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arXiv:2506.05084 · Physical Review Research

Efficient characterization of N -beam Gaussian fields through photon-number measurements: Quantum universal invariants

Nazarii Sudak, Artur Barasiński, Jan Peřina, Antonín Černoch

Published May 27, 2025• 1 citation
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Abstract

Quantum universal invariants of general N -beam Gaussian fields are investigated from the point of view of fields' intensity moments. A method that uniquely links these invariants, including the global and marginal fields' purities, to intensity moments is suggested. Determination of these invariants identifies the Gaussian states including their quantum correlations. In particular, the Peres-Horodecki separability criterion is reformulated in terms of quantum universal invariants, and consequently in terms of experimental intensity moments, offering a practical tool for determining the entanglement or separability of these states. The approach is experimentally demonstrated by determining the invariants of noisy symmetric three-beam Gaussian states using photon-number resolved measurements. Furthermore, their entanglement properties are analyzed and characterized.

DOI
10.1103/5hnr-6jbf

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Quantum universal invariants of general N -beam Gaussian fields are investigated from the point of view of fields' intensity moments. A method that uniquely links these invariants, including the global and marginal fields' purities, to intensity moments is suggested. Determination of these invariants identifies the Gaussian states including their quantum correlations. In particular, the Peres-Horodecki separability criterion is reformulated in terms of quantum universal invariants, and consequently in terms of experimental intensity moments, offering a practical tool for determining the entanglement or separability of these states. The approach is experimentally demonstrated by determining the invariants of noisy symmetric three-beam Gaussian states using photon-number resolved measurements. Furthermore, their entanglement properties are analyzed and characterized.
Quantum correlations of symmetric three-beam Gaussian states are analyzed using their quantum universal invariants. These invariants, one-, two-, and three-beam purities, are expressed in terms of the beams’ intensity moments up to sixth order. The three-beam symmetric Gaussian states with varying amounts of the noise are experimentally generated using entangled photon pairs from down-conversion, their invariants are determined, and their quantum correlations are quantified. The coexistence of bipartite and tripartite entanglement and genuine tripartite entanglement is observed in these states that resemble the noisy GHZ/W states.
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.
Identification, and subsequent quantification of quantum correlations, is critical for understanding, controlling, and engineering quantum devices and processes. We derive and implement a general method to quantify various forms of quantum correlations using solely the experimental intensity moments up to the fourth order. This is possible as these moments allow for an exact determination of the global and marginal impurities of two-beam Gaussian fields. This leads to the determination of steering, tight lower and upper bounds for the negativity, and the Kullback-Leibler divergence used as a quantifier of state nonseparability. The principal squeezing variances are determined as well using the intensity moments. The approach is demonstrated on the experimental twin beams with increasing intensity and the squeezed super-Gaussian beams composed of photon pairs. Our method is readily applicable to multibeam Gaussian fields to characterize their quantum correlations.
We present an experimental method for complete identification of the nonclassicality of Gaussian states in the whole phase space. Our method relies on nonclassicality witnesses written in terms of measured integrated intensity moments up to the third order, provided that appropriate local coherent displacements are applied to the state under consideration. The introduced approach, thus, only requires linear detectors for measuring intensities of optical fields, that is, very convenient and powerful from the experimental point of view. Additionally, we demonstrate that the proposed technique not only allows to completely identify the nonclassicality of the Gaussian states, but also to quantify it.
We introduce a new quantity for describing nonclassicality of an arbitrary optical two-mode Gaussian state which remains invariant under any global photon-number preserving unitary transformation of the covariance matrix of the state. The invariant naturally splits into an entanglement monotone and local-nonclassicality quantifiers applied to the reduced states. This shows how entanglement can be converted into local squeezing and vice versa. Twin beams and their transformations at a beam splitter are analyzed as an example providing squeezed light. An extension of this approach to pure three-mode Gaussian states is given.