Quantifying non-Gaussianity of a quantum state by the negative entropy of quadrature distributions

Estimating Non-Gaussianity of a Quantum State by Measuring Orthogonal Quadratures

“…In this study, we show that the sum of the negentropies for two quadrature distributions with the maximum and minimum variances provides a lower bound for the QRE-based non-Gaussianity measure of a single-mode quantum state. We demonstrate that our lower bound can be greater than the maximum negentropy of quadrature distributions, i.e., the lower bound proposed in [ 46 ]. We also extend our method to estimate the non-Gaussianity of multimode quantum states with or without the help of a Gaussian unitary operation.…”

“…Probing non-Gaussianity in CV quantum information, it is essential to obtain a faithful quantifier for non-Gaussianity. Thus, non-Gaussianity measures for quantum states have been proposed by employing the quantum Hilbert–Schmidt distance [ 41 ], quantum relative entropy (QRE) [ 42 ], Wehrl entropy [ 43 ], quantum Rényi relative entropy [ 44 ], Wigner–Yanase skew information [ 45 ], and Kullback–Leibler divergence (KLD) [ 46 ]. Furthermore, quantum non-Gaussianity, i.e., a stronger form of non-Gaussianity, has been introduced to distinguish genuinely non-Gaussian states from classical mixtures of Gaussian states [ 47 ].…”

“…Although the non-Gaussianity measures are helpful in characterizing non-Gaussian quantum resources, resource-intensive quantum state tomography [ 68 ] is generally required to obtain the exact value of the measures in general. For the case of QRE-based non-Gaussianity measure, observable lower bounds have been developed to address this issue by using the information of the covariance matrix in conjunction with the photon number distribution [ 69 ] and the negentropy of a quadrature distribution [ 46 ]. The former, i.e., the lower bound in [ 69 ], works better than the latter, i.e., the lower bound in [ 46 ], especially for quantum states with rotational symmetry in phase space but demands two measurement setups, i.e., homodyne detection and photon-number-resolving detection, in general.…”

“…For the case of QRE-based non-Gaussianity measure, observable lower bounds have been developed to address this issue by using the information of the covariance matrix in conjunction with the photon number distribution [ 69 ] and the negentropy of a quadrature distribution [ 46 ]. The former, i.e., the lower bound in [ 69 ], works better than the latter, i.e., the lower bound in [ 46 ], especially for quantum states with rotational symmetry in phase space but demands two measurement setups, i.e., homodyne detection and photon-number-resolving detection, in general. If there is a priori information that the quantum state has rotational symmetry in phase space, the lower bound in [ 69 ] can be deduced from a quadrature distribution.…”

“…By contrast, the latter requires only homodyne detection always. In addition, the latter can be used to detect non-Gaussian entanglement in conjunction with partial transposition [ 46 ], which may be impossible for the former. Here, we investigate whether an improved lower bound can be obtained by exploiting the negentropy of more than one quadrature distribution, which may open the way to detect non-Gaussian entanglement untestable by the method proposed in [ 46 ].…”