*6.1. Implementation of Weak Measurement to Project* |*χ*

In Section 5.1, we stated the implementation of a weak measurement to project |*χ* conveniently onto <sup>|</sup>*χ*0 <sup>=</sup> <sup>|</sup>*β*0 or <sup>|</sup>*χ*1 <sup>=</sup> <sup>∑</sup><sup>3</sup> *<sup>i</sup>*=<sup>1</sup> *p i* |*βi*. Despite, in the experimental approach, there are certain differences due to the resources been used. In this section, we present how to afford the weak measurement stated in (40). We use an ancilla qubit |0*a* to do the measurement minimizing the impact on |*χ* as is desired. In this implementation, we will use as a central resource the Toffoli gate. In order to prepare the |*χ* stated properly for such measurement, we combine it with the ancilla. Then, we send the combined system into the circuit presented in Figure 16a. This circuit employs the Toffoli gate T1,2,*<sup>a</sup>* on channels 1, 2 for <sup>|</sup>*χ* and *<sup>a</sup>* for <sup>|</sup>0*a* together with the *<sup>C</sup>*1*Not*<sup>2</sup> gate (already developed for ions [25,26] and photons [27]). In fact, it is well-known that Toffoli gate can be performed using *CNOT* gates and single-qubit gates [14] o by means of the Sleator–Weinfurter construction [28], despite other more efficient developments are known for ions [29] and photons [30]. Some single-qubit gates as Hadamard (H) and Not (<sup>X</sup> ) are also used. In the following development, we write <sup>|</sup>*χ* <sup>=</sup> <sup>∑</sup><sup>3</sup> *i*=0 *p*<sup>∗</sup> *<sup>i</sup>* |*βi* as the imperfect entangled resource (be aware that ∗ not means complex conjugation). Thus, all necessary quantum gates have been experimentally developed in our days at least in quantum optics.

**Figure 16.** (**a**) Quantum circuit generating the weak measurement on |*χ*, and (**b**) contour plots for the map on the region (*p*1, *p*2, *p*3) between those probabilities and (*p*<sup>∗</sup> <sup>1</sup>, *p*<sup>∗</sup> <sup>2</sup>, *p*<sup>∗</sup> 3 ).

A direct calculation shows that this circuit performs the following transformation on |*ψ*0 = |*χ*⊗|0*a* into:

$$\left|\psi\_{1}\right\rangle \quad = \sqrt{p\_{0}}|\beta\_{0}\rangle \otimes |1\_{\mathfrak{a}}\rangle + \left(\sqrt{p\_{1}}|\beta\_{1}\rangle + \sqrt{p\_{2}}|\beta\_{2}\rangle + \sqrt{p\_{3}}|\beta\_{3}\rangle\right) \otimes |0\_{\mathfrak{a}}\rangle \tag{47}$$

$$\begin{aligned} \text{with}: \qquad \sqrt{2p\_0} &= \sqrt{p\_0^\*} - \sqrt{p\_{3'}^\*} & \sqrt{2p\_1} &= \sqrt{p\_1^\*} - \sqrt{p\_{2'}^\*}\\ \sqrt{2p\_2} &= \sqrt{p\_0^\*} + \sqrt{p\_{3'}^\*} & \sqrt{2p\_3} &= \sqrt{p\_1^\*} + \sqrt{p\_2^\*} \end{aligned} \tag{48}$$

Just before of the projective measurement on the qubit *a* shown in the Figure 16a. Clearly, after measurement, two possible outcomes arise in the qubit *a*, |1*a*, |0*a* while on qubits 1, 2 the outcomes are <sup>|</sup>*χ*0 <sup>=</sup> <sup>|</sup>*β*0, <sup>|</sup>*χ*1 <sup>=</sup> <sup>∑</sup><sup>3</sup> *i*=1 *p i* |*βi* respectively as in the Section 5.1, thus completing the weak measurement. The only difference with respect our previous development is that those coefficients are not the original {*p*<sup>∗</sup> *<sup>i</sup>* }. Despite this, in the event that such coefficients are unknown, this fact is not important, the really outstanding outcome is that this procedure projects the state into the perfect Bell state to perform the teleportation |*β*0 or otherwise on the frontal face if this resource is planned to be used under indefinite causal order and measurement (as it was previously depicted in the procedure of Section 5.1). Anyway, Figure 16b shows the contour plots of *p*∗ <sup>1</sup> (red), *p*<sup>∗</sup> <sup>2</sup> (green) and *p*∗ <sup>3</sup> (blue) in the region (*p*1, *p*2, *p*3) as a reference of the involved geometric transformations.

