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Physical Review A

Four types of non-Gaussian squeezed states and their entanglement properties

Zhang Da, Yanwen Zhang, Juan Gao

Published Aug 14, 2025• 2 citations
Publisher

Abstract

We investigate the coupling of the fully degenerate triple-photon state generated by three-mode spontaneous parametric down-conversion with the vacuum at a beam splitter via numerical modeling. By using fully degenerate triple-photon states with distinct pump phases as input states, four different squeezed states are extracted at the output, which can be referred to as zero-phase, $\ensuremath{\pi}/2$-phase, $\ensuremath{\pi}$-phase, and $3\ensuremath{\pi}/2$-phase positive Wigner squeezed states. The non-Gaussianity evolution during the coupling process is quantitatively analyzed via quantum relative entropy, and the entanglement between the two output modes is confirmed from an entropic perspective. Furthermore, the entanglement properties of the zero-phase and $\ensuremath{\pi}$-phase positive Wigner squeezed states at the third- and sixth-order moments are revealed by applying the partial transpose positive criterion based on the higher-order covariance matrix. In particular, it is found that the steerability of the two output modes at the third- and sixth-order moments can be dynamically controlled by adjusting the intensity of the input light and the transmission of the beam splitter. These results enrich the repertoire of squeezed states in quantum optics and offer a feasible approach for the deterministic preparation of non-Gaussian entanglement via coupling fully degenerate triple-photon states with vacuum.

