Quantum repeaters for continuous variables

Purpose of the study: systematization and critical analysis of existing approaches to error suppression in quantum repeaters, as well as to assess their advantages and limitations for the implementation of scalable quantum networks and the quantum internet. Methods of research: the paper provides a detailed analysis of modern literature, including a comparison of various quantum repeater schemes, as well as an assessment of resource costs and performance. Result(s): quantum repeaters can be divided into three generations, each of which demonstrates optimal efficiency under certain conditions. The first generation, implementing probabilistic error suppression by generating heralded entanglement and two-way entanglement distillation, is easy to implement and provides the basic functionality of a quantum network, but requires significant time due to the need to exchange classical signals and long-term storage of quantum states. The second generation combines probabilistic entanglement generation with deterministic operation error suppression using quantum error correction, which reduces the requirements for long-term quantum memory, although the exchange of classical signals between neighboring nodes remains mandatory. The third generation relies entirely on deterministic error suppression using one-way classical communication, which allows for a significant reduction in time delays and high entanglement generation rates, despite the need for a denser arrangement of repeaters and high-quality local gates. In addition, the review covers new areas such as memoryless repeaters and all-photonic repeaters. The conducted comparative analysis of resource costs demonstrates that optimization of repeater operating parameters is a key factor for the implementation of scalable quantum networks, which is of direct importance for quantum key distribution, quantum metrology, and distributed quantum computing. Scientific novelty: the scientific novelty lies in the integration of disparate approaches to the implementation of quantum repeaters into a single holistic representation, which allows for an objective assessment of their efficiency by key parameters. The review highlights the potential for new classes of repeaters, such as memoryless repeaters and all-photonic circuits, to become important elements in the development of a quantum internet.

Quantum teleportation is one of the most basic quantum protocols, which transfers an unknown quantum state from one location to another through local operation and classical communication by using shared quantum entanglement without physical transfer of the information carrier. And it has been widely used in various quantum information protocols such as entanglement swapping, quantum repeaters, quantum gate teleportation, quantum computation based on measurement, and quantum teleportation networks, which have important application value in quantum computation and quantum information. Quantum teleportation is a naturally bipartite process, in which an unknown quantum state can only be transmitted from one node to another. With the further development of quantum information research, it is necessary to transfer quantum states or quantum information among more and more nodes. Multipartite quantum protocols are expected to form fundamental components for larger-scale quantum communication and computation. A bipartite quantum teleportation should be extended to a multipartite protocol known as a quantum teleportation network. In this paper, a multifunctional quantum teleportation network is proposed theoretically. We first propose a special method of constructing four-partite quantum resources in continuous variables (CVs), and based on this, construct two different types of CV quantum teleportation networks. One type of network contains just one quantum teleportation process consisting of a sender, a receiver and two controllers. In this type of network, the unknown quantum state can be recovered at any other node according to the requirement after the measurement in the input node, which enriches the transfer direction and transfer mode of the unknown quantum state. And meanwhile, the two controllers can control the transfer of a quantum state from the sender to the receiver by restricting the sender and receiver’s access to their information, which makes the quantum teleportation network controllable. The other type of network has two quantum teleportation processes, each containing only a sender, a receiver and no controllers, which increases the number of quantum states that can be transmitted. Then we analyze the dependence of the fidelity of each quantum teleportation network on different physical parameters, and compare the characteristics, advantages and disadvantages among different types of quantum teleportation networks. The scheme for constructing a multifunctional quantum teleportation network in this paper shows some advantages, such as the greater number of quantum nodes, diversity of types, simple operation procedure. And all these advantages provide a broader application prospect for establishing larger and more complex quantum information networks in the future and quicken the pace of the application of quantum information.

