Abstract
Quantum teleportation plays a key role in modern quantum technologies. Thus, it is of much interest to generate alternative approaches or representations that are aimed at allowing us a better understanding of the physics involved in the process from different perspectives. With this purpose, here an approach based on graph theory is introduced and discussed in the context of some applications. Its main goal is to provide a fully symbolic framework for quantum teleportation from a dynamical viewpoint, which makes explicit at each stage of the process how entanglement and information swap among the qubits involved in it. In order to construct this dynamical perspective, it has been necessary to define some auxiliary elements, namely virtual nodes and edges, as well as an additional notation for nodes describing potential states (against nodes accounting for actual states). With these elements, not only the flow of the process can be followed step by step, but they also allow us to establish a direct correspondence between this graph-based approach and the usual state vector description. To show the suitability and versatility of this graph-based approach, several particular teleportation examples are examined in detail, which include bipartite, tripartite, and tetrapartite maximally entangled states as quantum channels. From the analysis of these cases, a general protocol is devised to describe the sharing of quantum information in presence of maximally entangled multi-qubit system.
Highlights
Quantum entanglement is the cornerstone of modern quantum technologies [1]
Quantum teleportation, the capability to transfer the information encoded in the quantum state of a physical system to another distant physical system, plays an important role, as its remarkable theoretical and experimental development has shown over the last quarter of a century [2]
A record was established in 2012, when the experiments were performed between two laboratories separated a distance of 143 km [6], with the purpose to demonstrate the feasibility of the phenomenon at distances of the order of a low Earth orbit and, to implement quantum key distribution protocols at the global level
Summary
Quantum entanglement is the cornerstone of modern quantum technologies [1]. Within them, quantum teleportation, the capability to transfer the information encoded in the quantum state of a physical system (e.g., an atom or a photon) to another distant physical system, plays an important role, as its remarkable theoretical and experimental development has shown over the last quarter of a century [2]. The subsequent sub-polynomials and sub-links that arise after partially tracing with respect to every qubit provide us with valuable information regarding the entanglement that still remains and, which is going to characterize every reduced subsystem It follows that the original polynomial contains all the monomials associated with all possible maximally entangled subsystems, which can be determined by means of the Peres–Horodecki separability criterion [20,21] (see [22,23]). In order to better appreciate the monomial structure, i.e., the two- and three-variable linking, bipartite entangled structures have been surrounded by yellow and red circles, respectively As it can be noticed, a topological description in terms of links is rather convenient to specify entanglement relationships, for it provides us with a general picture of this quantum property without making any explicit reference to a particular physical substrate (material particles, photons, or, in general, different types of degrees of freedom). It is clear that we need an alternative representation, which may introduce or describe similar elements, but that, in turn, becomes more versatile, providing us with a more suitable framework to deal with the notion of flow of entanglement and information
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