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

par Hertz, Anaëlle ;Oreshkov, Ognyan ;Cerf, Nicolas
Référence Physical Review A, 100, page (052112)
Publication Publié, 2019-11-19

Article révisé par les pairs

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Résumé :

We define an uncertainty observable acting on several replicas of a continuous-variable state, whose measurement induces phase-space uncertainty relations for a single copy of the state. By exploiting the Schwinger representation of angular momenta in terms of bosonic operators, this observable can be constructed so as to be 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 replicas of) any pure Gaussian state centered at the origin. The non-negativity of its variance translates into the Schrödinger-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 all Gaussian unitaries makes this observable a natural tool to capture the phase-space uncertainty (or the deviation from pure Gaussianity) of continuous-variable bosonic states. In particular, it suggests that the Shannon entropy associated with the measurement of this observable provides a symplectic-invariant entropic measure of uncertainty in position-momentum phase space.