With their ability to process and transfer quantum information, large-scale entanglement-based quantum networks could be at the heart of a new age of quantum information, enabling fundamentally new applications such as distributed quantum computation, quantum communication, and quantum enhanced sensing. Due to their long spin coherence, controllable local qubit registers and optically active spins, color centers in diamond are prime candidates for nodes of such a network, and have enabled some of the most advanced quantum network demonstrations to date. These demonstrations include the distribution of a 3-node GHZ state across a quantum network, entanglement swapping, entanglement distillation, and memory-enhanced quantum communication. To move beyond current proof-of-principle networks, a further increase of entanglement generation rates is crucial. This thesis presents theoretical and experimental work on enhancing the spin-photon interface of color centers in diamond to achieve this goal, making use of the Purcell effect. The discussed work follows two main directions: embedding of color centers in open, tuneable Fabry-Perot micro-cavities, and in all-diamond photonic crystal cavities. First, we describe theoretical and experimental progress towards cavity-enhanced quantum networks based on nitrogen-vacancy (NV) centers in diamond. Due to their first order sensitivity to electric fields, we choose to embed NV centers in microns-thin diamond membranes that can be integrated into open fiber-based micro-cavities. We develop analytical methods to optimize the design of such open cavity systems for maximum Purcell enhancement of embedded color centers. We demonstrate a method to fabricate optically coherent NV centers in microns-thin diamond membranes, and use such structures to demonstrate the resonant excitation and detection of coherent, Purcell enhanced NV emission. A theoretical model in excellent agreement with our results suggests our system can improve entanglement rates between distant NV centers by two orders of magnitude with near-term improvements to the setup. Second, we describe progress towards an efficient spin-photon interface of group-IV color centers in diamond by coupling them to photonic nanostructures, allowing for large-scale integration. Due to their first-order insensitivity to electric fields, group-IV color centers can be brought in close proximity to surfaces (~ 100 nm), allowing for sub-wavelength mode volumes and thus very high Purcell factors. We numerically optimize photonic crystal cavity designs to maximize the Purcell enhancement of embedded emitters, test the robustness of our designs to real-world fabrication imperfections, and devise a method to efficiently interface nanophotonic structures. We then proceed to fabricate all-diamond photonic crystal cavities, making use of a dry etching technique that is selective to different crystallographic directions, and characterize the resulting optical quality factors. Finally, we marry the developed fabrication methods to fabricate microns-sized diamond platelets that can be transferred from a holding frame in a controlled fashion. This capability could be crucial for the realization of hybrid photonic circuits that we expect to be at the heart of future large-scale quantum networks.

Light orbital angular momentum (OAM) has been recognized as a new promising resource for classical and quantum information applications. In contrast to the spin angular momentum, the OAM is an inherently multidimensional. Thus, the information can be encoded in the higher-dimensional OAM alphabets. Recently, Marrucci et al. have invented a new device named q-plate (QP), made of liquid crystal cell patterned in such a way to introduce a topological charge q at the transverse plane, which is able to generate a well-defined values of photon OAM. My research has been aimed at investigating the physics of the QP, of the optical fields that it generates and of its possible applications for optical communication and quantum information. We studied both theoretically and experimentally a novel set of non-orthogonal but over-complete paraxial modes, named Hypergeometric-Gaussian modes, that is typically associated with OAM-carrying optical fields. We have also found the light propagation kernel inside the QP and we have shown analytically that if small losses due to reflection, absorption, and scattering are neglected, the QP can convert the photon spin into OAM with up to 100 \% efficiency. We implemented a technique to control the QP optical retardation by tuning the QP temperature. At the optimal temperature, the QP can generate a beam with up to 97\% efficiency. Moreover, the OAM state generated by QP can be rotated easily in the 2D OAM Hilbert space by proper manipulation of the input polarization state, a fact which opens a new way of beam shape controlling in MHz scale. We also performed a novel way to encode and read two bits information on the 4D OAM space by using only one QP. Finally, we experimentally demonstrated the transfer of quantum information from spin to OAM and vice versa, including the case of bi-photon states having quantum correlations. Furthermore, by exploiting these quantum information transfer devices, we demonstrated the Hong-Ou-Mandel effect and the optimal quantum cloning with OAM-carrying photons.

