“…Continuous variable systems, which are quantum systems with infinite dimensional Hilbert spaces, have been an important platform for quantum cryptography, quantum information and quantum computation [Edwards & Belavkin, 2005, van Handel, Stockton, & Mabuchi, 2005, James, Nurdin, & Petersen, 2008, Wiseman & Milburn, 2010, Nurdin, Petersen, & James, 2012, Pan, Zhang, & James, 2016, Ma, Woolley, Petersen, & Yamamoto, 2018, Gao, Zhang, & Petersen, 2020, Gao, Dong, Petersen, & Ding, 2020,Ghalaii, Ottaviani, Kumar, Pirandola, & Razavi, 2021, Zhang & Pan, 2020, Gao, Dong, Petersen, & Ding, 2021, Guo, Peng, Liao, & Wang, 2021. Gaussian states, which include a wide and important class of quantum states such as vacuum states, squeezed vacuum, squeezed coherent states, quasi-free states and ground states of some free Hamiltonians, are the basis for various continuous variable quantum information processing [De ⋆ The material in this paper was not presented at any conference.…”