state-dependent and state-independent forms. A few examples are discussed as well. Using the Pegg--Barnett formalism and the Riesz theorem, we obtain a nontrivial inequality between norm-like functionals of generated probability distributions in finite dimensions. The principal point is that we take the infinite-dimensional limit right for this inequality. Hence number-phase uncertainty relations with finite phase resolutions are expressed in terms of the unified entropies. Especially important case of multiphoton coherent states is separately considered. We also give some entropic bounds in which the corresponding integrals of probability density functions are involved.

Quantum Information and ComputationSep 1, 201210 citations

n

-tuples of canonically conjugate variables

({x}_{1},{x}_{2},...,{x}_{n})

and

({p}_{1},{p}_{2},...,{p}_{n})

in terms of Shannon differential entropy. Here we consider the generalization to variables that are not canonically conjugate and derive an entropic uncertainty relation expressing the balance between any two

n

-variable Gaussian projective measurements. The bound on entropies is expressed in terms of the determinant of a matrix of commutators between the measured variables. This uncertainty relation also captures the complementarity between any two incompatible linear canonical transforms, the bound being written in terms of the corresponding symplectic matrices in phase space. Finally, we extend this uncertainty relation to R\'enyi entropies and also prove a covariance-based uncertainty relation which generalizes the Robertson relation.