Quantum cryptographic systems have been commercially available, with a striking advantage over classical systems that their security and ability to detect the presence of eavesdropping are provable based on the principles of quantum mechanics. On the other hand, quantum protocol designers may commit much more faults than classical protocol designers since human intuition is much better adapted to the classical world than the quantum world. To offer formal techniques for modeling and verification of quantum protocols, several quantum extensions of process algebra have been proposed. One of the most serious issues in quantum process algebra is to discover a quantum generalization of the notion of bisimulation, which lies in a central position in process algebra, preserved by parallel composition in the presence of quantum entanglement, which has no counterpart in classical computation. Quite a few versions of bisimulation have been defined for quantum processes in the literature, but in the best case they are only proved to be preserved by parallel composition of purely quantum processes where no classical communications are involved. Many quantum cryptographic protocols, however, employ the LOCC (Local Operations and Classical Communications) scheme, where classical communications must be explicitly specified. So, a notion of bisimulation preserved by parallel composition in the circumstance of both classical and quantum communications is crucial for process algebra approach to verification of quantum cryptographic protocols. In this paper we introduce a novel notion of bisimulation for quantum processes and prove that it is congruent with respect to various process algebra combinators including parallel composition even when both classical and quantum communications are present. We also establish some basic algebraic laws for this bisimulation. In particular, we prove uniqueness of the solutions to recursive equations of quantum processes, which provides a powerful proof technique for verifying complex quantum protocols.

The distribution of quantum entanglement is essential for quantum communication. Contrary to classical communication, the signal in quantum communication is impossible to copy due to the basic principles of quantum mechanics, i.e., quantum state collapse and no-cloning theorem. However, the communication range is limited by inevitable photon loss. To counteract this issue, one can insert several repeater stations between two remote end nodes to divide the full distance into several shorter segments and successively operate entanglement swapping between adjacent repeater stations to enable long-distance quantum communication. Up to now, one can classify quantum repeaters (QRs) into three generations based on the methods of reducing errors resulted from photon loss and imperfect operations. In this paper, we will briefly review advances for each generation of QRs and discuss the advantages and challenges of each generation. Moreover, we also discuss some efficient proposals of building blocks to overcome the bottlenecks for QRs.

We present a simple and general simulation technique that transforms any black-box quantum algorithm (a la Grover's database search algorithm) to a quantum communication protocol for a related problem, in a way that fully exploits the quantum parallelism. This allows us to obtain new positive and negative results. The positive results are novel quantum communication protocols that are built from nontrivial quantum algorithms via this simulation. These protocols, combined with (old and new) classical lower bounds, are shown to provide the first asymptotic separation results between the quantum and classical (probabilistic) two-party communication complexity models. In particular, we obtain a quadratic separation for the bounded-error model, and an exponential separation for the zero-error model. The negative results transform known quantum communication lower bounds to computational lower bounds in the black-box model. In particular, we show that the quadratic speed-up achieved by Grover for the OR function is impossible for the PARITY function or the MAJORITY function in the bounded-error model, nor is it possible for the OR function itself in the exact case. This dichotomy naturally suggests a study of bounded-depth predicates (i.e. those in the polynomial hierarchy) between OR and MAJORITY. We present black-box algorithms that achieve near quadratic speed up for all such predicates.

Quantum information science (QIS) aims at controlling quantum coherence effects in light and matter in order to enable new information processing capabilities that are not possible with classical resources. Quantum information can be encoded in photons that are used as flying quantum bits that can be sent over long distances, and in atoms that are used as stationary quantum bits to store and process the information. The efficient transfer of quantum information between single photons and atomic systems is an important requirement for QIS. It enables the realization of a quantum memory for light, which is required for many applications in QIS. In particular, it is necessary for scalable implementations of quantum information networks. It is also a crucial element of quantum repeaters that have been proposed to overcome losses in the transport of photonic quantum information over long distances.In this talk, I will concentrate on our latest results (obtained at the University of Geneva), where we demonstrate the quantum storage of photonic entanglement in a Neodymium doped crystal. We have developed a narrow band non-degenerate photon pair source adapted for quantum repeater architectures. One photon is in the telecom band and the other one is compatible with the spectral requirement of a Nd doped crystal QM (35 MHz bandwidth). Entanglement between the photon at telecommunication wavelength and a collective atomic excitation stored in the crystal is demonstrated by mapping the twin photon onto the crystal, using an atomic frequency comb protocol. The successful entanglement mapping is demonstrated by the violation of a Bell inequality. These results represent an important step towards quantum communication technologies based on solid-state devices.

