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.
Highlights
Production of non-Gaussian quantum states of light, and all-optical realization of non-Gaussian quantum unitary operations, are critical for most applications of photonic quantum information processing, e.g., universal photonic quantum computation [1], quantum-enhanced receivers for optical communications [2,3], all-optical quantum repeaters for long-distance entanglement distribution [4–6], and quantumenhanced optical sensing [7–12].Gaussian states and Gaussian unitaries, produced by the action of linear and quadratic Hamiltonians on the vacuum state, have efficient and complete mathematical representations [13–15]
Partial Photon number resolving (PNR) is the new trend for non-Gaussian bosonic state engineering because essentially it circumvents the technical difficulties of constructing non-Gaussian optical unitary operations
We recognized that zero displacement restricts the parity of the observed PNR pattern and it restricts the Fock expansion coefficients one should sum up to derive a fidelity upper bound, yielding a hard 1/2 upper bound for target states with nonzero mean-field amplitude such as the |+, |−, and |coherent-cat basis cluster states (CCCS) states
Summary
Production of non-Gaussian quantum states of light, and all-optical realization of non-Gaussian quantum unitary operations, are critical for most applications of photonic quantum information processing, e.g., universal photonic quantum computation [1], quantum-enhanced receivers for optical communications [2,3], all-optical quantum repeaters for long-distance entanglement distribution [4–6], and quantumenhanced optical sensing [7–12]. Since fully programmable linear optical circuits have been realized on-chip [49], scalable generation of arbitrary multimode zero-mean Gaussian states is well within the reach of modern technology This is the reason why we focus in this paper, on evaluating whether arbitrary non-Gaussian states can be prepared just by partial-PNR detection on zero-mean Gaussian states. For the coherent-cat basis cluster states (CCCS), our fidelity upper bound evaluates to 1/2, showing such a state cannot be prepared by partial PNR on a zero-mean Gaussian state.
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