DOI
10.1103/1j44-q3bw

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We investigate the coupling of the fully degenerate triple-photon state generated by three-mode spontaneous parametric down-conversion with the vacuum at a beam splitter via numerical modeling. By using fully degenerate triple-photon states with distinct pump phases as input states, four different squeezed states are extracted at the output, which can be referred to as zero-phase, \ensuremathπ/2\ensuremath{\pi}/2\ensuremathπ/2-phase, \ensuremathπ\ensuremath{\pi}\ensuremathπ-phase, and 3\ensuremathπ/23\ensuremath{\pi}/23\ensuremathπ/2-phase positive Wigner squeezed states. The non-Gaussianity evolution during the coupling process is quantitatively analyzed via quantum relative entropy, and the entanglement between the two output modes is confirmed from an entropic perspective. Furthermore, the entanglement properties of the zero-phase and \ensuremathπ\ensuremath{\pi}\ensuremathπ-phase positive Wigner squeezed states at the third- and sixth-order moments are revealed by applying the partial transpose positive criterion based on the higher-order covariance matrix. In particular, it is found that the steerability of the two output modes at the third- and sixth-order moments can be dynamically controlled by adjusting the intensity of the input light and the transmission of the beam splitter. These results enrich the repertoire of squeezed states in quantum optics and offer a feasible approach for the deterministic preparation of non-Gaussian entanglement via coupling fully degenerate triple-photon states with vacuum.
We theoretically demonstrate through numerical methods that the triple-photon state generated by three-photon spontaneous parametric down-conversion is a non-Gaussian Greenberger-Horne-Zeilinger state, showing super-Gaussian statistics. Interestingly, the degree of entanglement between the modes of the triple-photon state is stronger than that corresponding to the two-mode squeezed vacuum state produced by a quadratic Hamiltonian with the same parameters. Furthermore, we propose a model to prepare two-mode sub-Gaussian entangled states with a tunable negative Wigner function based on quadrature projection measurements. We find that these Gaussian projection measurements with outcomes $X\ensuremath{\ge}1$ not only improve the entanglement of the residual two modes but also introduce a Gaussian component, resulting in the coexistence of Gaussian and non-Gaussian entanglement.
Triple-photon states generated by three-mode spontaneous parametric down-conversion are the paradigm of unconditional non-Gaussian states, essential assets for quantum advantage. How to fully characterize their non-Gaussian entanglement remains however elusive. We propose here a set of sufficient and necessary conditions for separability of the broad family of spontaneously generated three-mode non-Gaussian states. We further derive state-of-the-art conditions for genuine tripartite non-Gaussian entanglement, the strongest class of entanglement. We apply our criteria to triple-photon states revealing that they are fully inseparable and genuinely entangled in moments of order 3n. Our results establish a systematic framework for characterizing the entanglement of triple-photon states and thus fostering their application in quantum information protocols.
Abstract In this work, we present a theoretical scheme for generating a new kind of tunable non-Gaussian entangled state, by means of beam splitters and multi-photon conditional measurement with two-mode squeezed vacuum state (TMSVS) inputs. The output state is proven to be two-variable Hermite polynomial excited squeezed vacuum states, whose success probability of detection relates to the Jacobi polynomials. We then mainly study the entanglement properties of the output state for the symmetric case quantified by the von Neumann entropy and Einstein–Podolsky–Rosen correlation. It is shown that the entropy of entanglement is valid for verifying the improvement of entanglement within a certain range of squeezing parameter and transmissivity, and the resulting state case may possess a stronger correlation than the initial TMSVS by increasing the transmissivity or photon number of detection. In addition, the violation of the Clauser–Horne–Shimony–Holt (CHSH) inequality for the output state is also evaluated in terms of Wigner representation in phase space. It is found that the negative region of Wigner function distribution becomes more visible with the increase of the photon number of detection, and the degree of the violation of the CHSH inequality significantly increases as the photon number of detection for the symmetrical case increases. The resulting state with highly nonclassical characteristics will provide a useful application in the fields of quantum information processing.
Triple-photon states generated by three-mode spontaneous parametric down-conversion are the paradigm of unconditional non-Gaussian states, essential assets for quantum advantage. How to fully characterize their non-Gaussian entanglement remains however elusive. We propose here a hierarchy of sufficient and necessary conditions for separability of the broad family of spontaneously-generated three-mode non-Gaussian states. We further derive state-of-the-art conditions for genuine tripartite non-Gaussian entanglement, the strongest class of entanglement. We apply our criteria to triple-photon states revealing that they are fully inseparable and genuinely entangled in moments of order 3n. Our results establish a systematic framework for characterizing the entanglement of triple-photon states and thus fostering their application in quantum information protocols.
Very recently, strongly non-Gaussian states have been observed via a direct three-mode spontaneous parametric down-conversion in a superconducting cavity [Phys. Rev. X 10, 011011 (2020)]. The created multiphoton non-Gaussian correlations are attractive and useful for various quantum information tasks. However, how to detect and classify multipartite non-Gaussian entanglement has not yet been completely understood. Here, we present an experimentally practical method to characterize continuous-variable multipartite non-Gaussian entanglement, by introducing a class of nonlinear squeezing parameters involving accessible higher-order moments of phase-space quadratures. As these parameters can depend on arbitrary operators, we consider their analytical optimization over a set of practical measurements, in order to detect different classes of multipartite non-Gaussian entanglement ranging from fully separable to fully inseparable. We demonstrate that the nonlinear squeezing parameters act as an excellent approximation to the quantum Fisher information within accessible third-order moments. The level of the nonlinear squeezing quantifies the metrological advantage provided by those entangled states. Moreover, by analyzing the above-mentioned experiment, we show that our method can be readily used to confirm fully inseparable tripartite non-Gaussian entangled states by performing a limited number of measurements without requiring full knowledge of the quantum state.
Quantum universal invariants of a Gaussian state's covariance matrix, which can be derived from intensity correlation moments, have been adopted to characterize the entanglement properties of Gaussian states via the positive partial transpose criterion, also known as the Peres-Horodecki separability criterion. Such intensity correlation moments enable the extraction of information about the covariance matrix without the need for a coherent local oscillator. Here, we experimentally detect the entanglement properties of multimode Gaussian states using high-order\,(up to sixth-order) intensity correlation moments. These multimode Gaussian states are prepared via spontaneous and cascaded parametric down-conversion pumped by a high-peak-energy pulsed laser. Their intensity correlation moments are measured using a pseudo-photon-number-resolving detector constructed through spatial multiplexing of 32 threshold superconducting nanowire single photon detectors. This method is successfully demonstrated for two-mode and three-mode Gaussian states and can be extended to NNN-mode Gaussian states with N>3N>3N>3.