#### *6.2. An Insight View about Teleportation Implementing Indefinite Causal Orders Experimentally with Light*

Formula (1) regards the teleportation algorithm as a quantum communication channel. Despite this formula being a useful simplification for the theoretical analysis, it expresses the teleportation channel with the input and output through the same system, which is not precisely the real experimental situation. Then, as it was true for the original implementation of the original teleportation proposal [9] in [31], the deployment should be modified to have a correct approach to the theory. In this section we discuss an insight view into the experimental deployment together with indefinite causal order based on current techniques and experimental developments.

A possible implementation with light should to consider an initial state with at least three initial photons exhibiting each one at least a pair of quantum variables as polarization, frequency or spatial localization (**k**-vector state) among others (as in the original experimental teleportation proposal [31]): |*ψ*0 = |*v*<sup>1</sup> ⊗ |*v<sup>a</sup>* ⊗ |*vb*, using polarization in the vertical direction as instance. Those photons should then be converted into five photons by splitting the last two into entangled pairs using Spontaneous Parametric Down Conversion (SPDC) [32] as instance, while the first state is arbitrarily rotated by a quartz polarization rotator (QPR) [33] -to generate the state to teleportate-: |*ψ*1 = (*α*|*v*<sup>1</sup> <sup>+</sup> *<sup>β</sup>*|*h*1) <sup>⊗</sup> <sup>√</sup><sup>1</sup> 2 (|*v*2|*h*<sup>3</sup> <sup>+</sup> <sup>|</sup>*h*2|*v*3) <sup>⊗</sup> <sup>√</sup><sup>1</sup> 2 (|*v*4|*h*<sup>5</sup> + |*h*4|*v*5). After, five photons should be sent together into two alternative directions (through a dichroic beamsplitter—a splitting wavelength dependent—instead a polarization beamsplitter) coincidentally, not independently (it means five photons will travel through corresponding paths labeled by *pA* or *pB*). This beamsplitter (BS) works as our control state superposing the two path states (the two causal orders further). Last effect should be solved based on the frequency of original photons which should be quantum generated to let a quantum splitting of all beams (or otherwise based on the previous generation of a *GHZ* state [34]). This necessary beamsplitter is still a cutting-edge technology. Such spatial quantization introduces an additional quantum variable thus converting the initial state into (removing the tensor product symbols for the sake of simplicity):

$$|\psi\_2\rangle = \frac{1}{\sqrt{8}} \left( (a|v\rangle\_1 + \beta|h\rangle\_1)|p\_A\rangle\_1 (|v\rangle\_2|h\rangle\_3 + |h\rangle\_2|v\rangle\_3) |p\_A\rangle\_2 |p\_A\rangle\_3 (|v\rangle\_4|h\rangle\_5 + |h\rangle\_4|v\rangle\_5) |p\_A\rangle\_4 |p\_A\rangle\_5 \right)$$

$$+ (a|v\rangle\_1 + \beta|h\rangle\_1)|p\_B\rangle\_1 (|v\rangle\_2|h\rangle\_3 + |h\rangle\_2|v\rangle\_3)|p\_B\rangle\_2 |p\_B\rangle\_3 (|v\rangle\_4|h\rangle\_5 + |h\rangle\_4|v\rangle\_5)|p\_B\rangle\_4 |p\_B\rangle\_5 \right) \tag{49}$$

If additionally we introduce certain optical distortion in the SPDC, we get imperfect entangled states then changing each <sup>√</sup><sup>1</sup> 2 (|*v<sup>i</sup>* |*h<sup>j</sup>* + |*h<sup>i</sup>* |*v<sup>j</sup>* ) by |*χij*. In the following, we will change *v*, *h* by 0, 1 respectively for simplicity.

Note that teleportation is, in a certain sense, automatically generated due to the non-locality of the resource |*β*0 (or imperfectly by |*χ*), then collapsed on four adequate outcomes involving an additional correction as a function of those outcomes using classical communication (Figure 1a). In addition, for two sequential teleportation channels, the process can be achieved by post-measurement at the end of both processes. Nevertheless, the implementation of indefinite causal order in teleportation introduces additional challenges due to the connectivity of paths and measurements. In the process, it will be required the implementation of the *SWAP* gate, which has already been experimentally performed in optics [35,36].

