Quantum teleportation is one of the important manifestations of quantum mechanics. We consider how quantum information is transferred via various entangled states of continuous variables including asymmetrically mixed Gaussian states and coherent entangled states.
Very recently, we took part in a new development of quantum information, the so-called continuous variable (CV) quantum information theory. Such a further development is mainly due to the experimental and theoretical advantages offered by CV systems, i.e., quantum systems described by a set of observables, like position and momentum, which have a continuous spectrum of eigenvalues. According to this novel trend, quantum information protocols like quantum teleportation have been suitably extended to the CV framework. Here, we briefly review some mathematical tools relative to CV systems, and we consequently develop the concepts of quantum entanglement and teleportation in the CV framework by analogy with the qubit-based approach. Some connections between teleportation fidelity and entanglement properties of the underlying quantum channel are inspected. Next, we address the study of CV quantum teleportation networks where more users share a multipartite state and an arbitrary pair of them performs quantum teleportation. In this context, we show alternative protocols, and we investigate the optimal strategy that maximizes the performance of the network.
The theoretical toolbox we have developed as the book unfolded will now be put to use to describe a series of tasks, related to the transmission or extraction of information, which may be implemented with quantum continuous variable systems. We will open with quantum teleportation, the transmission of a quantum state through shared quantum entanglement and classical communication, and determine certain thresholds it must attain to be considered genuinely quantum; we shall then proceed to establish the maximum classical communication capacity of a class of Gaussian channels; next, the optimal estimation of a dynamical parameter using Gaussian states as probes will be considered; we will then show that Gaussian states may be used for the secure distribution of a cryptographic key; finally, our enquiry will close with the illustration of a sampling algorithm aimed at achieving a ‘quantum advantage’ over competing classical methods. It is worth mentioning that, in discussing teleportation thresholds and communication capacities, although the optimal schemes we shall identify will turn out to be Gaussian, we will confront full optimisations at the Hilbert space level, and will thus introduce methods that go beyond the Gaussian description. Gaussian quantum key distribution has already been commercialised, Gaussian boson sampling has already been applied to claim a quantum advantage, and all the other techniques we will touch on in this chapter are gradually approaching the status of full-blown engineering technologies, which would arguably form the most convincing empirical evidence one may have in support of the underlying physical theory.
Bennett and his team pioneered the quantum teleportation protocol, a breakthrough where Alice transmits quantum information to Bob through a pre-shared entangled quantum channel. Remarkably, this transmission occurs without physically moving the information, relying on a combination of classical communication and quantum entanglement. This protocol not only demonstrates the essence of quantum entanglement but also underscores its potential in the realm of information transfer. The success of quantum teleportation experiments extends across various physical systems, with a particular emphasis on achieving effectiveness over long distances. Looking forward, future developments in this field may lead to multi-party protocols, thereby advancing both quantum communication and computing capabilities. A crucial component in quantum teleportation networks is controlled quantum teleportation (CQT), holding significant promise for the emerging concept of the quantum internet. In the preceding chapter, our exploration centered on the practical application of quantum teleportation protocols, with a specific focus on two key approaches: standard quantum teleportation and CQT. Our examination delved into the fundamental principles of quantum teleportation, encompassing critical concepts such as quantum entanglement, non-cloning, superposition, and the importance of measurements like fidelity. Throughout the chapter, we dedicated special attention to both the theoretical and experimental aspects of quantum teleportation, considering the intricacies of systems with continuous variables (CVs) as well as those with discrete variables (DVs).
The structural aspects of tripartite entanglement in three-mode Gaussian states of continuous variable systems have been studied in [Adesso G, Serafini A and Illuminati F 2006 Phys. Rev. A {\\bf 73} 032345]. Here we focus our attention on the usefulness of such states in the context of realistic processing of continuous-variable quantum information. We introduce and discuss in detail several examples of pure and mixed three-mode states that stand out for their informational and/or entanglement properties. We then describe practical schemes to engineer such states with linear optics. In particular, we introduce a simple procedure -- based on passive optical elements -- to produce pure three-mode Gaussian states with {\\em arbitrary} entanglement structure (upon availability of an initial single-mode squeezed state). We analyze in detail the properties of distributed entanglement, showing that the promiscuity of entanglement sharing is a feature peculiar to symmetric Gaussian states that survives even in the presence of significant degrees of mixedness and decoherence. Next, we discuss the suitability of the considered tripartite entangled states to the implementation of quantum information and communication protocols with continuous variables. This will lead to a feasible experimental proposal to test the promiscuous sharing of continuous-variable entanglement, in terms of the optimal fidelity of teleportation networks with Gaussian resources. As a byproduct, optimal resource states for various communication protocols (among which asymmetric telecloning) will be exactly determined.
We introduce a simple procedure - based on passive optical elements - to produce pure three-mode Gaussian states with arbitrary entanglement structure (upon availability of an initial single-mode squeezed state). We analyze in detail the properties of distributed entanglement, showing that the promiscuity of entanglement sharing is a feature peculiar to symmetric Gaussian states that survives even in the presence of significant degrees of mixedness and decoherence. Next, we discuss the suitability of the considered tripartite entangled states to the implementation of quantum information and communication protocols with continuous variables. This will lead to a feasible experimental proposal to test the promiscuous sharing of continuous-variable entanglement, in terms of the optimal fidelity of teleportation networks with Gaussian resources. As a byproduct, optimal resource states for various communication protocols (among which asymmetric telecloning) will be exactly determined.