Multi-copy uncertainty observable inducing a symplectic-invariant uncertainty relation in position and momentum phase space

Abstract: We define an uncertainty observable, acting on several replicas of a continuous-variable bosonic state, whose trivial uncertainty lower bound induces nontrivial phase-space uncertainty relations for a single copy of the state. By exploiting the Schwinger representation of angular momenta in terms of bosonic operators, we construct such an observable that is invariant under symplectic transformations (rotation and squeezing in phase space). We first design a two-copy uncertainty observable, which is a discrete-spectrum operator vanishing with certainty if and only if it is applied on (two copies of) any pure Gaussian state centered at the origin. The non-negativity of its variance translates into the Schr\"odinger-Robertson uncertainty relation. We then extend our construction to a three-copy uncertainty observable, which exhibits additional invariance under displacements (translations in phase space) so that it vanishes on every pure Gaussian state. The resulting invariance under Gaussian unitaries makes this observable a natural tool to measure the phase-space uncertainty -- or the deviation from pure Gaussianity -- of continuous-variable bosonic states. In particular, it suggests that the Shannon entropy of this observable provides a symplectic-invariant entropic measure of uncertainty in position and momentum phase space.