Thus, Figure 17 depicts a possible implementation for two teleportation processes assisted by indefinite causal order. The first photon goes to the QRP and then the five photons go through the coordinated BS. The proposed process can be then followed in the Figure 17 with paths labeled by *pA* in green and *pB* in red. For simplicity, teleportation processes are assumed to perform measurements on the Bell states basis as in Figure 1b, thus avoiding the use of *H* and *CNOT* gates in the analysis. Due to the above construction (post-measurement and measurement assumed on the Bell basis), almost no gates are present in the process, just two *SWAP* gates stating the causal connections. At the end of each path, a semi-transparent mirror should mix again the paths (but not the polarization) by pairs into the basis |±*<sup>i</sup>* <sup>=</sup> <sup>1</sup> <sup>2</sup> (|*pA<sup>i</sup>* ± |*pB<sup>i</sup>* ) for each photon *i*, in order to erase the information of the path followed. We labeled each path (or the information being carried on it) by *M<sup>k</sup> ij* (in case that photon carries the information of one of the complementary systems not containing the output of teleportation) remarking the path type followed *k* = *A*, *B*, +, −; the final belonging teleportation process *i* = 1, 2; and the number of the sequential qubit to be measured there: *j* = 1 for the former input and *j* = 2 for the correspondent to the first qubit of the original entangled resource. Instead, the final outputs are labeled by *<sup>S</sup><sup>k</sup>* (*<sup>k</sup>* <sup>=</sup> *<sup>A</sup>*, *<sup>B</sup>*, <sup>+</sup>, <sup>−</sup>). By following the color, the reader should easily identify each path considering additionally the effect of the intermediate use of *SWAP* gates which is discussed below.

**Figure 17.** Diagram for implementation of teleportation with causal ordering as it is discussed in the text. Photons are split on two different set of paths to superpose the two causal orders of two sequential teleportation process.

By ignoring first the *SWAP* gates in the Figure 17, we can realize that the circuit has not any effect. We have indicated each optical element described before. The dotted line connecting the *BS*'s denotes the not independent functioning, all together should send the five photons on the green paths or on the red ones. States |*ψ* and |*χ* are remarked on photons 1 and 2, 3, 4, 5 respectively. Each path

(green or red) is labeled from 1 to 5 in agreement with the photon carried out. Blue arrow remarks the group of photons involved in each teleportation process T<sup>1</sup> or T<sup>2</sup> on each path (the first subscript in *M<sup>k</sup> ij*): 1, 2, 3 and 3, 4, 5 respectively for the green paths, and 1, 4, 5 and 5, 2, 3 respectively for the red ones. On each path, we reported the associated label for each system *S<sup>k</sup>* or *M<sup>k</sup> ij* as it was depicted before. Note that brown labels correspond to the information being carried before of *SWAP* gates, while black labels are the final states reported there at the end of the path but before of the recombining in the semi-transparent mirrors. The reason for the *SWAP* gate between the paths 3 and 5 should be clear, we need to get the teleportation outputs on the same photon to generate the superposition of information. The *SWAP* gate on the red paths 2 and 4 exchanges the information on those paths in order to generate the superposition at the end among path information *M*<sup>1</sup> *ij* and *<sup>M</sup>*<sup>2</sup> *<sup>i</sup> <sup>j</sup>* with *i* = *i* , *j* = 1, 2 thus mixing both. Note that the set of states in *M<sup>k</sup> ij* are those to be measured in the teleportation process (here in the Bell basis by pairs) in order to correct the output states. The reader should advise this process does not reproduce exactly that depicted by (1) because such formula assumes the measurements are internal operations generating a mixed state coming from the corresponding projections and corrections. In this approach, we have the possibility to measure only four qubits instead of eight. Despite, we will note this procedure still reproduces some of the main previous features analyzed. At the end of the process, each semi-transparent mirror (diagonal in grey) mixes the information on the states |±*<sup>i</sup>* for each photon *i* on the red and green edges (with information *M*<sup>±</sup> *ij* or *S*± respectively -red and green-, not represented in the Figure 17). On the red edges, a detector first decides if the photon exits through them (they are the projective measurement on |*ϕ*<sup>±</sup> *<sup>m</sup>* states in our development). In addition, a Bell measurement is then performed on each pair 1, 3 and 2, 4 in order to inquire the information codified in the output *S*+.

