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Optimized detection of unfaithful high-dimensional entanglement

Xiao‐Min Hu, Wen-Bo Xing, Yu Jing Guo, Mirjam Weilenmann, Edgar A. Aguilar, Xiaoqin Gao, Bi‐Heng Liu, Yun‐Feng Huang, Chuan‐Feng Li, Guang‐Can Guo, Zizhu Wang, Miguel Navascués

Published Nov 4, 2020• 1 citation
OpenAlex

Abstract

Entanglement detection is one of the most conventional tasks in quantum information processing. While most experimental demonstrations of high-dimensional entanglement rely on fidelity-based witnesses, these are powerless to detect entanglement within a large class of entangled quantum states, the so-called unfaithful states. In this paper, we introduce a highly flexible automated method to construct optimal tests for entanglement detection given a bipartite target state of arbitrary dimension, faithful or unfaithful, and a set of local measurement operators. By restricting the number or complexity of the considered measurement settings, our method outputs the most convenient protocol to test the entanglement of the target state in the lab. Using the $3$-setting protocols generated by our method, we experimentally certify $2$- and $3$-dimensional entanglement in $4$-dimensional unfaithful states.

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Entanglement detection is one of the most conventional tasks in quantum information processing. While most experimental demonstrations of high-dimensional entanglement rely on fidelity-based witnesses, these are powerless to detect entanglement within a large class of entangled quantum states, the so-called unfaithful states. In this paper, we introduce a highly flexible automated method to construct optimal tests for entanglement detection given a bipartite target state of arbitrary dimension, faithful or unfaithful, and a set of local measurement operators. By restricting the number or complexity of the considered measurement settings, our method outputs the most convenient protocol to test the entanglement of the target state in the lab. Using the 333-setting protocols generated by our method, we experimentally certify 222- and 333-dimensional entanglement in 444-dimensional unfaithful states.
Entanglement detection is one of the most conventional tasks in quantum information processing. While most experimental demonstrations of high-dimensional entanglement rely on fidelity-based witnesses, these are powerless to detect entanglement within a large class of entangled quantum states, the so-called unfaithful states. In this Letter, we introduce a highly flexible automated method to construct optimal tests for entanglement detection given a bipartite target state of arbitrary dimension, faithful or unfaithful, and a set of local measurement operators. By restricting the number or complexity of the considered measurement settings, our method outputs the most convenient protocol which can be implemented using a wide range of experimental techniques such as photons, superconducting qudits, cold atoms, or trapped ions. With an experimental quantum optics setup that can prepare and measure arbitrary high-dimensional mixed states, we implement some three-setting protocols generated by our method. These protocols allow us to experimentally certify two- and three-unfaithful entanglement in four-dimensional photonic states, some of which contain well above 50% of noise.
One of the most widespread methods to determine if a quantum state is entangled, or to quantify its entanglement dimensionality, is by measuring its fidelity with respect to a pure state. In this Letter, we find a large class of states whose entanglement cannot be detected in this manner; we call them unfaithful. We find that unfaithful states are ubiquitous in information theory. For small dimensions, we check numerically that most bipartite states are both entangled and unfaithful. Similarly, numerical searches in low dimensions show that most pure entangled states remain entangled but become unfaithful when a certain amount of white noise is added. We also find that faithfulness can be self-activated, i.e., there exist instances of unfaithful states whose tensor powers are faithful. To explore how the fidelity approach limits the quantification of entanglement dimensionality, we generalize the notion of an unfaithful state to that of a D unfaithful state, one that cannot be certified as D-dimensionally entangled by measuring its fidelity with respect to a pure state. For describing such states, we additionally introduce a hierarchy of semidefinite programming relaxations that fully characterizes the set of states of Schmidt rank at most D.
Constructing optimal entanglement witnesses for arbitrary quantum systems remains a formidable challenge, hindered by the structural complexity of high-dimensional quantum state spaces. In this work, we overcome this barrier by introducing a unified framework for constructing geometrically optimal witnesses capable of identifying arbitrary entanglement structures. By leveraging the geometric properties of a target quantum state relative to a specific entanglement structure in Hilbert-Schmidt space, these witnesses are constructed in an efficient and effective manner. Among all nontrivial witnesses, they provide maximal distinguishability between the target state and the set of states without the desired entanglement structure. As a result, our approach enables the universal detection of bipartite, multipartite, and high-dimensional entanglement. To demonstrate its experimental feasibility, we apply the framework to a four-qubit nuclear spin quantum system. Compared to existing methods, our approach offers enhanced noise robustness, broader detection universality, and improved construction efficiency, providing a powerful and versatile tool for advancing the characterization and understanding of entanglement in quantum systems.
Entanglement witness is an effective method to detect entanglement in unknown states without doing full tomography. One of the most widespread schemes of witnessing entanglement is measuring its fidelity with respect to a pure entangled state. Recently, a large class of states whose entanglement can not be detected with the fidelity witness has been discovered in Phys.Rev.Lett \textbf{124},200502(2020). They are called unfaithful states. In this paper we propose a new way to detect entanglement by calculating the lower bound of entanglement using measurement results. Numerical simulation shows our method can detect entanglement in unfaithful states with a small number of measurements. Moreover, we generalize our scheme to multipartite states and show that it can tolerate higher noise than previous entanglement witness operators with same number of measurement settings.
Abstract Certifying entanglement for unknown quantum states experimentally is a fundamental problem in quantum computing and quantum physics. Because of being easy to implement, a most popular approach for this problem in modern quantum experiments is detecting target quantum states with fidelity-based entanglement witnesses. Specifically, if the fidelity between a target state and an entangled pure state exceeds a certain value, the target state can be guaranteed to be entangled. Recently, however, it has been realized that there exist so-called unfaithful quantum states, which can be entangled, but their entanglement cannot be certified by any fidelity-based entanglement witnesses. In this paper, by specific examples, we show that if one makes a slight modification to fidelity-based entanglement witnesses by combining multiple fidelities together, it is still possible to certify entanglement for unfaithful quantum states with this popular technique. Particularly, we will analyze the mathematical structure of the modified entanglement witnesses, and propose an algorithm that can search for the optimal designs for them.
High-dimensional entanglement (HDE) is a valuable resource in quantum information processing, and efficient certification of HDE is crucial to many applications. In this work, we propose a simple and general framework for certifying HDE in general bipartite pure states under restricted operations, such as local operations and classical communication (LOCC). On this basis we show that HDE in general bipartite pure states can be certified efficiently. Moreover, the sample cost for certifying a given degree of HDE even decreases monotonically with the local dimensions. In addition, for a general two-qubit pure state, we construct an optimal entanglement certification strategy based on separable operations, which can be realized by LOCC when the target state has sufficiently high entanglement. The core concept of our framework is versatile and can be extended to certify a wide range of critical resources under restricted operations.