  Consider two bosonic modes which are prepared in one of two possible Gaussian
states with the same local energy: either a tensor-product thermal state (with
zero correlations) or a separable Gaussian state with maximal correlations
(with both classical and quantum correlations, the latter being quantified by
quantum discord). For the discrimination of these states, we compare the
optimal joint coherent measurement with the best local measurement based on
single-mode Gaussian detections. We show how the coherent measurement always
strictly outperforms the local detection strategy for both single- and
multi-copy discrimination. This means that using local Gaussian measurements
(assisted by classical communication) is strictly suboptimal in detecting
discord. A better performance may only be achieved by either using non Gaussian
measurements (non linear optics) or coherent non-local measurements.
