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Applied Sciences

Missed Detection of Entanglement in Two-Mode Squeezed States Based on the Inseparability Criterion

Chunxiao Cai, Wenting Diao, Fupan Feng, Wenhai Yang, Xinyu Su, Weigang Zhao, Jinman Ge, Long Ma, Nan Huo, Yanhong Liu, Chongdi Duan

Published Jan 21, 2025• 0 citations

Abstract

The inseparability criterion provides a straightforward and efficient method for identifying and quantifying two-mode Gaussian quantum entanglement, making it a crucial tool in quantum optics experiments. However, it is crucial to recognize that the inseparability criterion serves only as a sufficient condition for entanglement assessment, thereby posing a risk of missed detection during evaluation. This paper investigates the use of the inseparability criterion in assessing two-mode squeezed states, with a particular focus on examining missed entanglement detection due to entanglement asymmetry. The results show that when decoherence symmetrically affects both modes, the inseparability criterion effectively detects entanglement. In contrast, when this symmetry is broken, the criterion may fail to identify entanglement, with the likelihood of missed detection increasing with increasing asymmetry. By comparing these results with the positive partial transpose criterion, which serves as a necessary and sufficient condition, the occurrence of missed detections by the inseparability criterion is confirmed. Our research not only provides valuable insights into the application of the inseparability criterion in quantum information tasks but also deepens the understanding of its operational principles and limitations.

DOI: 10.3390/app15031013

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Entanglement, or quantum inseparability, is a crucial resource in quantum information applications, and therefore the experimental generation of separated yet entangled systems is of paramount importance. Experimental demonstrations of inseparability with light are not uncommon, but such demonstrations in physically well-separated massive systems, such as distinct gases of atoms, are new and present significant challenges and opportunities. Rigorous theoretical criteria are needed for demonstrating that given data are sufficient to confirm entanglement. Such criteria for experimental data have been derived for the case of continuous-variable systems obeying the Heisenberg-Weyl (position-momentum) commutator. To address the question of experimental verification more generally, we develop a sufficiency criterion for arbitrary states of two arbitrary systems. When applied to the recent study by Julsgaard, Kozhekin, and Polzik [Nature 413, 400 (2001)] of spin-state entanglement of two separate, macroscopic samples of atoms, our criterion confirms the presence of spin entanglement.

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.