Multidimensional entropic uncertainty relation based on a commutator matrix in position and momentum spaces

The uncertainty relation for continuous variables due to Byałinicki-Birula and Mycielski [I. Białynicki-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 Rényi entropies and also prove a covariance-based uncertainty relation which generalizes the Robertson relation.

Figure 1

Schematic of two $n$-modal Gaussian projective measurements applied onto state $|\psi \rangle$, resulting in the $n$ quadratures ${\widehat{y}}_i$ or ${\widehat{z}}_i$. These measurements can be implemented by applying a Gaussian unitary ($U_A$ or $U_B$) onto $|\psi \rangle$ and measuring the $\widehat{x}$ quadratures of the $n$ modes. Our uncertainty relation (32) expresses the complementarity between the ${\widehat{y}}_i$ and ${\widehat{z}}_i$ or, equivalently, between the linear canonical transforms associated with $U_A$ and $U_B$.