A direct but large calculation to expand (49) then applying the *SWAP* gates and projecting on |+*i*, *i* = 1, ..., 5 was performed. Finally, this output was written in terms of |*βi*1,3 ⊗ |*βj*2,4, *i*, *j* = 0, ..., 3 to ease the identification of final successful measurements. If *p*<sup>0</sup> = 1 or *p*<sup>0</sup> = 0, upon the measurement of <sup>|</sup>*βi*1,3 ⊗ |*βj*2,4 and then the application of *<sup>σ</sup>iσ<sup>j</sup>* as correction, the output *<sup>S</sup>*<sup>+</sup> becomes <sup>|</sup>*ψ* faithfully in the following cases:


This clearly resembles our main outcomes. For the second case, other measurement outcomes give imperfect teleportation thus rearranging the success probabilities with respect of those in our theoretical development. Thus, alternative experimental proposals should be developed to approach them into the ideal case considered in our theoretical results.

#### **7. Conclusions**

Quantum teleportation has an important role in quantum processing for the transmission of quantum information, nevertheless, there are possible issues on the entangled resource assisting the teleportation process mainly related to its maintenance and precise generation. It introduces imprecision in the teleported state. In this work, the implementation of indefinite causal order has been studied in order to propose an improved scheme to tackle such imprecision on the entangled state when it is combined with the measurement of the control assessing it.

The analysis for the redundant case where quantum channels are simply applied sequentially (assumed as identical) shows that the number *N* of channels applied, rapidly decreases the fidelity converging to the maximal depolarization of the teleported state thus obtaining <sup>F</sup>*N*→<sup>∞</sup> <sup>=</sup> <sup>1</sup> 2 . By modifying the process under indefinite causal order for two or more teleportation channels as it was proposed by [7] and later discussed in [22], we advise advantages on the quantum fidelity of the teleported state for the first values *N* of sequential teleportation channels. From the outcomes, a categorization was performed to analyze the effects on the entangled state, thus obtaining a surprising

enhancement for the most imperfect entangled resource, *p*<sup>0</sup> = 0 with the absence of the ideal entangled resource |*β*0, and still for near regions of it with *p*<sup>0</sup> ≈ 0 when *N* increases. Notably, in the first case, it is possible to obtain a perfect teleportation process with F*<sup>N</sup>* = 1. However, when *N* increases, the principal downside is the reduction of the probability of successful measurement P*m*, which decreases drastically as *N* increases.

In order to improve the global probability of success, we have proposed the combined use of weak measurement to first projecting the entangled resource to either *p*<sup>0</sup> = 1 with *p*1, *p*2, *p*<sup>3</sup> = 0 or *p*<sup>1</sup> + *p*<sup>2</sup> + *p*<sup>3</sup> = 1 with *p*<sup>0</sup> = 0, where the indefinite causal order generates the most notable enhancements. In such cases, F = 1 is obtained always and P*<sup>m</sup>* is improved. Those notable processes are possible as for pure as for mixed states [8]. A remarkable aspect is that for such a notable case the outcome is independent of the teleported state.

Finally, a more detailed process for the weak measurement (first barely discussed in the initial presentation) is after detailed and oriented to the practical implementation in terms of the current experimental developments for light and matter. The development of a Toffoli gate is advised as central in the implementation. In addition, an introductory analysis for a possible experimental implementation has been included for the teleportation process under indefinite causal order using two teleportation channels. Such an approach is still imperfect and not optimal. Despite this, it reproduces the main features found in our development. In the proposal, recent experiments and technological developments in optics become central, particularly the implementation of the *SWAP* gate and the generation of |*GHZ* states. A valuable aspect being noticed is the use of post-measurement in the teleportation process. Additional theoretical and experimental developments should still improve the vast possibilities of indefinite causal order in the teleportation research field.

**Author Contributions:** C.C.-I. performed the research related with the analysis of indefinite causal order using 2 channels setting the structure for the further research. Both authors performed the computer algorithms to reproduce the output states after of indefinite causal orders. F.D. performed the theoretical development of indefinite causal order using *N* channels. C.C.-I. contributed in the development of computer simulations and computer graphics in the entire manuscript. C.C.-I. perform the research about the feasibility of indefinite causal order using the current technologies. F.D. developed the experimental proposal to implement indefinite causal order in Section 6.2. All authors contributed evenly in the writing of the manuscript. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Acknowledgments:** F.D. acknowledges to Jesus Ramírez Joachín his taught and thoroughly discussions in combinatorics during 1982, without which some parts of this work will not be possible. Both authors acknowledge the economic support to publish this article to the School of Engineering and Science from Tecnologico de Monterrey. The support of CONACYT is also acknowledged.

**Conflicts of Interest:** The authors declare no conflict of interest.