Distributing quantum entanglement between distant parties is a significant but difficult task in quantum information science, as it can enable numerous applications but suffers from exponential decay in the quantum channel. Quantum repeaters are one of the most promising approaches towards this goal. In a quantum repeater protocol, it is essential that the entanglement generation speed within each elementary link is faster than the memory decoherence rate, and this stringent requirement has not been implemented over a fiber of metropolitan scale so far. As a step towards this challenging goal, in this work we experimentally realize multiplexing-enhanced generation of heralded atom-photon quantum correlation over a 12 km fiber. We successively generate 280 pairs of atom-photon quantum correlations with a train of photonic time-bin pulses filling the long fiber, and read out the excited memory modes on demand with either fixed or variable storage time after successful heralding. With the multiplexing enhancement, the heralding rate of atom-photon correlation can reach 1.95 kHz, and the ratio between the quantum correlation generation rate to memory decoherence rate can be improved to 0.46 for a fiber length of 12 km. This work therefore constitutes an important step towards the realization of a large-scale quantum repeater network.

Quantum communications promises reliable transmission of quantum information, efficient distribution of entanglement and generation of completely secure keys. For all these tasks, we need to determine the optimal point-to-point rates that are achievable by two remote parties at the ends of a quantum channel, without restrictions on their local operations and classical communication, which can be unlimited and two-way. These two-way assisted capacities represent the ultimate rates that are reachable without quantum repeaters. Here, by constructing an upper bound based on the relative entropy of entanglement and devising a dimension-independent technique dubbed ‘teleportation stretching’, we establish these capacities for many fundamental channels, namely bosonic lossy channels, quantum-limited amplifiers, dephasing and erasure channels in arbitrary dimension. In particular, we exactly determine the fundamental rate-loss tradeoff affecting any protocol of quantum key distribution. Our findings set the limits of point-to-point quantum communications and provide precise and general benchmarks for quantum repeaters.

Quantum communication relies on the existence of entanglement between two nodes of a network. However, due to its fragile nature, it is nearly impossible to establish entanglement at large distances through the direct transmission of qubits. Quantum repeaters have been proposed to solve this problem, which split-up the network to create small-scale entangled links and then connect them up to create the large-scale link. As researchers race to establish entanglement over larger and larger distances, it becomes essential to gauge the performance and robustness of the different protocols that have been proposed to design a quantum repeater, before deploying them in real life. Currently available noisy quantum computers are ideal for this task, as they can emulate the noisy environment in a quantum communication channel, and provide a measure for how the protocols will perform on real-life hardware. In this paper, we report the circuit-level implementation of the complete architecture of a quantum repeater, and benchmark this protocol on IBM's cloud quantum computer - IBMQ. Our experiments indicate a 26% fidelity of shared bell-pairs for a complete on-chip quantum repeater with a yield of 49%. We also compare these results with simulation data from IBM Qiskit. The results of our experiments provide a quantitative measure for the fidelity of entanglement that currently available repeaters can establish. In addition, the proposed circuit-implementation provides a robust benchmark for state-of-the-art quantum computing hardware.

