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Acta Physica Sinica

Experimentally characterizing multiparticle entanglement based on measuring quantum Fisher information

Ran Liu, Ze Wu, Yuchen Li, Yuquan Chen, Xinhua Peng

Published Jan 1, 2023• 0 citations
Publisher

Abstract

Quantum Fisher information plays a vital role in the field of quantum metrology and quantum information, because it not only quantifies the ultimate precision bound of parameter estimation but also provides criteria for entanglement detection. Nevertheless, experimentally extracting quantum Fisher information is intractable. Quantum state tomography is a typical approach to obtaining the complete information about a quantum system and extract quantum Fisher information. However it becomes infeasible for large-scale quantum systems owing to the exponentially growing complexity. In this paper, we present a general relationship between quantum Fisher information and the overlap of quantum states. Specifically, we show that for pure states, the quantum Fisher information can be exactly extracted from the overlap, whereas for mixed states, only the lower bound can be obtained. We also develop a protocol for measuring the overlap of quantum states, which only requires one additional auxiliary qubit and a single measurement for pure state. Our protocol is more efficient and scalable than previous approaches because it requires less time and fewer measurements. We use this protocol to characterize the multiparticle entanglement in a three-body interaction system undergoing adiabatic quantum phase transition, and experimentally demonstrate its feasibility for the first time in a nuclear magnetic resonance quantum system. We conduct our experiment on a 4-qubit nuclear magnetic resonance quantum simulator, three of which are used to simulate the quantum phase transition in a three-body interaction system, and the remaining one is used as the auxiliary qubit to detect the overlap of the quantum state. We use gradient ascent pulse engineering pulses to implement the process of evolution. By measuring the auxiliary qubit, the experimental results of quantum Fisher information are obtained and match well with the theoretical predictions, thus successfully characterizing the multiparticle entanglement in a practical quantum system. We further confirm our results by performing quantum state tomography on some quantum states in the adiabatic process. The experimentally reconstructed quantum states are close to the corresponding instantaneous ground states.

