**Carlos Cardoso-Isidoro † and Francisco Delgado \*,†**

School of Engineering and Sciences, Tecnologico de Monterrey, Atizapan 52926, Mexico; A01750267@itesm.mx

**\*** Correspondence: fdelgado@tec.mx; Tel.: +52-55-5864-5670 † These authors contributed equally to this work.

Received: 10 October 2020; Accepted: 16 November 2020; Published: 20 November 2020

**Abstract:** Quantum teleportation has had notorious advances in the last decade, being successfully deployed in the experimental domain. In other terrains, the understanding of indefinite causal order has demonstrated a valuable enhancement in quantum communication to correct channel imperfections. In this work, we address the symmetries underlying imperfect teleportation when it is assisted by indefinite causal order to correct the use of noisy entangled resources. In the strategy being presented, indefinite causal order introduces a control state to address the causal ordering. Then, by using post-selection, it fulfills the teleportation enhancement to recover the teleported state by constructive interference. By analysing primarily sequential teleportation under definite causal order, we perform a comparison basis for notable outcomes derived from indefinite causal order. After, the analysis is conducted by increasing the number of teleportation processes, thus suggesting additional alternatives to exploit the most valuable outcomes in the process by adding weak measurement as a complementary strategy. Finally, we discuss the current affordability for an experimental implementation.

**Keywords:** teleportation; indefinite causal order; weak measurement; quantum algorithm

#### **1. Introduction**

Quantum communication has always looked for improvements and new outstanding approaches. Particularly, it has been shown that certain enhancements in information transmission can be reached through the superposition of quantum communication channels. That enhancement has shown that the interference of causal orders using sequential extreme imperfect depolarizing channels surprisingly produces a transparent quantum channel due to constructive superposition in the components of the state being transmitted [1]. Since that discovery, a growing interest in indefinite causal order has emerged boosting a deep study of this topic. Experimental implementations have been proposed in order to find, to understand, and to control their advantages [2].

### *1.1. Background of Indefinite Causal Order in Communication*

In quantum communication with extremely noisy channels, only limited information can be transmitted. If we continue applying such quantum channels sequentially, no information becomes transmitted, obtaining the so-called depolarizing quantum channel. Despite, it has been shown that when such channels are applied in a superposition of causal orders, we can still transmit information, and notably, the quality of information transmitted becomes improved while more channels are applied under this scheme. Concretely for the case of two quantum channels, some works considering controllable strengths of depolarization have shown that combining a superposition of causal orders, it is still possible to transmit information (instead of worsening it as it obviously happens for the simpler sequential case) [1,3]. The success of the causal orders superposition has been experimentally verified for two channels transmitting information [4].

Following such a trend in communication, it has been found the possibility to extrapolate the increasing number of causal orders superposed (with more than two channels) by developing a combinatoric approach to the problem [5,6]. As a matter of fact, it has been shown that the amount of information transmitted, in comparison with the two-channel scenario, increases for the three-channels scenario [5]. Therefore, it has been concluded that the amount of classical information transmitted becomes higher if the number of causal orders increases.

Some notorious approaches regarding the indefiniteness of causal orders have been explored, exhibiting the capability to transmit information in a more efficient way. It highlights the importance to extend this approach on teleportation, as a genuine communication process [7,8].

#### *1.2. Approaches to Teleportation under Causal Order Schemes*

Information can be transmitted from one party to another as a quantum state if it is prepared in combination with an Einstein–Podolsky–Rosen state [9]. Such a quantum communication process is called quantum teleportation. It plays an important role related to quantum information and quantum communication. Teleportation algorithm for one single qubit is performed using one entangled Bell state and one channel for classical communication in order to achieve it [10]. Symmetries in the conformation of such quantum entangled state automatically transfer a state into another party if post-measurement is applied. The same algorithm has also been useful to teleport states of larger systems if they are composed of two-level systems [11]. The teleportation algorithm has been widely studied and new approaches have been discovered, as well as variants on the algorithm in order to make it either more efficient in terms of the quantum resources used [12] or more adaptive to some specific quantum systems [13–15]. Additionally, several successful tests have been experimentally performed in order to prove the feasibility of teleportation when the distance increases [16–18]. Tests with larger multidimensional states rather than qubits have been performed successfully [19]. Recently, a new approach has shown that the assistance of indefinite causal order in teleportation improves its performance when imperfect entangled resources become involved [7], which is equivalent to a quantum noisy communication channel.

