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Gaussian states and measurements collectively are not powerful-enoughresources for quantum computing, as any Gaussian dynamics can be simulatedefficiently, classically. However, it is known that any one non-Gaussianresource -- either a state, a unitary operation, or a measurement -- togetherwith Gaussian unitaries, makes for universal quantum resources. Photon numberresolving (PNR) detection, a readily-realizable non-Gaussian measurement, hasbeen a popular tool to try and engineer non-Gaussian states for universalquantum processing. In this paper, we consider PNR detection of a subset of themodes of a zero-mean pure multi-mode Gaussian state as a means to herald atarget non-Gaussian state on the undetected modes. This is motivated from theease of scalable preparation of Gaussian states that have zero mean, usingsqueezed vacuum and passive linear optics. We calculate upper bounds on thefidelity between the actual heralded state and the target state. We find thatthis fidelity upper bound is $1/2$ when the target state is a multi-modecoherent cat-basis cluster state, a resource sufficient for universal quantumcomputing. This proves that there exist non-Gaussian states that are notproducible by this method. Our fidelity upper bound is a simple expression thatdepends only on the target state represented in the photon-number basis, whichcould be applied to other non-Gaussian states of interest.

Non-producibility of arbitrary non-Gaussian states using zero-mean Gaussian states and partial photon number resolving detection