DOI
10.7498/aps.72.20230356

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Quantum Fisher information plays a vital role in the field of quantum metrology and quantum information, because it not only quantifies the ultimate precision bound of parameter estimation but also provides criteria for entanglement detection. Nevertheless, experimentally extracting quantum Fisher information is intractable. Quantum state tomography is a typical approach to obtaining the complete information about a quantum system and extract quantum Fisher information. However it becomes infeasible for large-scale quantum systems owing to the exponentially growing complexity. In this paper, we present a general relationship between quantum Fisher information and the overlap of quantum states. Specifically, we show that for pure states, the quantum Fisher information can be exactly extracted from the overlap, whereas for mixed states, only the lower bound can be obtained. We also develop a protocol for measuring the overlap of quantum states, which only requires one additional auxiliary qubit and a single measurement for pure state. Our protocol is more efficient and scalable than previous approaches because it requires less time and fewer measurements. We use this protocol to characterize the multiparticle entanglement in a three-body interaction system undergoing adiabatic quantum phase transition, and experimentally demonstrate its feasibility for the first time in a nuclear magnetic resonance quantum system. We conduct our experiment on a 4-qubit nuclear magnetic resonance quantum simulator, three of which are used to simulate the quantum phase transition in a three-body interaction system, and the remaining one is used as the auxiliary qubit to detect the overlap of the quantum state. We use gradient ascent pulse engineering pulses to implement the process of evolution. By measuring the auxiliary qubit, the experimental results of quantum Fisher information are obtained and match well with the theoretical predictions, thus successfully characterizing the multiparticle entanglement in a practical quantum system. We further confirm our results by performing quantum state tomography on some quantum states in the adiabatic process. The experimentally reconstructed quantum states are close to the corresponding instantaneous ground states.
The quantum Fisher information (QFI) represents a fundamental concept in quantum physics. It quantifies the metrological potential of quantum states in quantum parameter estimation measurements, and is intrinsically related to quantum geometry and multipartite entanglement of many-body systems. Using a nitrogen-vacancy center spin in diamond, we experimentally demonstrate a randomized-measurement method to extract the QFI of the qubit, for both pure and mixed states. We then apply this scheme to a 4-qubit state, using a superconducting quantum computer, and show that it provides access to the sub-QFI, which sets a lower bound on the QFI for general mixed states. We numerically study the scaling of statistical error, considering NNN-qubit states, to illustrate the advantage of our randomized-measurement approach in estimating the QFI and multipartite entanglement. Our results highlight the general applicability of our method to different quantum platforms, including solid-state spin systems, superconducting quantum computers, and trapped ions.
Quantum state tomography is a technique in quantum information science used to reconstruct the density matrix of an unknown quantum state, providing complete information about the quantum state. It is of significant importance in fields such as quantum computation, quantum communication, and quantum simulation. However, as the size of the quantum system increases, the number of measurement settings and sampling requirements for quantum state tomography grow exponentially with the number of qubits. This not only makes experimental design and implementation more complex, but also exacerbates the consumption of experimental resources. These limitations severely hinder the application of state tomography in large-scale quantum systems. To reduce measurement settings and improve sampling efficiency, this study proposes a state tomography method based on auxiliary systems. This method can be implemented through either entanglement between the quantum system to be measured and a quantum auxiliary system or through correlation between the quantum system and a probabilistic classical auxiliary system. Measurements on the entire joint system enable more efficient extraction of information about the quantum state to be measured. This method relies on standard quantum gate operations and requires only two measurement settings, with a total sampling complexity of O(d2)O(d^2)O(d2), significantly simplifying experimental operations and measurement processes. Additionally, this study provides two schemes for measuring purity based on the proposed circuit, one of which achieves measurement precision at the Heisenberg limit. This study validates the effectiveness of the proposed method through a detailed theoretical analysis, a series of numerical simulations, and experiments.
Quantum Fisher information (QFI) plays a vital role in quantum precision measurement, quantum information, many-body physics, and other domains. Obtaining the QFI from experiment for a quantum state reveals insights such as the limits of estimation accuracy for a certain parameter, the degree of entanglement, and the geometric characteristics of the quantum state. Nonetheless, the measurement complexity of the QFI and its lower bound hinges on the dimension of the quantum state. Consequently, reducing the complexity of measurement is a significant challenge. This paper presents a methodology for evaluating the QFI of high-dimensional systems by transferring information to an auxiliary system and measuring its sub-QFI, while also offering conditions to diminish the dimension of auxiliary system to be measured without affecting the amount of information obtained by it.
Quantum tomography is one of the major challenges of large-scale quantum information research due to the exponential time complexity. In this Letter, we develop and apply a Bayesian state estimation method to experimentally demonstrate quantum overlapping tomography [Phys. Rev. Lett. 124, 100401 (2020)PRLTAO0031-900710.1103/PhysRevLett.124.100401], a scheme intent on characterizing critical information of a many-body quantum system in logarithmic time complexity. By comparing the measurement results of full-state tomography and overlapping tomography, we show that overlapping tomography gives accurate information of the system with much fewer state measurements than full-state tomography.
We propose a versatile Loschmidt echo protocol to detect and quantify multiparticle entanglement. It allows us to extract the quantum Fisher information for arbitrary pure states, and finds direct application in quantum metrology. In particular, the protocol applies to states that are generally difficult to characterize, as non-Gaussian states, and states that are not symmetric under particle exchange. We focus on atomic systems, including trapped ions, polar molecules, and Rydberg atoms, where entanglement is generated dynamically via long-range interaction, and show that the protocol is stable against experimental detection errors.
Abstract Entanglement is a key ingredient for quantum technologies and a fundamental signature of quantumness in a broad range of phenomena encompassing many-body physics, thermodynamics, cosmology and life sciences. For arbitrary multiparticle systems, entanglement quantification typically involves nontrivial optimisation problems, and it may require demanding tomographical techniques. Here, we develop an experimentally feasible approach to the evaluation of geometric measures of multiparticle entanglement. Our framework provides analytical results for particular classes of mixed states of N qubits, and computable lower bounds to global, partial, or genuine multiparticle entanglement of any general state. For global and partial entanglement, useful bounds are obtained with minimum effort, requiring local measurements in just three settings for any N. For genuine entanglement, a number of measurements scaling linearly with N are required. We demonstrate the power of our approach to estimate and quantify different types of multiparticle entanglement in a variety of N-qubit states useful for quantum information processing and recently engineered in laboratories with quantum optics and trapped ion setups.