In the domain of quantum communication, one of the main challenges is the distribution of entanglement over long distances. A scheme, the so-called quantum repeater, has been proposed in order to overcome the problem of channel losses. In this thesis we adresse different problems related to quantum repeaters, both theoretically and experimentally. We present an alternative scheme of quantum repeater based on atomic ensembles and linear optics. It is based on the use of a qubit amplifier and two photon interference. The main features of the protocol are that the distributed entangled state does not need post selection and the entanglement distribution rates approach the optimal rates one might expect from quantum repeaters using probabilistic swapping. An important concern of the whole class of quantum repeaters based on one photon interference is the phase stability in the communication channels. We performed an experiment evaluating the realistic phase noise in installed optical fibers in the Geneva urban area. We found fluctuations of order of 0.1 rad on the time scale of order of 100 $mu$s. The feasibility of the phase stabilization is discussed. One of the essential building blocks of quantum repeater is a quantum memory. We focus our attention on quantum memories based on solid state atomic ensembles and photon echo like techniques. In particular, we show, that the atomic coherence in eryso can be controlled to a very high degree using an external electric field. This allowed us to implement the CRIB protocol in this material and realize the first storage of light at the single photon level at telecommunication wavelength. A necessary step for the implementation of CRIB and AFC storage protocols, in order to achieve long storage times and unit efficiencies, is the use of control light pulses. We developped a numerical Maxwell-Bloch simulator allowing us to study a coherent control of atomic excitations. In particular, we investigate the use of adiabatic chirped pulses in the context of AFC protocol. We study the relationship between the pulse characteristics and the echo properties. The comparison with the conventional $pi$ pulses and the impacts on the multimode capacity of the AFC protocol are also discussed.

The exploration of distributed quantum computing (DQC) represents a significant frontier in quantum information science, aiming to harness the unique properties of quantum mechanics to solve complex computational problems effectively. This paper discusses the diverse architectures and models that facilitate the distribution of quantum computing tasks across multiple quantum nodes, thereby addressing the limitations inherent in single quantum devices. DQC systems exploit the principles of superposition and entanglement to enhance computational capabilities beyond what classical computing systems can achieve. We examine various DQC architectures, including both quantum-classical hybrid systems and fully quantum distributed systems, highlighting their respective benefits and challenges. Key issues such as coherence, communication overhead, and error correction are discussed in detail, and we analyze specific use cases where DQC demonstrates superior performance relative to classical computing approaches, including optimization problems in logistics and finance, as well as quantum simulations for material science. Furthermore, we identify future research directions aimed at overcoming existing barriers to the practical implementation of DQC systems. By assessing the current landscape and future possibilities of DQC, this paper underscores the transformative potential of distributed quantum computing in various fields, paving the way for realizing more complex quantum algorithms and applications.

Quantum internet will enable a number of revolutionary applications, such as distributed quantum computing, large scale quantum communication and cooperative operation of atomic clocks. Cold atomic ensembles are a very promising approach for quantum internet, featuring long coherence time and collectively enhanced interaction with single photons. The current central theme in this direction is to improve the performance of fundamental building blocks and to construct small-scale networks which go beyond state of the art. In this talk, I will present our experimental works in this direction, such as the improvement of single-node performance with cavity enhancement, optical lattice, and Rydberg nonlinearity. I will also talk about the extension of atom-atom entanglement distance via quantum frequency conversion, and the establishment of entanglement among three quantum nodes via three-photon interference etc.

Quantum error correcting codes have been shown to have the ability of making quantum information resilient against noise. Here we show that we can use quantum error correcting codes as diagnostics to characterize noise. The experiment is based on a three-bit quantum error correcting code carried out on a three-qubit nuclear magnetic resonance (NMR) quantum information processor. Utilizing both engineered and natural noise, the degree of correlations present in the noise affecting a two-qubit subsystem was determined. We measured a correlation factor of $c=0.5\ifmmode\pm\else\textpm\fi{}0.2$ using the error correction protocol, and $c=0.3\ifmmode\pm\else\textpm\fi{}0.2$ using a standard NMR technique based on coherence pathway selection. Although the error correction method demands precise control, the results demonstrate that the required precision is achievable in the liquid-state NMR setting.