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We generalise a non-classicality test described by Kot et al. [Phys. Rev.Lett. 108, 233601 (2010)], which can be used to rule out any classicaldescription of a physical system. The test is based on measurements ofquadrature operators and works by proving a contradiction with the classicaldescription in terms of a probability distribution in phase space. As opposedto the previous work, we generalise the test to include states withoutrotational symmetry in phase space. Furthermore, we compare the performance ofthe non-classicality test with classical tomography methods based on theinverse Radon transform, which can also be used to establish the quantum natureof a physical system. In particular, we consider a non-classicality test basedon the so-called filtered back-projection formula. We show that the generalnon-classicality test is conceptually simpler, requires less assumptions on thesystem and is statistically more reliable than the tests based on the filteredback-projection formula. As a specific example, we derive the optimal test fora quadrature squeezed single photon state and show that the efficiency of thetest does not change with the degree of squeezing.

Elementary test for non-classicality based on measurements of position and momentum