Kavli Affiliate: David Gross
| First 5 Authors: Filippo Girardi, Filippo Girardi, , ,
| Summary:
Gaussian states are widely regarded as one of the most relevant classes of
continuous-variable (CV) quantum states, as they naturally arise in physical
systems and play a key role in quantum technologies. This motivates a
fundamental question: given copies of an unknown CV state, how can we
efficiently test whether it is Gaussian? We address this problem from the
perspective of representation theory and quantum learning theory,
characterizing the sample complexity of Gaussianity testing as a function of
the number of modes. For pure states, we prove that just a constant number of
copies is sufficient to decide whether the state is exactly Gaussian. We then
extend this to the tolerant setting, showing that a polynomial number of copies
suffices to distinguish states that are close to Gaussian from those that are
far. In contrast, we establish that testing Gaussianity of general mixed states
necessarily requires exponentially many copies, thereby identifying a
fundamental limitation in testing CV systems. Our approach relies on
rotation-invariant symmetries of Gaussian states together with the recently
introduced toolbox of CV trace-distance bounds.
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