Quantum correlation is a fundamental feature of quantum mechanics that distinguishes quantum systems from the classical macroscopic world. Quantum correlation can be categorized into spatial and temporal quantum correlations. A well-known example of spatial quantum correlation is quantum entanglement, which has played a crucial role in advancing our understanding of quantum mechanics while also serving as a key resource for cutting-edge quantum technologies, such as quantum communication, quantum computing, and quantum sensing. Research on quantum correlations has recently extended to the temporal domain, exploring the characteristics of quantum correlations across different time points and their potential applications in quantum technologies.

A research team led by Associate Professor Dawei Lu from the Department of Physics at the Southern University of Science and Technology (SUSTech), in collaboration with Tsinghua University and CNRS@CREATE Singapore, has made significant progress in the field of quantum temporal correlation detection. Their research utilizes the measurement properties of nuclear magnetic resonance (NMR) ensembles to efficiently detect temporal correlations in quantum systems.

The findings have been published in Physical Review Letters under the title “Certifying Quantum Temporal Correlation via Randomized Measurements: Theory and Experiment”.

The concept of temporal correlation has intrigued quantum mechanics pioneers since the early days of the field. However, its systematic study and practical applications in quantum information science only began after Leggett and Garg introduced the famous Leggett-Garg inequality (LGI) in 1985. The LGI is used to test whether a physical system adheres to the principle of macroscopic realism, which consists of two fundamental assumptions. The first, macroscopic realism, states that a macroscopic object exists in one of multiple distinct states at any given time. The second, noninvasive measurement, asserts that it is possible, in principle, to measure the state of the system without disturbing its current state or affecting its future dynamics.

In classical physics, all systems satisfy the LGI. However, certain quantum processes violate the LGI, demonstrating that quantum systems cannot be described using classical macroscopic realism.

Research on quantum temporal correlation not only deepens our understanding of the fundamental nature of the physical world but also holds promise for various quantum technology applications. For instance, quantum temporal correlation has potential uses in quantum key distribution, environmental dimension estimation, quantum channel capacity estimation, quantum timing systems, and quantum causal inference. However, efficient methods for detecting quantum temporal correlation remain a challenge in current experimental systems.

Figure 1. (a) The PDO diagram is constructed by sequentially measuring the quantum system before and after the quantum channel; (b) The quantum circuit diagram for the ‘virtual’ preparation of PDO at a single time slice, followed by randomized measurements to estimate its second-order moment.

To address this issue, Professor Dawei Lu’s team and collaborators have successfully designed and experimentally validated a novel approach for efficiently detecting quantum temporal correlations. Their method is based on the Pseudo Density Operator (PDO) theoretical framework, which extends density matrices to the temporal domain. Unlike conventional density matrices, PDOs allow for negative eigenvalues, which indicate the presence of quantum temporal correlations. Detecting negative eigenvalues thus confirms temporal correlation in quantum systems.

Traditional methods for detecting temporal correlations using PDOs rely on tomographic reconstruction, which requires extensive quantum resources and increases experimental complexity. To overcome this limitation, the research team innovatively integrated the techniques of quasiprobability decomposition and randomized measurements. This approach enabled the “virtual” preparation of a two-time PDO at a single time slice. By performing randomized measurements, they efficiently estimated the second-order statistical moments of the PDO, ultimately allowing for the detection of negative eigenvalues with high precision.

Figure 2. Ensemble NMR system. The sample contains a large number of identical molecules, all of which participate in the experiment, and an ensemble-averaged statistical measurement is performed on all molecules at the end.

The proposed method offers several advantages. Although it measures second-order moments of the PDO, it only requires operations on a single quantum system, reducing the complexity and scale of quantum devices. Additionally, the number of required measurement bases is independent of system size, making it particularly suitable for ensemble-based measurement techniques. These include NMR systems, cold atomic systems, and nitrogen-vacancy centers in diamonds, where a single measurement basis can execute exponentially large projection measurements, significantly improving efficiency.

The team successfully demonstrated their method using an NMR platform, validating the theoretical predictions and showcasing the feasibility and high accuracy of the approach in thermodynamic quantum systems.

Figure 3. (a) Experimental validation of the feasibility and accuracy of the ‘virtual’ preparation of PDO before randomized measurements; (b-c) Eigenvalue analysis of the PDO; (d-e). Estimation of the second-order moment of the PDO through randomized measurements.

Their research not only provides a novel method for detecting quantum temporal correlations but also offers valuable insights for future developments in quantum technologies. The exceptional capability of the NMR system in measuring density matrix elements further paves the way for exploring other quantum experimental techniques.

The first authors of the paper are Ph.D. students Hongfeng Liu from SUSTech and Zhenhuan Liu from Tsinghua University. The corresponding authors are Associate Professor Dawei Lu, Dr. Xiangjing Liu from CNRS@CREATE Singapore, and Assistant Professor Xinfang Nie from SUSTech. SUSTech is the primary affiliation of the paper.

Paper link: https://doi.org/10.1103/PhysRevLett.134.040201