Teleportation assisted by indefinite causal order and measurement has been introduced in [7] by pointing out that teleportation is a quantum channel itself (here, entanglement distribution is assumed to be performed through a transparent communication channel). The last proposal has been criticized in [20] arguing the entanglement distribution in teleportation is a critical aspect not being considered there (due to the large distances and communication issues involved). Instead, as in [7], the most recent work [20] interestingly has analysed the use of indefinite causal order in the form of a quantum switch for the entanglement distribution process as a part of the teleportation algorithm, thus making an analysis to quantify the performance gained by such a switch. Nevertheless, nowadays teleportation has been achieved through kilometers in the free space or through optical fiber, with still high fidelities [21] without considerable deformation in the entangled resource other than that the introduced in its imperfect generation. Thus, we believe both approaches are still valuable in the quest of understanding creative ways to implement indefinite causal order in teleportation. Both approaches show interesting features in the quantification of indefinite causal order issues applied to teleportation.

In [7], the quantum teleportation uses imperfect singlets showing that despite those noisy singlets make impossible a faithful teleportation, there is still a stochastic possibility of teleporting perfectly the state by applying indefinite causal order as the superposition of two teleportation channels. Such teleportation process has been conducted considering two identical teleportation channels with the same imperfect entangled resources, but in a superposition of causal orders through an evenly quantum control system. Finally, the outcome is measured on a specific basis in order to improve the fidelity of the teleportation process in the best possible way by recovering the symmetrical composition of the teleported state. Following this analysis and considering the same two imperfect channels but with an arbitrary initialized quantum control system, it has been also found the possibility to get again the highest possible transmission by post-selecting the appropriate

outputs under alternative scenarios [22]: a proper selection of the post-measurement state on the control system, thus extending the interesting outcomes obtained in [7]. In addition, it has been shown that for the less noisy cases, the effect becomes still limited [7].

In teleportation, the traditional algorithm [9] is entirely represented as a quantum channel *T* in Figure 1a. In order to carry out the teleportation, it is necessary an entangled resource shown as |*χ*. When this resource is the Bell state <sup>|</sup>*β*00 <sup>=</sup> <sup>√</sup><sup>1</sup> 2 (|00 + |11), a perfect teleportation is then achieved, but if such state is imperfect (it can be generally expressed as a mixture of all Bell states), teleportation process does not work properly. In Figure 1b, an alternative (but still equivalent) circuit is presented assuming that Bell states measurements could be performed. In such a case, no gates are required, due to teleportation is just reached due to the non-locality of the entangled resource |*β*00 (or imperfectly, |*χ*). This fact will be useful at the end of the article for a tentative experimental proposal.

**Figure 1.** (**a**) Traditional teleportation circuit *T* where |*ψ* = *α* |0 + *β* |1 and ideally |*χ* is the Bell state <sup>|</sup>*β*00 <sup>=</sup> <sup>√</sup><sup>1</sup> 2 (|00 + |11). Measurements refer to one single qubit measurement and the double line to classical communication channels. (**b**) Modified teleportation circuit considering a Bell states measurement (which are generated by enclosing the gates on (**a**) within the measurement gadget).

Still, applying a sequence of two imperfect teleportation channels, the outcome worsens. In [7], it has been shown that for the worst deformed case of |*χ*, the fidelity of single teleportation goes down. However, if two teleportation channels are used in an indefinite causal order with the superposition ruled by a quantum control system, surprisingly the previous worst-case arises with fidelity equal to 1. The analysis has been extended in [22] considering a wider kind of measurements required in the original approach. In this sense, the use of indefinite causal order improves the teleportation process. Thus, it is possible to correct this lack of fidelity working with the worst entangled state by applying indefinite causal order, together, with some appropriate selection in the control used and in the measurement performed, making possible to reach perfect teleportation.

In the current work, we deal with an extended version of the algorithm presented in [7,22] by using several sequential channels in order to benchmark the outcomes obtained by increasing the number of channels [5]. Section 2 develops the case of sequential channels in a definite causal order as a comparison basis. Section 3 develops the same situation but considering an indefinite causal order superposition using *N* channels. Section 4 uses the last formalism with more than two teleportation channels under indefinite causal order widening the spectrum of analysis. Section 5 revisits the problem but implementing additionally weak measurement proposing an improved procedure. Finally, Section 6 discusses the affordability of a possible experimental implementation for two teleportation channels under indefinite causal order using the current experimental developments. The last section gives the conclusions and future work to extend our findings.
