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

TL;DR

This paper proves that certain non-Gaussian states, including multi-mode coherent cat-basis cluster states, cannot be generated by zero-mean Gaussian states combined with partial photon number resolving detection, highlighting fundamental limitations.

Contribution

It establishes a fundamental upper bound on the fidelity of heralded non-Gaussian states from Gaussian states with PNR detection, demonstrating some states are non-producible by this method.

Findings

Fidelity upper bound is 1/2 for multi-mode coherent cat-basis cluster states.

Certain non-Gaussian states cannot be generated from Gaussian states with PNR detection.

The fidelity bound depends only on the target state's photon-number basis representation.

Abstract

Gaussian states and measurements collectively are not powerful-enough resources for quantum computing, as any Gaussian dynamics can be simulated efficiently, classically. However, it is known that any one non-Gaussian resource -- either a state, a unitary operation, or a measurement -- together with Gaussian unitaries, makes for universal quantum resources. Photon number resolving (PNR) detection, a readily-realizable non-Gaussian measurement, has been a popular tool to try and engineer non-Gaussian states for universal quantum processing. In this paper, we consider PNR detection of a subset of the modes of a zero-mean pure multi-mode Gaussian state as a means to herald a target non-Gaussian state on the undetected modes. This is motivated from the ease of scalable preparation of Gaussian states that have zero mean, using squeezed vacuum and passive linear optics. We calculate upper bounds on the fidelity between the actual heralded state and the target state. We find that this fidelity upper bound is $1/2$ when the target state is a multi-mode coherent cat-basis cluster state, a resource sufficient for universal quantum computing. This proves that there exist non-Gaussian states that are not producible by this method. Our fidelity upper bound is a simple expression that depends only on the target state represented in the photon-number basis, which could be applied to other non-Gaussian states of interest.