# Shared Quantum Key Distribution Based on Asymmetric Double Quantum Teleportation
## 1 Introduction  
With the development of quantum applications, particularly quantum cryptography [1], new cryptosystems intended to be unconditionally secure are being developed. Such cryptosystems are commonly composed of a sender and a receiver assuming to share an Encryption and a Decryption key [2]. Then, a message can be encrypted and transmitted from the sender's end to the receiver's end. Along the way, an eavesdropper can try to steal the key intended to be transmitted between them. For instance, experimental implementations are led using imperfect photon detectors, thus allowing the loss of some photons [3] and allowing an intruder to tamper these imperfect devices to obtain advantages against the security of the protocol [4, 5]. Due to this feasibility, it is necessary to strengthen the security in all cryptosystems. With this purpose, new research aiming to obtain better unbreakable ways of key distribution between two parties has been conducted. Such development has boosted technology implementing Quantum Key Distribution (QKD). There, two entities (sender and receiver) can communicate securely to set codification keys. The peculiarity of quantum cryptography is the use of fundamental aspects of quantum mechanics such as theuncertainty principle [6], entanglement [7], and the quantum measurement theory [8] to provide a set of constraints on the communication channel to make it safer [9]. The generation of this quantum key can be distributed through many protocols developed for this purpose [10]. Some existing QKD protocols include BB84 [11], B92 [12], SARG04 [13], and E91 [14]. In this sense, quantum cryptography has become a leading development for the secure transmission of data [15]. After the last-mentioned QKD protocols, quantum cryptography has been refining its methods and complexity to keep off quantum hacking as a counterpart [16]. Thus, post-quantum cryptography pursues cryptography algorithms being secure against cryptanalytic attacks performed by quantum computers [17]. Otherwise, QKD can be made unconditionally secure over arbitrarily long distances against attacks by an eavesdropper [18]. Thus, quantum cryptography is requiring more complex procedures including quantum processing to enhance security. In another trend, for the development of communications, quantum teleportation has played a central role in communication enhancements. Various approaches seeking experimental implementations of such algorithms soon emerged [19; 20]. Since then, the great importance of the development of quantum teleportation has boosted applications in quantum communication to a large extent. Some of them include the creation of quantum networks [21], cryptography applications regarding quantum computing systems [22], settlement of photonic quantum computing [23], and particularly teleportation-based quantum cryptography protocols [24] as complementary scaffolding procedures improving its efficiency and security. Improvements in the quality of teleportation involve new approaches. Some of them for long-distance quantum teleportation with the use of a fiber-delayed Bell state measurement (BSM) [25] and others using optical fiber to avoid using large-aperture optics and other complex techniques [26; 27]. Teleportation is being combined with quantum strategies as a causal order [28] to remove some underlying noisy effects. Recently, an analysis for a double teleportation process for the same input state has been presented in [29; 30]. In such a scenario, one main party ([PERSON]) has prepared the input state and then shared two entangled resources with another two parties (here called [PERSON]\({}_{0}\) and [PERSON]\({}_{1}\)) keeping one qubit of each pair. A central resource, in principle accessible for the three parties, works as a control to decide who of the [PERSON]'s will receive the teleported state. With such a scheme, cryptography protocols can be performed to set secure authentication methods [30].