Serge Deside, Vojtěch Kala and Nicolas J. Cerf of École polytechnique de Bruxelles have developed a new method for certifying entanglement in continuous-variable quantum systems. The researchers introduced separability criteria based on the second-order moments of an operator created by realigning a quantum state’s density operator, finding this level of moment sufficient to detect entanglement across a wide range of optical states.

This approach uses a linear interferometer coupling identical copies of the state, mirroring a technique previously used for partial-transpose moments as detailed in Phys. Rev. Research 8, 033064 (2026), and successfully detects entanglement even in challenging lossy NOON states.

Entanglement Detection with Realignment: Continuous-Variable Systems

This approach offers a potentially simpler method for verifying entanglement in continuous-variable quantum systems, which have historically presented challenges for experimental verification. The work, originating from the Centre for Quantum Information and Communication at École polytechnique de Bruxelles, focuses on developing a dimension-independent technique for entanglement detection.

Researchers achieved this by introducing a family of separability criteria based on singular and weak moments of the operator created by applying the realignment map to the density operator, a mathematical transformation of the quantum state. The team demonstrated that the separability criterion derived from the second-order moments is effective for both Gaussian and non-Gaussian states, including those experiencing loss, a common impediment in real-world quantum systems.

The ability to discern entanglement, a uniquely quantum phenomenon, relies on experimentally accessible criteria, and a newly detailed approach bypasses limitations of existing methods by focusing on the singular and weak moments of a realigned operator. Unlike the original realignment criterion, which proves difficult to measure in continuous-variable (CV) systems, these moment-based criteria offer a pathway to experimentally verify entanglement without demanding full state tomography.

Researchers found that the criterion based on the second-order moments of the realigned operator is already sufficient to witness the entanglement of a wide range of quantum optical states, streamlining the process of entanglement verification. This advancement addresses a key challenge in CV quantum information science: the difficulty of directly measuring the realigned operator. The team’s approach shows how singular and weak moments relate, offering a means to assess separability, the condition of non-entanglement, through readily measurable quantities.

Singular Moments and Weak Moments of the Realigned Operator

The second-moment realignment criterion proves sufficient to detect entanglement in quantum states that elude the weaker, original realignment criterion, including lossy NOON states. Researchers have demonstrated that this second-moment criterion surpasses the performance of previously used weak criteria while maintaining a level of experimental accessibility important for practical application. This advancement offers a pathway to witness entanglement without requiring full state tomography, a demanding process for continuous-variable quantum systems.

Research 8, 033064 (2026)*. Specifically, the first and second weak moments, denoted as w₁ and w₂, can be readily accessed using this interferometric approach combined with homodyne measurements. The purity, r₂, is also measurable through the same multicopy technique, supplemented by photon-number measurement. The second-moment realignment criterion, expressed as ρ separable ⇒ w₂ + |w₂-w₁|² ≤ 1, provides a relaxed condition for determining separability, meaning it can reliably identify entangled states. This criterion’s effectiveness extends to states where the singular weak criterion fails to detect entanglement, particularly when assessing the purity of the state.

Even when direct measurement of r₂ proves challenging, the criterion can be adapted to rely solely on w₁ and w₂, enhancing its versatility. While higher-order criteria can be derived, accessing them experimentally requires additional bounds due to the limitations in directly measuring even-order singular moments with the same ease as the second moment.

Detecting Entanglement in Gaussian Quantum States

The ability to reliably detect entanglement is important for advancing continuous-variable quantum systems, and a new family of separability criteria offers a potentially simpler route than full state tomography. This approach bypasses the need for accessing high-order singular values, a common limitation in previous realignment-based methods. Research 8, 033064 (2026)*. This connection allows for efficient determination of a key parameter in the entanglement criterion without requiring complete state reconstruction.

When this equation holds true, entanglement is definitively certified. The method’s success extends to lossy NOON states, a type of non-Gaussian state often used in quantum metrology, demonstrating its versatility beyond ideal scenarios. The ability to detect entanglement in these states is particularly significant, as they are susceptible to photon loss, a common source of error in real-world quantum experiments.

The researchers illustrate that even with loss, the second-order realignment criterion can still reliably identify entanglement. The work’s focus on moments of the realigned operator provides a practical pathway for entanglement certification, potentially streamlining the process for complex quantum optical states. The researchers’ findings offer a valuable addition to the toolkit for certifying entanglement in continuous-variable quantum information science.

Realignment Criterion Applied to Non-Gaussian States (NOON, Schrödinger Cat)

The second-order realignment criterion surpasses the weak realignment method while demanding a similar level of experimental resources, a key advantage for practical quantum information processing. The method’s accessibility relies on a quantum optical scheme mirroring techniques previously used to measure the moments of partially transposed states, as detailed in Phys. Rev. Research 8, 033064 (2026).

👉 More information
🗞 Entanglement detection from moments of the realigned state: application to continuous-variable quantum systems
✍️ Serge Deside, Vojtěch Kala and Nicolas J. Cerf (Centre for Quantum Information and Communication, École polytechnique de Bruxelles, CP 165, Université libre de Bruxelles, Brussels, Belgium)
🧠 ArXiv: https://arxiv.org/abs/2611.17021v1

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