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arXiv:2502.14700 · Physical Review A

Interferometric detection of continuous-variable entanglement using two states

Elena Callus, Martin Gärttner, Tobias Haas

Published Aug 21, 2025• 3 citations
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Abstract

The efficient witnessing and certification of entanglement is necessitated by its ubiquitous use in various aspects of quantum technologies. In the case of continuous-variable bipartite systems, the Shchukin-Vogel hierarchy gives necessary conditions for separability in terms of moments of the mode operators. In this work we derive mode-operator-based witnesses for continuous-variable bipartite entanglement relying on the interference of two states. Specifically, we show how one can access higher moments of the mode operators, crucial for detecting entanglement of non-Gaussian states, using a single beamsplitter with variable phase and photon-number-resolving detectors. We demonstrate that the use of an entangled state paired with a suitable reference state is sufficient to detect entanglement in, e.g., two-mode squeezed vacuum, NOON states, and mixed entangled cat states. We also take into account experimental noise, including photon loss and detection inefficiency, as well as finite measurement statistics.

DOI
10.1103/4wzg-fc7b

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The efficient witnessing and certification of entanglement is necessitated by its ubiquitous use in various aspects of quantum technologies. In the case of continuous-variable bipartite systems, the Shchukin-Vogel hierarchy gives necessary conditions for separability in terms of moments of the mode operators. In this work we derive mode-operator-based witnesses for continuous-variable bipartite entanglement relying on the interference of two states. Specifically, we show how one can access higher moments of the mode operators, crucial for detecting entanglement of non-Gaussian states, using a single beamsplitter with variable phase and photon-number-resolving detectors. We demonstrate that the use of an entangled state paired with a suitable reference state is sufficient to detect entanglement in, e.g., two-mode squeezed vacuum, NOON states, and mixed entangled cat states. We also take into account experimental noise, including photon loss and detection inefficiency, as well as finite measurement statistics.
We present several measurement schemes for accessing separability criteria for continuous-variable bipartite quantum systems. Starting from moments of the bosonic mode operators, criteria suitable to witness entanglement are expressed in terms of multimode spin observables via the Jordan-Schwinger map. These observables are typically defined over a few replicas of the state of interest and can be transformed into simple photon number measurements by passive optical circuits. Our measurement schemes require only a handful of measurements, thereby allowing one to efficiently detect entanglement without the need for costly state tomography as illustrated for a variety of physically relevant states (Gaussian, mixed Schr\"odinger cat, and NOON states). The influence of typical experimental imperfections is shown to be moderate.
We use quantum entanglement witnesses derived from Gaussian operators to study the separable criteria of continuous variable states. We transform the validity of a Gaussian witness to a Bosonic Gaussian channel problem. It follows that the maximal means of two-mode and some four-mode Gaussian operators over product pure states are achieved by vacuum (or coherent states and squeezed states) according to the properties of Bosonic Gaussian channels. Then we have necessary and sufficient criteria of separability not only for Gaussian quantum states, but also for non-Gaussian states prepared by photon adding to or/and subtracting from Gaussian states. The criterion can be further explicitly expressed with covariance matrix of the Gaussian state or covariance matrix of Gaussian kernel of the non-Gaussian state. This opens a way for precise detection of non-Gaussian entanglement.
The dimensionality of entanglement in high-dimensional quantum systems is quantified by the Schmidt number, whose large value signals potential advantages for quantum communication and quantum computation. A commonly used approach to practical Schmidt-number detection in finite-dimensional systems is based on the estimation of fidelities to pure states with large entanglement dimensionality or lower bounds to these fidelities. However, for mode-entangled continuous-variable systems, the corresponding measurements are typically difficult to realize. Here, we introduce and assess fidelity bounds for Schmidt-number detection based on practically more accessible measurements of the first and second statistical moments of arbitrary continuous-variable quantum states. We provide analytical formulas that allow evaluating these bounds and test them for families of Gaussian and non-Gaussian states, including two-mode squeezed thermal states and photon-added displaced two-mode squeezed vacuum states. We compare our bounds to other methods for the certification of high-dimensional entanglement based on the same measurements.
We use quantum entanglement witnesses derived from Gaussian operators to study the separable criteria of continuous variable states. For bipartite system, we transform the validity of a Gaussian witness to a bosonic Gaussian channel problem. It follows that the maximal means of two-mode and some four-mode Gaussian operators over product pure states are achieved by vacuum (or coherent states and squeezed states) according to the properties of bosonic Gaussian channels. We demonstrate necessary and sufficient criteria of separability for two-mode and some four-mode Gaussian quantum states. We also propose multipartite Gaussian witnesses based on numeric evidence. The entanglement and multipartite entanglement detecting powers of the Gaussian witness criteria are illustrated by the mixture of two Gaussian states. We apply the Gaussian witness criterion to a Gaussian Fock state, which is prepared by passing a product Fock state through bosonic Gaussian channel. We show that the necessary criterion of separability for a Gaussian Fock state does not rely on the initial product Fock state, it relies on the channel.
Detection of entanglement in quantum states is one of the most important problems in quantum information processing. However, it is one of the most challenging tasks to find a universal scheme which is also desired to be optimal to detect entanglement for all states of a specific class--as always preferred by experimentalists. Although, the topic is well studied at least in case of lower dimensional compound systems, e.g., two-qubit systems, but in the case of continuous variable systems, this remains as an open problem. Even in the case of two-mode Gaussian states, the problem is not fully solved. In our work, we have tried to address this issue. At first, a limited number of Hermitian operators is given to test the necessary and sufficient criterion on the covariance matrix of separable two-mode Gaussian states. Thereafter, we present an interferometric scheme to test the same separability criterion in which the measurements are being done via Stokes-like operators. In such case, we consider only single-copy measurements on a two-mode Gaussian state at a time and the scheme amounts to the full state tomography. We further analyze the robustness of the proposed detection method against experimentally relevant imperfections and demonstrate that the separability test remains reliable under moderate levels of detection inefficiency. Although this latter approach is a linear optics-based one, nevertheless it is not an economic scheme. Resource-wise a more economical scheme than the full state tomography is obtained if we consider measurements on two copies of the state at a time. However, optimality of the scheme is not yet known.
Efficiently certifying non-Gaussian entanglement in continuous-variable quantum systems is a central challenge for advancing quantum information processing, photonic quantum computing, and metrology. Here, we put forward continuous-variable counterparts to the recently introduced entanglement criteria based on moments of the partially transposed state, together with simple readout schemes that require only a few replicas of the state, passive linear optics, and particle-number measurements. We develop a multicopy method that enables the detection of strongly non-Gaussian entanglement for various relevant state families overlooked by standard approaches, which include the entire class of NOON states. Further, it is robust against realistic experimental constraints (losses, noise, and finite statistics), which we demonstrate by extensive numerical simulations.