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Chinese Physics Letters

Asymptotical Locking Tomography of High-Dimensional Entanglement*

Ling‐Jun Kong, Rui Liu, Wen-Rong Qi, Zhou-Xiang Wang, Shuang‐Yin Huang, Chenghou Tu, Yongnan Li, Hui‐Tian Wang

Published Mar 1, 2020• 12 citations
Publisher

Abstract

High-dimensional (HD) entanglement provides a very promising way of transcending the limitations of the two-dimensional entanglement between qubits for increasing channel capacity in many quantum protocols. In the pursuit of capitalizing on the HD entangled states, one of the central issues is to unambiguously and comprehensively quantify and reconstruct them. The full quantum state tomography is a unique solution, but it is undesirable and even impractical because the measurements increase rapidly in d 4 for a bipartite d-dimensional quantum state. Here we present a very efficient and practical tomography method—asymptotical locking tomography (ALT), which can harvest full information of bipartite d-dimensional entangled states by very few measurements less than 2d 2 only. To showcase the validity and reasonableness of our ALT, we carry out the test with the two-photon spin-orbital angular momentum hyperentangled states in a four-dimensional subspace. Besides high-efficiency and practicality, our ALT is also universal and can be generalized into multipartite HD entanglement and other quantum systems.

DOI
10.1088/0256-307x/37/3/034204

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High-dimensional (HD) entanglement provides a very promising way of transcending the limitations of the two-dimensional entanglement between qubits for increasing channel capacity in many quantum protocols. In the pursuit of capitalizing on the HD entangled states, one of the central issues is to unambiguously and comprehensively quantify and reconstruct them. The full quantum state tomography is a unique solution, but it is undesirable and even impractical because the measurements increase rapidly in d 4 for a bipartite d-dimensional quantum state. Here we present a very efficient and practical tomography method—asymptotical locking tomography (ALT), which can harvest full information of bipartite d-dimensional entangled states by very few measurements less than 2d 2 only. To showcase the validity and reasonableness of our ALT, we carry out the test with the two-photon spin-orbital angular momentum hyperentangled states in a four-dimensional subspace. Besides high-efficiency and practicality, our ALT is also universal and can be generalized into multipartite HD entanglement and other quantum systems.
High-dimensional (HD) entanglement enables an encoding of more bits than in the two-dimensional case and promises to increase communication capacity over quantum channels and to improve robustness to noise. In practice, however, one of the central challenges is to devise efficient methods to quantify the HD entanglement explicitly. Full quantum state tomography is a standard technology to obtain all the information about the quantum state, but it becomes impractical because the required measurements increase exponentially with the dimension in HD systems. Hence, it is highly anticipated that a new method will be found for characterizing the HD entanglement with as few measurements as possible and without introducing unwarranted assumptions. Here, we present and demonstrate a scan-free tomography method independent of dimension, which only requires two measurements for the characterization of two-photon HD orbital angular momentum (OAM) entanglement. Taking Laguerre-Gaussian modes of photons as an example, the density matrices of OAM entangled states are experimentally reconstructed with very high fidelity. Our method is also generalized to the mixed HD OAM entanglement. Our results provide realistic approaches for quantifying more complex OAM entanglement in many scientific and engineering fields such as multiphoton HD quantum systems and quantum process tomography.
Exploiting photonic high-dimensional entanglement allows one to expand the bandwidth of quantum communication and information processing protocols through increased photon information capacity. However, the characterization of entanglement in higher dimensions is experimentally challenging because the number of measurements required scales unfavourably with dimension. While bounds can be used to certify high-dimensional entanglement, they do not quantify the degree to which quantum states are entangled. Here, we propose a quantitative measure that is both an entanglement measure and a dimension witness, which we coin the P-concurrence. We derive this measure by requiring entanglement to extend to qubit subspaces that constitute the high-dimensional state. The computation of the P-concurrence is not contingent on reconstructing the full density matrix, and requires less measurements compared to standard quantum state tomography by orders of magnitude. This allows for faster and more efficient characterization of high-dimensional quantum states.
Large-scale quantum systems have advanced rapidly via the exploration of more particles and higher dimensions, offering great potential for developing quantum technologies. However, their characterization becomes prohibitive with increasing local dimensionality and particle number. Here we propose high-dimensional quantum overlapping tomography based on a graph-theoretic formulation, which allows one to efficiently reconstruct few-body marginals of multipartite high-dimensional quantum systems. We experimentally realize it on a photonic four-party entangled state in a 4×4×2×24 \times 4 \times 2 \times 24×4×2×2 system. Using measurements in mutually unbiased bases, we reconstruct all six two-body marginals with only 25 projective measurement settings, compared with 94 and 225 settings for independent tomography of all two-body reduced states and full state tomography, respectively. The reconstructed marginals reveal a layered entanglement structure vital for high-dimensional quantum networks. We further show that these marginals enable more noise-resilient certification of multipartite high-dimensional entanglement than the fidelity-based criterion. Our work thus offers a scalable route for learning multidimensional quantum systems.
High-dimensional encoding of quantum information provides a promising method of transcending current limitations in quantum communication. One of the central challenges in the pursuit of such an approach is the certification of high-dimensional entanglement. In particular, it is desirable to do so without resorting to inefficient full state tomography. Here, we show how carefully constructed measurements in two bases (one of which is not orthonormal) can be used to faithfully and efficiently certify bipartite high-dimensional states and their entanglement for any physical platform. To showcase the practicality of this approach under realistic conditions, we put it to the test for photons entangled in their orbital angular momentum. In our experimental setup, we are able to verify 9-dimensional entanglement for a pair of photons on a 11-dimensional subspace each, at present the highest amount certified without any assumptions on the state.
Quantum state tomography is an important tool for quantum communication, computation, metrology, and simulation. Efficient quantum state tomography on a high dimensional quantum system is still a challenging problem. Here, we propose a novel quantum state tomography method, auxiliary Hilbert space tomography, to avoid pre-rotations before measurement in a quantum state tomography experiment. Our method requires projective measurements in a higher dimensional space that contains the subspace that includes the prepared states. We experimentally demonstrate this method with orbital angular momentum states of photons. In our experiment, the quantum state tomography measurements are as simple as taking a photograph with a camera. We experimentally verify our method with near-pure- and mixed-states of orbital angular momentum with dimension up to d=13d=13d=13, and achieve greater than 95 % state fidelities for all states tested. This method dramatically reduces the complexity of measurements for full quantum state tomography, and shows potential application in various high dimensional quantum information tasks.
High-dimensional photonic entanglement holds significant promise for advancing quantum communication, computation, and metrology. For example, large-alphabet quantum communication protocols are known to benefit from enhanced noise resilience and information capacity via multi-bit time-bin encoding. Yet, characterizing high-dimensional entangled states is challenging, as full state tomography becomes prohibitively costly and often requires unrealizable measurements. Here, we demonstrate a scan-free method to characterize high-dimensional entanglement in the time-frequency domain. Our reconstruction achieves a record 5.70±0.075.70\pm0.075.70±0.07 ebits and a fidelity of 65.4±0.4%65.4\pm0.4\%65.4±0.4% with the maximally entangled state of local dimension 102110211021, certifying the presence of 668668668-dimensional entanglement. We further prove the attainability of a secure key rate of 15.615.615.6 kB/s in a composable finite-size, entanglement-based protocol, and show that in continuous operation, the setup can quickly approach asymptotic key rates. Using commercial telecom components and state-of-the-art low-jitter single-photon detectors, our scalable architecture offers a practical path towards high-rate, noise-resilient quantum communication testbeds.